problem stringlengths 11 5.47k | reasoning stringlengths 708 124k | deepseek_solution stringlengths 0 10.3k | ground_truth_solution stringlengths 97 9.16k |
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Let \( a, b, c \) be positive real numbers. Prove that
$$
\frac{1}{a(1+b)}+\frac{1}{b(1+c)}+\frac{1}{c(1+a)} \geq \frac{3}{1+abc},
$$
and that equality occurs if and only if \( a = b = c = 1 \). | Okay, so I need to prove this inequality: \(\frac{1}{a(1+b)}+\frac{1}{b(1+c)}+\frac{1}{c(1+a)} \geq \frac{3}{1+abc}\), where \(a, b, c\) are positive real numbers, and equality holds if and only if \(a = b = c = 1\). Hmm, let me try to think through this step by step.
First, I remember that inequalities often involve ... | To prove the inequality
\[
\frac{1}{a(1+b)} + \frac{1}{b(1+c)} + \frac{1}{c(1+a)} \geq \frac{3}{1+abc}
\]
for positive real numbers \(a, b, c\) with equality if and only if \(a = b = c = 1\), we proceed as follows:
### Key Steps and Proof:
1. **Substitution for \(abc = 1\):**
- Assume \(abc = 1\). Let \(a = \fr... | 1. Consider the given inequality:
\[
\frac{1}{a(1+b)}+ \frac{1}{b(1+c)}+ \frac{1}{c(1+a)} \geq \frac{3}{1 + abc}
\]
To simplify, we add \( \frac{3}{1 + abc} \) to both sides. The new inequality becomes:
\[
\frac{1}{a(1+b)} + \frac{1}{b(1+c)} + \frac{1}{c(1+a)} + \frac{3}{1 + abc} \geq \frac{6}{1 + abc}
\]
2. Let's ... |
A set consists of five different odd positive integers, each greater than 2. When these five integers are multiplied together, their product is a five-digit integer of the form $AB0AB$, where $A$ and $B$ are digits with $A \neq 0$ and $A \neq B$. (The hundreds digit of the product is zero.) In total, how many different... | Okay, so I need to figure out how many different sets of five different odd positive integers multiply together to give a product of the form AB0AB, where A and B are digits, A isn't zero, and A isn't equal to B. Hmm, let's break this down step by step.
First, the product is a five-digit number with the pattern AB0AB.... | To determine the number of different sets of five different odd positive integers whose product is a five-digit number of the form \(AB0AB\), we analyze the prime factorization and possible combinations:
1. **Understanding the Product**:
- The product \(AB0AB\) can be written as \(1001 \times (10A + B)\), where \(1... |
1. **Observe the Structure of \( N \)**:
Let \( N = AB0AB \) and let \( t \) be the two-digit integer \( AB \).
We recognize that \( N = 1001 \cdot t \), where \( 1001 = 11 \cdot 91 = 11 \cdot 7 \cdot 13 \).
Thus,
\[
N = t \cdot 7 \cdot 11 \cdot 13
\]
2. **Formulate the Problem**:
We n... |
Given real numbers \( a, b, c \) and a positive number \( \lambda \) such that the polynomial \( f(x) = x^3 + a x^2 + b x + c \) has three real roots \( x_1, x_2, x_3 \), and the conditions \( x_2 - x_1 = \lambda \) and \( x_3 > \frac{1}{2}(x_1 + x_2) \) are satisfied, find the maximum value of \( \frac{2 a^3 + 27 c - ... | Okay, let's try to tackle this problem step by step. The problem is about finding the maximum value of the expression (2a³ + 27c - 9ab)/λ³ given some conditions on the roots of the cubic polynomial f(x) = x³ + ax² + bx + c. The roots are real, with x₂ - x₁ = λ and x₃ > (x₁ + x₂)/2.
First, I need to recall that for a ... | Given real numbers \(a, b, c\) and a positive number \(\lambda\) such that the polynomial \(f(x) = x^3 + ax^2 + bx + c\) has three real roots \(x_1, x_2, x_3\) with the conditions \(x_2 - x_1 = \lambda\) and \(x_3 > \frac{1}{2}(x_1 + x_2)\), we need to find the maximum value of \(\frac{2a^3 + 27c - 9ab}{\lambda^3}\).
... |
We begin by analyzing the function \( f(x) = x^3 + a x^2 + b x + c \), which has three real roots \( x_1, x_2, x_3 \). We are given the following conditions:
1. \( x_2 - x_1 = \lambda \)
2. \( x_3 > \frac{1}{2} (x_1 + x_2) \)
We aim to find the maximum value of \( \frac{2a^3 + 27c - 9ab}{\lambda^3} \).
1. **Transfor... |
In triangle $ABC$, $CA = CB$, and $D$ is the midpoint of $AB$. Line $EF$ passes through point $D$ such that triangles $ABC$ and $EFC$ share the same incenter. Prove that $DE \cdot DF = DA^2$. | Okay, so I have this geometry problem here. Let me try to understand it step by step. The problem says: In triangle ABC, CA equals CB, so it's an isosceles triangle with CA = CB. Point D is the midpoint of AB. There's a line EF passing through D such that triangles ABC and EFC share the same incenter. I need to prove t... | In triangle \(ABC\), \(CA = CB\), making it isosceles with \(D\) as the midpoint of \(AB\). Line \(EF\) passes through \(D\) such that triangles \(ABC\) and \(EFC\) share the same incenter. We need to prove that \(DE \cdot DF = DA^2\).
1. **Coordinate Setup**:
- Place \(D\) at the origin \((0, 0)\), \(A\) at \((-a,... | 1. **Identify Key Elements**: Consider \( \triangle ABC \) where \( CA = CB \) and \( D \) is the midpoint of \( AB \). Line \( \mathrm{EF} \) passes through \( D \) such that \( \triangle ABC \) and \( \triangle \mathrm{EFC} \) share the same incircle (inscribed circle).
2. **Given Common Incircle**: Let the shared i... |
Let \( p = 2^{3009}, q = 3^{2006}, \) and \( r = 5^{1003} \). Which of the following statements is true?
(A) \( p < q < r \)
(B) \( p < r < q \)
(C) \( q < p < r \)
(D) \( r < p < q \)
(E) \( q < r < p \) | Okay, so I need to figure out which of the given options is correct by comparing the sizes of p, q, and r where p = 2^3009, q = 3^2006, and r = 5^1003. Hmm, comparing exponents with different bases and exponents... This might be tricky. Let me think about how to approach this.
First, I remember that comparing numbers ... | To determine which of the given statements is true, we need to compare the sizes of \( p = 2^{3009} \), \( q = 3^{2006} \), and \( r = 5^{1003} \).
First, we express each number with the same exponent by noting that:
- \( p = 2^{3009} = (2^3)^{1003} = 8^{1003} \)
- \( q = 3^{2006} = (3^2)^{1003} = 9^{1003} \)
- \( r =... | Given the values:
\[ p = 2^{3009}, \quad q = 3^{2006}, \quad r = 5^{1003} \]
1. Express \( p \) and \( q \) in terms of powers of the same base:
\[ p = 2^{3009} = 2^{3 \times 1003} = (2^3)^{1003} = 8^{1003} \]
\[ q = 3^{2006} = 3^{2 \times 1003} = (3^2)^{1003} = 9^{1003} \]
Note: \( r = 5^{1003} \) is already... |
When \( a < -1 \), the nature of the roots for the equation
$$
\left(a^{3}+1\right) x^{2}+\left(a^{2}+1\right) x-(a+1)=0
$$
is:
(A) Two negative roots.
(B) One positive root and one negative root, with the absolute value of the negative root being larger.
(C) One positive root and one negative root, with the absolute v... | Okay, so I need to figure out the nature of the roots for the quadratic equation \((a^3 + 1)x^2 + (a^2 + 1)x - (a + 1) = 0\) when \(a < -1\). The options are given as (A) two negative roots, (B) one positive and one negative root with the negative root having a larger absolute value, (C) similar to B but the negative r... | To determine the nature of the roots for the quadratic equation \((a^3 + 1)x^2 + (a^2 + 1)x - (a + 1) = 0\) when \(a < -1\), we analyze the discriminant, sum, and product of the roots.
1. **Discriminant Analysis**:
- The discriminant \(D\) is given by:
\[
D = (a^2 + 1)^2 - 4(a^3 + 1)(-(a + 1))
\]
... |
Given the quadratic equation:
\[
(a^3 + 1) x^2 + (a^2 + 1) x - (a + 1) = 0
\]
where \( a < -1 \).
1. **Identify the coefficients and analyze their signs:**
- \( a^3 + 1 \)
- \( a^2 + 1 \)
- \( -(a+1) \)
2. **Signs of the coefficients under the condition \( a < -1 \):**
- Since \( a < -1 \), we ha... |
Suppose that $A, B, C, D$ are four points in the plane, and let $Q, R, S, T, U, V$ be the respective midpoints of $AB, AC, AD, BC, BD, CD$. If $QR = 2001$, $SU = 2002$, and $TV = 2003$, find the distance between the midpoints of $QU$ and $RV$. | Okay, let's see. So, we have four points A, B, C, D in a plane. Then there are midpoints of the segments AB, AC, AD, BC, BD, CD, named Q, R, S, T, U, V respectively. The lengths QR = 2001, SU = 2002, and TV = 2003. We need to find the distance between the midpoints of QU and RV. Hmm, okay.
First, I need to visualize ... | Given four points \(A, B, C, D\) in the plane, we consider the midpoints \(Q, R, S, T, U, V\) of the segments \(AB, AC, AD, BC, BD, CD\) respectively. We are given the distances \(QR = 2001\), \(SU = 2002\), and \(TV = 2003\), and we need to find the distance between the midpoints of \(QU\) and \(RV\).
First, we expre... | To find the distance between the midpoints of $Q U$ and $R V$, let's break down the given information and analyze each step.
1. **Identify the midpoints**:
- $Q$, $R$, $S$, $T$, $U$, $V$ are midpoints of segments $AB$, $AC$, $AD$, $BC$, $BD$, $CD$ respectively.
2. **Recall Midpoint theorem**:
- The Midpoint T... |
Given that \(1 \leq x, y, z \leq 6\), how many cases are there in which the product of natural numbers \(x, y, z\) is divisible by 10? | Okay, let's see. The problem is asking how many ordered triples (x, y, z) there are, where each of x, y, z is a natural number between 1 and 6 inclusive, such that their product x*y*z is divisible by 10. Hmm, so I need to count all possible triples where the product is a multiple of 10.
First, let me recall that a nu... | To determine the number of ordered triples \((x, y, z)\) where \(1 \leq x, y, z \leq 6\) and their product is divisible by 10, we use the principle of inclusion-exclusion. A product is divisible by 10 if it contains at least one factor of 2 and one factor of 5.
1. **Total number of triples**: \(6 \times 6 \times 6 = ... | Given the constraints \(1 \leq x, y, z \leq 6\), we are to find the number of natural number combinations \((x, y, z)\) such that their product can be divided exactly by 10.
To begin, we observe:
1. The total number of combinations of \(x, y, z\) is \(6^3\):
\[
6^3 = 216
\]
2. To be divisible by 10, the pro... |
Prove that for each \( n \geq 2 \), there is a set \( S \) of \( n \) integers such that \((a - b)^{2}\) divides \( ab \) for every distinct \( a, b \in S \). | Okay, so I need to prove that for each integer n ≥ 2, there exists a set S of n integers such that for any two distinct elements a and b in S, the square of their difference (a - b)² divides their product ab. Hmm, interesting problem. Let me try to break this down and figure out a way to approach it.
First, let me und... | To prove that for each \( n \geq 2 \), there exists a set \( S \) of \( n \) integers such that \((a - b)^2\) divides \( ab \) for every distinct \( a, b \in S \), we construct such a set explicitly using the following approach:
**Construction:**
For each \( n \geq 2 \), define the set \( S = \{0, 2, 3, 4, \ldots, n... |
Given the problem:
> Prove that for each \( n \geq 2 \), there is a set \( S \) of \( n \) integers such that \( (a-b)^2 \) divides \( ab \) for every distinct \( a, b \in S \).
We aim to show that such a set \( S \) exists for all \( n \geq 2 \). To do this, we rely on the direct corollary of a more general result p... |
Find all prime numbers \( p \) such that for any prime number \( q < p \), if \( p = kq + r \) with \( 0 \leq r < q \), then there does not exist an integer \( a > 1 \) such that \( a^2 \) divides \( r \). | Okay, let's try to tackle this problem. So, we need to find all prime numbers \( p \) such that for any prime number \( q < p \), when we divide \( p \) by \( q \) and get a remainder \( r \) (so \( p = kq + r \) with \( 0 \leq r < q \)), there's no integer \( a > 1 \) where \( a^2 \) divides \( r \). Hmm, that's a bit... | To find all prime numbers \( p \) such that for any prime number \( q < p \), if \( p = kq + r \) with \( 0 \leq r < q \), then there does not exist an integer \( a > 1 \) such that \( a^2 \) divides \( r \), we need to ensure that the remainder \( r \) when \( p \) is divided by any smaller prime \( q \) is square-fre... |
**Step 1:** Identify the prime numbers \( p \) that meet the given conditions:
For every prime \( q < p \), \( p = k q + r \) where \( 0 \leq r < q \). It must hold that no integer \( a > 1 \) exists such that \( a^2 \mid r \).
**Step 2:** Check small prime numbers:
- For \( p = 2 \):
- \( p = 2 \) and there are... |
Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy for all \( x, y \in \mathbb{R} \):
\[
f(y-f(x)) = f(x) - 2x + f(f(y))
\] | Okay, so I need to find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy the functional equation:
\[
f(y - f(x)) = f(x) - 2x + f(f(y))
\]
for all real numbers \( x \) and \( y \). Hmm, functional equations can be tricky, but let me try to break this down step by step.
First, I remember that fu... | To find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy the functional equation:
\[
f(y - f(x)) = f(x) - 2x + f(f(y))
\]
for all \( x, y \in \mathbb{R} \), we proceed as follows:
1. **Assume a linear function**: Suppose \( f(x) = ax + b \). Substituting this into the equation and equating coe... |
Nous voulons trouver toutes les fonctions \( f: \mathbb{R} \rightarrow \mathbb{R} \) qui satisfont l'équation fonctionnelle suivante pour tous \( x, y \in \mathbb{R} \):
\[
f(y - f(x)) = f(x) - 2x + f(f(y))
\]
1. **Injectivité de \( f \)**:
Pour prouver que \( f \) est injective, supposons que \( f(a) = f(b... |
Points $M$ and $N$ are taken on the sides $AB$ and $BC$ respectively of triangle $ABC$. It turns out that the perimeter of $\triangle AMC$ is equal to the perimeter of $\triangle CNA$, and the perimeter of $\triangle ANB$ is equal to the perimeter of $\triangle CMB$. Prove that $\triangle ABC$ is isosceles. | Okay, so I need to prove that triangle ABC is isosceles given some conditions about the perimeters of smaller triangles. Let me try to understand the problem first.
We have triangle ABC, with points M on AB and N on BC. The perimeters of triangles AMC and CNA are equal. Also, the perimeters of triangles ANB and CMB ar... | Given triangle \(ABC\) with points \(M\) on \(AB\) and \(N\) on \(BC\), we know the perimeters of triangles \(AMC\) and \(CNA\) are equal, and the perimeters of triangles \(ANB\) and \(CMB\) are equal. We need to prove that \(\triangle ABC\) is isosceles.
