source stringclasses 7
values | problem stringlengths 19 6.18k | generation stringlengths 345 68.1k |
|---|---|---|
olympiads | 3. (17 points) In a $4 \times 5$ grid, 5 crosses need to be placed such that each row and each column contains at least one cross. How many ways can this be done? | <think>
Okay, so I need to figure out how many ways to place 5 crosses in a 4x5 grid where each row and each column has at least one cross. Hmm, let me start by understanding the problem better.
First, the grid has 4 rows and 5 columns. We need to place exactly 5 crosses, one in each column, but also ensuring that eac... |
olympiads | 3. Find the number of natural numbers $k$, not exceeding 454500, such that $k^{2}-k$ is divisible by 505. | <think>
Okay, so I need to find the number of natural numbers k, up to 454500, such that k² - k is divisible by 505. Hmm, let's start by breaking this down.
First, divisibility by 505. I know that 505 can be factored into prime numbers. Let me check: 505 divided by 5 is 101, right? So 505 = 5 × 101. Both 5 and 101 ar... |
olympiads | 7.1. Is it possible to place rational numbers at the vertices of a square $A B C D$ such that the sum of the numbers at vertices $A$ and $B$ is 1, at vertices $B$ and $C$ is 2, at vertices $C$ and $D$ is 3, and at vertices $A$ and $D$ is 4? | <think>
Okay, so the problem is asking if we can assign rational numbers to the vertices of a square ABCD such that the sums of certain pairs of vertices are given. Specifically, the sum at A and B should be 1, B and C should be 2, C and D should be 3, and A and D should be 4. Hmm, let's see. I need to figure out if su... |
olympiads | $$
14 \cdot 32 \text { Let } S=[\sqrt{1}]+[\sqrt{2}]+\cdots+[\sqrt{1988}] \text {. Find }[\sqrt{S}] \text {. }
$$
(Advanced Class for Science Experiment Mathematics Admission Test, 1988) | <think>
Okay, so I have this problem where I need to find the sum S, which is the sum of the integer parts of the square roots of each number from 1 to 1988. Then, after summing all those integer parts, I have to find the integer part of the square root of that sum S. That sounds a bit complicated, but maybe I can brea... |
olympiads | 6. (7 points) Petya showed Vasya 37 identical-looking cards laid out in a row. He said that on the hidden sides of the cards, all numbers from 1 to 37 are written (each exactly once) such that the number on any card starting from the second is a divisor of the sum of the numbers written on all preceding cards. Then Pet... | <think>
Okay, let's try to figure out this problem step by step. So, we have 37 cards laid out in a row, each with a unique number from 1 to 37 on the hidden side. The key condition here is that starting from the second card, each number must be a divisor of the sum of all the numbers before it. Petya told Vasya that t... |
olympiads | Determine all pairs of real numbers $(x, y)$ for which $\left(4 x^{2}-y^{2}\right)^{2}+(7 x+3 y-39)^{2}=0$. | <think>
Okay, let's see. I need to find all pairs of real numbers (x, y) that satisfy the equation (4x² - y²)² + (7x + 3y - 39)² = 0. Hmm, both terms are squared, so they're each non-negative. The sum of two non-negative numbers equals zero only if each of them is zero. So, this means both (4x² - y²)² = 0 and (7x + 3y ... |
olympiads | Concerning the Homothety of Tangent Circles helps to solve the problem, $\quad]$
In a right triangle $ABC$, angle $C$ is a right angle, and side $CA=4$. A point $D$ is taken on the leg $BC$, such that $CD=1$. A circle with radius $\frac{\sqrt{5}}{2}$ passes through points $C$ and $D$ and is tangent at point $C$ to the... | <think>
Okay, so I have this geometry problem here. Let me try to parse it step by step. It's about a right triangle ABC where angle C is the right angle. The length of side CA is 4. There's a point D on leg BC such that CD is 1. Then there's a circle with radius sqrt(5)/2 that passes through points C and D and is tang... |
olympiads | I1.3 If $R$ is the unit digit of the value of $8^{Q}+7^{10 Q}+6^{100 Q}+5^{1000 Q}$, find the value of $R$. | <think>
Okay, so I need to find the unit digit of this expression: 8^Q + 7^(10Q) + 6^(100Q) + 5^(1000Q). Hmm, unit digit problems usually involve looking at patterns in the last digits of powers. Let me recall how these cycles work. For example, powers of numbers cycle every certain number. Like 2, 4, 8, 6 for powers o... |
olympiads | ## Problem Statement
Calculate the definite integral:
$$
\int_{0}^{5 / 2} \frac{x^{2} \cdot d x}{\sqrt{25-x^{2}}}
$$ | <think>
Okay, so I need to calculate the definite integral from 0 to 5/2 of x² divided by the square root of (25 minus x²) dx. Hmm, let's think about how to approach this. Integrals involving square roots in the denominator often suggest a trigonometric substitution, especially when there's a term like a² minus x². The... |
olympiads | 4. Calculate the value of the expression
$$
\left(\frac{1+i}{\sqrt{2}}\right)^{n+4}+\left(\frac{1-i}{\sqrt{2}}\right)^{n+4}+\left(\frac{1+i}{\sqrt{2}}\right)^{n}+\left(\frac{1-i}{\sqrt{2}}\right)^{n},
$$
( i is the imaginary unit). | <think>
Okay, let me see. I need to calculate the value of this expression:
$$
\left(\frac{1+i}{\sqrt{2}}\right)^{n+4}+\left(\frac{1-i}{\sqrt{2}}\right)^{n+4}+\left(\frac{1+i}{\sqrt{2}}\right)^{n}+\left(\frac{1-i}{\sqrt{2}}\right)^{n}
$$
Hmm, so first, I remember that complex numbers can often be expressed in polar f... |
cn_contest | $$
\begin{array}{l}
\text { II. (40 points) Let } f(x)=x^{2}+a, \text { and define } \\
f^{1}(x)=f(x), \\
f^{n}(x)=f\left(f^{n-1}(x)\right)(n=2,3, \cdots) .
\end{array}
$$
Find the set
$$
M=\left\{a \in \mathbf{R}|| f^{n}(0) \mid \leqslant 2, n \in \mathbf{Z}_{+}\right\}
$$ | <think>
Okay, so I need to find the set M of real numbers a such that when we iterate the function f(x) = x² + a starting at 0, the absolute value of the nth iterate is always less than or equal to 2 for all positive integers n. That sounds like a problem related to bounded orbits in dynamical systems, maybe similar to... |
olympiads | Problem 4. Calculate the area of the square inscribed in a right-angled triangle with hypotenuse $13 \mathrm{~cm}$ and leg $5 \mathrm{~cm}$. | <think>
Okay, so I have this problem here: Calculate the area of the square inscribed in a right-angled triangle with hypotenuse 13 cm and leg 5 cm. Hmm, right. Let's see. First, let me visualize the problem. There's a right-angled triangle, and inside it, there's a square inscribed. I need to find the area of that squ... |
olympiads | 11. (3b,9-11) In the conditions of a chess match, the winner is declared as the one who outperforms the opponent by two wins. Draws do not count. The probabilities of winning for the opponents are equal. The number of decisive games in such a match is a random variable. Find its mathematical expectation. | <think>
Okay, so I need to find the mathematical expectation (which is the expected value) of the number of decisive games in a chess match where the winner is the first to be two wins ahead, and draws don't count. The probability of each player winning a game is equal. Hmm, let's break this down.
