source stringclasses 7
values | problem stringlengths 19 6.18k | generation stringlengths 345 68.1k |
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olympiads | 2. Around a table are sitting 4 people, each of whom can either be a knight (who always tells the truth) or a knave (who always lies). Each of those present states: "Of the other three people sitting at this table with me, the knaves are exactly two." What is the total number of knaves sitting at the table?
(A) none
(B... | <think>
Okay, so there's this problem where four people are sitting around a table, and each one can be either a knight or a knave. Knights always tell the truth, and knaves always lie. Each person says, "Of the other three people sitting at this table with me, the knaves are exactly two." The question is asking how ma... |
olympiads | Example 2.34. $I=$ V.p. $\int_{-\infty}^{\infty} \frac{(1+x) d x}{1+x^{2}}$. | <think>
Alright, so I need to compute the principal value integral of (1 + x)/(1 + x²) from -∞ to ∞. Hmm, okay. Let me start by recalling what a principal value integral is. If there are any singularities in the integrand within the limits of integration, the principal value is a way to assign a finite value to the int... |
olympiads | Two unit-radius circles intersect at points $A$ and $B$. One of their common tangents touches the circles at points $E$ and $F$. What can be the radius of a circle that passes through points $E, F$, and $A$? | <think>
Okay, so I need to find the possible radius of a circle that passes through points E, F, and A. The problem states that there are two unit-radius circles intersecting at points A and B, and they have a common tangent that touches them at points E and F. Let me visualize this first.
First, I should probably dra... |
olympiads | 6. Let $\omega=\cos \frac{\pi}{5}+i \sin \frac{\pi}{5}$, then the equation with roots $\omega, \omega^{3}, \omega^{7}, \omega^{9}$ is
(A) $x^{4}+x^{3}+x^{2}+x+1=0$
(B) $x^{4}-x^{3}+x^{2}-x+1=0$
(C) $x^{4}-x^{3}-x^{2}+x+1=0$
(D) $x^{4}+x^{3}+x^{2}-x-1=0$ | <think>
Okay, let's see. The problem says that ω is equal to cos(π/5) + i sin(π/5). So, ω is a complex number on the unit circle, right? Because cosine squared plus sine squared is 1. So, it's a complex number of modulus 1. And then, the equation with roots ω, ω³, ω⁷, ω⁹. The options are given from A to D, and I need t... |
cn_contest | 24. Points $A_{1}, \cdots, A_{5}$ are on a sphere with radius 1, what is the maximum value of $\min _{1 \leqslant \mathrm{i}, \mathrm{j} \leqslant 5} A_{1} A_{\mathrm{j}}$? Determine all cases where the maximum value is achieved. | <think>
Okay, so I need to figure out the maximum value of the minimum distance between any two points among five points on a unit sphere. In other words, I want to place five points on a sphere with radius 1 such that the smallest distance between any two points is as large as possible. And then also determine all cas... |
olympiads | 3、Clock at 5 o'clock strikes 5 times using 8 seconds, then at 10 o'clock strikes 10 times using how many seconds? | <think>
Okay, let's see. The problem is about a clock that strikes a certain number of times at different hours and how long it takes. So, at 5 o'clock, it strikes 5 times and takes 8 seconds. Then, at 10 o'clock, it strikes 10 times, and we need to figure out how many seconds that would take. Hmm, seems like a classic... |
olympiads | 37. (10-11 grades) How many planes are equidistant from four points that do not lie in the same plane? | <think>
Okay, so I need to figure out how many planes are equidistant from four points that don't all lie in the same plane. Hmm, let's start by understanding what the problem is asking.
First, equidistant from a point to a plane means that the distance from the point to the plane is the same. So, if a plane is equid... |
aops_forum | Let $a_{i}$ and $b_{i}$, where $i \in \{1,2, \dots, 2005 \}$, be real numbers such that the inequality $(a_{i}x-b_{i})^{2} \ge \sum_{j=1, j \not= i}^{2005} (a_{j}x-b_{j})$ holds for all $x \in \mathbb{R}$ and all $i \in \{1,2, \dots, 2005 \}$. Find the maximum possible number of positive numbers amongst $a_{i}$ and $b_... | <think>
Okay, let's try to tackle this problem step by step. The problem states that for each i from 1 to 2005, we have real numbers a_i and b_i. The inequality given is (a_i x - b_i)^2 ≥ sum_{j≠i} (a_j x - b_j) for all real numbers x and for all i. We need to find the maximum possible number of positive numbers among ... |
olympiads | 3.1. Along the groove, there are 100 multi-colored balls arranged in a row with a periodic repetition of colors in the following order: red, yellow, green, blue, purple. What color is the ball at the $78-$th position?
$$
\text { (4-5 grades) }
$$ | <think>
Okay, let's see. The problem is about finding the color of the 78th ball in a row where the colors repeat every five balls: red, yellow, green, blue, purple. So the order is red, yellow, green, blue, purple, then red again, yellow, and so on. There are 100 balls in total, but we only need to figure out the colo... |
olympiads | Find all natural numbers $x$, satisfying the conditions: the product of the digits of the number $x$ is equal to $44x - 86868$, and the sum of the digits is a cube of a natural number. | <think>
Okay, let's tackle this problem step by step. The question is asking to find all natural numbers x such that two conditions are satisfied:
1. The product of the digits of x is equal to 44x - 86868.
2. The sum of the digits of x is a cube of a natural number.
First, let's parse what this means. We need to find... |
olympiads | 33. Find the largest positive integer $n$ such that $\sqrt{n-100}+\sqrt{n+100}$ is a rational number. | <think>
Alright, so I need to find the largest positive integer n such that the expression √(n - 100) + √(n + 100) is a rational number. Hmm, okay. Let me start by breaking this down.
First, since we have square roots involved, the expressions under the roots must be non-negative. So, n - 100 ≥ 0 and n + 100 ≥ 0. The ... |
aops_forum | Define the Fibonacci numbers via $F_0=0$, $f_1=1$, and $F_{n-1}+F_{n-2}$.
Olivia flips two fair coins at the same time, repeatedly, until she has flipped a tails on both, not necessarily on the same throw. She records the number of pairs of flips $c$ until this happens (not including the last pair, so if on the last f... | <think>
Okay, so I need to find the expected value of F_c, where c is the number of pairs of flips Olivia makes until she gets both tails. The Fibonacci sequence is defined here with F_0=0, F_1=1, and then each subsequent term is the sum of the two previous ones. Hmm, okay. Let me first try to understand the problem st... |
olympiads | What could be the largest prime divisor of a number in the decimal system of the form $\overline{x y x y x y}$? | <think>
Okay, so I need to find the largest prime divisor of a number that's written in the form xyxyxy in the decimal system. That is, a six-digit number where the digits alternate between x and y. For example, if x is 1 and y is 2, the number would be 121212. The question is asking for the largest possible prime divi... |
olympiads | 6. Let $f(x)$ be a quadratic function, $g(x)=2^{x} f(x), g(x+1)-g(x)=2^{x+1} \cdot x^{2}$, then $f(x)=$ | <think>
Okay, so I need to find the quadratic function f(x) given that g(x) = 2^x times f(x), and the difference between g(x+1) and g(x) is equal to 2^{x+1} times x squared. Alright, let's break this down step by step.