1. **Perimeter Conditions**:
- Perimeter of \(\triangle AMC\... |
1. Denote the perimeter with the letter $P$. According to the conditions of the problem, we know the following equalities:
\[
P(\triangle AMC) = P(\triangle CAN)
\]
and
\[
P(\triangle ANB) = P(\triangle CMB)
\]
2. This implies that:
\[
P(\triangle AMC) + P(\triangle CMB) = P(\triangle C... |
Find all real numbers \( x, y, z \) such that
\[ x + y + z = 3, \quad x^2 + y^2 + z^2 = 3, \quad x^3 + y^3 + z^3 = 3 \] | Okay, so I have this problem where I need to find all real numbers x, y, z that satisfy three equations:
1. x + y + z = 3
2. x² + y² + z² = 3
3. x³ + y³ + z³ = 3
Hmm, let me think about how to approach this. I remember that when dealing with equations involving sums of powers, symmetric polynomials and maybe Newton's... | Given the system of equations:
\[
x + y + z = 3, \quad x^2 + y^2 + z^2 = 3, \quad x^3 + y^3 + z^3 = 3
\]
We start by using the elementary symmetric polynomials. Let \( S_1 = x + y + z \), \( S_2 = xy + yz + zx \), and \( S_3 = xyz \). From the given equations, we know:
1. \( S_1 = 3 \)
2. Using the identity \( x^2 +... |
1. **Identify and Define the Given Conditions:**
We are given the following equations involving the real numbers $x, y,$ and $z$:
\[
x + y + z = 3, \quad x^{2} + y^{2} + z^{2} = 3, \quad x^{3} + y^{3} + z^{3} = 3
\]
2. **Apply Newton's Sums:**
Use the relationships provided by Newton's sums on ... |
Let \( a, b, c \) be given complex numbers, with \( |a+b| = m \) and \( |a-b| = n \). Given that \( m \cdot n \neq 0 \), prove that:
$$
\max \{|ac+b|, |a+bc|\} \geqslant \frac{mn}{\sqrt{m^2 + n^2}}
$$ | Okay, so I need to prove that the maximum of |ac + b| and |a + bc| is at least mn over the square root of (m² + n²), where m is |a + b| and n is |a - b|, and a, b, c are complex numbers with m and n not zero. Hmm, let's start by understanding what each part means.
First, since a, b, c are complex numbers, all these ex... | Given complex numbers \(a, b, c\) with \(|a + b| = m\) and \(|a - b| = n\) where \(m \cdot n \neq 0\), we need to prove that:
$$
\max \{|ac + b|, |a + bc|\} \geq \frac{mn}{\sqrt{m^2 + n^2}}
$$
### Key Steps and Proof:
1. **Express \(m^2\) and \(n^2\) in terms of \(a\) and \(b\)**:
- \(m^2 = |a + b|^2 = |a|^2 + |b|... | Given that \( a, b, c \) are complex numbers with \( |a+b|=m \) and \( |a-b|=n \), and \( mn \neq 0 \), we need to prove:
\[
\max \{|ac+b|, |a+bc|\} \geq \frac{mn}{\sqrt{m^2+n^2}}
\]
#### Proof Method 1:
1. Considering the inequality:
\[
\max \{|ac+b|, |a+bc|\} \geq \frac{|b| \cdot |ac+b| + |a| \cdot |a+bc|}{|... |
Write the first $n$ natural numbers in decimal form on a (very long) strip of paper, then cut the strip so that each piece contains exactly one digit. Place these pieces in a box, mix them up, and draw one at random. Let $p_{n}$ denote the probability that the drawn piece of paper has the digit 0 on it. Determine the l... | Okay, so I need to find the limit of the probability p_n as n approaches infinity. The problem is about writing the first n natural numbers in decimal, cutting each digit into separate pieces, mixing them up, and then finding the probability that a randomly drawn piece has a 0. Then, we need to determine what happens t... | To determine the limit of the probability \( p_n \) as \( n \) approaches infinity, we need to analyze the distribution of the digit 0 in the decimal representations of the first \( n \) natural numbers.
1. **Total Number of Digits**:
- Numbers from 1 to \( n \) have varying digit lengths. For large \( n \), the t... |
1. **Define Variables and Notation:**
Let's analyze the problem by denoting the number of digits in the number \(n\) by \(k\). Each of the numbers less than \(n\) is padded with leading zeros to construct \(k\)-digit numbers. Suppose we consider all numbers from \(0\) to \(n\) with exactly \(k\) digits. The number ... |
Participants in a chess tournament played one game with each other. For each participant \( A \), the number of points they earned was calculated (1 point for a win, \( \frac{1}{2} \) point for a draw, and 0 points for a loss). Additionally, a strength coefficient was determined using the formula: the sum of the points... | Okay, so I need to figure out whether all participants in a chess tournament can have strength coefficients greater than 0 in part (a), and less than 0 in part (b). Let me start by understanding the problem thoroughly.
First, the setup: It's a chess tournament where each participant plays every other participant exact... | To determine whether the strength coefficients of all participants in a chess tournament can be greater than 0 (part a) or less than 0 (part b), we analyze various tournament setups and their resulting strength coefficients.
### Part (a): Can all strength coefficients be greater than 0?
1. **Initial Analysis**:
- ... | ### Problem
The participants of a chess tournament each played one game with every other participant. For each participant \( A \), the points they scored were counted (1 point for a win, \( -\frac{1}{2} \) point for a draw, and 0 points for a loss), as well as their strength coefficient, calculated by: the sum of the... |
How many plums will balance one pear, given that 3 apples and one pear weigh as much as 10 plums, and one apple and 6 plums balance one pear? Assume that fruits of the same kind have the same weight. | Okay, let's see. I need to figure out how many plums balance one pear. The problem gives me two equations involving apples, pears, and plums. All fruits of the same type weigh the same. Hmm, let's break it down step by step.
First, let me restate the information given. The problem says:
1. 3 apples and one pear weigh... | To determine how many plums balance one pear, we set up the following equations based on the problem statement:
1. \(3A + P = 10M\)
2. \(A + 6M = P\)
**Step 1: Substitute \(P\) from Equation 2 into Equation 1**
From Equation 2, \(P = A + 6M\). Substituting this into Equation 1:
\[
3A + (A + 6M) = 10M
\]
Sim... |
Given the problem, we are to find how many plums (sliv) balance with one pear (grusha). We have the following information:
- 3 apples (yabloka) and 1 pear (grusha) together balance 10 plums (sliv).
- 1 apple (yabloko) and 6 plums (slivy) balance 1 pear (grusha).
We begin by setting up equations based on the balance ... |
Let \(a \star b = ab - 2\). Compute the remainder when \((((579 \star 569) \star 559) \star \cdots \star 19) \star 9\) is divided by 100. | Okay, so I need to compute this expression where the operation a star b is defined as ab - 2. Then, after performing this operation repeatedly starting from 579 and 569, then taking the result and starring it with 559, and so on all the way down to 9, I have to find the remainder when the final number is divided by 100... | To solve the problem, we need to compute the remainder when the expression \((((579 \star 569) \star 559) \star \cdots \star 19) \star 9\) is divided by 100, where the operation \(a \star b = ab - 2\).
1. **Understanding the Operation**:
- The operation \(a \star b\) is defined as \(ab - 2\).
- We need to comput... |
Given the operation \(a \star b = ab - 2\), we need to compute the remainder when \(\left( ( ( (579 \star 569) \star 559) \star \cdots \star 19) \star 9 \right)\) is divided by 100.
We start by examining the operation \(\star\):
\[ (10a + 9) \star (10b + 9) = (10a + 9)(10b + 9) - 2 \]
\[ = 100ab + 90a + 90b + 81 - 2 ... |
Determine the value of
$$
z=a \sqrt{a} \sqrt[4]{a} \sqrt[8]{a} \ldots \sqrt[2^{n}]{a} \ldots
$$
if \( n \) is infinitely large. | Okay, so I need to find the value of z, which is this infinite product of a multiplied by the square root of a, then the fourth root, then the eighth root, and so on, each time the exponent being 1 over 2 to the nth power as n goes to infinity. Hmm, let's break this down step by step.
First, let's write out the expres... | The value of \( z \) is determined by recognizing the infinite product as a geometric series in the exponents.
**Step-by-Step Explanation:**
1. **Express Each Term as a Power of \( a \):**
- The given product is \( z = a \cdot \sqrt{a} \cdot \sqrt[4]{a} \cdot \sqrt[8]{a} \cdots \).
- Rewrite each root using ex... |
1. We begin by rewriting the given infinite product expression in a more manageable form. The given product is:
\[
z = a \sqrt{a} \sqrt[4]{a} \sqrt[8]{a} \cdots \sqrt[2^{n}]{a} \cdots
\]
2. Observe that each term in the product is of the form \( a^{1/2^k} \) where \( k \) is a non-negative integer. Hen... |
Give the value of \(0 - 1 + 2 - 3 + 4 - 5 + \ldots - 49 + 50\). Only a numerical answer is expected. | Okay, let me try to figure out the value of this series: 0 - 1 + 2 - 3 + 4 - 5 + ... - 49 + 50. Hmm, that's an alternating series where the signs alternate between negative and positive, starting with 0 minus 1. Let me write out the first few terms to see the pattern better.
So the series is: 0, then -1, +2, -3, +4, -... | The given series is \(0 - 1 + 2 - 3 + 4 - 5 + \ldots - 49 + 50\). To solve this, observe the alternating pattern and group terms in pairs:
\[
(-1 + 2) + (-3 + 4) + \ldots + (-49 + 50)
\]
Each pair simplifies to \(1\). There are \(25\) such pairs (since the numbers from \(1\) to \(49\) contain \(25\) odd numbers), lea... |
To find the value of the series \(0 - 1 + 2 - 3 + 4 - 5 + \ldots - 49 + 50\), we group the terms in pairs:
1. Group terms in pairs:
\[
(0 - 1) + (2 - 3) + (4 - 5) + \ldots + (48 - 49) + 50
\]
2. Each of these pairs, except for the last term 50, simplifies as follows:
\[
(0 - 1) = -1, \quad (2 - 3) = -1... |
Verify that 2 is a solution of the equation:
$$
(x+1)^{3}+(x+2)^{3}+(x+3)^{3}=(x+4)^{3}
$$
Does this equation have any other integer solutions? | Okay, so I need to verify if 2 is a solution to the equation (x+1)^3 + (x+2)^3 + (x+3)^3 = (x+4)^3. Then, I also have to check if there are any other integer solutions. Let me start by plugging in x=2 into the equation and see if both sides are equal.
First, let's compute each term when x=2.
Left side:
(2+1)^3 = 3^3... | To verify if \( x = 2 \) is a solution to the equation:
\[
(x+1)^3 + (x+2)^3 + (x+3)^3 = (x+4)^3
\]
**Step 1: Substitute \( x = 2 \) into both sides.**
- **Left Side:**
\[
(2+1)^3 + (2+2)^3 + (2+3)^3 = 3^3 + 4^3 + 5^3 = 27 + 64 + 125 = 216
\]
- **Right Side:**
\[
(2+4)^3 = 6^3 = 216
\]
Both sides e... |
The problem requires us to verify if \(x = 2\) is a solution to the equation \((x+1)^3 + (x+2)^3 + (x+3)^3 = (x+4)^3\), and to determine if there are other integer solutions.
Let's break down the detailed steps:
1. **Substitute \(x = 2\) into the equation**:
\[
(2+1)^3 + (2+2)^3 + (2+3)^3 = (2+4)^3
\]
... |
The sequence $\left\{a_{n}\right\}$ is defined as follows: $a_{1}=\frac{1}{2}$, and $a_{n+1}=\frac{a_{n}^{2}}{a_{n}^{2}-a_{n}+1}$ for $n=1,2,\ldots$. Prove that for every positive integer $n$, the inequality $a_{1}+a_{2}+\cdots+a_{n}<1$ holds. | Okay, so I need to prove that for every positive integer n, the sum of the first n terms of this sequence is less than 1. The sequence is defined by a₁ = 1/2, and then recursively by a_{n+1} = (a_n²)/(a_n² - a_n + 1). Hmm, let me start by understanding the sequence better. Maybe compute the first few terms to see if th... | To prove that for every positive integer \( n \), the inequality \( a_1 + a_2 + \cdots + a_n < 1 \) holds for the sequence defined by \( a_1 = \frac{1}{2} \) and \( a_{n+1} = \frac{a_n^2}{a_n^2 - a_n + 1} \), we start by examining the sequence and transforming it into a more manageable form.
First, we compute the firs... |
To prove that for every positive integer \( n \), we have \( a_{1} + a_{2} + \cdots + a_{n} < 1 \).
1. **Initialization**: Given the sequence \(\{a_n\}\) defined by:
\[
a_1 = \frac{1}{2}
\]
and
\[
a_{n+1} = \frac{a_n^2}{a_n^2 - a_n + 1}, \quad n=1,2,\ldots
\]
2. **Positivity of \(a_n\)**: Observ... |
The number of solutions to the equation \(\sin 12 x = x\) in the interval \([0, \pi)\) is | Okay, so I need to figure out how many solutions the equation sin(12x) = x has in the interval [0, π). Hmm, let's start by understanding the problem. We have two functions here: sin(12x) and x. The equation is asking where these two functions intersect between 0 and π.
First, let me recall that the sine function osci... | The equation \(\sin(12x) = x\) in the interval \([0, \pi)\) can be analyzed as follows:
1. **Interval Restriction**: Since \(\sin(12x)\) oscillates between \(-1\) and \(1\), valid solutions for \(x\) must lie in \([0, 1]\) (as \(x \geq 0\) and \(\sin(12x) \leq 1\)).
2. **Behavior of Functions**:
- \(\sin(12x)\) ha... |
To solve the equation $\sin(12x) = x$ in the interval $[0, \pi)$, we need to find the number of solutions. Here's the step-by-step approach:
1. **Understand the functions:**
- The function $y = \sin(12x)$ is a sinusoidal function with a frequency that is 12 times the standard sine function.
- The function $y = ... |
If the integer \( n \) is neither a multiple of 2 nor a multiple of 5, prove that \( n \) must be able to divide a number composed entirely of the digit 1. | Okay, so I need to prove that if an integer n isn't a multiple of 2 or 5, then there's some number made up entirely of 1's (like 1, 11, 111, etc.) that n divides. Hmm, interesting. Let me try to break this down.
First, let's restate the problem to make sure I understand it. If n is an integer such that n is not divisi... | To prove that if an integer \( n \) is neither a multiple of 2 nor 5, then \( n \) divides a number composed entirely of the digit 1 (a repunit), we proceed as follows:
**Proof:**
1. **Repunit Representation:** A repunit with \( k \) digits is \( R_k = \frac{10^k - 1}{9} \).
2. **Coprimality:** Since \( n \) is not ... |
1. **Understanding the Problem:**
- We need to prove that if an integer \( n \) is neither a multiple of 2 nor a multiple of 5, then \( n \) must divide some number that consists entirely of the digit 1.