First, let me make s... |
olympiads | 3. 1. 10 * Given a positive integer $n$ and a positive number $M$. For all arithmetic sequences $a_{1}, a_{2}, a_{3}, \cdots$ satisfying the condition $a_{1}^{2}+a_{n+1}^{2} \leqslant M$, find the maximum value of $S=a_{n+1}+a_{n+2}+\cdots+a_{2 n+1}$. | <think>
Okay, let's tackle this problem. So, we need to find the maximum value of S, which is the sum of terms from a_{n+1} to a_{2n+1} in an arithmetic sequence. The condition given is that a₁² + a_{n+1}² ≤ M, where n is a positive integer and M is a positive number. Hmm, arithmetic sequences, so they have a common di... |
olympiads | 10.023. Given a square, two of its vertices lie on a circle of radius $R$, and the other two lie on a tangent to this circle. Find the length of the diagonal of the square. | <think>
Okay, so I have this problem where there's a square. Two of its vertices are on a circle with radius R, and the other two are on a tangent to this circle. I need to find the length of the diagonal of the square. Hmm, let me try to visualize this.
First, let me draw a circle with radius R. Let's say the circle ... |
olympiads | 4.007. When the ninth term of an arithmetic progression is divided by the second term, the quotient is 5, and when the thirteenth term is divided by the sixth term, the quotient is 2 and the remainder is 5. Find the first term and the common difference of the progression. | <think>
Okay, so I need to solve this problem about an arithmetic progression. Let me recall what an arithmetic progression is. It's a sequence where each term after the first is obtained by adding a constant difference. So, if the first term is \( a \) and the common difference is \( d \), then the \( n \)-th term is ... |
olympiads | Example 10 Polynomial $(1-z)^{b_{1}} \cdot\left(1-z^{2}\right)^{b_{2}} \cdot\left(1-z^{3}\right)^{b_{3}} \cdots\left(1-z^{32}\right)^{b_{32}}$, where $b_{i}(i=1,2, \cdots, 32)$ are positive integers, and this polynomial has the following remarkable property: when expanded, and the terms of $z$ with powers higher than 3... | <think>
Okay, so I need to figure out the value of \( b_{32} \) in the polynomial product \( (1 - z)^{b_1} \cdot (1 - z^2)^{b_2} \cdot \ldots \cdot (1 - z^{32})^{b_{32}} \). The key here is that when this product is expanded and all terms with powers higher than 32 are removed, the result is exactly \( 1 - 2z \).
Fir... |
olympiads | 4. Find all natural numbers $n$ for which the number $n^{7}+n^{6}+n^{5}+1$ has exactly three natural divisors. (O. Nechaeva, I. Rubanov) | <think>
Okay, so I need to find all natural numbers n where n⁷ + n⁶ + n⁵ + 1 has exactly three natural divisors. Hmm. First, I remember that a number has exactly three natural divisors if and only if it's the square of a prime number. Because the divisors would be 1, p, and p². So, the number must be a prime squared. T... |
olympiads | 1. From point A to point B, which are 12 km apart, a pedestrian and a bus set out simultaneously. Arriving at point B in less than one hour, the bus, without stopping, turned around and started moving back towards point A at a speed twice its initial speed. After 12 minutes from its departure from point B, the bus met ... | <think>
Okay, let's try to figure out this problem step by step. So, we have a pedestrian and a bus starting at the same time from point A to point B, which are 12 km apart. The bus gets to point B in less than one hour, then immediately turns around and starts heading back towards point A at twice its initial speed. T... |
olympiads | 3. Calculate the value of the product $\frac{2^{3}-1}{2^{3}+1} \cdot \frac{3^{3}-1}{3^{3}+1} \cdot \frac{4^{3}-1}{4^{3}+1} \cdot \ldots \cdot \frac{200^{3}-1}{200^{3}+1}$. | <think>
Okay, let me try to figure out how to calculate this product. The problem is to find the value of the product from n=2 to n=200 of (n³ - 1)/(n³ + 1). So, each term in the product is (n³ - 1) divided by (n³ + 1). Hmm, that seems a bit complicated, but maybe there's a pattern or a formula I can use to simplify ea... |
olympiads | 4[ Numerical inequalities. Comparisons of numbers.]
What is greater: $\log _{3} 4$ or $\log _{4} 5$ ? | <think>
Okay, so I need to figure out whether log base 3 of 4 is greater than log base 4 of 5, or the other way around. Hmm, comparing logarithms with different bases can be tricky. Let me think about the methods I know for comparing logarithms.
One approach might be to convert both logarithms to the same base so I ca... |
cn_contest | 8. As shown in Figure 3, in the Cartesian coordinate system $x O y$,
the vertices of the polygon $O A B C D E$ are
$O(0,0), A(0,6)$,
$B(4,6), C(4,4)$,
$D(6,4), E(6,0)$. If
the line $l$ passes through the point $M(2,3)$, and divides the polygon $O A B C D E$
into two parts of equal area, then the function expressi... | <think>
Okay, let's tackle this problem step by step. So, we have a polygon with vertices O(0,0), A(0,6), B(4,6), C(4,4), D(6,4), and E(6,0). We need to find the equation of a line l that passes through the point M(2,3) and divides the polygon into two regions of equal area.
First, I need to visualize the polygon. Le... |
olympiads | If $P_{1} P_{2} \ldots P_{100}$ is a regular 100 -gon, what is the measure of the angle $\angle P_{20} P_{2} P_{1}$ in degrees? | <think>
Okay, let's see. I need to find the measure of angle ∠P20P2P1 in a regular 100-gon. Hmm, regular 100-gon means all sides and angles are equal, right? So each vertex is equally spaced around a circle.
First, I should probably visualize this. A regular polygon with 100 sides is inscribed in a circle. All the po... |
olympiads | 1. Find all integers $a$ for which the modulus of the number $\left|a^{2}-3 a-6\right|$ is a prime number. Answer. $-1 ; 4$. | <think>
Okay, let's try to figure out how to solve this problem. The question is asking for all integers \( a \) such that the absolute value of \( a^2 - 3a - 6 \) is a prime number. The answer given is -1 and 4, but I need to work through this step by step to understand why those are the solutions.
First, let's resta... |
olympiads | 7th Irish 1994 Problem B2 p, q, r are distinct reals such that q = p(4-p), r = q(4-q), p = r(4-r). Find all possible values of p+q+r. | <think>
Okay, let me try to work through this problem. So, we have three distinct real numbers p, q, r. The relationships between them are given by:
q = p(4 - p)
r = q(4 - q)
p = r(4 - r)
And we need to find all possible values of p + q + r.