First, let's recall that a quadratic function f(x) generally has the form f(x) = ax² + bx + c, where... |
olympiads | ## Problem Statement
Calculate the limit of the function:
$$
\lim _{x \rightarrow 2} \frac{\ln (x-\sqrt[3]{2 x-3})}{\sin \left(\frac{\pi x}{2}\right)-\sin ((x-1) \pi)}
$$ | <think>
Alright, so I need to find the limit as x approaches 2 of [ln(x - cube root(2x - 3))] divided by [sin(πx/2) - sin((x - 1)π)]. Hmm, okay. Let me start by understanding the components of this function.
First, let's check if plugging in x = 2 directly works. If both the numerator and denominator approach 0 or bot... |
olympiads | 3 [Pythagorean Theorem in Space]
Find the distance between the midpoints of the non-parallel sides of different bases of a regular triangular prism, all edges of which are equal to 2.
# | <think>
Okay, so I need to find the distance between the midpoints of the non-parallel sides of different bases of a regular triangular prism, where all edges are equal to 2. Hmm, let me start by visualizing the prism. A regular triangular prism has two congruent equilateral triangles as bases and three rectangles conn... |
olympiads | [7] $\triangle A B C$ is right angled at $A . D$ is a point on $A B$ such that $C D=1 . A E$ is the altitude from $A$ to $B C$. If $B D=B E=1$, what is the length of $A D$ ? | <think>
Okay, so I need to solve this geometry problem. Let me try to visualize it first. There's a right triangle ABC, right-angled at A. Point D is on AB such that CD = 1. AE is the altitude from A to BC. Also, BD = BE = 1. The question is asking for the length of AD. Hmm, right. Let me break this down step by step.
... |
olympiads | 1. Given the functions
$$
f(x)=x^{2}-2 x, \quad g(x)=\left\{\begin{array}{ll}
x-2, & x \geqslant 1, \\
-x, & x<1 .
\end{array}\right.
$$
Then the solution set of the inequality $f(x) \leqslant 3 g(x)$ is $\qquad$ (Supplied by An Zhenping) | <think>
Okay, so I need to solve the inequality f(x) ≤ 3g(x), where f(x) is given as x² - 2x and g(x) is a piecewise function. Let me write down both functions again to make sure I have them right.
f(x) = x² - 2x
g(x) is defined as:
- x - 2 when x is greater than or equal to 1
- -x when x is less than 1
So, the ineq... |
olympiads | $28 \cdot 30$ Divide the 8 numbers $12,30,42,44,57,91,95,143$ into two groups so that the product of the numbers in each group is equal. The correct grouping is
(A) $12,42,57,143$ and $30,44,91,95$.
(B) $12,30,95,143$ and $42,44,57,91$.
(C) $12,42,95,143$ and $30,44,57,91$.
(D) $12,44,95,143$ and $30,42,57,91$.
(3rd "F... | <think>
Okay, let's try to figure out how to divide these eight numbers into two groups with equal products. The options are given, so maybe we can check each one. But first, maybe there's a smarter way than multiplying all numbers in each group for every option. Let me think.
First, let me list the numbers: 12, 30, 4... |
aops_forum | How many ways can the integers from $-7$ to $7$ inclusive be arranged in a sequence such that the absolute value of the numbers in the sequence does not decrease? | <think>
Okay, so I need to figure out how many ways the integers from -7 to 7 inclusive can be arranged in a sequence such that the absolute value of the numbers does not decrease. Let me break this down.
First, the numbers involved are from -7 to 7, which includes zero. So that's 15 numbers in total: -7, -6, ..., -1... |
olympiads | Example 2. Solve the equation
$$
(0.4)^{x-1}=(6.25)^{6x-5}
$$ | <think>
Okay, so I need to solve the equation (0.4)^{x-1} = (6.25)^{6x-5}. Hmm, let's see. I remember that when dealing with exponential equations, if the bases can be made the same, that might be helpful. Otherwise, taking logarithms is a good approach. Let me check if 0.4 and 6.25 can be expressed with the same base.... |
olympiads | Vassilyev N.B.
In a round glass, the axial cross-section of which is the graph of the function $y=x^{4}$, a cherry - a sphere of radius $r$ - is dropped. For what largest $r$ will the sphere touch the lowest point of the bottom? (In other words, what is the maximum radius $r$ of a circle lying in the region $y \geq x^... | <think>
Alright, so I need to figure out the largest radius r of a sphere (which is a circle in this 2D cross-section) that can fit inside the glass defined by y = x⁴ and still touch the origin, which is the lowest point of the bottom. Let me break this down step by step.
First, let's visualize the problem. The glass ... |
olympiads | Task 1. Martin is older than his sisters Ana, Ivana, and Elena by 8, 10, and 13 years, respectively. In one year, the sum of the ages of Martin's sisters will be 39 years more than Martin's age. How old is Martin today? | <think>
Okay, let's see. So the problem is about Martin and his three sisters: Ana, Ivana, and Elena. Martin is older than each of them by 8, 10, and 13 years respectively. Then, in one year, the sum of the ages of the sisters will be 39 years more than Martin's age at that time. We need to find Martin's current age.
... |
olympiads | 8. (10 points) When withdrawing money from an $A T M$ machine, one needs to input the bank card password to proceed to the next step. The password is a 6-digit number ranging from 000000 to 999999. A person forgot the password but remembers that it contains the digits 1, 3, 5, 7, 9 and no other digits. If there is no l... | <think>
Okay, so I need to figure out how many different 6-digit passwords someone might have to try if they remember that the password contains the digits 1, 3, 5, 7, 9 and no other digits. The password is a 6-digit number, so each position can be one of those five digits, right? Wait, but hold on, the problem says "c... |
olympiads |
NT3 Let $s(a)$ denote the sum of digits of a given positive integer $a$. The sequence $a_{1}, a_{2}, \ldots a_{n}, \ldots$ of positive integers is such that $a_{n+1}=a_{n}+s\left(a_{n}\right)$ for each positive integer $n$. Find the greatest possible $n$ for which it is possible to have $a_{n}=2008$.
| <think>
Okay, let's see. I need to find the greatest possible n such that there's a sequence where each term is the previous term plus the sum of its digits, and the nth term is 2008. So, starting from some number a₁, we add the sum of its digits to get a₂, then add the sum of a₂'s digits to get a₃, and so on until we ... |
aops_forum | There are $n$ distinct lines in three-dimensional space such that no two lines are parallel and no three lines meet at one point. What is the maximal possible number of planes determined by these $n$ lines?
We say that a plane is determined if it contains at least two of the lines. | <think>
Okay, so the problem is about finding the maximal number of planes determined by n distinct lines in three-dimensional space. The conditions are that no two lines are parallel and no three lines meet at one point. Hmm, interesting. Let me try to think through this step by step.