2. **Formulation of the Sequence of Numbers:**
- Consider the sequence of numbers given by 1, 11, 111, ..., ... |
Rectangle \(ABCD\) is made up of six squares. The areas of two of the squares are shown. The perimeter of rectangle \(ABCD\), in centimetres, is:
(A) 50
(B) 44
(C) 46
(D) 52
(E) 48 | Okay, let's see. I need to find the perimeter of rectangle ABCD, which is made up of six squares. Two of the squares have their areas shown, which are 25 cm² and 16 cm². The options are given from 50 to 48, so I need to figure out the dimensions of the rectangle first to compute the perimeter.
First, since the areas o... | The rectangle is composed of six squares, including two with areas 25 cm² (side 5 cm) and 16 cm² (side 4 cm). To determine the perimeter, we deduce the rectangle's dimensions by analyzing the arrangement of squares.
1. **Identify Key Squares**: The 5 cm and 4 cm squares must align such that their sides contribute to ... | 1. Identify the side lengths of the known squares:
- Given: The area of the top-left square is \(36 \, \mathrm{cm^2}\).
- Calculate the side length:
\[
\text{Side length} = \sqrt{36 \, \mathrm{cm^2}} = 6 \, \mathrm{cm}
\]
- Given: The area of the bottom-left square is \(25 \, \mathrm{cm^2}\).
... |
A game begins with seven coins lined up on a table, all showing heads up. To win the game, you need to flip some coins such that, in the end, two adjacent coins always show different faces. The rule of the game is to flip two adjacent coins in each move. What is the minimum number of moves required to win the game? | Okay, let's try to figure out the minimum number of moves required to win this coin flipping game. The starting point is seven coins all showing heads. The goal is to have two adjacent coins always showing different faces, which means we need an alternating pattern like H-T-H-T-H-T-H or T-H-T-H-T-H-T. Since we start wi... | To determine the minimum number of moves required to achieve an alternating pattern (H-T-H-T-H-T-H or T-H-T-H-T-H-T) starting with seven heads (H-H-H-H-H-H-H), we analyze the problem through the following steps:
1. **Objective**: Flip coins such that no two adjacent coins have the same face. Since the starting configu... |
1. **Initial Setup**:
- We start with 7 coins, all showing heads (H). Assign the value \(1\) to heads (H) and \(-1\) to tails (T).
- Initially, the sum of values is \(7 \times 1 = 7\).
2. **Objective**:
- The goal is to flip some coins such that no two adjacent coins are heads.
- This means we should achi... |
The least integer that is greater than \((2+\sqrt{3})^{2}\) is:
(A) 13
(B) 14
(C) 15
(D) 16
(E) 17 | Okay, let me try to figure out this problem. The question is asking for the least integer that is greater than (2 + √3)². The options are from 13 to 17. Hmm, I need to compute (2 + √3) squared and then see which integer is the smallest one that's bigger than that value.
First, I remember that expanding a binomial squa... | To determine the least integer greater than \((2 + \sqrt{3})^2\), we start by expanding the expression using the binomial formula:
\[
(2 + \sqrt{3})^2 = 2^2 + 2 \cdot 2 \cdot \sqrt{3} + (\sqrt{3})^2
\]
Calculating each term separately:
- \(2^2 = 4\)
- \(2 \cdot 2 \cdot \sqrt{3} = 4\sqrt{3}\)
- \((\sqrt{3})^2 = 3\)
C... | 1. First, compute \((2+\sqrt{3})^2\):
\[
(2 + \sqrt{3})^2 = (2 + \sqrt{3})(2 + \sqrt{3}) = 2^2 + 2 \cdot 2 \cdot \sqrt{3} + (\sqrt{3})^2
\]
Breaking this down step-by-step:
\[
2^2 = 4
\]
\[
2 \cdot 2 \cdot \sqrt{3} = 4\sqrt{3}
\]
\[
(\sqrt{3})^2 = 3
\]
Adding these results toge... |
Find the function $F(x)$, whose graph passes through the point $M_{0}(3 ;-2)$, given that $F'(x) = 4x^{2} + 9x^{-2}$. | Okay, so I need to find the function F(x) whose graph passes through the point M₀(3, -2), and we know that the derivative F'(x) is 4x² + 9x⁻². Hmm, let's start by recalling that to find F(x), we need to integrate F'(x) with respect to x. Then, we can use the point M₀ to determine the constant of integration. Let me bre... | To find the function \( F(x) \) whose graph passes through the point \( M_0(3, -2) \) given that \( F'(x) = 4x^2 + 9x^{-2} \), we need to integrate \( F'(x) \) and then determine the constant of integration using the given point.
1. **Integrate \( F'(x) \)**:
- The integral of \( 4x^2 \) is \( \frac{4}{3}x^3 \).
... | To find the function \( F(x) \) whose graph passes through the point \( M_0(3, -2) \) and given that the derivative \( F'(x) \) is \( 4x^2 + 9x^{-2} \):
1. **Integrate the derivative**:
We know that:
\[
F'(x) = 4x^2 + 9x^{-2}
\]
To find \( F(x) \), we need to integrate \( F'(x) \):
\[
F(x) =... |
To traverse a straight path 10 meters long, a flea uses the following strategy: each day, it covers half of the remaining distance. Thus, it covers 5 meters on the first day, 2.5 meters on the second, and so on (the size of the flea can be disregarded).
(a) How many meters will it have covered by the end of the sevent... | Okay, let's tackle part (a) first. The problem says that a flea is trying to traverse a 10-meter path by covering half the remaining distance each day. So, on the first day, it goes 5 meters, leaving 5 meters. Then the next day, half of the remaining 5 meters, which is 2.5 meters, so total covered is 5 + 2.5 = 7.5 mete... | To solve the problem, we analyze the flea's movement using a geometric series and logarithmic calculations.
### Part (a)
The flea covers half of the remaining distance each day. This forms a geometric series with the first term \( a = 5 \) meters and common ratio \( r = 0.5 \). The sum of the first \( n \) terms of a... |
**Part (a):**
1. **Day-by-day analysis:**
The problem states that each day, the flea covers half the remaining distance. Thus:
- On the 1st day:
\[
10 \times \left(\frac{1}{2}\right) = 5 \text{ meters}
\]
- On the 2nd day:
\[
10 \times \left(\frac{1}{2}\right)^2 = 10 \times \f... |
In an isosceles trapezoid with bases \(a = 21\), \(b = 9\) and height \(h = 8\), find the radius of the circumscribed circle. | Okay, so I need to find the radius of the circumscribed circle around an isosceles trapezoid with bases a = 21, b = 9, and height h = 8. Hmm, first, let me recall some properties of trapezoids and circumscribed circles.
Wait, not all trapezoids can have a circumscribed circle. In fact, a trapezoid is cyclic (meaning ... | To find the radius of the circumscribed circle of an isosceles trapezoid with bases \(a = 21\), \(b = 9\), and height \(h = 8\), we follow these steps:
1. **Verify the trapezoid is cyclic**: Since it is isosceles, it is cyclic.
2. **Calculate the lengths of the non-parallel sides (legs)**:
- The difference between ... |
1. **Identify Given Data and Setup**:
- The given isosceles trapezoid \(ABCD\) has bases \(AD\) and \(BC\) with lengths \(a = 21\) and \(b = 9\) respectively, and height \(h = 8\).
- We need to find the radius \(R\) of the circumscribed circle around \(ABCD\).
2. **Drop Perpendicular and Calculate Lengths**:
... |
Given two linear functions \( f(x) \) and \( g(x) \) such that the graphs \( y = f(x) \) and \( y = g(x) \) are parallel lines that are not parallel to the coordinate axes. Find the minimum value of the function \( 3(g(x))^2 + 2 f(x) \), given that the minimum value of the function \( 3(f(x))^2 + 2 g(x) \) is \( -\frac... | Okay, so I need to find the minimum value of the function \( 3(g(x))^2 + 2 f(x) \), given that the minimum value of \( 3(f(x))^2 + 2 g(x) \) is \( -\frac{19}{6} \). Both \( f(x) \) and \( g(x) \) are linear functions whose graphs are parallel lines not parallel to the coordinate axes. Hmm, let me break this down step b... | Given two linear functions \( f(x) \) and \( g(x) \) such that their graphs are parallel lines not parallel to the coordinate axes, we need to find the minimum value of the function \( 3(g(x))^2 + 2 f(x) \), given that the minimum value of the function \( 3(f(x))^2 + 2 g(x) \) is \( -\frac{19}{6} \).
1. **Identify the... | :
Given are two linear functions \( f(x) \) and \( g(x) \) such that the graphs of \( y = f(x) \) and \( y = g(x) \) are parallel lines, and not parallel to the coordinate axes. We need to find the minimum value of the function \( 3(g(x))^2 + 2f(x) \), given that the minimum value of the function \( 3(f(x))^2 + 2g(x) ... |
Buratino calculated the time accurately and left Papa Carlo's house at 13:40 to reach the Field of Miracles and plant 4 coins exactly at sunset. If he had walked $25 \%$ faster, he would have arrived at the Field of Miracles 1.5 hours earlier and waited. At what time did Buratino arrive at the Field of Miracles? | Okay, let me try to figure out this problem. So, Buratino left Papa Carlo's house at 13:40 and wanted to plant 4 coins at sunset. The key information here is that if he had walked 25% faster, he would have arrived 1.5 hours earlier and had to wait. We need to find out the actual time he arrived at the Field of Miracles... | Buratino left Papa Carlo's house at 13:40 and needed to plant coins at sunset. If he walked 25% faster, he would arrive 1.5 hours earlier. Let \( v \) be his original speed and \( t \) be the original time taken.
When walking 25% faster, his speed is \( 1.25v \) and the time taken is \( t - 1.5 \). The distance is th... | 1. Let \( t \) be the time it takes Buratino to walk from Papa Carlo's house to the Field of Wonders at his usual speed.
2. According to the problem, if Buratino increases his speed by \( 25\% \), his new speed becomes:
\[
1.25 \text{ (usual speed)}
\]
3. The time taken at this increased speed would therefor... |
Given five distinct real numbers, consider all their pairwise sums. The smallest two sums are 20 and 200, while the largest two sums are 2014 and 2000. Determine the range of the smallest number, \( a \), among these five real numbers. | Okay, let's tackle this problem step by step. So, we have five distinct real numbers, and we're told about their pairwise sums. The smallest two sums are 20 and 200, and the largest two sums are 2014 and 2000. We need to find the range of the smallest number, a, among these five. Hmm, okay.
First, let me recall that w... | The range of the smallest number \( a \) among the five distinct real numbers is determined by analyzing the given pairwise sums and the constraints they impose. Denote the numbers as \( a < b < c < d < e \). The key steps are as follows:
1. **Identify the smallest and largest sums**:
- Smallest sums: \( a + b = 20... |
1. We start with the five distinct real numbers from smallest to largest: \( a, b, c, d, e \).
2. We know from the problem statement that the smallest sums of these numbers are:
\[
a + b = 20 \quad \text{and} \quad a + c = 200,
\]
and the largest sums are:
\[
d + e = 2014 \quad \text{and} \quad c + e... |
On a board, nine quadratic trinomials are written: \(x^{2}+a_{1} x+b_{1}, x^{2}+a_{2} x+b_{2}, \ldots, x^{2}+a_{9} x+b_{9}\). It is known that the sequences \(a_{1}, a_{2}, \ldots, a_{9}\) and \(b_{1}, b_{2}, \ldots, b_{9}\) are arithmetic progressions. It turns out that the sum of all nine trinomials has at least one ... | Okay, so there's this problem about nine quadratic trinomials on a board. Each trinomial is of the form x² + a_i x + b_i, where the coefficients a_i and b_i form arithmetic progressions. The sum of all nine trinomials has at least one root, and we need to find the maximum number of the original trinomials that may not ... | The problem involves nine quadratic trinomials \(x^2 + a_i x + b_i\), where \(a_i\) and \(b_i\) form arithmetic progressions. The sum of these trinomials has at least one root, and we aim to determine the maximum number of original trinomials that may not have roots.
### Key Steps and Analysis:
1. **Discriminant Condi... |
To solve this problem, we need to analyze the given quadratic trinomials and take into account the fact that their coefficients form arithmetic progressions.
1. **Define the Quadratic Polynomials:**
We have nine quadratic trinomials:
\[
P_i(x) = x^2 + a_i x + b_i \quad \text{for} \quad i = 1, 2, \ldots, 9
... |
Find the variance of the discrete random variable $X$ that is distributed according to the Poisson distribution:
$$
\begin{array}{ccccccc}
X & 0 & 1 & 2 & \cdots & k & \cdots \\
P & \mathrm{e}^{-\lambda} & \lambda \mathrm{e}^{-\lambda} / 1! & \lambda^{2} \mathrm{e}^{-\lambda} / 2! & \cdots & \lambda^{k} \mathrm{e}^{-\... | Okay, so I need to find the variance of a discrete random variable X that follows a Poisson distribution. Hmm, let's start by recalling what the Poisson distribution is. From what I remember, the Poisson distribution gives the probability of a certain number of events happening in a fixed interval of time or space, ass... | To find the variance of a discrete random variable \( X \) that follows a Poisson distribution, we start by noting that the probability mass function (PMF) of \( X \) is given by:
\[
P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \quad \text{for } k = 0, 1, 2, \ldots
\]
The variance of \( X \) is calculated using the f... |
To find the variance $D(X)$ of a discrete random variable $X$ which is Poisson-distributed with parameter $\lambda$, we can use the formula for the variance:
\[ D(X) = M\left(X^2\right) - [M(X)]^2 \]
1. **Expectation \( M(X) \)**:
For a Poisson-distributed random variable $X$ with parameter $\lambda$, the expec... |
Prove that
$$
\int_{0}^{\pi}|\ln \sin x| \, dx \leq \frac{\pi^{3}}{12}
$$
where $\ln$ denotes the natural logarithm with base $e = 2.71\ldots$. | Okay, I need to prove that the integral of the absolute value of the natural logarithm of sine x from 0 to pi is less than or equal to pi cubed over 12. Hmm, let me start by understanding the integral and the function involved.
First, the integrand is |ln sin x|. Since sine x is between 0 and 1 for x in (0, pi), the n... | To prove that
$$
\int_{0}^{\pi}|\ln \sin x| \, dx \leq \frac{\pi^{3}}{12},
$$
we proceed as follows:
**Step 1: Simplify the integral**
Since $\sin x \in (0, 1]$ for $x \in (0, \pi)$, $\ln \sin x \leq 0$. Thus, $|\ln \sin x| = -\ln \sin x$, and the integral becomes:
$$
\int_{0}^{\pi} -\ln \sin x \, dx.
$$
**Step 2: U... |
1. **Understanding the Integral:**
We start by noting that for \(x\) in the interval \([0, \pi]\), \(\sin x\) ranges from 0 to 1. Given this, \(\ln \sin x\) is negative (since \(\ln(\text{number between 0 and 1})\) is negative). Thus, \(|\ln \sin x| = -\ln \sin x\).
2. **Symmetry Utilization and Transformation:**
... |
Prove that in a right-angled triangle, the distance between the incenter and the circumcenter is at least $(\sqrt{2}-1)$ times the radius of the circumcircle. | Okay, so I need to prove that in a right-angled triangle, the distance between the incenter and the circumcenter is at least (√2 - 1) times the radius of the circumcircle. Hmm, let me start by recalling some properties of right-angled triangles, incenter, and circumcenter.