Hmm, this seems like a system of equations with three variables. Since ea... |
olympiads | 21. When the young fishermen were asked how many fish each of them had caught, the first one answered: "I caught half the number of fish that my friend caught, plus 10 fish." The second one said: "And I caught as many as my friend, plus 20 fish." How many fish did the fishermen catch? | <think>
Okay, so there are two fishermen, right? Let's call them Fisherman A and Fisherman B. The problem says when they were asked how many fish each caught, the first one (let's say Fisherman A) answered: "I caught half the number of fish that my friend caught, plus 10 fish." Then the second one (Fisherman B) said: "... |
cn_contest | Example 12 Let $m>0, n>0$, and real numbers $a, b, c, d$ satisfy $a+b+c+d=m, ac=bd=n^{2}$. Try to find the value of $\sqrt{(a+b)(b+c)(c+d)(d+a)}$. (Express the result in terms of $m, n$) | <think>
Okay, so I need to find the value of this expression: √[(a+b)(b+c)(c+d)(d+a)] given that a + b + c + d = m and ac = bd = n². Hmm, let's start by understanding what's given and what needs to be found.
First, the variables a, b, c, d are real numbers, and they add up to m. Also, the products ac and bd are both e... |
olympiads | 32. Let $A$ and $B$ be points that lie on the parabola $y=x^{2}$ such that both are at a distance of $8 \sqrt{2}$ units from the line $y=-x-4$. Find the square of the distance between $A$ and $B$. | <think>
Okay, so I need to solve this problem where points A and B are on the parabola y = x², and both are at a distance of 8√2 units from the line y = -x - 4. Then I have to find the square of the distance between A and B. Hmm, let me try to break this down step by step.
First, I remember that the distance from a po... |
olympiads | Example 2. Calculate the line integral
$$
\int_{L}(x-y) d x+d y+z d z
$$
from point $M(2,0,4)$ to point $N(-2,0,4)(y \geq 0)$ along the curve $L$, formed by the intersection of the paraboloid $z=x^{2}+y^{2}$ and the plane $z=4$, | <think>
Okay, so I need to calculate this line integral ∫_L (x - y) dx + dy + z dz from point M(2,0,4) to point N(-2,0,4) along the curve L, which is the intersection of the paraboloid z = x² + y² and the plane z = 4, with the condition that y ≥ 0. Hmm, let's break this down step by step.
First, let me visualize the c... |
olympiads | 2. Given two linear functions $f(x)$ and $g(x)$ such that the graphs $y=f(x)$ and $y=g(x)$ are parallel lines, not parallel to the coordinate axes. Find the minimum value of the function $(g(x))^{2}+$ $2 f(x)$, if the minimum value of the function $(f(x))^{2}+2 g(x)$ is 5. | <think>
Okay, let's tackle this problem. Hmm, so we have two linear functions f(x) and g(x) whose graphs are parallel lines, not parallel to the coordinate axes. That means their slopes are equal, right? Since they're linear functions, they must be of the form f(x) = ax + b and g(x) = ax + c, where a, b, c are constant... |
olympiads | 8.3. Specify the integer closest to the smaller root of the equation
$$
\operatorname{arctg}\left(\left(\frac{7 x}{10}-\frac{5}{14 x}\right)^{2}\right)-\operatorname{arctg}\left(\left(\frac{7 x}{10}+\frac{5}{14 x}\right)^{2}\right)=-\frac{\pi}{4}
$$ | <think>
Okay, let's see. I have this equation involving arctangent functions, and I need to find the integer closest to the smaller root of it. Hmm. The equation is:
$$
\operatorname{arctg}\left(\left(\frac{7 x}{10}-\frac{5}{14 x}\right)^{2}\right)-\operatorname{arctg}\left(\left(\frac{7 x}{10}+\frac{5}{14 x}\right)^{... |
olympiads | A square board of three rows by three columns contains nine cells. In how many different ways can we write the three letters A, B, and $\mathbf{C}$ in three different cells, so that in each row exactly one of these three letters is written? | <think>
Okay, so I have this problem here about a 3x3 grid, and I need to figure out how many different ways I can place the letters A, B, and C in three different cells such that each row has exactly one of these letters. Let me try to break this down step by step.
First, let me visualize the board. It's a 3x3 grid, ... |
aops_forum | Alice and Bob play the following game. They alternate selecting distinct nonzero digits (from $1$ to $9$) until they have chosen seven such digits, and then consider the resulting seven-digit number by concatenating the digits in the order selected, with the seventh digit appearing last (i.e. $\overline{A_1B_2A_3B_4A_6... | <think>
Okay, so Alice and Bob are playing this game where they take turns picking distinct nonzero digits (from 1 to 9) until they have seven digits. The resulting number is formed by the order they picked them, with Alice going first, then Bob, and so on. So the number would look like A1, B2, A3, B4, A5, B6, A7. Alic... |
olympiads | 15. (15 points) 100 people participate in a quick calculation test, with a total of 10 questions. The number of people who answered each question correctly is shown in the table below:
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline Question Number & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline \begin{tabular}{l}
... | <think>
Okay, let's see. The problem is about determining the minimum number of people who passed a test based on the number of correct answers each question received. There are 100 participants, and each answered 10 questions. To pass, someone needs to get at least 6 questions right. The table shows how many people go... |
olympiads | Example 2 Solve the equation $5^{2 x}-6 \cdot 5^{x}+5=0$.
Translate the text above into English, keeping the original text's line breaks and format, and output the translation result directly.
Example 2 Solve the equation $5^{2 x}-6 \cdot 5^{x}+5=0$. | <think>
Alright, so I need to solve this equation: 5^(2x) - 6*5^x + 5 = 0. Hmm, okay. Let me think about how to approach this. I remember that when dealing with equations where the exponent has a variable, a common method is substitution. Maybe I can let y equal 5^x, right? Because if I do that, then 5^(2x) would be (5... |
olympiads | 3. The graph of the function $f(x)=A \cos (\omega x+\phi)$ is symmetric with respect to the origin if and only if ( )(where $k \in \mathbf{Z}$ ).