First, I need to recall some bas... |
olympiads | 125. There are 60 squares, each with a side length of 1 cm. From these squares, all possible rectangles are formed, each using all 60 squares. Find the sum of the areas of all these rectangles.
## Answers. Solutions | <think>
Alright, let's tackle this problem. So, we have 60 squares, each 1 cm in side length. We need to form all possible rectangles using all 60 squares and find the sum of the areas of all these rectangles. Hmm, okay. Let me break this down step by step.
First, I need to understand what exactly the problem is askin... |
olympiads | ## Task Condition
Calculate the volumes of bodies bounded by surfaces.
$$
z=x^{2}+4 y^{2}, z=2
$$ | <think>
Okay, so I need to find the volume bounded by the surfaces z = x² + 4y² and z = 2. Hmm, let me visualize this. The first surface is a paraboloid that opens upward, right? Because z is equal to x squared plus 4 times y squared. So, in the x-direction, it's a standard parabola, but in the y-direction, it's squash... |
olympiads | $4 \cdot 20$ There is a $\triangle A B C, a 、 b 、 c$ are the sides opposite to $\angle A 、 \angle B 、 \angle C$ respectively. If $b$ is the arithmetic mean of $a 、 c$, and $\operatorname{tg} \frac{B}{2}$ is the geometric mean of $\operatorname{tg} \frac{A}{2} 、 \operatorname{tg} \frac{C}{2}$, try to form a quadratic eq... | <think>
Alright, so I need to form a quadratic equation whose roots are tg(A/2) and tg(C/2) given some conditions in triangle ABC. Let me jot down what's given:
1. Triangle ABC, with sides a, b, c opposite angles A, B, C respectively.
2. b is the arithmetic mean of a and c. So, that means b = (a + c)/2.
3. tg(B/2) is ... |
olympiads | 1. [5 points] Point $D$ lies on side $A C$ of triangle $A B C$. The circle with diameter $B D$ intersects sides $A B$ and $B C$ at points $P$ and $T$ respectively. Points $M$ and $N$ are the midpoints of segments $A D$ and $C D$ respectively. It is known that $P M \| T N$.
a) Find the angle $A B C$.
b) Suppose additi... | <think>
Okay, so I have this geometry problem here. Let me try to parse through it step by step. The problem is about triangle ABC, with point D on side AC. There's a circle with diameter BD that intersects sides AB and BC at points P and T respectively. Then, M and N are the midpoints of AD and CD. It's given that PM ... |
olympiads | 4. Given the vector $\boldsymbol{a}=(1,1)$. Then the vector $\boldsymbol{b}=$ $\left(\frac{1-\sqrt{3}}{2}, \frac{1+\sqrt{3}}{2}\right)$ is obtained by rotating vector $a$ ( ).
(A) clockwise by $60^{\circ}$
(B) clockwise by $120^{\circ}$
(C) counterclockwise by $60^{\circ}$
(D) counterclockwise by $120^{\circ}$ | <think>
Okay, so I have this problem here where I'm given vector a as (1,1) and vector b as ((1 - sqrt(3))/2, (1 + sqrt(3))/2). The question is asking me to determine the angle and direction (clockwise or counterclockwise) that vector a was rotated to get vector b. The options are 60 or 120 degrees, clockwise or counte... |
olympiads | 8. (10 points) Given that the product of 4 prime numbers is 11 times their sum, then their sum is ( )
A. 46
B. 47
C. 48
D. There are no numbers that meet the condition | <think>
Okay, let's tackle this problem step by step. The question says that the product of 4 prime numbers is 11 times their sum, and we need to find their sum from the given options or determine if there's no solution. The options are A. 46, B. 47, C. 48, D. No numbers meet the condition.
First, let's restate the pr... |
olympiads | G2.2 If $R$ is the remainder of $1^{6}+2^{6}+3^{6}+4^{6}+5^{6}+6^{6}$ divided by 7 , find the value of $R$. | <think>
Okay, let me try to figure out this problem. So, we need to find the remainder R when the sum of 1⁶ + 2⁶ + 3⁶ + 4⁶ + 5⁶ + 6⁶ is divided by 7. Hmm, okay. Let me start by recalling some math concepts that might help here. Maybe modular arithmetic? Yeah, since we're dealing with remainders, that's probably the way... |
olympiads_ref | 8. 1b.(TUR 5) Find the smallest positive integer $n$ such that (i) $n$ has exactly 144 distinct positive divisors, and (ii) there are ten consecutive integers among the positive divisors of $n$. | <think>
Okay, so I need to find the smallest positive integer n that has exactly 144 distinct positive divisors and also has ten consecutive integers among its divisors. Hmm, let me try to break this down step by step.
First, I know that the number of divisors of a number relates to its prime factorization. If n is fa... |
olympiads | 19.2.15 ** Let $s$ be the set of all rational numbers $r$ that satisfy the following conditions: $0<r<1$, and $r$ can be expressed as a repeating decimal of the form $0 . a b c a b c a b c \cdots=0 . \dot{a} b \dot{c}$, where the digits $a, b, c$ do not necessarily have to be distinct. Write the elements of $s$ as redu... | <think>
Okay, so I have this problem here where I need to find the number of different numerators when all the elements of set S are written as reduced fractions. Set S consists of all rational numbers r between 0 and 1 such that r can be expressed as a repeating decimal of the form 0.abcabcabc..., where a, b, c are di... |
olympiads | 10. $[\mathbf{8}]$ Let $f(n)=\sum_{k=1}^{n} \frac{1}{k}$. Then there exists constants $\gamma, c$, and $d$ such that
$$
f(n)=\ln (n)+\gamma+\frac{c}{n}+\frac{d}{n^{2}}+O\left(\frac{1}{n^{3}}\right),
$$
where the $O\left(\frac{1}{n^{3}}\right)$ means terms of order $\frac{1}{n^{3}}$ or lower. Compute the ordered pair $(... | <think>
Okay, so I need to find the constants c and d in the asymptotic expansion of the harmonic series f(n) = H_n, which is the sum from k=1 to n of 1/k. The given expansion is H_n = ln(n) + γ + c/n + d/n² + O(1/n³). I remember that the harmonic series is related to the natural logarithm and the Euler-Mascheroni cons... |
olympiads | Problem 5.3. A cuckoo clock is hanging on the wall. When a new hour begins, the cuckoo says "cuckoo" a number of times equal to the number the hour hand points to (for example, at 19:00, "cuckoo" sounds 7 times). One morning, Maxim approached the clock when it was 9:05. He started turning the minute hand until he advan... | <think>
Okay, so the problem is about a cuckoo clock. Maxim turns the minute hand forward by 7 hours starting at 9:05 in the morning. We need to figure out how many times the cuckoo sounds during this time. Let me try to break this down step by step.
First, let's understand how the cuckoo clock works. It says "cuckoo"... |
olympiads | (12) The line $y=b$ intersects the graph of the function $y=|x^2-4x+3|$ at least at three points, then the range of the real number $b$ is | <think>
Okay, so I need to find the range of real numbers b such that the horizontal line y = b intersects the graph of y = |x² - 4x + 3| at least three times. Let me think step by step.