First, in a right-angled triangle, the circum... | To prove that in a right-angled triangle, the distance between the incenter and the circumcenter is at least \((\sqrt{2} - 1)\) times the radius of the circumcircle, we start by considering a right-angled triangle \(ABC\) with the right angle at \(C\). Let the legs be \(a\) and \(b\), and the hypotenuse be \(c\). The c... |
1. **Identification of Geometric Elements**:
- Let's consider a right triangle \(ABC\), where \(\angle C = 90^\circ\).
- Let \(O_1\) be the center of the incircle of \( \triangle ABC \) with radius \( r \).
- Let \( O_2 \) be the center of the circumcircle of \( \triangle ABC \). Since \( \triangle ABC \) is ... |
Calculate the volume of the body bounded above by the surface \(z = xy^2\) and below by the rectangle \(0 \leq x \leq 1\), \(0 \leq y \leq 2\). | Okay, so I need to find the volume bounded above by the surface z = xy² and below by the rectangle defined by 0 ≤ x ≤ 1 and 0 ≤ y ≤ 2. Hmm, volume under a surface over a rectangular region. I remember that this involves a double integral, right? Let me recall how that works.
First, the volume under a surface z = f(x, ... | To calculate the volume of the body bounded above by the surface \(z = xy^2\) and below by the rectangle \(0 \leq x \leq 1\), \(0 \leq y \leq 2\), we need to compute the double integral of \(xy^2\) over the given rectangular region.
First, we set up the double integral. Since the region is a rectangle, we can integrat... |
1. **Identify the problem:**
To compute the volume of the solid bounded above by the surface \(z = x y^2\) and below by the rectangle \(0 \leq x \leq 1, 0 \leq y \leq 2\).
2. **Setup the double integral:**
The volume \(V\) of the solid can be computed using the double integral over the given region \(D\):
\[... |
Vasya cut a triangle out of cardboard and numbered its vertices with the digits $1, 2, 3$. It turned out that if Vasya rotates the triangle 12 times clockwise around its vertex numbered 1 by an angle equal to the angle at this vertex, it will return to its original position. If Vasya rotates the triangle 6 times clock... | Okay, let's try to figure out this problem. So, Vasya has a triangle with vertices numbered 1, 2, 3. When he rotates the triangle around vertex 1 twelve times, it returns to its original position. Similarly, rotating around vertex 2 six times brings it back. We need to find the minimal n such that rotating around verte... | Vasya cut a triangle out of cardboard with vertices numbered 1, 2, and 3. The problem states that rotating the triangle 12 times clockwise around vertex 1 by its angle returns it to its original position, and rotating it 6 times clockwise around vertex 2 by its angle also returns it. We need to find the minimal \( n \)... |
1. We observe that if turning Vasya's triangle 12 times around the first vertex returns it to its original position, the sum total of the angles turned equals \(360^\circ\). This tells us that the angle at the first vertex must be at least \(360^\circ / 12 = 30^\circ\).
2. Similarly, rotating the triangle 6 times aro... |
The tower in King Arthur's castle is topped with a roof that represents a triangular pyramid in which all plane angles at the apex are right angles. The three roof slopes are painted in different colors. The red roof slope is inclined to the horizontal at an angle $\alpha$, and the blue slope at an angle $\beta$. Find ... | Okay, let's try to tackle this problem. So, we have a tower with a roof shaped like a triangular pyramid where all the plane angles at the apex are right angles. That probably means the three edges meeting at the apex are perpendicular to each other. The three roof slopes are red, blue, and green. We know the angles of... | To solve the problem of finding the probability that a raindrop falls onto the green slope of a triangular pyramid roof with all plane angles at the apex being right angles, we need to consider the following steps:
1. **Understanding the Geometry**: The roof is a trirectangular tetrahedron with three mutually perpendi... | 1. **Understanding the Problem**:
We are given a triangular pyramid (tetrahedron) where each face is perpendicular to the adjacent face. The faces are painted in three different colors: red, blue, and green. We need to find the probability that a raindrop falling vertically on the roof lands on the green face.
2. ... |
It is known that the length of the segment representing the product of two segments depends on the choice of the unit segment. How does the length of the product segment change if the unit segment is halved? | Okay, let's tackle this problem step by step. The question is about how the length of a product segment changes when the unit segment is halved. Hmm, first, I need to make sure I understand what a "product of two segments" means. I remember from geometry that you can represent the product of two line segments using ano... | When the unit segment is halved, the numerical length of the product segment **doubles**. Here's the systematic breakdown:
1. **Original Unit (U):** Let the product of segments \(a\) and \(b\) be \(ab\) in units of \(U\).
2. **New Unit (\(U'\)):** \(U' = \frac{1}{2}U\), so \(1U = 2U'\).
3. **Conversion:** The physical... |
1. Let the two given segments be \(a\) and \(b\). Denote the unit segment with \(e\).
2. We have the following diagram illustrating the relationship of the segments:
3. According ... |
A firecracker was thrown vertically upward with a speed of \(20 \text{ m/s}\). One second after the flight began, it exploded into two fragments of equal mass. The first fragment flew horizontally immediately after the explosion with a speed of 48 m/s. Find the magnitude of the speed of the second fragment (in m/s) imm... | Okay, let's try to solve this problem step by step. So, there's a firecracker thrown upward with 20 m/s. Then, after one second, it explodes into two equal mass fragments. One fragment flies horizontally at 48 m/s. We need to find the speed of the second fragment right after the explosion. Hmm, okay, let's break this d... | First, we determine the velocity of the firecracker just before the explosion. Using the equation \( v = u + at \), where \( u = 20 \, \text{m/s} \), \( a = -10 \, \text{m/s}^2 \), and \( t = 1 \, \text{s} \):
\[
v = 20 \, \text{m/s} - 10 \, \text{m/s}^2 \cdot 1 \, \text{s} = 10 \, \text{m/s}
\]
The firecracker is mo... |
1. **Initial vertical velocity of the firecracker:**
The firecracker is thrown vertically upward with an initial velocity of \( v_0 = 20 \ \mathrm{m/s} \).
2. **Vertical velocity after 1 second:**
After 1 second, the vertical velocity \( v_y \) can be found using the kinematic equation:
\[
v_y = v_0 - gt
... |
Find the sum \( S_{n} = \sin \alpha \cdot \sin ^{2} \frac{\alpha}{2} + 2 \sin \frac{\alpha}{2} \cdot \sin ^{2} \frac{\alpha}{4} + \cdots + 2^{n-1} \cdot \sin \frac{\alpha}{2^{n-1}} \cdot \sin ^{2} \frac{\alpha}{2^{n}} \). | Okay, let me try to figure out how to find the sum \( S_n = \sin \alpha \cdot \sin^2 \frac{\alpha}{2} + 2 \sin \frac{\alpha}{2} \cdot \sin^2 \frac{\alpha}{4} + \cdots + 2^{n-1} \cdot \sin \frac{\alpha}{2^{n-1}} \cdot \sin^2 \frac{\alpha}{2^{n}} \). Hmm, this looks like a trigonometric series with each term involving a ... | To find the sum \( S_n = \sin \alpha \cdot \sin^2 \frac{\alpha}{2} + 2 \sin \frac{\alpha}{2} \cdot \sin^2 \frac{\alpha}{4} + \cdots + 2^{n-1} \cdot \sin \frac{\alpha}{2^{n-1}} \cdot \sin^2 \frac{\alpha}{2^{n}} \), we start by analyzing the general term of the series.
The k-th term is given by \( 2^{k-1} \cdot \sin \f... |
1. The given series is:
\[
S_{n} = \sin \alpha \cdot \sin^2 \frac{\alpha}{2} + 2 \sin \frac{\alpha}{2} \cdot \sin^2 \frac{\alpha}{4} + 2^{2-1} \sin \frac{\alpha}{2^2} \cdot \sin^2 \frac{\alpha}{2^3} + \cdots + 2^{n-1} \sin \frac{\alpha}{2^{n-1}} \cdot \sin^2 \frac{\alpha}{2^n}
\]
2. We will use trigonometri... |
On the sides of a convex centrally symmetric hexagon \( A B C D E F \), equilateral triangles are constructed externally. Prove that the midpoints of the segments connecting the vertices of adjacent triangles form a regular hexagon. | Okay, so I need to prove that if we take a convex centrally symmetric hexagon ABCDEF, build equilateral triangles on each of its sides externally, and then connect the vertices of adjacent triangles with segments, the midpoints of those segments form a regular hexagon. Hmm, that sounds a bit complex, but let's break it... | To prove that the midpoints of the segments connecting the vertices of adjacent equilateral triangles constructed externally on a convex centrally symmetric hexagon form a regular hexagon, we proceed as follows:
**Step 1: Represent the Hexagon with Central Symmetry**
Let the hexagon be \( A, B, C, D, E, F \), with \( ... | 1. **Given:**
- A convex centrally symmetric hexagon \( ABCDEF \) with equilateral triangles constructed externally on its sides.
- We need to prove that the midpoints of the segments connecting the vertices of neighboring triangles form a regular hexagon.
2. **Vertices of Equilateral Triangles:**
- Let \( K,... |
A person writes \( n \) letters for \( n \) individuals, each receiving one letter, and prepares \( n \) envelopes labeled with the recipients' names and addresses. How many different ways are there to place the letters into the envelopes (one letter per envelope) such that no letter is placed in the envelope designate... | Okay, let's try to figure out this problem. So, we have n letters and n envelopes, each labeled for a specific person. The question is asking how many ways we can put the letters into the envelopes so that no letter ends up in its corresponding envelope. Hmm, this seems familiar. I think it's called something like dera... | The number of ways to place \( n \) letters into \( n \) envelopes such that no letter is in its designated envelope is given by the derangement number \( D(n) \). This can be calculated using the formula:
\[
D(n) = n! \left(1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \cdots + (-1)^n \frac{1}{n!}\right)
\]
This ... |
1. Let's denote the set of all different ways to distribute the letters as $S$.
2. $A_{i}$ represents the set of distributions where the $i$-th letter is correctly placed in the envelope addressed to the $i$-th person.
3. Clearly, the total number of permutations of $n$ letters is $|S| = n!$.
4. We next calcula... |
Given $x$ and $y$ as positive real numbers such that $x \leq y$, show that $\frac{1}{y^{2}} \leq \frac{1}{x^{2}} | Okay, so I need to show that if x and y are positive real numbers and x is less than or equal to y, then 1 over y squared is less than or equal to 1 over x squared. Let me start by understanding the problem step by step.
First, we know that both x and y are positive, so neither of them is zero or negative. That’s impo... | Given that \( x \) and \( y \) are positive real numbers with \( x \leq y \), we aim to show \( \frac{1}{y^2} \leq \frac{1}{x^2} \).
**Step-by-Step Explanation:**
1. **Start with the given inequality:**
Since \( x \leq y \) and both are positive, taking reciprocals reverses the inequality:
\[
\frac{1}{x}... |
1. **Given**: We have two positive real numbers \(x\) and \(y\) such that \(x \leq y\).
2. **Squaring both sides**:
\[
x \leq y \implies x^2 \leq y^2
\]
This step uses the fact that the function \(f(z) = z^2\) is monotonically increasing for non-negative real numbers.
3. **Taking reciprocals**:
\[
... |
Find the smallest number \( n \) such that given any \( n \) distinct numbers from \(\{1, 2, 3, \ldots, 999\}\), one can choose four different numbers \( a, b, c, d \) such that \( a + 2b + 3c = d \). | Okay, so I need to find the smallest number n such that if I pick any n distinct numbers from the set {1, 2, 3, ..., 999}, there will always be four distinct numbers a, b, c, d in that selection where a + 2b + 3c = d. Hmm, this sounds like a problem from combinatorics, maybe related to the pigeonhole principle or Ramse... | To find the smallest number \( n \) such that any \( n \) distinct numbers from \(\{1, 2, 3, \ldots, 999\}\) must contain four distinct numbers \( a, b, c, d \) satisfying \( a + 2b + 3c = d \), we need to consider the structure of the set and the equation.
1. **Analysis of the Equation**:
- The equation \( a + 2b ... | ### Problem:
Find the smallest number \( n \) such that given any \( n \) distinct numbers from \(\{1, 2, 3, \ldots, 999\}\), one can choose four different numbers \( a, b, c, d \) such that \( a + 2b + 3c = d \).
1. **Choosing \( n \) and defining the possible sums \( a + 2b + 3c \):**
Consider the set \( S = \{16... |
On February 1, it was 16.2°C outside Jacinta's house at 3:00 p.m. On February 2, it was -3.6°C outside Jacinta's house at 2:00 a.m. If the temperature changed at a constant rate between these times, the rate at which the temperature decreased was
(A) 1.1°C / h
(B) 1.8°C / h
(C) 2.2°C / h
(D) 3.6°C / h
(E) 4.4°C / h | Okay, so the problem is about figuring out the rate at which the temperature decreased between two specific times. Let me start by understanding the given information.
First, on February 1, the temperature was 16.2°C at 3:00 p.m. Then, on February 2, it was -3.6°C at 2:00 a.m. The question states that the temperature ... | To determine the rate at which the temperature decreased, follow these steps:
1. **Calculate the total time elapsed**:
From February 1, 3:00 p.m. to February 2, 2:00 a.m.:
- 3:00 p.m. → 12:00 a.m. (midnight): **9 hours**
- 12:00 a.m. → 2:00 a.m.: **2 hours**
Total time = \( 9 + 2 = 11 \, \text{hour... |
1. **Determine the total change in temperature:**
To find the total decrease in temperature from \( 16.2^{\circ}\mathrm{C} \) to \( -3.6^{\circ}\mathrm{C} \), calculate:
\[
\Delta T = 16.2^{\circ}\mathrm{C} - (-3.6^{\circ}\mathrm{C}) = 16.2^{\circ}\mathrm{C} + 3.6^{\circ}\mathrm{C} = 19.8^{\circ}\mathrm{C}
... |
If \( n \) is a positive integer, the notation \( n! \) (read " \( n \) factorial") represents the product of the integers from 1 to \( n \) inclusive. For example, \( 5! = 1 \times 2 \times 3 \times 4 \times 5 = 120 \). Which of the following is equal to a perfect square?
(A) \(\frac{(20!)(19!)}{1}\)
(B) \(\frac{(20!)... | Okay, so I need to figure out which of the options (A) through (E) is equal to a perfect square. Each option is the product of 20! and 19! divided by a different integer from 1 to 5. Let me start by recalling that a perfect square is an integer that can be expressed as some integer squared, which means all the exponent... | To determine which of the given options is a perfect square, we start by analyzing the expression \((20!)(19!)\). This can be rewritten as \(20 \times (19!)^2\). We need to check the effect of dividing this product by each of the integers 1, 2, 3, 4, and 5 on the prime factorization, ensuring all exponents remain even.... |
1. Recall the definition of a factorial. For a positive integer $n$, the notation $n!$ is used to represent the product of the integers from 1 to $n$ inclusive. For example, $5! = 1 \times 2 \times 3 \times 4 \times 5 = 120$.
2. To solve the given problem, we need to determine which expression is equal to a perfect... |
Find all positive integers \( n \) such that the set \( \{ n, n+1, n+2, n+3, n+4, n+5 \} \) can be partitioned into two subsets so that the product of the numbers in each subset is equal. | Okay, so I need to find all positive integers n such that the set {n, n+1, n+2, n+3, n+4, n+5} can be partitioned into two subsets with equal products. Hmm, this seems like a number theory problem involving partitioning and products. Let me start by understanding the problem step by step.
