source stringclasses 7
values | problem stringlengths 19 6.18k | generation stringlengths 345 68.1k |
|---|---|---|
olympiads | II. (40 points) Given the function $f: \mathbf{R}_{+} \rightarrow \mathbf{R}_{+}$ satisfies $f(f(n))=2016 n-215 f(n)$.
Find $f(n)$. | <think>
Okay, so I need to find the function f(n) where f maps from the positive real numbers to the positive real numbers, and it satisfies the equation f(f(n)) = 2016n - 215f(n). Hmm, functional equations can be tricky, but maybe I can assume a form for f(n) and solve for the coefficients. Let me think.
Since the eq... |
olympiads | 38. In a tournament where each pair of teams played each other twice, 4 teams participated. For each win, two points were awarded, for a draw - one, and for a loss - 0. The team that finished in last place scored 5 points. How many points did the team that finished in first place score? | <think>
Alright, let me try to figure out this problem. So, there are 4 teams in a tournament, and each pair plays each other twice. That means each team plays 6 matches, right? Because there are 3 other teams, and they play each of them twice. So, 3 opponents times 2 games each equals 6 matches per team.
Now, for eac... |
olympiads | Task 2. (10 points) In triangle $ABC$ with sides $AB=9$, $AC=3$, $BC=8$, the bisector $AK$ is drawn, and a point $M$ is marked on side $AC$ such that $AM: CM=3: 1$. Point $N$ is the intersection of $AK$ and $BM$. Find $KN$.
# | <think>
Okay, let's see. I need to find the length of KN in triangle ABC where AK is the angle bisector, M divides AC in a 3:1 ratio, and N is the intersection of AK and BM. Hmm. Let me start by drawing a rough sketch of the triangle to visualize the problem. AB is 9 units, AC is 3 units, and BC is 8 units. Wait, that ... |
olympiads | 2. In a computer game, a turtle moves across a grid on the computer screen, which contains 11 columns and 5 rows. Initially, it is located at the bottom-left corner of the screen - on the cell with coordinates $(0,0)$. If the program instructs the turtle to move off the screen, it reappears on the opposite side - for e... | <think>
Alright, let's try to figure out where the turtle ends up after all those moves. The grid has 11 columns and 5 rows, so the coordinates should be (x, y) where x is between 0 and 10 (since there are 11 columns) and y is between 0 and 4 (since there are 5 rows). The turtle starts at (0,0).
Each time the turtle m... |
olympiads | Example 5. Find the derivative of the function $y=\arcsin \frac{1}{x}$. | <think>
Okay, so I need to find the derivative of y = arcsin(1/x). Hmm, let's see. I remember that the derivative of arcsin(u) is 1 over sqrt(1 - u squared) times the derivative of u, right? So maybe I can use the chain rule here. Let me write that down.
First, let me recall the formula: d/dx [arcsin(u)] = u' / sqrt(1... |
olympiads | 3. Two people simultaneously step onto an escalator from opposite ends, which is moving downward at a speed of $u=1.5 \mathrm{~m} / \mathrm{s}$. The person moving downward has a speed of $v=3 \mathrm{~m} / \mathrm{s}$ relative to the escalator, while the person moving upward has a speed of $2 v / 3$ relative to the esc... | <think>
Okay, let me try to figure this problem out. So there are two people stepping onto an escalator from opposite ends at the same time. The escalator itself is moving downward at 1.5 m/s. One person is moving downward relative to the escalator at 3 m/s, and the other is moving upward relative to the escalator at 2... |
olympiads | 8. The integers from 1 to $n$ are written in increasing order from left to right on a blackboard. David and Goliath play the following game: starting with David, the two players alternate erasing any two consecutive numbers and replacing them with their sum or product. Play continues until only one number on the board ... | <think>
Okay, let's try to figure out this problem. So, we have numbers from 1 to n written in order, and two players take turns replacing two consecutive numbers with either their sum or product. The game ends when there's one number left. If it's odd, David wins; if even, Goliath wins. We need to find the 2011th smal... |
olympiads | ## Task A-3.6.
Calculate the product
$$
\left(1-\frac{\cos 61^{\circ}}{\cos 1^{\circ}}\right)\left(1-\frac{\cos 62^{\circ}}{\cos 2^{\circ}}\right) \ldots\left(1-\frac{\cos 119^{\circ}}{\cos 59^{\circ}}\right)
$$ | <think>
Okay, let's see. I need to calculate the product from k=1 to k=59 of (1 - cos(60° + k°)/cos(k°)). Wait, let me check. The problem starts with 61° over cos1°, so that's when k=1: 60+1=61°, over cos1°, right? And it goes up to 119°, which is 60+59. So the product is from k=1 to k=59 of [1 - cos(60° + k°)/cos(k°)]... |
olympiads | 5. (10 points) Among the divisors of a four-digit palindrome, exactly 3 are prime, and 39 are not prime. The value of the four-digit palindrome is
| <think>
Okay, let's try to figure out this problem. So, we need to find a four-digit palindrome number that has exactly 3 prime divisors and 39 non-prime divisors. Hmm, okay. Let me break this down step by step.