First, the function inside the absolute value is a quadratic: x² - 4x + 3. Let me graph that quadratic first before considering the ... |
olympiads | 4. Let $z_{1}, z_{2}$ be complex numbers, and $\left|z_{1}\right|=3,\left|z_{2}\right|=5,\left|z_{1}+z_{2}\right|=7$, then the value of $\arg \left(\frac{z_{2}}{z_{1}}\right)^{3}$ is . $\qquad$ | <think>
Okay, let's see. I need to find the argument of (z₂/z₁)³, given the magnitudes of z₁, z₂, and their sum. Hmm. Alright, let's start by recalling some properties of complex numbers. The modulus of a complex number z is |z|, and the argument is the angle it makes with the positive real axis. When dividing two comp... |
olympiads_ref | Determine all integers $a>0$ for which there exist strictly positive integers $n, s, m_{1}, \cdots, m_{n}, k_{1}, \cdots, k_{s}$ such that
$$
\left(a^{m_{1}}-1\right) \cdots\left(a^{m_{n}}-1\right)=\left(a^{k_{1}}+1\right) \cdots\left(a^{k_{s}}+1\right)
$$ | <think>
Okay, so I have this problem here: I need to find all integers a > 0 such that there exist strictly positive integers n, s, m1, ..., mn, k1, ..., ks where the product of (a^{m1} - 1) * ... * (a^{mn} - 1) equals the product of (a^{k1} + 1) * ... * (a^{ks} + 1). Hmm, interesting. Let me try to wrap my head around... |
olympiads | A square number in the decimal system has the form: $\overline{a b 1 a b}$. Which is this square number? | <think>
Okay, so I need to find a square number that has the form \(\overline{a b 1 a b}\) in the decimal system. That means the number is a five-digit number where the first two digits are 'a' and 'b', followed by a 1, then again 'a' and 'b'. So the structure is like: a, b, 1, a, b. Each letter represents a digit, rig... |
olympiads | 99(1209). How many consecutive natural numbers, starting from 1, need to be added to get a three-digit number written with identical digits? | <think>
Okay, so I need to figure out how many consecutive natural numbers starting from 1 I have to add together to get a three-digit number where all the digits are the same. Hmm, three-digit numbers with identical digits are numbers like 111, 222, up to 999. So the possible targets are 111, 222, 333, 444, 555, 666, ... |
olympiads | To be calculated
$$
a^{3}+b^{3}+3\left(a^{3} b+a b^{3}\right)+6\left(a^{3} b^{2}+a^{2} b^{3}\right)
$$
value, if $a+b=1$. | <think>
Okay, so I need to find the value of this expression:
$$
a^{3} + b^{3} + 3(a^{3}b + ab^{3}) + 6(a^{3}b^{2} + a^{2}b^{3})
$$
given that \( a + b = 1 \). Hmm, let's see. I remember that when dealing with expressions involving \( a + b \), it might be helpful to use substitution or factorization. Maybe I can ex... |
olympiads | [ Skeletons of Polyhedral Figures ]
A toy factory produces wireframe cubes, with small colorful beads located at their vertices. According to the GOST standard, each cube must use beads of all eight colors (white and the seven colors of the rainbow). How many different models of cubes can the factory produce?
# | <think>
Okay, let's see. The problem is about counting how many different models of wireframe cubes a toy factory can produce. Each cube has beads at its vertices, and they must use all eight colors: white and the seven rainbow colors (which I guess are red, orange, yellow, green, blue, indigo, violet). The key here is... |
olympiads | 30. Find the number of subsets $\{a, b, c\}$ of $\{1,2,3,4, \ldots, 20\}$ such that $a<b-1<c-3$. | <think>
Okay, so I need to find the number of subsets {a, b, c} from the set {1,2,3,...,20} such that a < b-1 < c-3. Hmm, let me parse this condition.
First, the subset has three elements: a, b, c. They must satisfy a < b - 1 and b - 1 < c - 3. Let me rewrite these inequalities to make them clearer.
Starting with a... |
olympiads | Find the $(x, y)$ natural integers such that $x^{2}=y^{2}+7 y+6$ | <think>
Okay, let's try to solve this equation: find natural integers (x, y) such that x² = y² + 7y + 6. Hmm, natural integers, so x and y are both positive integers, right? Let me think step by step.
First, maybe I can rearrange the equation to make it easier to handle. If I subtract y² from both sides, I get x² - y²... |
olympiads | ## Task 1 - 090731
Imagine all natural numbers from 1 to 2555, each written exactly once. Determine the total number of the digit 9 that would need to be written! | <think>
Okay, so I need to figure out how many times the digit 9 appears when writing all natural numbers from 1 to 2555. Hmm, let me think. This seems like a problem where I have to count the occurrences of a specific digit in each position—units, tens, hundreds, and thousands. Yeah, that makes sense. Let me break it ... |
olympiads | 3. A robot is located in one of the cells of an infinite grid paper, to which the following commands can be given:
- up (the robot moves to the adjacent cell above);
- down (the robot moves to the adjacent cell below);
- left (the robot moves to the adjacent cell to the left);
- right (the robot moves to the adjacent ... | <think>
Okay, so I need to figure out how many different sequences of 8 commands will bring the robot back to its starting position. The commands are up, down, left, right. Each command moves the robot one cell in that direction. A sequence of commands is a series of these moves, and we want sequences where, after 8 mo... |
olympiads | ## Task 2 - 060612
At the beginning of the school year, Heinz bought two different types of notebooks, one costing 8 Pf, the other 15 Pf per piece. He paid 1.31 MDN for 12 notebooks in total.
How many notebooks did he buy of each type? | <think>
Okay, let's see. Heinz bought two types of notebooks. One costs 8 Pf each and the other 15 Pf each. He bought a total of 12 notebooks and spent 1.31 MDN. Hmm, I need to find out how many of each type he bought.
First, I should probably convert all the amounts to the same unit. Since the notebooks are priced i... |
olympiads | 9. $[7]$ Let $a_{0}=\frac{6}{7}$, and
$$
a_{n+1}=\left\{\begin{array}{ll}
2 a_{n} & \text { if } a_{n}<\frac{1}{2} \\
2 a_{n}-1 & \text { if } a_{n} \geq \frac{1}{2}
\end{array}\right.
$$
Find $a_{2008}$. | <think>
Okay, let's tackle this problem step by step. So, we have a sequence defined by a₀ = 6/7, and then each subsequent term is determined by the rule: if aₙ is less than 1/2, multiply it by 2; if it's 1/2 or more, subtract 1 after multiplying by 2. The question is asking for a₂₀₀₈. Hmm, that's a pretty large index.... |
aops_forum | $a, b, c$ are positive real numbers such that $$(\sqrt {ab}-1)(\sqrt {bc}-1)(\sqrt {ca}-1)=1$$
At most, how many of the numbers: $$a-\frac {b}{c}, a-\frac {c}{b}, b-\frac {a}{c}, b-\frac {c}{a}, c-\frac {a}{b}, c-\frac {b}{a}$$ can be bigger than $1$? | <think>
Alright, let's tackle this problem step by step. We have three positive real numbers \(a\), \(b\), and \(c\) satisfying the equation:
\[
(\sqrt{ab} - 1)(\sqrt{bc} - 1)(\sqrt{ca} - 1) = 1
\]
We need to determine the maximum number of the following expressions that can be greater than 1:
\[
a - \frac{b}{c}, \q... |
cn_contest | 3. In an equilateral triangle with side length 1, there is a point $P$, the distances from point $P$ to the three sides are $a, b, c$, respectively, then $a+b+c=$ $\qquad$ - | <think>
Okay, let's see. I have an equilateral triangle with side length 1. There's a point P inside it, and the distances from P to the three sides are a, b, c. The question is asking what a + b + c equals. Hmm. I remember something about this... Maybe it's related to the height of the triangle? Let me think.