A. $\phi=2 k \pi+\frac{\pi}{2}$
B. $\frac{\varphi}{\omega}=2 k \pi \pm \frac{\pi}{2}$
C. $\phi=k \pi+\frac{\pi}{2}$
D. $\phi=k \pi+\frac{\pi}{2}$ or $\omega x=k \pi$ | <think>
Okay, so I need to figure out when the graph of f(x) = A cos(ωx + φ) is symmetric with respect to the origin. Symmetry about the origin means that the function is odd, right? An odd function satisfies f(-x) = -f(x) for all x. So, maybe I should start by checking what condition makes A cos(ωx + φ) an odd functio... |
amc_aime | Given the four equations:
$\textbf{(1)}\ 3y-2x=12 \qquad\textbf{(2)}\ -2x-3y=10 \qquad\textbf{(3)}\ 3y+2x=12 \qquad\textbf{(4)}\ 2y+3x=10$
The pair representing the perpendicular lines is:
$\textbf{(A)}\ \text{(1) and (4)}\qquad \textbf{(B)}\ \text{(1) and (3)}\qquad \textbf{(C)}\ \text{(1) and (2)}\qquad \textbf{(D)}... | <think>
Okay, so I need to figure out which pair of the given equations represents perpendicular lines. The options are from A to E, each pairing two of the four equations. Let me recall that two lines are perpendicular if the product of their slopes is -1. So, first, I should probably find the slopes of each line give... |
olympiads | 8. (10 points) There is a special calculator. When a natural number between 10 and 49 is input, the calculator first multiplies this number by 2, then reverses the tens and units digits of the result, and finally adds 2 to display the final result. Therefore, among the following four options, ( ) could be the final dis... | <think>
Okay, let me try to figure out this problem step by step. So, we have a special calculator that takes a natural number between 10 and 49. Then it multiplies that number by 2, reverses the digits of the result, and adds 2 to get the final displayed result. We need to determine which of the options A (44), B (43)... |
olympiads | Problem 3. Answer: $\frac{1}{64}$.
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly.
Note: The provided translation is already in the requested format and directly represents the translation of the given text. However, the not... | <think>
Okay, I need to translate the given text from Chinese to English while keeping the original line breaks and format. Let me first understand the original content. The user provided a problem statement with a line break between "Problem 3. Answer:" and the mathematical fraction. The note mentions that the transla... |
olympiads | $\underline{\text { Folklore }}$
In triangle $ABC$, the median drawn from vertex $A$ to side $BC$ is four times smaller than side $AB$ and forms an angle of $60^{\circ}$ with it. Find angle $A$. | <think>
Okay, let's see. I need to find angle A in triangle ABC. The problem says that the median from vertex A to side BC is four times smaller than side AB, and it forms a 60-degree angle with AB. Hmm, let me break this down step by step.
First, let me recall what a median is. A median in a triangle is a line segmen... |
olympiads | 11.1. Solve the equation $\arccos \frac{x+1}{2}=2 \operatorname{arctg} x$. | <think>
Okay, so I have this equation to solve: arccos[(x + 1)/2] = 2 arctg(x). Hmm, let's see. I need to find all real numbers x that satisfy this equation. Alright, let's start by recalling some properties of the arccos and arctg functions.
First, the domain of arccos is [-1, 1], right? So whatever (x + 1)/2 is, it ... |
olympiads | 2. Non-zero numbers $a$ and $b$ are roots of the quadratic equation $x^{2}-5 p x+2 p^{3}=0$. The equation $x^{2}-a x+b=0$ has a unique root. Find $p$. Justify your solution. | <think>
Okay, let's try to solve this problem step by step. Hmm, so we have non-zero numbers a and b that are roots of the quadratic equation x² - 5p x + 2p³ = 0. Then, the equation x² - a x + b = 0 has a unique root. We need to find p.
First, since a and b are roots of the first equation, by Vieta's formulas, the su... |
olympiads | 10. (3 points) A turtle and a rabbit are racing on a 1000-meter track. The rabbit's speed is 15 times that of the turtle. However, the rabbit took a rest during the race. When it woke up, it found that the turtle had just reached the finish line, and at that moment, the rabbit was still 100 meters away from the finish ... | <think>
Okay, let's see. So, there's a turtle and a rabbit racing on a 1000-meter track. The rabbit's speed is 15 times that of the turtle. But the rabbit took a rest during the race. When the rabbit woke up, the turtle had just finished the race, and the rabbit was still 100 meters away from the finish line. The quest... |
olympiads | 1. On a certain day, there were several nuts in the bag. The next day, as many nuts were added to the bag as there already were, but eight nuts were taken away. On the third day, again as many nuts were added as there already were, but eight were taken away. The same thing happened on the fourth day, and after that, th... | <think>
Okay, so I need to figure out how many nuts were in the bag originally. Let me try to understand the problem step by step.
First, there's a bag with some nuts on day one. Let's call the initial number of nuts "x". Then, the next day, they add as many nuts as there already are. So if there are x nuts on day on... |
olympiads | 449. Ten Barrels. A merchant had 10 barrels of sugar, which he arranged into a pyramid as shown in the figure. Each barrel, except one, was marked with its own number. It turned out that the merchant had accidentally placed the barrels in such a way that the sum of the numbers along each row was 16.
 errors. Random measurement errors $X$ follow a normal distribution with a root mean square deviation $\sigma=10$ mm. Find the probability that the measurement will be performed with an error not exceeding $15 \mathrm{mm}$ in ab... | <think>
Okay, so I have this problem here about finding the probability that a measurement error doesn't exceed 15 mm. Let me read it again carefully. It says that the shaft diameter measurements are done without systematic errors, meaning there's no bias—so the errors are just random. These random errors, denoted by X... |
olympiads | 11. Given positive real numbers $a, b, c$ satisfying $a b+b c+c a=1$, find the minimum value of $\sqrt{a^{2}+a b+b^{2}}+\sqrt{b^{2}+b c+c^{2}}+\sqrt{c^{2}+c a+a^{2}}$. | <think>
Okay, so I have this problem where I need to find the minimum value of the sum of three square roots: √(a² + ab + b²) + √(b² + bc + c²) + √(c² + ca + a²), given that a, b, c are positive real numbers and ab + bc + ca = 1. Hmm, interesting. Let me try to think through this step by step.
First, I remember that e... |
olympiads | 1. Option 1. To a confectionery factory for cake packaging, 5 rolls of ribbon, each 60 m long, were delivered. How many cuts need to be made to get pieces of ribbon, each 1 m 50 cm long? | <think>
Okay, let's see. So, the problem is about a confectionery factory that received 5 rolls of ribbon. Each roll is 60 meters long. They need to cut these ribbons into pieces that are 1 meter 50 centimeters long. The question is asking how many cuts need to be made to achieve this.
First, let me make sure I unders... |
olympiads | 5. Find the fraction $\frac{p}{q}$ with the smallest possible natural denominator, for which $\frac{1}{2014}<\frac{p}{q}<\frac{1}{2013}$. Enter the denominator of this fraction in the provided field | <think>
Okay, so I need to find a fraction p/q that's between 1/2014 and 1/2013, right? And among all such fractions, I need the one with the smallest possible natural denominator. Then, just give the denominator. Hmm, let's think about this step by step.
First, let me recall that fractions between two numbers can be ... |
olympiads | In a convex quadrilateral $A B C D A B=B C$. The rays $B A$ and $C D$ intersect at point $E$, and the rays $A D$ and $B C$ intersect at point $F$. It is also known that $B E=B F$ and
$\angle D E F=25^{\circ}$. Find the angle $E F D$. | <think>
Okay, let's see. I need to find the angle EFD in this convex quadrilateral ABCD where AB equals BC. There are intersecting rays: BA and CD intersect at E, and AD and BC intersect at F. Also, BE equals BF, and angle DEF is 25 degrees. Hmm, okay, let me try to visualize this first.