First, partitioning the set i... | To solve the problem of finding all positive integers \( n \) such that the set \( \{ n, n+1, n+2, n+3, n+4, n+5 \} \) can be partitioned into two subsets with equal products, we analyze the conditions required for such a partition.
### Key Observations:
1. **Product Must Be a Perfect Square**: For the product of the ... | To solve this problem, we need to find all positive integers \( n \) such that the set \(\{n, n+1, n+2, n+3, n+4, n+5\}\) can be partitioned into two subsets with equal product.
1. **Identify the Possible Prime Factors:**
Since we want the product of the numbers in each subset to be equal, we need to consider the p... |
The real numbers \( x_{1}, x_{2}, \ldots, x_{1996} \) have the following property: for any polynomial \( W \) of degree 2, at least three of the numbers \( W(x_{1}), W(x_{2}), \ldots, W(x_{1996}) \) are equal. Prove that at least three of the numbers \( x_{1}, x_{2}, \ldots, x_{1996} \) are equal. | Okay, so I need to prove that if you have 1996 real numbers x₁, x₂, ..., x₁₉₉₆ such that for any quadratic polynomial W, at least three of the values W(x₁), W(x₂), ..., W(x₁₉₉₆) are equal, then at least three of the original numbers x₁, x₂, ..., x₁₉₉₆ must be equal. Hmm, interesting. Let me try to break this down step ... | To prove that at least three of the numbers \( x_1, x_2, \ldots, x_{1996} \) are equal, we proceed by contradiction.
**Assumption**: Suppose no three \( x_i \) are equal. Then, each value in the multiset \( \{x_1, x_2, \ldots, x_{1996}\} \) appears at most twice. This implies there are at least \( 1996/2 = 998 \) dist... |
1. **Introduction of Variable $m$:**
Let us define \( m \) as the minimum of the given real numbers \( x_1, x_2, \ldots, x_{1996} \):
\[
m = \min \{x_1, x_2, \ldots, x_{1996}\}.
\]
2. **Consider a Specific Polynomial:**
Consider the polynomial \( W(x) = (x - m)^2 \).
3. **Properties of the Polynomial:... |
Given that $O$ is the circumcenter of an acute-angled triangle $ABC$, and $\angle A = \theta$. If $\frac{\cos B}{\sin C} AB + \frac{\cos C}{\sin B} AC = 2m \cdot AO$, then find the value of $m$. | Okay, so I need to find the value of \( m \) given the equation involving the circumcenter \( O \) of an acute-angled triangle \( ABC \) with angle \( A = \theta \). The equation is:
\[
\frac{\cos B}{\sin C} AB + \frac{\cos C}{\sin B} AC = 2m \cdot AO
\]
First, let me recall some properties of triangles and circumcen... | Given that \( O \) is the circumcenter of an acute-angled triangle \( ABC \) with \( \angle A = \theta \), we need to find the value of \( m \) such that:
\[
\frac{\cos B}{\sin C} AB + \frac{\cos C}{\sin B} AC = 2m \cdot AO
\]
1. **Using the Law of Sines**:
- \( AB = 2R \sin C \)
- \( AC = 2R \sin B \)
- \( ... |
Given: \(O\) is the circumcenter of the acute-angled triangle \( \triangle ABC \) and \( \angle A = \theta \). We know:
\[
\frac{\cos B}{\sin C} AB + \frac{\cos C}{\sin B} AC = 2 m AO
\]
We are to find \( m \) in terms of \( \sin \theta \).
1. Consider the triangle \( \triangle ABC \) with circumcenter \( O \). T... |
What is the largest result we can find when we divide a 2-digit number by the sum of its digits? | Okay, so I need to find the largest result when dividing a two-digit number by the sum of its digits. Let me start by understanding the problem. A two-digit number can be represented as 10a + b, where a is the tens digit and b is the units digit. The sum of its digits would then be a + b. The task is to maximize the va... | The largest result when dividing a two-digit number by the sum of its digits is **10**. This occurs for numbers where the units digit is 0 (e.g., 10, 20, ..., 90). For these numbers, the ratio simplifies as follows:
\[
\frac{10a + 0}{a + 0} = \frac{10a}{a} = 10
\]
**Key Steps:**
1. **Representation**: Let the two-dig... | 1. Let's define the problem more formally. We want to find the maximum value of the expression:
\[
\frac{10a + b}{a + b}
\]
where \(a\) and \(b\) are digits of a two-digit number (1 through 9 for \(a\) and 0 through 9 for \(b\)) and not both zero.
2. We start by simplifying the expression. Observe that:
\[
\frac{10... |
Given an isosceles triangle with a base \(a\), a leg \(b\), and a vertex angle equal to \(12^{\circ}\). Prove that \(b < 5a\). | Okay, so I need to prove that in an isosceles triangle with base \(a\), legs \(b\), and a vertex angle of \(12^\circ\), the leg \(b\) is less than five times the base \(a\), that is, \(b < 5a\). Let me start by recalling some basic properties of triangles and trigonometric relationships. Since it's an isosceles triangl... | To prove that \(b < 5a\) in an isosceles triangle with base \(a\), legs \(b\), and a vertex angle of \(12^\circ\), follow these steps:
1. **Apply the Law of Sines**:
\[
\frac{a}{\sin 12^\circ} = \frac{b}{\sin 84^\circ}
\]
Solving for \(b\):
\[
b = a \cdot \frac{\sin 84^\circ}{\sin 12^\circ}
\]
2.... | 1. **Construct an equilateral triangle:**
- Start with the given isosceles triangle \( \triangle ABC \), where the base is \( a \), the sides are \( b \), and the vertex angle is \( 12^\circ \).
- To simplify things, construct the triangles and put together until form an equilateral triangle.
2. **Create additio... |
In a class of \( n \geq 3 \) students, the teacher has \( m = \frac{1}{2} n(n-1) \) cards, each with a unique number from \( 1, 2, 3, \ldots, m \). The teacher gives each student one card. For each pair of students, the sum of the numbers on their cards is calculated and written on the board. It turns out that the numb... | Okay, let me try to work through this problem step by step. So, the problem is about a class of n students, each receiving a unique card numbered from 1 to m, where m is equal to n(n-1)/2. Then, for every pair of students, the sum of their cards is calculated, and all these sums give different remainders when divided b... | To prove that if \( m \) is even, then \( n \) must be a perfect square, we analyze the given conditions systematically:
1. **Definitions and Initial Setup**:
- \( m = \frac{n(n-1)}{2} \), which is even.
- Each pair of students' card sums yields distinct remainders modulo \( m \).
2. **Key Observations**:
- ... |
1. **Problem Setup and Known Information:**
We start by noting that there are $n \geq 3$ students in the class. The teacher has $m = \frac{1}{2}n(n-1)$ cards, each labeled with a unique number from 1 to $m$. Each student receives one card, and all pairwise sums of these cards leave distinct remainders when divided ... |
Show that for every positive integer \( n \), there exists a nonzero multiple of \( n \) that has at most \( n \) digits, all of which are either 0 or 1. | Okay, so I need to show that for every positive integer n, there's a nonzero multiple of n made up only of 0s and 1s, and it has at most n digits. Hmm, interesting problem. Let me think about how to approach this.
First, let me understand the problem statement. For any n, like 2, 3, 4, etc., there should be some multi... | To show that for every positive integer \( n \), there exists a nonzero multiple of \( n \) that has at most \( n \) digits, all of which are either 0 or 1, we can use the pigeonhole principle as follows:
1. **Consider the sequence of numbers**: For each positive integer \( n \), consider the numbers \( a_1, a_2, \ldo... |
1. **Select the Objects and Define the Problem:**
Consider the sequence of numbers:
\[
a_0 = 0, \quad a_1 = 1, \quad a_2 = 11, \quad a_3 = 111, \quad \ldots, \quad a_n = \underbrace{111 \ldots 1}_{n \text{ ones}}
\]
These numbers \(a_i\) are represented with digits that are either 0 or 1. Specifically,... |
A rhombus with a side length of 6 cm is inscribed in a right triangle with a $60^{\circ}$ angle, such that the $60^{\circ}$ angle is common to both shapes and all vertices of the rhombus lie on the sides of the triangle. Find the sides of the triangle. | Okay, let's see. I need to find the sides of a right triangle that has a 60-degree angle, and a rhombus with side length 6 cm is inscribed in it. The rhombus shares the 60-degree angle with the triangle, and all its vertices lie on the sides of the triangle. Hmm, this sounds a bit complex, but let's break it down step ... | Given a rhombus with a side length of 6 cm inscribed in a right triangle with a \(60^\circ\) angle, where the \(60^\circ\) angle is common to both shapes and all vertices of the rhombus lie on the sides of the triangle, we need to find the sides of the triangle.
1. **Identify the Triangle Type**:
- The triangle is ... | 1. Consider the right-angled triangle $\triangle ABC$ with the angle $\angle BAC = 60^\circ$. A rhombus $FBDE$ is inscribed such that one of its angles coincides with the $60^\circ$ angle of the triangle, and each of its sides has a length of 6 cm.
2. Observe that $\triangle FBD$ is a right-angled triangle with $\a... |
Let \( F(x) \) be the antiderivative of the function \( f(x) \) on the interval \([a, b]\), and let \( \varphi(y) \) be a differentiable function on the interval \([p, q]\). Suppose \( a \leq \varphi(y) \leq b \) for all \( y \) in the interval \([p, q]\), and for any point \( y_0 \) in the interval \([p, q]\), there e... | Okay, so I need to prove that F(φ(y)) is an antiderivative of f(φ(y))φ'(y). Let me start by recalling some calculus concepts. An antiderivative of a function is another function whose derivative is the original function. Here, F is the antiderivative of f, so by definition, F'(x) = f(x). The problem wants me to show th... | To prove that \( F(\varphi(y)) \) is an antiderivative of \( f(\varphi(y)) \varphi'(y) \), we proceed as follows:
**Step 1: Fundamental Property of Antiderivatives**
By definition, \( F(x) \) is an antiderivative of \( f(x) \) on \([a, b]\), meaning:
\[
F'(x) = f(x) \quad \forall x \in [a, b].
\]
**Step 2: Differ... | 1. Given that \( F(x) \) is the antiderivative of \( f(x) \) on the interval \([a, b]\), and \( \varphi(y) \) is a differentiable function on the interval \([p, q]\), we know that \( a \leq \varphi(y) \leq b \) for all \( y \in [p, q] \).
2. Define the function \( G(y) = F(\varphi(y)) \). We need to show that \( G(y) \... |
The probability of an event $A$ occurring is $v=0.1$. We conduct $n=998$ trials. Which is more likely; that $A$ occurs 100 times, or that it occurs 99 times? | Okay, let's try to figure out which is more likely: event A occurring 100 times or 99 times out of 998 trials when the probability of A is 0.1. Hmm, this seems like a probability question involving binomial distributions. Let me recall what I know about binomial probabilities.
The probability of getting exactly k succ... | To determine whether event \( A \) is more likely to occur 100 times or 99 times in 998 trials with probability \( v = 0.1 \), we analyze the binomial probabilities \( P(100) \) and \( P(99) \).
**Key Steps:**
1. **Binomial Probability Ratio:**
The ratio of probabilities for \( k+1 \) and \( k \) successes is: ... |
1. **Understanding the Probability Formula**:
- The probability that an event $A$ occurs exactly $k$ times out of $n$ trials, where the probability of success in each trial is $v$, is given by the binomial distribution formula:
\[
V_{n/k} = \binom{n}{k} v^k (1-v)^{n-k}
\]
- Here, $\binom{n}{k}$ re... |
A person is practicing target shooting, starting at a distance of 100 meters from the target for the first shot. If the first shot misses, they move back 50 meters for the second shot and continue this pattern; each time they miss, they move back 50 meters and take another shot until they hit the target. Given that the... | Okay, let's try to figure out this probability problem. So, the person is shooting at a target starting from 100 meters. If they miss, they move back 50 meters each time and try again. The probability of hitting on the first shot is 1/4, and the probability is inversely proportional to the square of the distance. We ne... | The person starts target shooting at 100 meters and moves back 50 meters each time they miss. The probability of hitting the target is inversely proportional to the square of the distance. Given the probability of hitting on the first shot is \(\frac{1}{4}\), we need to find the probability they will eventually hit the... |
Given a person practicing archery, starting at a distance of 100 meters from the target and moving back 50 meters after each miss, we need to determine the probability that this person hits the target eventually. The probability of hitting the target for the first time is given as \(\frac{1}{4}\), and this probability... |
\((\cos x - \sin x)^2 + \cos^4 x - \sin^4 x = 0.5 \sin 4x\) | Okay, so I have this trigonometric equation to solve: \((\cos x - \sin x)^2 + \cos^4 x - \sin^4 x = 0.5 \sin 4x\). Hmm, let me try to figure out how to approach this step by step.
First, I need to simplify both sides of the equation to see if I can make them look similar or find some common terms. Let's start with th... | To solve the trigonometric equation \((\cos x - \sin x)^2 + \cos^4 x - \sin^4 x = 0.5 \sin 4x\), we start by simplifying both sides.
First, expand and simplify the left-hand side (LHS):
\[
(\cos x - \sin x)^2 = \cos^2 x - 2 \sin x \cos x + \sin^2 x = 1 - \sin 2x
\]
\[
\cos^4 x - \sin^4 x = (\cos^2 x - \sin^2 x)(\cos^2... | Given the equation:
$$(\cos x - \sin x)^2 + \cos^4 x - \sin^4 x = 0.5 \sin 4x.$$
We start by expanding and simplifying the left-hand side (LHS) of the equation:
1. Expand $(\cos x - \sin x)^2$ using the binomial expansion:
\[
(\cos x - \sin x)^2 = (\cos^2 x - 2 \cos x \sin x + \sin^2 x).
\]
2. We recognize ... |
The line \( l \) intersects \( \triangle ABC \)'s side \( AB \) at \( B_1 \) and \( AC \) at \( C_1 \), with the centroid \( G \) of \( \triangle ABC \) and point \( A \) on the same side of \( l \). Prove that \( S_{B_1GC_1} + S_{C_1GB_1} \geq \frac{4}{9} S_{\triangle ABC} \). | Okay, so I need to prove that the sum of the areas of triangles \( B_1GC_1 \) and \( C_1GB_1 \) is at least \( \frac{4}{9} \) of the area of triangle \( ABC \). Wait, hold on, the problem says \( S_{B_1GC_1} + S_{C_1GB_1} \), but that seems like the same area twice. Maybe that's a typo? Let me check the original proble... | The problem statement likely contains an error or misinterpretation, as demonstrated through coordinate analysis and area calculations. The sum \( S_{B_1GC_1} + S_{C_1GB_1} \) can be shown to approach zero under the given conditions, contradicting the stated inequality. Therefore, the original problem may require revis... | To understand this problem and solution, we need to break down the geometry and the calculations. We seek to prove that:
\[ S_{B B_1 G C_1} + S_{C C_1 G B_1} \geq \frac{4}{9} S_{\triangle ABC} \]
Here is the solution in a detailed step-by-step format.
1. **Define Midpoint D**:
Let \( D \) be the midpoint of side \( ... |
In the regular hexagonal prism \(ABCDEF - A'B'C'D'E'F'\), \(M\) is the midpoint of \(DE\). Determine the shape of the cross-section passing through the points \(A'\), \(C\), and \(M\).