First, ... |
olympiads | 1. Circle $k_{1}$ with radius 1 has an external tangency $\mathrm{s}$ with circle $k_{2}$ with radius 2. Each of the circles $k_{1}, k_{2}$ has an internal tangency with circle $k_{3}$ with radius 3. Calculate the radius of circle $k$, which has external tangency $\mathrm{s}$ with circles $k_{1}, k_{2}$ and internal ta... | <think>
Okay, so there's this geometry problem about circles. Let me try to visualize it first. We have three circles: k₁ with radius 1, k₂ with radius 2, and k₃ with radius 3. The problem states that k₁ and k₂ are externally tangent to each other. Both k₁ and k₂ are internally tangent to k₃. Then, we need to find the ... |
olympiads | 9. Given triangle $A B C$, let $D$ be a point on side $A B$ and $E$ be a point on side $A C$. Let $F$ be the intersection of $B E$ and $C D$. If $\triangle D B F$ has an area of $4, \triangle B F C$ has an area of 6 , and $\triangle F C E$ has an area of 5 , find the area of quadrilateral $A D F E$. | <think>
Okay, let me try to figure out this geometry problem. So we have triangle ABC, with points D on AB and E on AC. Then BE and CD intersect at point F. We're given the areas of three triangles: DBF is 4, BFC is 6, and FCE is 5. We need to find the area of quadrilateral ADFE.
Hmm, okay. Let me start by drawing a r... |
olympiads | 4. Vasya remembers that his friend Petya lives on Kurchatovskaya Street, house number 8, but he forgot the apartment number. In response to a request to clarify the address, Petya replied: “My apartment number is a three-digit number. If you rearrange the digits, you can get five other three-digit numbers. So, the sum ... | <think>
Okay, so the problem is about finding Petya's apartment number, which is a three-digit number. When you rearrange its digits, you can get five other three-digit numbers, and the sum of those five numbers is 2017. I need to figure out what the original number is.
First, let's parse the problem. The apartment nu... |
olympiads | Tairova
A father and son are skating in a circle. From time to time, the father overtakes the son. After the son changed the direction of his movement to the opposite, they started meeting 5 times more frequently. How many times faster does the father skate compared to the son? | <think>
Okay, so there's this problem about a father and son skating in a circle. When they're both going the same direction, the father overtakes the son every so often. Then the son changes direction, so they're moving in opposite directions, and now they meet 5 times more frequently. The question is asking how many ... |
olympiads | 3.2.8* The sequence $\left\{a_{n}\right\}$ satisfies: $a_{1}=1, a_{2}=2, a_{n+2}+a_{n+1}+a_{n}=0$. Find the general term formula $a_{n}$. | <think>
Okay, so I need to find the general term formula for the sequence {a_n} where a_1 = 1, a_2 = 2, and the recurrence relation is a_{n+2} + a_{n+1} + a_n = 0. Hmm, this looks like a linear recurrence relation with constant coefficients. I remember that for such recursions, we can find the characteristic equation a... |
olympiads | 10. A kilogram of one type of candy is 80 k more expensive than another. Andrey and Yura bought 150 g of candies, which included both types of candies, with Andrey having twice as much of the first type as the second, and Yura having an equal amount of each. Who paid more for their purchase and by how much?
14 | <think>
Okay, let me try to figure out this problem step by step. So, the problem is about two types of candy. The first type is 80 kopeks more expensive per kilogram than the second type. Then we have two people, Andrey and Yura, who each bought 150 grams of candies, which include both types. The details are: Andrey b... |
olympiads | 1. Compare the fractions $f_{1}=\frac{a+125^{725}}{a+625^{544}}$ and $f_{2}=\frac{b+121^{1007}}{b+343^{671}}$, where $a$ and $b$ are natural numbers. | <think>
Alright, so I need to compare these two fractions: f₁ = (a + 125^725)/(a + 625^544) and f₂ = (b + 121^1007)/(b + 343^671), where a and b are natural numbers. Hmm, okay. Let me try to figure out how to approach this.
First, maybe I should express the bases of the exponents in terms of their prime factors? That ... |
olympiads | In a tournament, each participant plays a match against every other participant. The winner of a match earns 1 point, the loser 0 points, and if the match is a draw, both players earn half a point. At the end of the tournament, the participants are ranked according to their score (in case of a tie, the order is arbitra... | <think>
Okay, so I have this problem here about a tournament where each participant plays against every other participant. The scoring is straightforward: a win gives 1 point, a loss gives 0, and a draw gives half a point each. After the tournament, they rank everyone based on their scores, and if there's a tie, the or... |
olympiads | 1. It is known that for three consecutive natural values of the argument, the quadratic function $f(x)$ takes the values 6, 14, and 14, respectively. Find the greatest possible value of $f(x)$. | <think>
Alright, let's tackle this problem step by step. The problem states that a quadratic function f(x) takes the values 6, 14, and 14 for three consecutive natural numbers as arguments. We need to find the greatest possible value of f(x). Hmm, okay. Let me break this down.
First, since f(x) is a quadratic function... |
cn_contest | Example 2 Divide the numbers $1,2, \cdots, 200$ into two groups arbitrarily, each containing 100 numbers. Arrange one group in ascending order (denoted as $a_{1}<a_{2}<\cdots<a_{100}$) and the other in descending order (denoted as $b_{1}>b_{2}>\cdots>b_{100}$). Try to find
$$
\left|a_{1}-b_{1}\right|+\left|a_{2}-b_{2}\... | <think>
Okay, so I need to figure out the value of this expression where we split the numbers from 1 to 200 into two groups of 100 each, sort one group in ascending order and the other in descending order, and then sum up the absolute differences between corresponding elements. Hmm, let's break this down step by step.
... |
olympiads | 1. Petya and Vasya competed in a 60 m race. When Petya finished, Vasya was 9 m behind him. During the second race, Petya started exactly 9 m behind Vasya. Who finished first in the second race and by how many meters did he outpace his opponent? (Assume that each boy ran at a constant speed both times). | <think>
Okay, so there's this problem about Petya and Vasya racing twice. Let me try to figure it out step by step. First, I need to understand the details given.