So, quadrilateral ABCD is conv... |
olympiads | 1. (8 points) The calculation result of the expression $20.15 \div(1.2+1.9) \times\left(4^{2}-3^{2}+2^{2}-1^{2}\right)$ is | <think>
Okay, let's see. I need to calculate this expression: 20.15 divided by (1.2 plus 1.9) multiplied by (4 squared minus 3 squared plus 2 squared minus 1 squared). Hmm, order of operations is crucial here. Let me break it down step by step.
First, according to PEMDAS, Parentheses come first. So I should handle the... |
olympiads | 2. (15 points) Find the range of the similarity coefficient of triangles with side lengths $x, y, z$ and $y, z, p$. In your answer, specify the nearest integers between which the found interval lies. | <think>
Okay, so I have this problem here: Find the range of the similarity coefficient of triangles with side lengths x, y, z and y, z, p. Then I need to specify the nearest integers between which this interval lies. Hmm, similarity coefficient... right, that's the ratio of corresponding sides in similar triangles. Bu... |
olympiads | 3. Solve the equation $9^{1+\sqrt{x}}+3^{1-2 \sqrt{x}}=28$.
# | <think>
Okay, let's try to solve this equation: 9^{1+√x} + 3^{1−2√x} = 28. Hmm, first, I need to remember the properties of exponents. Maybe I can rewrite the terms so they have the same base.
The first term is 9^{1+√x}. Since 9 is 3 squared, I can rewrite this as (3^2)^{1+√x}, which simplifies to 3^{2(1+√x)}. That w... |
olympiads | 13.296. Two excavator operators must complete a certain job. After the first one worked for 15 hours, the second one starts and finishes the job in 10 hours. If, working separately, the first one completed $1 / 6$ of the job, and the second one completed $1 / 4$ of the job, it would take an additional 7 hours of their ... | <think>
Okay, let me try to figure out this problem. It's about two excavator operators working together and separately to complete a job. There are some specific details given, so I need to parse them carefully.
First, the problem states: "After the first one worked for 15 hours, the second one starts and finishes th... |
olympiads | 8.1. Solve the equation $x^{x+y}=y^{y-x}$ in natural numbers. In your answer, specify the sum $x+y$ for the solution $(x, y)$, where $y$ is the smallest, exceeding 1500. | <think>
Okay, so I need to solve the equation \( x^{x+y} = y^{y-x} \) where \( x \) and \( y \) are natural numbers. The problem asks for the sum \( x + y \) for the solution where \( y \) is the smallest possible value exceeding 1500. Hmm, this looks pretty challenging. Let me start by understanding the equation bette... |
olympiads_ref | 20. N1 (POL) Let \( m \) be a fixed integer greater than 1. The sequence \( x_{0}, x_{1}, x_{2}, \ldots \) is defined as follows:
\[ x_{i}= \begin{cases}2^{i}, & \text { if } 0 \leq i \leq m-1 \\ \sum_{j=1}^{m} x_{i-j}, & \text { if } i \geq m\end{cases} \]
Find the greatest \( k \) for which the sequence contains ... | <think>
Okay, let me try to figure out this problem. So, we have a sequence defined in two parts: for the first m terms (from x₀ to x_{m-1}), each term is just 2^i. Then, starting from x_m onwards, each term is the sum of the previous m terms. We need to find the greatest k such that there are k consecutive terms in th... |
cn_contest | Example 2 The equation $2 x^{2}+5 x y+2 y^{2}=2007$ has $\qquad$ different integer solutions.
(2007, National Junior High School Mathematics League Sichuan Preliminary Competition) | <think>
Okay, so I need to figure out how many different integer solutions exist for the equation \(2x^2 + 5xy + 2y^2 = 2007\). Hmm, let's start by understanding the structure of this equation. It's a quadratic in both x and y, and it has cross terms (the 5xy part). Since it's symmetric in x and y to some extent, maybe... |
olympiads | 125. Dear club. Ten people decided to found a club. If there were 5 more people, each would contribute 100 dollars less. How much money did each member contribute? | <think>
Okay, so here's the problem: Ten people decided to found a club. If there were 5 more people, each would contribute 100 dollars less. How much money did each member contribute?
Hmm, let me think. Alright, so there are 10 original members, and if there were 15 members instead (since 10 + 5 = 15), each person wo... |
olympiads | A parallelogram's interior angle bisectors define a quadrilateral whose area is one-third of the parallelogram's area. Determine the ratio of the adjacent sides of the parallelogram. | <think>
Okay, let's tackle this problem. So, we have a parallelogram where the interior angle bisectors form a quadrilateral, and the area of this quadrilateral is one-third of the original parallelogram's area. We need to find the ratio of the adjacent sides of the parallelogram. Hmm, okay. Let's break this down step ... |
olympiads_ref | On a $(4 n+2) \times(4 n+2)$ square grid, a turtle can move between squares sharing a side. The turtle begins in a corner square of the grid and enters each square exactly once, ending in the square where she started. In terms of $n$, what is the largest positive integer $k$ such that there must be a row or column that... | <think>
Okay, so I have this problem about a turtle moving on a (4n+2)×(4n+2) grid. The turtle starts at a corner, visits every square exactly once, and ends where it started. The question is asking for the largest integer k such that there must be at least one row or column that the turtle has entered at least k disti... |
olympiads | 2. It is known that $\frac{\cos x-\sin x}{\sin y}=\frac{2 \sqrt{2}}{5} \tan \frac{x+y}{2}$ and $\frac{\sin x+\cos x}{\cos y}=-\frac{5}{\sqrt{2}} \cot \frac{x+y}{2}$. Find all possible values of the expression $\tan(x+y)$, given that there are at least three. | <think>
Okay, so I need to solve for tan(x+y) given these two equations. Let me write them down again to make sure I have them right:
1. (cos x - sin x)/sin y = (2√2)/5 * tan[(x+y)/2]
2. (sin x + cos x)/cos y = -5/√2 * cot[(x+y)/2]
Hmm, the goal is to find all possible values of tan(x+y), and it's mentioned there are... |
olympiads | 7. In the expansion of $(\sqrt{x}+2)^{2 n+1}$, the sum of the coefficients of the terms where the power of $x$ is an integer is $\qquad$ | <think>
Okay, so I need to find the sum of the coefficients in the expansion of \((\sqrt{x} + 2)^{2n+1}\) where the power of \(x\) is an integer. Hmm, let's break this down step by step.