A. Triangle
B. Quadrilateral
C. Pentagon
D. Hexagon | Alright, so I need to figure out the shape of the cross-section that passes through the points A', C, and M in a regular hexagonal prism. The options are triangle, quadrilateral, pentagon, or hexagon. Let me start by visualizing the prism and the positions of these points.
First, a regular hexagonal prism has two cong... | To determine the shape of the cross-section passing through the points \(A'\), \(C\), and \(M\) in a regular hexagonal prism \(ABCDEF - A'B'C'D'E'F'\), we need to analyze the intersections of the plane with the prism's edges.
1. **Identify the Points**:
- \(A'\) is on the bottom base.
- \(C\) is on the top base.... |
To determine the shape of the cross-section in regular hexagonal prism \(A B C D E F-A^{\prime} B^{\prime} C^{\prime} D^{\prime} E^{\prime} F^{\prime}\) when cutting through points \(A^{\prime}\), \(C\), and \(M\) where \(M\) is the midpoint of \(DE\):
1. **Identify the plane and extend lines**:
- Extend \(MC\) an... |
Prove that for any positive integer \( n \), the following inequality holds:
\[ \lg n \geqslant k \lg 2 \]
where \( k \) is the number of distinct prime factors of \( n \). | Okay, so I need to prove that for any positive integer n, the logarithm of n (base 10, I assume) is greater than or equal to k times the logarithm of 2, where k is the number of distinct prime factors of n. Let me start by understanding the problem.
First, let's clarify terms. The number of distinct prime factors of n... | To prove that for any positive integer \( n \), the inequality \(\lg n \geqslant k \lg 2\) holds, where \( k \) is the number of distinct prime factors of \( n \), we can follow these steps:
1. **Understanding the Inequality**:
- The inequality can be converted to base 2 logarithms. If we let \(\lg\) denote the log... |
1. Suppose $n$ is an integer greater than 1, and let $p_1, p_2, \ldots, p_k$ be the distinct prime factors of $n$. We can express $n$ in terms of its prime factors as follows:
\[
n = p_1^{l_1} p_2^{l_2} \cdots p_k^{l_k}, \quad \text{where } l_i \geq 1 \text{ for } i = 1, 2, \ldots, k
\]
2. Since each $p_i$ is a pri... |
In a city, there are 10 parallel streets running from north to south, and 10 streets running from west to east, forming a $9 \times 9$ grid. There is a square at each intersection, totaling 100 squares. The distance between any two adjacent squares along a street is 1 km. A walking distance pharmacy is defined as a ph... | Alright, let me try to figure out this pharmacy problem. So, we have a 9x9 grid of streets, making 10 streets in each direction, right? That forms 100 intersections, each with a square. The goal is to place pharmacies such that anyone walking along any street, in any direction, will always be getting closer to a pharma... | To determine if 12 pharmacies suffice, we analyze the grid requirements:
1. **Grid Structure**: A 9×9 grid (10×10 intersections) with streets spaced 1 km apart. Each intersection is a square.
2. **Requirement**: Any person walking along any street must always get closer to a pharmacy within 3 km, regardless of direct... |
1. **Setup the Problem**:
- We have a grid of $9 \times 9$ formed by $10$ parallel streets running north-south and $10$ parallel streets running west-east.
- Each intersection on the grid corresponds to one of the $100$ squares (intersections) formed.
- The distance between two adjacent intersections is $1... |
How many convex polygons are there such that three consecutive vertices are the points $(0, 5)$, $(5, 5)$, and $(5, 0)$, and the coordinates of the other vertices are also non-negative integers? | Alright, so I need to figure out how many convex polygons there are with three consecutive vertices at (0,5), (5,5), and (5,0), and all other vertices have non-negative integer coordinates. Hmm, okay. Let me start by visualizing the problem.
First, the three given points: (0,5), (5,5), and (5,0). If I plot these, (0,5... | To determine the number of convex polygons with three consecutive vertices at \((0, 5)\), \((5, 5)\), and \((5, 0)\) and other vertices with non-negative integer coordinates, we analyze the possible configurations:
1. **Triangle**: The simplest convex polygon is the triangle formed by the three given points. This is a... |
1. **Identify the Given Points and Condition**: We are given three points of a convex polygon with coordinates $A(5, 0)$, $B(5, 5)$, and $C(0, 5)$. The remaining vertices must also have non-negative integer coordinates, and the polygon must remain convex, meaning all interior angles must be less than $180^\circ$.
2. ... |
Let \( a_{1}, a_{2}, \ldots, a_{2000} \) be real numbers in the interval \([0,1]\). Find the maximum possible value of
\[
\sum_{1 \leq i < j \leq 2000}(j - i) \left| a_{j} - a_{i} \right|
\] | Okay, so I need to find the maximum possible value of the sum Σ_{1 ≤ i < j ≤ 2000} (j - i) |a_j - a_i| where each a_k is in the interval [0,1]. Hmm, let's start by understanding the problem.
First, the variables a_1 to a_2000 are all between 0 and 1. The expression we need to maximize is a sum over all pairs i < j of ... | To find the maximum possible value of the sum
\[
\sum_{1 \leq i < j \leq 2000}(j - i) \left| a_{j} - a_{i} \right|
\]
where \(a_1, a_2, \ldots, a_{2000}\) are real numbers in the interval \([0, 1]\), we need to consider the optimal arrangement of these values to maximize the absolute differences and their weighted dist... | To find the maximum possible value of
\[ S = \sum_{1 \leq i < j \leq 2000} (j - i) \left| a_j - a_i \right|, \]
we start by noting the properties and characteristics of this sum.
1. **Convexity:**
Since \( |a_j - a_i| \) is convex and a sum of convex functions retains convexity, the sum \( S \) is convex as a func... |
A regular polygon \( A_{1} \ldots A_{n} \) is inscribed in a circle of radius \( R \) with center \( O \). Let \( X \) be an arbitrary point. Prove that \( A_{1} X^{2} + \ldots + A_{n} X^{2} = n \left( R^{2} + d^{2} \right) \), where \( d = OX \). | Okay, let's see. I need to prove that for a regular polygon inscribed in a circle with radius R, the sum of the squares of the distances from any point X to each vertex is equal to n times (R squared plus the square of the distance from X to the center O). Hmm, that seems like it might involve some coordinate geometry ... | To prove that the sum of the squares of the distances from an arbitrary point \( X \) to each vertex of a regular polygon inscribed in a circle of radius \( R \) is \( n(R^2 + d^2) \), where \( d = OX \), follow these steps:
**Step 1: Represent the Polygon and Point in Coordinate System**
Place the polygon’s center ... | 1. To prove the given equation, consider an arbitrary vertex \( A_i \) of the regular polygon \( A_1 \ldots A_n \) inscribed in a circle with radius \( R \) and center \( O \). Let \( X \) be any point in the plane, and let \( d = OX \) be the distance from \( O \) to \( X \).
2. We start by expressing \( A_i X^2 \) t... |
The force exerted by the airflow on a sail can be calculated using the formula:
\[ F = \frac{C S \rho (v_0 - v)^2}{2}, \]
where \(C\) is the coefficient of aerodynamic force, \(S\) is the area of the sail (\(S = 5 \, \text{m}^2\)), \(\rho\) is the air density, \(v_0\) is the wind speed (\(v_0 = 6 \, \text{m/s}\)), an... | Okay, so I need to find the speed of the sailboat when the instantaneous power from the wind is maximized. Hmm, let's start by recalling the formula given for the force on the sail: F = (C * S * ρ * (v₀ - v)²) / 2. The variables here are C, which is a coefficient, S is the sail area (5 m²), ρ is air density, v₀ is wind... | To determine the sailboat's speed when the instantaneous wind power is maximized, we start with the given force equation:
\[ F = \frac{C S \rho (v_0 - v)^2}{2} \]
Power \( P \) is the product of force and velocity:
\[ P = F \cdot v = \frac{C S \rho (v_0 - v)^2}{2} \cdot v \]
Simplifying, we focus on the function p... |
Given formula for the force acting on the sail due to wind flow is:
\[ F = \frac{C S \rho (v_{0} - v)^{2}}{2} \]
We are given the following:
- $S = 5 \, \text{m}^2$ (Area of the sail)
- $v_{0} = 6 \, \text{m/s}$ (Wind speed)
- $\rho$ (Density of air)
Our goal is to find the speed of the sailboat when the instantaneo... |
Jitka hiked a trail. After hiking 60% of the length of the trail, she had 8 km left to go. What is the length of the trail?
(A) 28 km
(B) 12.8 km
(C) 11.2 km
(D) 13 \(\frac{1}{3}\) km
(E) 20 km | Okay, let me try to figure out this problem. So, Jitka hiked a trail, and after hiking 60% of the trail's length, she still had 8 km left. We need to find the total length of the trail. The options are given, so maybe after solving, I can check which one matches.
Hmm, let's start by breaking down the information. If s... | The problem states that after hiking 60% of the trail, Jitka has 8 km remaining. This remaining distance corresponds to 40% of the total trail length. Let \( x \) be the total length.
**Key Steps:**
1. **Interpretation:**
60% completed ⇒ 40% remaining = 8 km.
Thus, \( 0.40x = 8 \).
2. **Solve for \( x \):*... |
1. Let's denote the total length of the trail by \( L \) kilometers.
2. According to the problem, Jitka has hiked \( 60\% \) of the trail, which means the remaining \( 40\% \) of the trail is left.
\[
100\% - 60\% = 40\%
\]
3. We are given that \( 40\% \) of the trail corresponds to \( 8 \) kilometers. Theref... |
In the expansion of \((a+b)^n\), there are \(n+1\) different terms. In the expansion of \((a+b+c)^{10}\), the number of different terms is:
(A) 11
(B) 33
(C) 55
(D) 66
(E) 132
(Provided by the 9th American High School Mathematics Examination, 1958) | Okay, let's see. The problem is asking about the number of different terms in the expansion of (a + b + c)^10. The options are given as (A) 11, (B) 33, (C) 55, (D) 66, (E) 132. Hmm, I need to figure out which one is correct.
First, I remember that when expanding a binomial like (a + b)^n, the number of distinct terms ... | To determine the number of different terms in the expansion of \((a + b + c)^{10}\), we need to find the number of non-negative integer solutions to the equation \(k + m + p = 10\), where \(k\), \(m\), and \(p\) are the exponents of \(a\), \(b\), and \(c\) respectively.
Using the stars and bars theorem in combinatori... |
We need to determine the number of distinct terms in the expansion of \((a+b+c)^{10}\).
1. **Understanding the problem with a simpler case:**
- Consider the expansion of \((a+b)^{n}\), which includes \(n+1\) distinct terms. Each term is derived from the binomial theorem:
\[
(a+b)^n = \sum_{k=0}^n \binom{... |
The sequence \( x_{n} \) has its first two elements as \( x_{1}=1001 \) and \( x_{2}=1003 \). For \( n \geq 1 \), the recurrence relation is given by:
\[ x_{n+2}=\frac{x_{n+1}-2004}{x_{n}}. \]
What is the sum of the first 2004 terms of the sequence? | Okay, so I need to figure out the sum of the first 2004 terms of this sequence. Let me start by writing down the given information to make sure I have everything straight.
The first two terms are x₁ = 1001 and x₂ = 1003. Then, for each n ≥ 1, the sequence follows the recurrence relation x_{n+2} = (x_{n+1} - 2004)/x_n.... | The sum of the first 2004 terms of the sequence is **1,338,004**.
**Step-by-Step Explanation:**
1. **Identify the Recurrence Pattern:**
- Given \( x_1 = 1001 \), \( x_2 = 1003 \), and \( x_{n+2} = \frac{x_{n+1} - 2004}{x_n} \).
- Compute initial terms to detect a pattern:
- \( x_3 = -1 \)
- \( x_4 = -... |
1. We start with the given sequence \( x_{n} \) where the initial terms are:
\[
x_{1}=1001, \quad x_{2}=1003
\]
and for \( n \geq 1 \):
\[
x_{n+2}=\frac{x_{n+1}-2004}{x_{n}}
\]
2. Notice that \( x_{1} + x_{2} = 1001 + 1003 = 2004 \). This relationship can be used to rewrite the recursion:
\[
... |
Can some of the four notable points of a triangle coincide with others? What can be said about a triangle in which the centers of the circumscribed and inscribed circles coincide? | Okay, so I have this geometry problem to tackle. The first question is whether any of the four notable points of a triangle can coincide. The second part asks about a triangle where the centers of the circumscribed (circumcenter) and inscribed (incenter) circles coincide. Let me start by recalling what the four notable... | The four notable points of a triangle (centroid, circumcenter, incenter, and orthocenter) can indeed coincide. This occurs in an **equilateral triangle**, where all four points merge into a single center due to the triangle's symmetry.
For a triangle where the circumcenter (O) and incenter (I) coincide, the following... |
1. To determine if some of the notable points of a triangle can coincide and what can be said about the triangle where the centers of the circumscribed and inscribed circles coincide, we start by recalling the four notable points: the centroid \(G\), the orthocenter \(H\), the circumcenter \(O\), and the incenter \(I\... |
Find any five consecutive natural numbers less than 100, whose product is divisible by 2014. | Alright, so I need to find five consecutive natural numbers less than 100 such that their product is divisible by 2014. Let me start by understanding the problem step by step.
First, let's break down the number 2014. If I can factorize it, that might help me figure out what conditions the five consecutive numbers must... | To find five consecutive natural numbers less than 100 whose product is divisible by 2014, we start by factorizing 2014:
\[ 2014 = 2 \times 19 \times 53. \]
The product of the five numbers must include these prime factors. Since the numbers are consecutive, at least one even number guarantees divisibility by 2. We ... |
To find five consecutive natural numbers less than 100 whose product is divisible by 2014, we start by analyzing the prime factorization of 2014.
#### Step 1: Prime Factorization of 2014
We write 2014 as a product of its prime factors:
\[
2014 = 2 \times 19 \times 53
\]
This indicates that the product of the five con... |
$A, B, C, D, E$ are seated in a train that consists of 5 carriages, with each carriage only holding one person. It is known that $D$ is seated in the last carriage, $A$ is immediately behind $E$, $B$ is in a carriage before $A$, and there is at least one person between $B$ and $C$. Who is seated in the middle position?... | Okay, let's try to solve this problem step by step. So we have five people: A, B, C, D, E, each sitting in separate carriages of a train. The carriages are in a row, and each person is in one. We need to figure out who is in the middle position, which would be the third carriage. Let's list out the given conditions and... | The middle position is occupied by **A**.
**Step-by-Step Explanation:**
1. **Fix D in Position 5**: Since D is in the last carriage (position 5), we only need to arrange A, B, C, E in positions 1–4.
2. **Determine E and A Positions**: A must be immediately behind E. Possible (E, A) pairs:
- (1,2): Invalid as it f... |
1. Label the sequence of seats in the train as \( \square \square \square \square \square \), where the first position corresponds to the first car and the fifth position to the last car.