In the first race, they both ran 60 meters. When Petya finished, Vasya was 9 meters behind. That means by the time Petya completed 60 meters, Vasya had only... |
cn_contest | 1. If $x=\frac{23+\sqrt{469}}{5}$, then $25 x^{4}-1996 x^{2}+144$ $=(\quad)$.
(A) 0
(B) 1
(C) 469
(D) 1996 | <think>
Okay, let's see. The problem is asking me to evaluate the expression \(25x^4 - 1996x^2 + 144\) where \(x = \frac{23 + \sqrt{469}}{5}\). The options are A through D with numbers 0, 1, 469, and 1996. Hmm, so I need to compute that expression with the given x. Let me think about how to approach this.
First, maybe... |
olympiads | 80. Given that $\mathrm{n}$ is a positive integer, and the tens digit of $n^{2}$ is 7, find $n-100\left[\frac{n}{100}\right]$. (Note: $[x]$ denotes the integer part of $\mathrm{x}$) | <think>
Okay, let's see. The problem says that n is a positive integer, and the tens digit of n squared is 7. We need to find n minus 100 times the integer part of n divided by 100. Hmm, that expression sounds like it's asking for n modulo 100, right? Because when you take n divided by 100, the integer part is how many... |
olympiads | Subject 2. Let $x \neq 1, \quad y \neq-2, \quad z \neq-3$ be rational numbers such that $\frac{2015}{x+1}+\frac{2015}{y+2}+\frac{2015}{z+3}=2014$. Calculate $\frac{x-1}{x+1}+\frac{y}{y+2}+\frac{z+1}{z+3}$. | <think>
Okay, let's see. I need to solve this problem where we have three rational numbers x, y, z, each not equal to certain values, and they satisfy the equation (2015)/(x+1) + (2015)/(y+2) + (2015)/(z+3) = 2014. Then, I need to calculate another expression involving x, y, z: (x-1)/(x+1) + y/(y+2) + (z+1)/(z+3). Hmm,... |
olympiads | (solved by Alice Héliou). Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying:
$$
f\left(x^{2}-y^{2}\right)=(x-y)(f(x)+f(y))
$$
for all real numbers $x$ and $y$. | <think>
Okay, so I need to find all functions f: R → R that satisfy the equation f(x² - y²) = (x - y)(f(x) + 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, maybe I should start by plugging in some specific values for x and y to see if... |
olympiads | 23. Let $a=\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$ and $b=\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$. Find the value of $a^{4}+b^{4}+(a+b)^{4}$. | <think>
Okay, so I need to find the value of \( a^{4} + b^{4} + (a + b)^{4} \), where \( a = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} \) and \( b = \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}} \). Hmm, this looks a bit complicated, but maybe there's a way to simplify \( a \) and \( b \) first before dealing w... |
olympiads | We create a random sequence using the digits $0, 1, 2$. For sequences of what length will the probability be at least $61 \%$ that all three digits appear in the sequence? | <think>
Okay, so the problem is: We create a random sequence using the digits 0, 1, 2. For sequences of what length will the probability be at least 61% that all three digits appear in the sequence?
Hmm, let me try to figure this out. So we need to find the smallest integer n such that the probability of a random sequ... |
cn_contest | 4. Given the sequence $\left\{x_{n}\right\}$ satisfies $x_{1}=1$, for $n \geqslant 1$, we have
$$
\begin{array}{l}
4\left(x_{1} x_{n}+2 x_{2} x_{n-1}+3 x_{3} x_{n-2}+\cdots+n x_{n} x_{1}\right) \\
=(n+1)\left(x_{1} x_{2}+x_{2} x_{3}+\cdots+x_{n} x_{n+1}\right) .
\end{array}
$$
Find the general term formula for $\left\... | <think>
Okay, let's see. So we have this sequence {x_n} where x₁ = 1, and for every n ≥ 1, there's this equation involving a sum of products of terms from the sequence. The left side is 4 times the sum of terms like k x_k x_{n - k + 1} for k from 1 to n, and the right side is (n + 1) times the sum of terms like x_k x_{... |
olympiads | 1.049. $\frac{2^{-2}+5^{0}}{(0.5)^{-2}-5(-2)^{-2}+\left(\frac{2}{3}\right)^{-2}}+4.75$. | <think>
Okay, let's tackle this problem step by step. The expression given is:
1.049. \(\frac{2^{-2}+5^{0}}{(0.5)^{-2}-5(-2)^{-2}+\left(\frac{2}{3}\right)^{-2}} +4.75\).
First, I need to simplify the complex fraction in the numerator and denominator before dealing with the addition of 4.75. Let me break it down into ... |
olympiads | 13. Vector $\vec{a}=(1, \sin \theta), \vec{b}=(\cos \theta, \sqrt{3}), \theta \in R$, then the range of $|\vec{a}-\vec{b}|$ is | <think>
Okay, so I need to find the range of the magnitude of the vector difference between vectors a and b. The vectors are given as a = (1, sinθ) and b = (cosθ, √3), where θ is a real number. The problem is asking for the possible values that |a - b| can take. Hmm, right. Let me think step by step.
First, I remember... |
olympiads | Two spheres of one radius and two of another are arranged so that each sphere touches three others and one plane. Find the ratio of the radius of the larger sphere to the radius of the smaller one.
# | <think>
Okay, so I have this geometry problem here: there are two spheres of one radius and two of another. They're arranged such that each sphere touches three others and one plane. I need to find the ratio of the radius of the larger sphere to the smaller one. Hmm, okay, let's try to visualize this.
First, let me pa... |
olympiads | G8.4 If $m n p, n m p, m m p$ and $n n p$ are numbers in base 10 composed of the digits $m, n$ and $p$, such that: $m n p-n m p=180$ and $m m p-n n p=d$. Find $d$. | <think>
Alright, let's try to figure out this problem step by step. So, we have these numbers in base 10: mnp, nmp, mmp, and nnp. Each of these is a three-digit number composed of the digits m, n, and p. The problem gives us two equations:
1. mnp - nmp = 180
2. mmp - nnp = d
We need to find the value of d.
First, le... |
olympiads | 44. In the kitchen, there are five light bulbs. How many lighting options exist?
Do you understand what lighting options mean? Each lamp can be on or off. Two options are considered different if they differ in the state of at least one lamp (see Fig. 7 and 8). | <think>
Okay, let's see. The problem is about five light bulbs in the kitchen, and we need to find out how many different lighting options there are. The user mentioned that each lamp can be either on or off, and two options are different if at least one lamp has a different state. Hmm, so I need to figure out all the ... |
olympiads | 7.042. $x(\lg 5-1)=\lg \left(2^{x}+1\right)-\lg 6$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
7.042. $x(\lg 5-1)=\lg \left(2^{x}+1\right)-\lg 6$. | <think>
Okay, let's see. I need to solve this equation: x times (log base 10 of 5 minus 1) equals the logarithm of (2 to the x power plus 1) minus the logarithm of 6. Hmm.