First, I remember that when you expand a binomial like \((a + b)^m\), each term is of the form \(\binom{m}{k}a^{m-k}b^k\). In this c... |
aops_forum | Let $p_{1}$, $p_{2}$, ..., $p_{k}$ be different prime numbers. Determine the number of positive integers of the form $p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}...p_{k}^{\alpha_{k}}$, $\alpha_{i}$ $\in$ $\mathbb{N}$ for which $\alpha_{1} \alpha_{2}...\alpha_{k}=p_{1}p_{2}...p_{k}$. | <think>
Okay, let's try to figure out this problem. We need to find the number of positive integers of the form \( p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}\dots p_{k}^{\alpha_{k}} \) where each \( \alpha_{i} \) is a natural number (so starting from 1), and the product of the exponents \( \alpha_{1}\alpha_{2}\dots\alpha_{k}... |
olympiads | $4 \cdot 37$ For real numbers $x$, the inequality $1 \leqslant|x-2| \leqslant 7$ is equivalent to
(A) $x \leqslant 1$ or $x \geqslant 3$.
(B) $1 \leqslant x \leqslant 3$.
(C) $-5 \leqslant x \leqslant 9$.
(D) $-5 \leqslant x \leqslant 1$ or $3 \leqslant x \leqslant 9$.
(E) $-6 \leqslant x \leqslant 1$ or $3 \leqslant x... | <think>
Okay, so I need to solve this inequality: \(1 \leqslant |x - 2| \leqslant 7\). The question is asking which of the given options (A through E) is equivalent to this inequality for real numbers x. Hmm, let's break this down step by step.
First, I remember that an absolute value inequality like \(|a| \leqslant b... |
olympiads | The numbers $a$ and $b$ are positive integers and satisfy $96 a^{2}=b^{3}$. What is the smallest value of $a$? | <think>
Okay, so I have this problem here: Find the smallest positive integer a such that 96 times a squared equals b cubed, where b is also a positive integer. Hmm, let me think about how to approach this.
First, I remember that equations involving exponents like this often require prime factorization. Maybe if I br... |
olympiads | 5. A natural number $n$ is called cubish if $n^{3}+13 n-273$ is a cube of a natural number. Find the sum of all cubish numbers. | <think>
Okay, let's tackle this problem step by step. So, we need to find all natural numbers n such that n³ + 13n - 273 is a perfect cube. Then, sum all those n's. Alright, let's start by understanding what the problem is asking.
First, let's restate the equation: n³ + 13n - 273 = k³, where k is some natural number. ... |
aops_forum | Let $0<c<1$ be a given real number. Determine the least constant $K$ such that the following holds: For all positive real $M$ that is greater than $1$, there exists a strictly increasing sequence $x_0, x_1, \ldots, x_n$ (of arbitrary length) such that $x_0=1, x_n\geq M$ and
\[\sum_{i=0}^{n-1}\frac{\left(x_{i+1}-x_i\ri... | <think>
Okay, so I need to find the least constant K such that for any M > 1, there's a strictly increasing sequence starting at 1 and ending at least M, where the sum of (x_{i+1} - x_i)^c divided by x_i^{c+1} is less than or equal to K. The constant c is between 0 and 1. Hmm.
First, let me try to understand the probl... |
olympiads | ## Task B-3.3.
To access a certain webpage, Matko has to choose a 4-digit PIN. Leading zeros are allowed, but there are some requirements (restrictions) on the PIN. No digit can be repeated three or more times in a row. For example, 0006 or 6666 are not allowed PINs, but 0030 is an allowed PIN. Additionally, no pair o... | <think>
Okay, let's see. Matko needs to choose a 4-digit PIN, and there are some restrictions. The PIN can't have any digit repeated three or more times in a row, so things like 0006 or 6666 are out. Also, no pair of digits can be repeated, meaning something like 1616 is invalid, but 1661 or 6611 are okay. Hmm, so I ne... |
olympiads | 5. Sixth-graders were discussing how old their principal is. Anya said: "He is older than 38 years." Borya said: "He is younger than 35 years." Vova: "He is younger than 40 years." Galya: "He is older than 40 years." Dima: "Borya and Vova are right." Sasha: "You are all wrong." It turned out that the boys and girls mad... | <think>
Okay, so we have this problem where six sixth-graders are discussing the age of their principal. Each of them made a statement, and then it's mentioned that the boys and girls made the same number of mistakes. We need to figure out the principal's age based on their statements. Let me try to break this down ste... |
olympiads | 12. An ant is at a vertex of a tetrahedron. Every minute, it randomly moves to one of the adjacent vertices. What is the probability that after one hour, it stops at the original starting point? $\qquad$ . | <think>
Okay, so I need to find the probability that an ant, starting at a vertex of a tetrahedron, returns to its original starting point after one hour, which is 60 minutes. The ant moves every minute to a randomly chosen adjacent vertex. Hmm, let's break this down step by step.
First, let me visualize a tetrahedron... |
olympiads | Task 5. (20 points) A three-meter gas pipe has rusted in two places. Determine the probability that all three resulting parts can be used as connections to gas stoves, if according to regulations, the stove should not be located closer than 75 cm to the main gas pipe.
# | <think>
Okay, let me try to figure out this probability problem. So, we have a three-meter gas pipe that's rusted in two places. We need to find the probability that all three resulting parts can be used as connections to gas stoves. The regulation says the stove shouldn't be closer than 75 cm to the main gas pipe. Hmm... |
aops_forum | For an integer $m\ge 3$, let $S(m)=1+\frac{1}{3}+…+\frac{1}{m}$ (the fraction $\frac12$ does not participate in addition and does participate in fractions $\frac{1}{k}$ for integers from $3$ until $m$). Let $n\ge 3$ and $ k\ge 3$ . Compare the numbers $S(nk)$ and $S(n)+S(k)$
. | <think>
Okay, so I need to compare S(nk) and S(n) + S(k) where S(m) is defined as the sum 1 + 1/3 + ... + 1/m, but skipping 1/2. Hmm, let's make sure I understand the problem correctly. For an integer m ≥ 3, S(m) is the sum of reciprocals of odd integers starting from 1 up to m, but wait, 1 is included, then 1/3, 1/5, ... |
olympiads | 8. (10 points) "Low-carbon living" starts now, starts with me. According to calculations, 1 hectare of deciduous broad-leaved forest can absorb 14 tons of carbon dioxide annually. If the cooling temperature of each air conditioner is adjusted from the national recommended $26^{\circ} \mathrm{C}$ to $27^{\circ} \mathrm{... | <think>
Okay, let me try to figure this problem out. So, the question is about calculating the number of households in a city based on the carbon dioxide emissions reduced by adjusting air conditioner temperatures. Hmm. Let me break it down step by step.
First, the problem says that 1 hectare of deciduous broad-leaved... |
olympiads | Subject (3). Consider the triangle $ABC$ with $\mathrm{m}(\angle BAC) = 90^{\circ}$. The bisector of $\angle ACB$ intersects the segment $(AB)$ at point $E$. If there exists $D \in (CE)$ such that $\mathrm{m}(\angle DAC) = \mathrm{m}(\angle BDE) = x^{\circ}$, then calculate $x$.
The statement in the problem is unfortu... | <think>
Okay, let's try to solve this geometry problem step by step. So we have triangle ABC where angle BAC is 90 degrees. The bisector of angle ACB intersects segment AB at point E. There's a point D on CE such that angles DAC and BDE are both equal to x degrees. We need to find the value of x.