2. From the condition that D is in the last car, we can place D at the fifth position:
\[
\square \square \square \square D
... |
Usually, two mechanisms work together to complete a certain task. Their efficiencies are not the same, and when working together, they complete the task in 30 hours. Once, the two mechanisms worked together for only 6 hours, after which the first mechanism was stopped, and the second mechanism completed the remaining p... | Okay, let's see. So, there are two mechanisms working together to complete a task. Their efficiencies are different. When they work together, they finish in 30 hours. But one time, they only worked together for 6 hours, then the first one stopped, and the second one took 40 hours alone to finish the rest. The question ... | Let's denote the time taken by the first mechanism to complete the task individually as \( A \) hours and the time taken by the second mechanism as \( B \) hours. Their respective work rates are \( \frac{1}{A} \) and \( \frac{1}{B} \) tasks per hour.
When working together, their combined work rate is \( \frac{1}{A} + ... |
Given that two mechanisms work together to complete a task in 30 hours, we aim to determine the individual times required for each mechanism to complete the task alone.
1. **Define Variables:**
- Let $x$ be the time (in hours) the first mechanism takes to complete the task alone.
- Let $y$ be the time (in hours... |
One way to pack a 100 by 100 square with 10,000 circles, each of diameter 1, is to put them in 100 rows with 100 circles in each row. If the circles are repacked so that the centers of any three tangent circles form an equilateral triangle, what is the maximum number of additional circles that can be packed?
(A) 647
(B... | Alright, so I need to figure out how many additional circles can be packed into a 100x100 square when they're arranged such that the centers of any three tangent circles form an equilateral triangle. The original packing is a grid with 100 rows and 100 columns, totaling 10,000 circles. The answer choices are given, so ... | To determine the maximum number of additional circles that can be packed into a 100x100 square when repacked such that the centers of any three tangent circles form an equilateral triangle, we need to consider hexagonal close packing.
1. **Original Packing**: The original packing is a square grid with 100 rows and 10... |
To solve this problem, we need to examine how we can more efficiently pack circles by changing their arrangement from squares to an arrangement where the centers of any three tangent circles form an equilateral triangle.
1. **Initial Square Packing Arrangement:**
The initial arrangement consists of 100 rows an... |
In the arithmetic sequence $\left\{a_{n}\right\}$, if $\frac{a_{11}}{a_{10}} < -1$ and the sum of its first $n$ terms $S_{n}$ has a maximum value, then when $S_{n}$ takes the smallest positive value, $n = (\quad$ ).
(A) 1
(B) 10
(C) 19
(D) 20 | Okay, let's try to solve this problem step by step. So, we have an arithmetic sequence {a_n}, and we know that the ratio a₁₁ / a₁₀ is less than -1. Also, the sum of the first n terms, S_n, has a maximum value. We need to find the smallest positive value of S_n and determine which n that corresponds to, with the options... | The problem involves an arithmetic sequence \(\{a_n\}\) where \(\frac{a_{11}}{a_{10}} < -1\) and the sum \(S_n\) has a maximum value. We need to determine \(n\) when \(S_n\) is the smallest positive value.
1. **Analyzing the Ratio \(\frac{a_{11}}{a_{10}} < -1\):**
- For an arithmetic sequence with common difference... |
#### Given Problem
In an arithmetic sequence $\left\{a_{n}\right\}$, if $\frac{a_{11}}{a_{10}} < -1$, and its sum of the first $n$ terms $S_{n}$ has a maximum value, then when $S_{n}$ takes the smallest positive value, $n = (\quad$ ).
#### Reference Solution:
The correct option is (C).
Let's provide a detailed brea... |
Do there exist rational numbers \(a, b, c, d\) that satisfy the equation
$$
(a + b \sqrt{2})^{2n} + (c + d \sqrt{2})^{2n} = 5 + 4 \sqrt{2}
$$
(where \(n\) is a natural number)? | Okay, let me try to figure out this problem. The question is whether there exist rational numbers a, b, c, d such that (a + b√2)^{2n} + (c + d√2)^{2n} = 5 + 4√2, where n is a natural number. Hmm, interesting. I need to check if such rational numbers exist for some natural number n. Let me start by breaking down the pro... | To determine whether there exist rational numbers \(a, b, c, d\) satisfying the equation:
\[
(a + b \sqrt{2})^{2n} + (c + d \sqrt{2})^{2n} = 5 + 4 \sqrt{2}
\]
for some natural number \(n\), we analyze the structure of the problem and use properties of algebraic integers.
### Key Steps and Analysis:
1. **Structure of ... |
1. **Uniqueness of Representation:**
We start by noting that representation of a number in the form \( p + q \sqrt{2} \), where \( p \) and \( q \) are rational numbers, is unique. That is, if \( p + q \sqrt{2} = p_1 + q_1 \sqrt{2} \) where \( p, q, p_1, \) and \( q_1 \) are rational, then \( p = p_1 \) and \( q = ... |
Given a positive integer \( m \), for a positive integer \( l \), let \( A_{l} = (4l + 1)(4l + 2) \cdots (4(5^{m} + 1)l) \). Prove that there exist infinitely many positive integers \( l \) such that \( 5^{5ml} \mid A_{l} \) and \( 5^{5ml + 1} + A_{l} \) is also an integer. Additionally, find the smallest value of \( l... | Okay, let's try to tackle this problem step by step. The problem is about proving that there are infinitely many positive integers \( l \) such that \( 5^{5ml} \) divides \( A_l \), and \( \frac{A_l}{5^{5ml + 1}} \) is an integer. Additionally, we need to find the smallest such \( l \).
First, let me make sure I unde... | To solve the problem, we analyze the exponent of 5 in \( A_l \) and ensure it meets the required conditions.
**Step-by-Step Explanation:**
1. **Define \( A_l \):** Given \( A_l = \prod_{k=4l+1}^{4(5^m + 1)l} k \), this is equivalent to \( \frac{(4(5^m + 1)l)!}{(4l)!} \).
2. **Legendre's Formula:** The exponent of 5 ... |
#### Step 1: Introduce and prove the lemma.
To begin, let \( n \) be a positive integer, and let \( p \) be a prime number. We use the notation \( S_p(n) \) to denote the sum of the digits of \( n \) when expressed in base \( p \). We also use \( v_p(m) \) to denote the $p$-adic valuation of a positive integer \( m \... |
Given that \(2^{96} - 1\) is divisible by two integers between 60 and 70, what are these two numbers?
(A) 61, 63
(B) 61, 65
(C) 63, 65
(D) 63, 67 | Okay, let me try to figure out this problem. The question says that \(2^{96} - 1\) is divisible by two integers between 60 and 70, and we need to pick which two they are from the given options. The options are pairs like 61 & 63, 61 & 65, etc. So, I need to check which numbers between 60 and 70 divide \(2^{96} - 1\).
... | To determine which two integers between 60 and 70 divide \(2^{96} - 1\), we analyze each candidate using modular arithmetic and factorization:
1. **Checking 63 (7×9):**
- **Modulo 7:** \(2^3 \equiv 1 \mod 7\). Since \(96 = 3 \times 32\), \(2^{96} \equiv 1^{32} \equiv 1 \mod 7\). Thus, \(2^{96} - 1 \equiv 0 \mod 7\)... | To determine which two integers between 60 and 70 exactly divide \(2^{96} - 1\), let us start by analyzing the expression of \(2^{96} - 1\):
1. We can rewrite \(2^{96} - 1\) using exponent laws:
\[
2^{96} - 1 = 2^{6 \times 16} - 1
\]
2. Now, express this as a difference of squares:
\[
2^{96} - 1 = \left(2^{6 \times 8... |
If \( m, n \) and \( p \) are positive integers with \( m+\frac{1}{n+\frac{1}{p}}=\frac{17}{3} \), the value of \( n \) is
(A) 3
(B) 4
(C) 1
(D) 17
(E) 13 | Okay, let me try to figure out this problem. So, we have m, n, and p as positive integers, and the equation is m + 1/(n + 1/p) = 17/3. We need to find the value of n from the options given. Hmm, let's break this down step by step.
First, I remember that when dealing with mixed numbers or continued fractions like this,... | To solve the equation \( m + \frac{1}{n + \frac{1}{p}} = \frac{17}{3} \) where \( m, n, p \) are positive integers:
1. **Express \( \frac{17}{3} \) as a mixed number**:
\( \frac{17}{3} = 5 + \frac{2}{3} \). Thus, \( m = 5 \), leaving \( \frac{1}{n + \frac{1}{p}} = \frac{2}{3} \).
2. **Solve for \( n + \frac{1}{p... | Let's solve the given equation \( m + \frac{1}{n + \frac{1}{p}} = \frac{17}{3} \).
1. **Identify the range of \( m \)**:
Since \( p \) is a positive integer, \( p \geq 1 \) which implies \( 0 < \frac{1}{p} \leq 1 \).
Therefore, for any positive integer \( n \), we have \( n + \frac{1}{p} > 1 \) implying \( 1 < n... |
There were 12 seats prepared for the jury at a round table, each with a name indicated on it. Nikolai Nikolaevich, who arrived first, absent-mindedly sat not in his designated seat but in the next one clockwise. Each jury member arriving at the table afterward would either sit in their designated seat, or, if it was al... | Okay, let me try to figure out this problem. So, there's a round table with 12 seats, each labeled for a jury member. Nikolai Nikolaevich arrives first but instead of sitting in his own seat, he sits in the next one clockwise. Then, each subsequent jury member will sit in their designated seat if it's free; if not, the... | The problem involves determining the number of distinct seating arrangements possible when jury members arrive in varying orders, given that Nikolai Nikolaevich initially sits in the next seat clockwise from his designated spot. The key steps are as follows:
1. **Initial Displacement**: Nikolai starts a chain by occup... |
To determine the number of possible distinct seating arrangements for the jury members, we need to consider the condition given: Nikolai Nikolaevich (NN), who arrives first, does not sit in his own place, but instead sits in the next available seat in a clockwise direction.
1. Let's consider a person sitting at the t... |
Every day, Patrick the dog chews up one slipper from the available stock at home. There is a strict 0.5 probability that Patrick wants to chew on the left slipper and a 0.5 probability that he wants to chew on the right slipper. If the desired slipper is not available, Patrick gets upset. How many identical pairs of sl... | Okay, let's try to figure out how many pairs of slippers Patrick's owners need to buy so that he doesn't get upset for a week with at least 0.8 probability. Hmm, Patrick chews one slipper each day, and each day he randomly chooses left or right with equal probability. If the desired slipper isn't available, he gets ups... | To determine how many pairs of slippers Patrick's owners need to buy to ensure he doesn't get upset for a week with at least 0.8 probability, we analyze his chewing behavior as a stochastic process. Each day, Patrick chooses to chew a left or right slipper with equal probability. If the desired slipper is unavailable, ... | 1. **Defining the problem and variables**:
Let $S$ be the number of left slippers Patrick wants to chew out of $7$ days and $7-S$ the number of right slippers. We need to find the smallest number $k$ of pairs of identical slippers so that the probability that Patrick is not disappointed on any of the $7$ days is at... |
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0} \frac{\arcsin 3x}{\sqrt{2+x}-\sqrt{2}}
$$ | Okay, so I need to find the limit as x approaches 0 of (arcsin(3x)) divided by (sqrt(2 + x) - sqrt(2)). Hmm, let me start by recalling some basic limit techniques. When dealing with limits that result in indeterminate forms like 0/0, L'Hôpital's Rule can be useful. But before jumping to that, maybe there's a way to sim... | The limit is calculated as follows:
1. **Recognize the indeterminate form**: Substituting \( x = 0 \) yields \( \frac{0}{0} \), prompting the use of techniques like rationalization or L'Hôpital's Rule.
2. **Rationalize the denominator**:
\[
\frac{\arcsin 3x}{\sqrt{2+x} - \sqrt{2}} \cdot \frac{\sqrt{2+x} + \sqrt... | 1. We need to compute the limit:
$$
\lim_{x \rightarrow 0} \frac{\arcsin(3x)}{\sqrt{2+x} - \sqrt{2}}
$$
2. Utilizing the equivalence of infinitesimals, we know that for small values of $x$:
$$
\arcsin(3x) \sim 3x
$$
when $x \to 0$.
3. Substitute this equivalence into the limit:
$$
\lim_{x \... |
List, in increasing order, each multiple of 3 whose sum with 1 is a perfect square:
$$
3, 15, 24, 48, \ldots
$$
a) What is the next number that will appear in this sequence after 48?
b) What is the eighth number in this sequence?
c) What number will appear in the 2013th position of this sequence? | Okay, let's try to tackle this problem step by step. The question is about finding multiples of 3 such that when you add 1 to them, the result is a perfect square. We need to answer three parts: the next number after 48, the eighth number in the sequence, and the 2013th term.
First, let me understand the given sequen... | To solve the problem, we need to find multiples of 3 such that when 1 is added to them, the result is a perfect square. We denote the multiple of 3 as \(3k\), so we have \(3k + 1 = n^2\), where \(n\) is an integer. This implies \(3k = n^2 - 1 = (n - 1)(n + 1)\). Therefore, \(n\) must be an integer not divisible by 3, i... |
Let's address each part of the problem in a step-by-step manner to ensure a comprehensive solution.
#### a) Determine the next number in the sequence after 48.
1. We start by noting that 48 is a multiple of 3 and satisfies the condition that \(48 + 1 = 49 = 7^2\).
2. To find the successor, we look for the next perfe... |
Prove that if \( P Q \| A C \), then triangle \( A B C \) is isosceles. | Okay, so I need to prove that if PQ is parallel to AC in triangle ABC, then triangle ABC is isosceles. Hmm, wait, the problem statement just says "if PQ is parallel to AC, then triangle ABC is isosceles." But wait, where exactly are points P and Q located? The problem doesn't specify. That seems important. Maybe they a... | To prove that if \( PQ \parallel AC \), then triangle \( ABC \) is isosceles, we need to consider the configuration and properties of the triangle and the line segment \( PQ \).
**Assumptions and Configuration:**
- Points \( P \) and \( Q \) are on sides \( AB \) and \( BC \) respectively.
- \( PQ \parallel AC \).
**... | :
Given that $PQ \parallel AC$, we are to prove that triangle $ABC$ is isosceles.
1. **Identify Key Bisectors:**
From the problem statement, since $PQ \parallel AC$, the lines $AN$ and $CM$ are the angle bisectors of triangle $ABC$. This implies that these lines pass through the incenter of the triangle (the ... |
Find the correlation function of the stationary random function $X(t)$, given its spectral density $s_{x}(\omega)=s_{0}$ in the interval $-\omega_{0} \leq \omega \leq \omega_{0}$, and outside this interval $s_{x}(\omega)=0$. | Okay, so I need to find the correlation function of a stationary random function X(t) given its spectral density. The spectral density is given as s_x(ω) = s₀ within the interval -ω₀ ≤ ω ≤ ω₀ and zero outside that interval. Hmm, right. I remember that for stationary processes, the correlation function is related to the... | The correlation function \( B_x(\tau) \) of the stationary random function \( X(t) \) with the given spectral density \( s_x(\omega) = s_0 \) for \( -\omega_0 \leq \omega \leq \omega_0 \) and zero elsewhere is derived using the Wiener-Khinchin theorem. This theorem relates the correlation function to the inverse Fourie... |
Given the spectral density function \( s_{x}(\omega) \), we need to find the correlation function \( k_{x}(\tau) \) of a stationary random process \( X(t) \). The spectral density is given by:
\[ s_{x}(\omega) = \begin{cases}
s_0 & \text{for } -\omega_0 \leq \omega \leq \omega_0 \\
0 & \text{otherwise}
\end{cases}
\]... |
In an acute-angled triangle \(ABC\), the altitudes are \(AD\), \(BE\), and \(CF\). Prove that the perimeter of triangle \(DEF\) does not exceed half the perimeter of triangle \(ABC\). | Okay, so I need to prove that in an acute-angled triangle ABC, the perimeter of the orthic triangle DEF (formed by the feet of the altitudes AD, BE, and CF) is at most half the perimeter of triangle ABC. Hmm, let's start by recalling some properties of orthic triangles and acute triangles.