First, let me write down the equation again to make sure I have it right. So, it's x(lg 5 - 1) = lg(2^x + 1) - lg 6. Alright. Maybe I can simplify ... |
olympiads | 6. In the Cartesian coordinate system $x O y$, the point set $K=\{(x, y) \mid x, y=-1,0,1\}$. If three points are randomly selected from $K$, then the probability that the distance between each pair of these three points is no more than 2 is $\qquad$ . | <think>
Okay, let's try to figure out this probability problem. So, we have a set K in the Cartesian coordinate system where each coordinate x and y can be -1, 0, or 1. That means K is a 3x3 grid of points, right? So, there are 9 points in total. We need to randomly select three points from these 9 and find the probabi... |
olympiads | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow \frac{\pi}{2}} \sqrt{3 \sin x+(2 x-\pi) \sin \frac{x}{2 x-\pi}}$ | <think>
Okay, so I need to find the limit of this function as x approaches π/2. The function is sqrt(3 sin x + (2x - π) sin (x / (2x - π))). Hmm, let me write that down to make sure I have it right:
Limit as x approaches π/2 of sqrt[3 sin x + (2x - π) sin(x / (2x - π))]
First, let me check if I can directly substitut... |
olympiads | 11.4 If $\sqrt{3-2 \sqrt{2}}=\sqrt{c}-\sqrt{d}$, find the value of $d$. | <think>
Okay, let's see. The problem is: If the square root of (3 minus 2 times the square root of 2) equals the square root of c minus the square root of d, find the value of d. Hmm. Alright, so I need to figure out what c and d are such that when I take the square root of c minus the square root of d, it equals the s... |
olympiads | $\underline{\text { Zhenedarov R.G. }}$
Two pirates, Bill and John, each having 74 gold coins, decided to play such a game: they will take turns laying coins on the table, one, two, or three at a time, and the one who places the hundredth coin in sequence will win. Bill starts. Who can win this game, regardless of how... | <think>
Okay, so there's this problem where two pirates, Bill and John, each have 74 gold coins. They take turns laying down 1, 2, or 3 coins on the table, and whoever places the 100th coin wins. Bill goes first. The question is, who can guarantee a win no matter how the other plays?
Hmm, let's think. This seems like ... |
olympiads | Ondra spent $\frac{2}{3}$ of his money on a trip and then gave $\frac{2}{3}$ of the remainder to a school for children from Tibet. He then spent $\frac{2}{3}$ of the new remainder on a small gift for his mother. From the remaining money, he lost $\frac{4}{5}$ due to a hole in his pocket, and when he gave half of what w... | <think>
Okay, let me try to figure out how much money Ondra started with. Hmm, the problem says he spent 2/3 of his money on a trip first. Then he gave 2/3 of the remainder to a school, then spent 2/3 of the new remainder on a gift for his mom. Then he lost 4/5 of what he had left due to a hole in his pocket, and after... |
olympiads | One of five brothers baked a pie for their mother. Andrey said: "It's Vitya or Tolya". Vitya said: "It wasn't me or Yura". Tolya said: "You are both joking". Dima said: "No, one of them is telling the truth, and the other is lying". Yura said: "No, Dima, you are wrong". Mother knows that three of her sons always tell t... | <think>
Okay, let's try to figure out who baked the pie. So, there are five brothers: Andrey, Vitya, Tolya, Dima, and Yura. One of them baked the pie, and their mother knows three of them always tell the truth. We need to determine who the baker is based on their statements.
First, let me list out the statements each ... |
aops_forum | The arithmetic mean of two different positive integers $x,y$ is a two digit integer. If one interchanges the digits, the geometric mean of these numbers is archieved.
a) Find $x,y$.
b) Show that a)'s solution is unique up to permutation if we work in base $g=10$, but that there is no solution in base $g=12$.
c) Give... | <think>
Okay, let's tackle part a) first. The problem says that the arithmetic mean of two different positive integers x and y is a two-digit integer. When we interchange the digits of this arithmetic mean, we get the geometric mean of x and y. So, we need to find x and y that satisfy these conditions.
First, let's pa... |
olympiads | Find all integer pairs $(x, y)$ for which
$$
(x+2)^{4}-x^{4}=y^{3}
$$ | <think>
Okay, let's try to figure out the integer pairs (x, y) that satisfy the equation (x + 2)^4 - x^4 = y³. Hmm, so we need to find integers x and y such that when you expand (x + 2)^4 and subtract x^4, the result is a perfect cube. Alright, let's start by expanding the left side of the equation to see what we're de... |
olympiads | 9 Real numbers $x, y$ satisfy $\left\{\begin{array}{l}x+\sin y=2008, \\ x+2008 \cos y=2007,\end{array}\right.$ where $0 \leqslant y \leqslant \frac{\pi}{2}$, then $x+y=$ | <think>
Okay, so I need to solve this system of equations where x and y are real numbers, and y is between 0 and π/2. The equations are:
1. x + sin y = 2008
2. x + 2008 cos y = 2007
And then find the value of x + y. Hmm, let's see. Both equations have x in them, so maybe I can subtract them or manipulate them to elim... |
olympiads | Task 3. (15 points) The function $f(x)$ satisfies the condition: for any real numbers $a$ and $b$, the equality $f\left(\frac{a+2 b}{3}\right)=\frac{f(a)+2 f(b)}{3}$ holds. Find the value of the function $f(2021)$, if $f(1)=1, f(4)=7$.
# | <think>
Okay, let's see. I have this problem where the function f(x) satisfies the condition that for any real numbers a and b, f((a + 2b)/3) = (f(a) + 2f(b))/3. And I need to find f(2021) given that f(1) = 1 and f(4) = 7. Hmm, interesting. So this is some kind of functional equation. I remember that functional equatio... |
olympiads | [ Perimeter of a Triangle ]
On side $A C$ of triangle $A B C$, point $E$ is marked. It is known that the perimeter of triangle $A B C$ is 25 cm, the perimeter of triangle $A B E$ is 15 cm, and the perimeter of triangle $B C E$ is 17 cm. Find the length of segment $B E$. | <think>
Okay, let me see. So, we have triangle ABC, and on side AC there's a point E. The perimeters of the three triangles are given: the whole triangle ABC has a perimeter of 25 cm, triangle ABE has a perimeter of 15 cm, and triangle BCE has a perimeter of 17 cm. We need to find the length of segment BE.