First, a four-digit palindrome is a number that reads the same forwards and backwards. So, it has the form ... |
olympiads | 10. (15 points) Kangaroo A and B are jumping in the area shown in the figure. A jumps in the order of $A B C D E F G H I A B C \cdots$, and B jumps in the order of $A B D E G H A B D \cdots$. If both kangaroos start from $A$ and this counts as the first time they jump together, how many times have they jumped together ... | <think>
Okay, so let's try to figure out this problem. We have two kangaroos, A and B, jumping around an area that's shown in a figure. Since I can't see the figure, I need to make some assumptions based on the order of their jumps.
First, Kangaroo A jumps in the order A, B, C, D, E, F, G, H, I, A, B, C, and so on. T... |
cn_contest | 11. (15 points) Given the function $f(x)=-2 x^{2}+b x+c$ has a maximum value of 1 at $x=1$, and $0<m<n$. When $x \in [m, n]$, the range of $f(x)$ is $\left[\frac{1}{n}, \frac{1}{m}\right]$. Find the values of $m$ and $n$. | <think>
Okay, let's tackle this problem step by step. So, we have the function f(x) = -2x² + bx + c. It has a maximum value of 1 at x = 1. Then, given 0 < m < n, when x is in [m, n], the range of f(x) is [1/n, 1/m]. We need to find m and n. Hmm, okay. Let me start by understanding the function first.
Since it's a quad... |
cn_contest | $$
\begin{array}{l}
M=\left\{(x, y) \left\lvert\, y \geqslant \frac{1}{4} x^{2}\right.\right\}, \\
N=\left\{(x, y) \left\lvert\, y \leqslant-\frac{1}{4} x^{2}+x+7\right.\right\}, \\
\left.D_{r}\left(x_{0}, y_{0}\right)=|(x, y)|\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2} \leqslant r^{2}\right\} .
\end{array}
$$
F... | <think>
Okay, let's see. I need to find the largest radius r such that the disk centered at (x₀, y₀) is entirely contained within the intersection of M and N. Hmm, M is the set of points where y is greater than or equal to (1/4)x², which is a parabola opening upwards. N is the set where y is less than or equal to -1/4 ... |
cn_contest | Let $n$ be a fixed integer, $n \geqslant 2$.
a) Determine the smallest constant $c$ such that the inequality
$$
\sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leqslant c\left(\sum_{i=1}^{n} x_{i}\right)^{4}
$$
holds for all non-negative real numbers $x_{1}, x_{2}, \cdots, x_{n} \geqsla... | <think>
Okay, so I need to solve part (a) and (b) of this inequality problem. Let me start with part (a). The question is to find the smallest constant c such that the inequality
$$
\sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leqslant c\left(\sum_{i=1}^{n} x_{i}\right)^{4}
$$
holds... |
olympiads | (4) Given that line $l$ forms a $45^{\circ}$ angle with plane $\alpha$, and line $m \subset \alpha$, if the projection of line $l$ on $\alpha$ also forms a $45^{\circ}$ angle with line $m$, then the angle formed by line $l$ and $m$ is $\qquad$ | <think>
Okay, so I need to find the angle between line l and line m. Let me start by understanding the problem step by step.
First, it says that line l forms a 45-degree angle with plane α. That means if I imagine plane α as a flat surface, line l is tilted at 45 degrees relative to this plane. I remember that the an... |
olympiads | Example 2. Find the third-order derivative of the function $y=x^{2}+3 x+2$. | <think>
Okay, so I need to find the third-order derivative of the function y = x² + 3x + 2. Hmm, derivatives. Let me remember. The first derivative is the slope of the function, the second derivative is the rate of change of the first derivative, and the third derivative would be the rate of change of the second deriva... |
olympiads | 4. Choose any two numbers from $2,4,6,7,8,11,12,13$ to form a fraction. Then, there are $\qquad$ irreducible fractions among these fractions. | <think>
Okay, so I have this problem here where I need to choose any two numbers from the list 2, 4, 6, 7, 8, 11, 12, 13 to form a fraction. Then, I have to figure out how many of those possible fractions are irreducible. Let me try to work through this step by step.
First, let me recall what an irreducible fraction i... |
olympiads | ## Problem Statement
Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.
$M_{1}(-2 ; 0 ;-4)$
$M_{2}(-1 ; 7 ; 1)$
$M_{3}(4 ;-8 ;-4)$
$M_{0}(-6 ; 5 ; 5)$ | <think>
Okay, so I need to find the distance from point M0 to the plane that passes through points M1, M2, and M3. Hmm, let me recall how to do this. I think the formula for the distance from a point to a plane is |ax + by + cz + d| divided by the square root of (a² + b² + c²), right? But first, I need to find the equa... |
olympiads | 2. Let $n$ be a three-digit positive integer that does not contain the digit 0. If any permutation of the units, tens, and hundreds digits of $n$ does not form a three-digit number that is a multiple of 4, find the number of such $n$.