First, in an acute-angled tr... | To prove that the perimeter of the orthic triangle \(DEF\) in an acute-angled triangle \(ABC\) does not exceed half the perimeter of \(ABC\), we analyze the relationship between their perimeters using trigonometric identities and geometric properties.
**Step 1: Express the Perimeter of the Orthic Triangle**
The orthic... |
To prove: the perimeter of triangle $DEF$ is not more than half the perimeter of triangle $ABC$. In other words, we need to show:
\[ DE + EF + FD \leq \frac{1}{2}(AB + BC + CA) \]
We'll demonstrate this in three steps:
1. Show that \( DE + DF \leq BC \)
2. Show that \( EF + FD \leq AB \)
3. Show that \( DE + EF \leq ... |
Show that
$$
f_{n}+f_{n-1}+\sum_{k=0}^{n-2} f_{k} 2^{n-2-k}=2^{n}
$$ | Okay, so I need to show that the equation \( f_{n} + f_{n-1} + \sum_{k=0}^{n-2} f_{k} 2^{n-2-k} = 2^{n} \) holds true. Hmm, first off, I need to make sure I understand what \( f_n \) represents here. The problem doesn't specify, but given the context of the equation, I might assume that \( f_n \) is the Fibonacci seque... | To prove the identity \( f_{n} + f_{n-1} + \sum_{k=0}^{n-2} f_{k} 2^{n-2-k} = 2^{n} \), where \( f_n \) follows the Fibonacci sequence with initial terms \( f_0 = 1 \) and \( f_1 = 1 \), we use mathematical induction.
**Base Case (n = 1):**
Left-hand side (LHS):
\( f_1 + f_0 = 1 + 1 = 2 \).
Right-hand side (RHS)... |
To prove the statement:
\[ f_{n}+f_{n-1}+\sum_{k=0}^{n-2} f_{k} 2^{n-2-k}=2^{n}, \]
we will use a combinatorial argument.
1. **Interpretation of \(2^n\):**
We interpret \(2^n\) as the number of binary sequences of length \(n\). There are exactly \(2^n\) such sequences.
2. **Binary Sequence Representation:**
... |
Sofia was taking some sweets to her grandmother: 7 blackberry sweets, 6 coconut sweets, and 3 chocolate sweets. On the way, the greedy Sofia eats 2 sweets. Which of the following situations is possible?
(A) Grandmother did not receive any chocolate sweets.
(B) Grandmother received fewer coconut sweets than chocolate ... | Okay, let's tackle this problem step by step. So, Sofia has 7 blackberry, 6 coconut, and 3 chocolate sweets. She eats 2 sweets on the way, and we need to figure out which of the given situations (A to E) is possible for the remaining sweets that the grandmother receives.
First, let me restate the problem to make sure... | **Step-by-Step Explanation:**
Sofia initially has 7 blackberry (B), 6 coconut (C), and 3 chocolate (Ch) sweets. After eating 2 sweets, we analyze the feasibility of each scenario:
1. **Option A: No chocolate sweets remaining.**
Sofia ate 2 sweets, but there are 3 chocolates. To eliminate all chocolates, she must... |
Let's analyze each of the proposed scenarios. Note that in the end, the grandmother received:
$$ 7 + 6 + 3 - 2 = 14 \text{ docinhos (sweets)} $$
1. **Scenario (A): Grandma received no chocolate sweets.**
Initially, there are 3 chocolate sweets. If Sofia eats 2 sweets, even if both are chocolate, at least 1 cho... |
Project the geographic network from the intersection point of the Earth's equator and the Prime Meridian onto the tangent plane at the antipodal point. Draw the map of the geographic network, displaying circles only at every 30 degrees. (A drawing and description of the construction are requested.) | Okay, so I need to figure out how to project the Earth's geographic network (like lines of latitude and longitude) from the point where the equator and Prime Meridian meet onto the tangent plane at the antipodal point. Then draw a map showing circles every 30 degrees. Hmm, let's break this down.
First, the intersectio... | To project the geographic network from the point (0°, 0°) onto the tangent plane at its antipodal point (0°, 180°), we use a **stereographic projection**. Here's the construction and description:
### **Construction Steps**:
1. **Define the Sphere and Tangent Plane**:
- Model Earth as a unit sphere. The projection c... |
1. **Identify the Point of Projection and the Tangent Plane:**
- The projection starts at the intersection of the Earth's equator and the prime meridian (0° longitude). This point is on the Earth's surface.
- The projection plane is tangent to the Earth at the point diametrically opposite to the projection cente... |
A group of children are arranged into two equal rows. Every child in the back row is taller than the child standing in front of him in the other row. Prove that this remains true if each row is rearranged so that the children increase in height from left to right. | Okay, let's see. I have this problem where there are two equal rows of children. The back row has each child taller than the one directly in front of them. Then, we need to rearrange each row so that both rows are in increasing order from left to right, and prove that the condition still holds—meaning each child in the... | To prove that rearranging both rows in increasing height maintains the condition where each back row child is taller than the front row child, consider the following reasoning:
**Step-by-Step Explanation:**
1. **Initial Setup:**
Let the original front row be \( F = [f_1, f_2, \dots, f_n] \) and the back row be \... | To solve this problem, we need to prove that if the children are rearranged in such a way that they increase in height from left to right in each row, the condition that every child in the back row is taller than the child standing directly in front of him still holds.
1. **Initial Arrangement**:
We start with two ... |
Find the total length of the intervals on the number line where the inequalities \( x < 1 \) and \( \operatorname{tg} \log _{4} x > 0 \) hold. | Okay, let's try to solve this problem step by step. The question is asking for the total length of the intervals on the number line where both inequalities \( x < 1 \) and \( \tan \log_4 x > 0 \) hold. Hmm, that seems a bit tricky, but let's break it down.
First, let's make sure I understand the problem correctly. We ... | To find the total length of the intervals on the number line where the inequalities \( x < 1 \) and \( \tan \log_4 x > 0 \) hold, we start by analyzing the condition \( \tan \log_4 x > 0 \). The tangent function is positive when its argument is in the intervals \( (k\pi, \frac{\pi}{2} + k\pi) \) for integer \( k \).
... | :
1. **Identify Relevant Range for \( x \)**:
We're interested in positive \( x \). The given inequalities are \( x < 1 \) and \( \operatorname{tg} \log_{4} x > 0 \).
2. **Transform the Inequality \( \operatorname{tg} \log_{4} x > 0 \)**:
For \( \operatorname{tg} \log_{4} x \) to be positive, \( \log_{4} x \) ... |
Given the sequences \(\{a_n\}\) and \(\{b_n\}\) where \(a_1 = b_1 = 10\), the recursive formulas are defined as:
\[
\begin{cases}
a_{n+1} = \frac{1}{a_n b_n} \\
b_{n+1} = a_n^4 b_n
\end{cases} \quad (n \in \mathbb{Z}^+)
\]
Find \(a_n\) and \(b_n\). | Okay, let me try to figure out how to solve this problem. We have two sequences, {a_n} and {b_n}, both starting at 10. The recursive formulas are a bit complicated. Let me write them down again to make sure I have them right:
a_{n+1} = 1 / (a_n * b_n)
b_{n+1} = a_n^4 * b_n
So each term depends on the previous terms i... | To solve the given recursive sequences \(\{a_n\}\) and \(\{b_n\}\), we start by converting the multiplicative relationships into additive ones using logarithms. Define \(x_n = \log_{10} a_n\) and \(y_n = \log_{10} b_n\). The recursions transform into a linear system:
\[
\begin{cases}
x_{n+1} = -x_n - y_n \\
y_{n+1} = ... |
1. Given sequences:
\[
\left\{ \begin{array}{l}
a_{1}=b_{1}=10, \\
a_{n+1} = \frac{1}{a_{n} b_{n}}, \\
b_{n+1} = a_{n}^4 b_{n}
\end{array} \right. \quad (\text{where } n \in \mathbb{Z}^{+}).
\]
2. Take the logarithm of both $a_{n}$ and $b_{n}$:
\[
\left\{ \begin{array}{l}
\log a_{n+1} = ... |
Arrange the numbers $1,2,3, \cdots, 37$ in a sequence $a_{1}, a_{2}, \cdots, a_{37}$, where $a_{1}=37$ and $a_{2}=1$, such that $a_{1}+a_{2}+\cdots+a_{k}$ is divisible by $a_{k+1}$ for $k=1,2, \cdots, 36$. Find $a_{37}$ and $a_{3}$. | Okay, so I have this problem where I need to arrange the numbers from 1 to 37 in a sequence. The sequence starts with a₁ = 37 and a₂ = 1. The main condition is that for each k from 1 to 36, the sum of the first k terms, which is S_k = a₁ + a₂ + ... + a_k, must be divisible by the next term a_{k+1}. I need to find what ... | Given the problem of arranging the numbers \(1, 2, 3, \ldots, 37\) in a sequence \(a_1, a_2, \ldots, a_{37}\) where \(a_1 = 37\) and \(a_2 = 1\), such that the sum \(a_1 + a_2 + \cdots + a_k\) is divisible by \(a_{k+1}\) for \(k = 1, 2, \ldots, 36\), we need to determine \(a_{37}\) and \(a_3\).
1. **Initial Conditions... |
1. Given that we need to arrange the numbers \(1, 2, 3, \ldots, 37\) in a sequence \(a_1, a_2, \ldots, a_{37}\), with \(a_1 = 37\) and \(a_2 = 1\), such that the sum of the first \(k\) terms can be divided by the \((k+1)\)-th term for \(k = 1, 2, \ldots, 36\).
2. We start by considering \(a_{37}\). Based on the condi... |
Prove that if \( \frac{a}{b} = \frac{b}{c} \), then \( a^{2} + c^{2} \geq 2 b^{2} \). | Okay, so I need to prove that if a/b = b/c, then a² + c² is greater than or equal to 2b². Hmm, let's start by understanding the given condition. The equation a/b = b/c implies a proportion, right? Maybe I can express one variable in terms of the others. Let me think.
If a/b = b/c, then cross-multiplying gives a*c = b²... | To prove that if \( \frac{a}{b} = \frac{b}{c} \), then \( a^{2} + c^{2} \geq 2b^{2} \), follow these steps:
### Step 1: Use the Given Proportion
From \( \frac{a}{b} = \frac{b}{c} \), cross-multiply to obtain:
\[
ac = b^{2}.
\]
### Step 2: Substitute \( a = \frac{b^{2}}{c} \) into \( a^{2} + c^{2} \)
Express \( a \) i... |
1. **Given the condition**: We are given \( a : b = b : c \). This implies that the ratio of \( a \) to \( b \) is the same as the ratio of \( b \) to \( c \).
2. **Expressing the ratio**: From the given proportion, we can express this as a fraction:
\[
\frac{a}{b} = \frac{b}{c}
\]
3. **Cross-multiplying to... |
We consider the two sequences $\left(a_{n}\right)_{n \geq 0}$ and $\left(b_{n}\right)_{n \geq 0}$ of integers, defined by $a_{0}=b_{0}=2$ and $a_{1}=b_{1}=14$ and by
$$
\begin{aligned}
a_{n} & =14 a_{n-1}+a_{n-2} \\
b_{n} & =6 b_{n-1}-b_{n-2}
\end{aligned}
$$
for $n \geq 2$.
Determine whether there are infinitely ma... | Okay, let's see. I need to figure out if there are infinitely many numbers that appear in both sequences \( a_n \) and \( b_n \). Both sequences start with the same initial terms, 2 and 14, but they have different recurrence relations. The \( a_n \) sequence is defined by \( a_n = 14a_{n-1} + a_{n-2} \), while the \( b... | To determine whether there are infinitely many numbers that occur in both sequences \( (a_n) \) and \( (b_n) \), we start by analyzing their recurrence relations and closed-form expressions.
The sequence \( (a_n) \) is defined by:
\[ a_0 = 2, \quad a_1 = 14, \quad a_n = 14a_{n-1} + a_{n-2} \]
The characteristic equati... |
To determine whether there are infinitely many whole numbers occurring in both sequences \( \left(a_{n}\right)_{n \geq 0} \) and \( \left(b_{n}\right)_{n \geq 0} \), we first analyze the recurrence relations and solutions for both sequences.
1. **Given Initial Conditions and Recurrence Relations:**
\[
a_{0} = b... |
Find all integers \(a\) such that the equation
$$
x^{2} + axy + y^{2} = 1
$$
has infinitely many integer solutions \((x, y)\). Prove your conclusion. | Okay, so I need to find all integers \( a \) such that the equation \( x^2 + a x y + y^2 = 1 \) has infinitely many integer solutions \( (x, y) \). Hmm, let's start by understanding the problem. The equation is a quadratic in two variables, and we want to know for which integer values of \( a \) there are infinitely ma... | To find all integers \(a\) such that the equation \(x^2 + axy + y^2 = 1\) has infinitely many integer solutions \((x, y)\), we analyze the equation through various methods:
1. **Case Analysis for Small Values of \(a\)**:
- For \(a = 0\), the equation becomes \(x^2 + y^2 = 1\), which has only four integer solutions.... | To solve for all integers \(a\) such that the equation
\[ x^2 + axy + y^2 = 1 \]
has infinitely many integer solutions \((x, y)\), we proceed as follows:
1. **Case \( a = 0 \):**
\[
x^2 + y^2 = 1
\]
The integer solutions to this equation can be obtained by checking the possible values:
\[
(x, y) = (\... |
What is the value of the expression \(x^{2000} + x^{1999} + x^{1998} + 1000x^{1000} + 1000x^{999} + 1000x^{998} + 2000x^{3} + 2000x^{2} + 2000x + 3000\) (where \(x\) is a real number), if \(x^{2} + x + 1 = 0\)? Vasya calculated the answer to be 3000. Is Vasya correct? | Okay, let's see. Vasya claims that the value of this big polynomial expression is 3000 when x satisfies the equation x² + x + 1 = 0. I need to check if that's correct. Hmm, so first, maybe I should recall that x² + x + 1 = 0 is a quadratic equation. Let me think about the roots of this equation. The discriminant would ... | Given the expression \(x^{2000} + x^{1999} + x^{1998} + 1000x^{1000} + 1000x^{999} + 1000x^{998} + 2000x^{3} + 2000x^{2} + 2000x + 3000\) where \(x\) satisfies \(x^2 + x + 1 = 0\), we need to determine if the value of the expression is 3000.
First, note that \(x\) is a root of the equation \(x^2 + x + 1 = 0\), which i... | ### Problem:
Determine the value of the expression
\[x^{2000} + x^{1999} + x^{1998} + 1000x^{1000} + 1000x^{999} + 1000x^{998} + 2000x^3 + 2000x^2 + 2000x + 3000\]
where \(x\) is a real number, given that \(x^2 + x + 1 = 0\).
1. **Identify the roots of the given quadratic equation \(x^2 + x + 1 = 0\)**:
\[
x^... |
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