First, I need to visu... |
olympiads | In quadrilateral $A B C D A B=B C=m, \angle A B C=\angle A D C=120^{\circ}$. Find $B D$. | <think>
Okay, so I need to find the length of BD in this quadrilateral ABCD where AB and BC are both equal to m, and angles ABC and ADC are 120 degrees. Hmm, let me visualize this first. Maybe drawing a diagram would help. Let me sketch quadrilateral ABCD with points A, B, C, D. Given that AB = BC = m, so points A, B, ... |
olympiads | 3. (3 points) The number 2016 is written on the board. Petya and Vasya are playing the following game: in one move, they can subtract any natural divisor of the number written on the board, except the number itself, and write the result of this subtraction on the board instead of the original number. Petya starts. The ... | <think>
Okay, so we have this game where the starting number is 2016. Petya and Vasya take turns subtracting a natural divisor of the current number, except the number itself, and replace the number with the result. The person who can't make a move loses. I need to figure out who has a winning strategy here.
First, l... |
olympiads | ## Task 11/61
A door opening of $90 \mathrm{~cm}$ in width is to be nailed shut with boards. Available are boards of suitable length and widths of $8 \mathrm{~cm}$, $10 \mathrm{~cm}$, and $12 \mathrm{~cm}$.
What are the possible combinations if no board is to be cut lengthwise? | <think>
Okay, let's see. I need to figure out the possible combinations of boards with widths 8 cm, 10 cm, and 12 cm that add up exactly to 90 cm. And no board can be cut lengthwise, so we can't split any boards; we have to use them whole. Hmm, okay. So this is like a coin change problem but with boards instead of coin... |
olympiads | Exercise 8. Determine the sum of all integers $a$ such that $a^{2}-82 a$ is a prime number. | <think>
Okay, so I need to find all integers a where a squared minus 82 times a is a prime number, and then sum those a's up. Let me start by writing down the equation: a² - 82a = prime. Hmm, prime numbers are numbers greater than 1 that have no positive divisors other than 1 and themselves. So, the result of a² - 82a ... |
olympiads | 32. If mom gives each of her children 13 notebooks, she will have 8 notebooks left; if she gives them 15 notebooks each, all the notebooks will be distributed. How many notebooks did mom have? | <think>
Okay, let me try to figure out this problem step by step. So, the problem is about a mom distributing notebooks to her children. There are two scenarios: one where she gives each child 13 notebooks and has 8 left over, and another where she gives each child 15 notebooks and uses up all the notebooks. We need to... |
olympiads | I2.1 If $x, y$ and $z$ are positive real numbers such that $\frac{x+y-z}{z}=\frac{x-y+z}{y}=\frac{-x+y+z}{x}$ and $a=\frac{(x+y) \cdot(y+z) \cdot(z+x)}{x y z}$, find the value of $a$.
I2.2 Let $u$ and $t$ be positive integers such that $u+t+u t=4 a+2$. If $b=u+t$, find the value of $b$. | <think>
Okay, so I have two problems here, I2.1 and I2.2. Let me start with I2.1. The problem says that x, y, and z are positive real numbers such that (x + y - z)/z = (x - y + z)/y = (-x + y + z)/x. Then we need to find the value of a, which is (x+y)(y+z)(z+x)/(xyz).
Hmm, all right. Let me parse this. We have three ... |
olympiads | ## Problem 3.
Let $a_{n}=1+n^{3}$ be the sequence $\{2,9,28,65, \ldots\}$ and $\delta_{n}=\operatorname{gcd}\left(a_{n+1}, a_{n}\right)$. Find the maximum value that $\delta_{n}$ can take. | <think>
Okay, let's see. I need to find the maximum value that δₙ can take, where δₙ is the greatest common divisor of aₙ₊₁ and aₙ, and the sequence aₙ is defined as 1 + n³. So, aₙ = 1 + n³, and δₙ = gcd(aₙ₊₁, aₙ). The problem asks for the maximum possible value of δₙ.
First, I should recall how to compute the gcd of ... |
olympiads |
Problem 2. Find all functions $f:(0, \infty) \rightarrow(0, \infty)$ such that
$$
f\left(f(x)+\frac{y+1}{f(y)}\right)=\frac{1}{f(y)}+x+1
$$
for all $x, y>0$.
(Dominik Burek)
| <think>
Alright, let's try to tackle this functional equation problem. So, we need to find all functions \( f: (0, \infty) \rightarrow (0, \infty) \) that satisfy the equation:
\[
f\left(f(x) + \frac{y + 1}{f(y)}\right) = \frac{1}{f(y)} + x + 1
\]
for all \( x, y > 0 \). Hmm, okay. Functional equations can be tricky,... |
olympiads |
09.2. On a faded piece of paper it is possible, with some effort, to discern the following:
$$
\left(x^{2}+x+a\right)\left(x^{15}-\ldots\right)=x^{17}+x^{13}+x^{5}-90 x^{4}+x-90
$$
Some parts have got lost, partly the constant term of the first factor of the left side, partly the main part of the other factor. It wo... | <think>
Okay, let me try to figure out this problem. So, the equation given is a product of two polynomials: (x² + x + a) and another polynomial that's mostly faded, except it starts with x¹⁵. When multiplied together, they equal x¹⁷ + x¹³ + x⁵ - 90x⁴ + x - 90. We need to find the constant term 'a' in the first polynom... |
olympiads | 6.5. On the board, there are 5 integers. By adding them in pairs, the following set of 10 numbers was obtained: $-1,5,8,9,11,12,14,18,20,24$. Determine which numbers are written on the board. In your answer, write their product. | <think>
Alright, let me try to figure out this problem. So, we have five integers on the board, and when we add them in pairs, we get these 10 numbers: -1, 5, 8, 9, 11, 12, 14, 18, 20, 24. Our job is to find the original five numbers and then calculate their product. Hmm, okay.
First, I remember that if you have five... |
olympiads | 5. For an equilateral triangle $ABC$ with side length $2 \sqrt{3}$, two moving points $P, M$ in its plane satisfy $|\overrightarrow{A P}|=1, \overrightarrow{P M}=\overrightarrow{M B}$, then the minimum value of $|\overrightarrow{C M}|$ is $\qquad$. | <think>
Okay, let's see. I need to find the minimum value of |CM| where C is a vertex of an equilateral triangle ABC with side length 2√3. The points P and M are moving points such that |AP| = 1 and PM = MB. Hmm. Let me break this down step by step.