Hmm. Let me... |
olympiads | 3. Let $x=-\sqrt{2}+\sqrt{3}+\sqrt{5}, y=\sqrt{2}-\sqrt{3}+\sqrt{5}$, and $z=\sqrt{2}+\sqrt{3}-\sqrt{5}$. What is the value of the expression below?
$$
\frac{x^{4}}{(x-y)(x-z)}+\frac{y^{4}}{(y-z)(y-x)}+\frac{z^{4}}{(z-x)(z-y)}
$$ | <think>
Alright, let me try to figure out this problem. So, we have three variables x, y, and z defined as follows:
x = -√2 + √3 + √5
y = √2 - √3 + √5
z = √2 + √3 - √5
And we need to compute the expression:
(x⁴)/[(x - y)(x - z)] + (y⁴)/[(y - z)(y - x)] + (z⁴)/[(z - x)(z - y)]
Hmm, this looks complicated, but mayb... |
olympiads | ## 84. And One More Case
- I remember another trial I attended as if it were today, - began the White Knight. - There were three defendants, and only one of them was guilty. I remember that the first defendant accused the second, but I completely forgot what the second and third defendants said. I also remember that l... | <think>
Okay, let's try to figure out who the guilty person is here. So, there are three defendants, and only one of them is guilty. The White Knight remembers that the first defendant accused the second one. He forgot what the second and third said, but he told the Black Queen an additional fact: either that the guilt... |
olympiads | 8. When $a$ and $b$ are two distinct positive numbers, among the following three algebraic expressions:
甲: $\left(a+\frac{1}{a}\right)\left(b+\frac{1}{b}\right)$,
乙: $\left(\sqrt{a b}+\frac{1}{\sqrt{a b}}\right)^{2}$,
丙: $\left(\frac{a+b}{2}+\frac{2}{a+b}\right)^{2}$
the one with the largest value is
(A) definitely 甲;... | <think>
Okay, let's try to figure out which of the three expressions 甲, 乙, or 丙 is the largest when a and b are two distinct positive numbers. Hmm, the options are given as (A) definitely 甲, (B) definitely 乙, (C) definitely 丙, or (D) it depends. So I need to compare these three expressions and see if one is always larg... |
olympiads | 2. Solve the equation $\sqrt{8 x+5}+2\{x\}=2 x+2$. Here $\{x\}$ is the fractional part of the number $\mathrm{x}$, i.e., $\{x\}=x-[x]$. In the answer, write the sum of all solutions. | <think>
Okay, let's see. I need to solve the equation √(8x + 5) + 2{x} = 2x + 2. Here, {x} is the fractional part of x, so {x} = x - [x], where [x] is the integer part of x. The problem asks for the sum of all solutions. Hmm, fractional part equations can be tricky because they involve both integer and fractional compo... |
cn_contest | 4. As shown in Figure 1, in the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, $P$ is a point on edge $A B$. A line $l$ is drawn through point $P$ in space such that $l$ makes a $30^{\circ}$ angle with both plane $A B C D$ and plane $A B C_{1} D_{1}$. The number of such lines is ( ).
(A) 1
(B) 2
(C) 3
(D) 4 | <think>
Okay, let's try to tackle this problem step by step. So, we have a cube ABCD-A₁B₁C₁D₁, and point P is on edge AB. We need to find the number of lines l passing through P that make a 30° angle with both the plane ABCD (which is the base of the cube) and the plane ABC₁D₁ (which is one of the side faces, I think).... |
olympiads | 3. The fields of a board with 25 rows and 125 columns are alternately colored black and white (like the fields on a chessboard), and knight figures are placed on it, which attack according to special rules. A knight on a white field attacks all white fields in the same row and all black fields in the same column, while... | <think>
Alright, let's tackle this problem step by step. So, we have a chessboard-like grid with 25 rows and 125 columns. The fields are alternately colored black and white. Knights are placed on this board, but their attacking rules are a bit different from regular chess.
First, let me make sure I understand the att... |
olympiads | In an acute-angled triangle, two altitudes are equal to 3 and $2 \sqrt{2}$, and their point of intersection divides the third altitude in the ratio 5:1, counting from the vertex of the triangle. Find the area of the triangle. | <think>
Okay, so I have this problem here about an acute-angled triangle with two altitudes equal to 3 and 2√2. The point where the altitudes intersect (the orthocenter) divides the third altitude in the ratio 5:1, starting from the vertex. I need to find the area of the triangle. Hmm, let's break this down step by ste... |
aops_forum | Fredek runs a private hotel. He claims that whenever $ n \ge 3$ guests visit the hotel, it is possible to select two guests that have equally many acquaintances among the other guests, and that also have a common acquaintance or a common unknown among the guests. For which values of $ n$ is Fredek right? (Acquaintance ... | <think>
Okay, let me try to figure out this problem. So, Fredek claims that whenever there are n ≥ 3 guests in his hotel, you can always find two guests who have the same number of acquaintances among the others. Moreover, these two guests must also have either a common acquaintance or a common unknown among the guests... |
olympiads | Problem 4. Using vectors.
$\overline{H K}=\overline{A K}-\overline{A H}=\frac{1}{2}(\overline{A Q}+\overline{A N})-\frac{1}{2}(\overline{A M}+\overline{A P})=$
$=\frac{1}{2}\left(\frac{1}{2}(\overline{A E}+\overline{A D})+\frac{1}{2}(\overline{A B}+\overline{A C})\right)-$
$-\frac{1}{2}\left(\frac{1}{2} \overline{A ... | <think>
Okay, let's see here. I need to figure out how HK is equal to 1.75 using vectors. The problem gives a series of steps, but I need to verify each part to make sure it's correct. Let me start by understanding the setup.
First, they state that HK is equal to AK minus AH. That makes sense because vectors can be su... |
olympiads | Let the unit area rectangle $ABCD$ have the incenter of the triangle determined by vertices $A, B, C$ as $O$. Consider the rectangle with $O$ and $D$ as opposite vertices, and the other two vertices on the perimeter of the original rectangle. What is the area of this rectangle? | <think>
Okay, so I have this problem here about a unit area rectangle ABCD. The incenter of the triangle formed by vertices A, B, C is called O. Then, we need to consider another rectangle with O and D as opposite vertices, and the other two vertices are on the perimeter of the original rectangle. The question is askin... |
olympiads | 3.15. Form the equation of a circle with center at point $P_{0}(1; -2)$ and radius $R=3$. | <think>
Okay, so I need to form the equation of a circle. The center is at P0(1, -2) and the radius is 3. Hmm, let me remember how the equation of a circle works. I think the general formula is something like (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius. Right?
Let me verify that. Yeah, ... |
olympiads | 7. Regulation: For $x \in \mathbf{R}$, if and only if $n \leqslant x<n+1\left(n \in \mathbf{N}^{*}\right)$, then $[x]=n$. Then the solution set of the inequality $4[x]^{2}-36[x]+45 \leqslant 0$ is $\qquad$. | <think>
Okay, let me try to figure out this inequality problem. So, the question is about solving the inequality 4[x]^2 -36[x] +45 ≤ 0, where [x] is the floor function, right? The floor function, as defined here, means that for any real number x, if n is a positive integer and n ≤ x < n+1, then [x] = n. So, basically, ... |
olympiads | 30.3. In a trapezoid with bases $x$ and 5, express the distance between the midpoints of the diagonals as a function of $x$.
$$
\text { (8-9 grades) }
$$ | <think>
Alright, so I have this trapezoid problem. The question is asking me to express the distance between the midpoints of the diagonals as a function of x, where the trapezoid has bases x and 5. Hmm, okay. Let me think about how to approach this.
First, let me recall some properties of trapezoids. A trapezoid is a... |
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