First, maybe I should visualize the problem. There's an equilateral t... |
number_theory | 2. Decipher the ciphertext message LFDPH LVDZL FRQTX HUHG that has been enciphered using the Caesar cipher. | <think>
Okay, so I need to figure out how to decipher this Caesar cipher text: LFDPH LVDZL FRQTX HUHG. Let me start by recalling what a Caesar cipher is. From what I remember, it's a type of substitution cipher where each letter in the plaintext is shifted a certain number of places down or up the alphabet. For example... |
olympiads_ref | Let $\mathbb{Z}_{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that
$$ m^{2}+f(n) \mid m f(m)+n $$
for all positive integers $m$ and $n$. (Malaysia) Answer. $f(n)=n$. | <think>
Okay, so I need to find all functions \( f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0} \) such that for all positive integers \( m \) and \( n \), the expression \( m^{2} + f(n) \) divides \( m f(m) + n \). Hmm, the problem states that the answer is \( f(n) = n \), but I need to verify that and make sure there... |
olympiads | Example 6. Solve the equation:
a) $\log _{x+1}\left(x^{2}-3 x+1\right)=1$
b) $\log _{x}\left(2 x^{2}-3 x-4\right)=2$. | <think>
Okay, so I need to solve these two logarithmic equations. Let me start with part a). The equation is log base (x+1) of (x² - 3x +1) equals 1. Hmm, logarithms can sometimes be tricky because of the base and the domain restrictions. Let me remember the basic definition of a logarithm. The equation log_b(a) = c me... |
olympiads | GS. 4 If the line $5 x+2 y-100=0$ has $d$ points whose $x$ and $y$ coordinates are both positive integers, find the value of $d$. | <think>
Okay, let's see. I need to find how many points on the line 5x + 2y - 100 = 0 have both x and y as positive integers. So, basically, I need to find all positive integer solutions (x, y) to the equation 5x + 2y = 100. Hmm, this seems like a Diophantine equation problem. Diophantine equations are equations where ... |
olympiads | 1.3. The base of an isosceles triangle is $4 \sqrt{2} \mathrm{~cm}$, and the median of the lateral side is $5 \mathrm{~cm}$. Find the lengths of the lateral sides. | <think>
Okay, so I need to find the lengths of the lateral sides of an isosceles triangle where the base is 4√2 cm and the median of the lateral side is 5 cm. Hmm, let me try to visualize this. An isosceles triangle has two equal sides (the lateral sides) and a base. The base is given as 4√2 cm. The median to a side is... |
olympiads | 12. (20 points) Given the ellipse $\frac{x^{2}}{6}+\frac{y^{2}}{2}=1$ with its right focus at $F$, the line $y=k(x-2)(k \neq 0)$ passing through $F$ intersects the ellipse at points $P$ and $Q$. If the midpoint of $PQ$ is $N$, and $O$ is the origin, the line $ON$ intersects the line $x=3$ at point $M$, find
(1) the siz... | <think>
Okay, so I need to solve this problem involving an ellipse, a line intersecting it, midpoints, and some distances and angles. Let me start by understanding all the components given.
First, the ellipse equation is given as x²/6 + y²/2 = 1. That's in standard form, right? So the standard form of an ellipse is x²... |
olympiads | 3. Chords $AB$ and $CD$ of a circle with center $O$ have a length of 5. The extensions of segments $BA$ and $CD$ beyond points $A$ and $D$ respectively intersect at point $P$, and $DP=13$. Line $PO$ intersects segment $AC$ at point $L$. Find the ratio $AL: LC$. | <think>
Okay, so I have this geometry problem here. Let me try to visualize it first. There's a circle with center O. Two chords, AB and CD, each have a length of 5. The extensions of BA and CD beyond points A and D meet at point P, and DP is 13 units long. Then, the line PO intersects the segment AC at point L, and I ... |
olympiads | 7. (2004 National College Entrance Examination - Beijing Paper) The smallest positive period of the function $f(x)=\cos 2 x-2 \sqrt{3} \sin x \cos x$ is | <think>
Okay, so I need to find the smallest positive period of the function f(x) = cos(2x) - 2√3 sinx cosx. Hmm, let me start by recalling that the period of a function is the smallest positive number T for which f(x + T) = f(x) for all x. Since this function is a combination of trigonometric functions, maybe I can si... |
olympiads | 2. The quadratic trinomial $x^{2}+u x-v$ has distinct non-zero roots $p$ and $q$, and the quadratic trinomial $x^{2}+p x-q-$ has distinct non-zero roots $u$ and $v$. Find all possible values of $p, q, u$ and $v$. | <think>
Okay, so I need to find all possible values of p, q, u, and v given these two quadratic trinomials. Let me start by writing down the information given.
First, the quadratic trinomial is x² + ux - v, and it has distinct non-zero roots p and q. From Vieta's formulas, I know that the sum of the roots is equal to ... |
olympiads | 13.137. Three machines from different systems perform a certain counting task. If the entire task is assigned to only the second or the first machine, then the second machine alone will take 2 minutes longer to complete the entire task than the first machine. The third machine can complete the entire task in a time twi... | <think>
Okay, so we've got this problem about three machines working on a task. Let me try to parse it step by step.
First, the problem says that if the entire task is assigned to only the second or the first machine, then the second machine alone takes 2 minutes longer than the first. So, let me note that down. Let'... |
olympiads | 13. Define $n!=1 \times 2 \times \ldots \times n$, for example, $5!=1 \times 2 \times 3 \times 4 \times 5$. If $\frac{n!\times(n+1)!}{2}$ (where $\mathbf{n}$ is a positive integer, and $1 \leq n \leq 100$) is a perfect square, for example, when $n=7$,
$\frac{n!\times(n+1)!}{2}=\frac{7 \times (7+1)!}{2}=\frac{7 \times 8... | <think>
Okay, so the problem is asking for all positive integers n between 1 and 100 inclusive such that (n! × (n+1)! ) / 2 is a perfect square. Then we need to sum all such n. The example given is n=7, and they showed that it simplifies to (7!)^2 × 4, which is a perfect square. Alright, so first, I need to understand ... |
aops_forum | A father and his son are skating around a circular skating rink. From time to time, the father overtakes the son. After the son starts skating in the opposite direction, they begin to meet five times more often. What is the ratio of the skating speeds of the father and the son?
(Tairova) | <think>
Okay, so there's this problem about a father and son skating around a circular rink. Initially, they're both going in the same direction, and the father overtakes the son every so often. Then, the son starts skating in the opposite direction, and now they meet five times more often. We need to find the ratio of... |
olympiads | A7. Two 2-digit multiples of 7 have a product of 7007 . What is their sum? | <think>
Okay, so the problem is: Two 2-digit multiples of 7 have a product of 7007. What is their sum?
Hmm, let me start by understanding the problem. We need to find two numbers, each of which is a two-digit multiple of 7, and when you multiply them together, the result is 7007. Then, we need to find the sum of these... |
olympiads | [ Quadrilaterals (extreme properties).]
Among all parallelograms of a given area, find the one for which the largest diagonal is minimal. | <think>
Okay, so I have this problem: Among all parallelograms of a given area, find the one for which the largest diagonal is minimal. Hmm. Let me try to figure this out step by step.
First, let's recall some properties of parallelograms. A parallelogram has opposite sides equal and parallel, and the diagonals bisec... |
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