(54th Ukrainian Mathematical Olympiad) | <think>
Okay, so I need to figure out how many three-digit numbers there are where none of the permutations of their digits result in a multiple of 4. The number itself can't have any zeros, either. Hmm, let me break this down step by step.
First, let's recall that a three-digit number is composed of hundreds, tens, a... |
olympiads | 1. Vernonia High School has 85 seniors, each of whom plays on at least one of the school's three varsity sports teams: football, baseball, and lacrosse. It so happens that 74 are on the football team; 26 are on the baseball team; 17 are on both the football and lacrosse teams; 18 are on both the baseball and football t... | <think>
Okay, so I need to find out how many seniors at Vernonia High School play all three sports: football, baseball, and lacrosse. The problem gives me a bunch of numbers, and there's a condition that twice the number of students playing all three sports are on the lacrosse team. Hmm, let me try to unpack this step ... |
olympiads | 6. The general term formula of the sequence $\left\{a_{n}\right\}$ is $a_{n}=\tan n \cdot \tan (n-1)$, and for any positive integer $n$, there is $a_{1}+a_{2}+\cdots+a_{n}=A \tan n+B n$ holds, then $A=$ $\qquad$ $. B=$ $\qquad$ . | <think>
Okay, so I need to find the values of A and B such that the sum of the sequence a_n = tan(n) * tan(n - 1) from 1 to n is equal to A * tan(n) + B * n. Hmm, let me think step by step.
First, let me write down what the problem says. The sequence is defined by a_n = tan(n) * tan(n - 1), and for any positive intege... |
olympiads | ## SUBJECT IV
Let the natural number $n=\overline{11 \ldots 1}+\overline{22 \ldots 2}+\ldots+\overline{88 \ldots 8}+\overline{99 \ldots 9}$, where each number of the form $\overline{a a \ldots a}$ contains 2015 digits of $a$. Determine how many digits of 9 the number $n$ contains.
## Note:
All subjects are mandatory... | <think>
Okay, let's see. The problem says that we have a natural number n which is the sum of numbers like 11...1, 22...2, up to 99...9, each with 2015 digits. We need to find how many digits of 9 the number n contains. Hmm. So each term in the sum is a number where all the digits are the same, from 1 to 9, each with 2... |
olympiads | 4. The sum $b_{6}+b_{7}+\ldots+b_{2018}$ of the terms of the geometric progression $\left\{b_{n}\right\}, b_{n}>0$ is 6. The sum of the same terms taken with alternating signs $b_{6}-b_{7}+b_{8}-\ldots-b_{2017}+b_{2018}$ is 3. Find the sum of the squares of the same terms $b_{6}^{2}+b_{7}^{2}+\ldots+b_{2018}^{2}$. | <think>
Okay, so I have this problem here about a geometric progression. Let me try to parse it step by step. The problem says:
We have a geometric progression {b_n} where all terms are positive (b_n > 0). The sum from the 6th term to the 2018th term, which is b₆ + b₇ + ... + b_{2018}, is equal to 6. Then, the same te... |
olympiads | Shapovalov A.V.
At a round table, 12 places were prepared for the jury with each place labeled with a name. Nikolai Nikolaevich, who arrived first, absent-mindedly sat not in his own place, but in the next one clockwise. Each subsequent jury member, approaching the table, would take their own place or, if it was alrea... | <think>
Alright, let's try to tackle this problem. So, we have a round table with 12 places, each labeled with a jury member's name. The first person to arrive, Nikolai Nikolaevich, didn't sit in his own seat but instead took the next one clockwise. Then, each subsequent person will take their own seat if it's availabl... |
olympiads | The probability that a purchased light bulb will work is 0.95.
How many light bulbs need to be bought so that with a probability of 0.99, there will be at least five working ones among them?
# | <think>
Okay, so I need to figure out how many light bulbs I should buy to ensure that there's at least a 99% chance that at least five of them work. Each bulb has a 95% chance of working. Let me start by understanding the problem.
First, this seems like a probability question involving binomial distributions. The bin... |
olympiads | 24. A polynomial of degree 10 has three distinct roots. What is the maximum number of zero coefficients it can have?
(A. Khryabrov) | <think>
Okay, so I need to figure out the maximum number of zero coefficients a degree 10 polynomial can have if it has three distinct roots. Hmm, let's start by recalling some basics about polynomials and their roots.
First, a polynomial of degree 10 will have 10 roots in total (counting multiplicities) according to... |
aops_forum | Two positive integers $a$ and $b$ are prime-related if $a = pb$ or $b = pa$ for some prime $p$. Find all positive integers $n$, such that $n$ has at least three divisors, and all the divisors can be arranged without repetition in a circle so that any two adjacent divisors are prime-related.
Note that $1$ and $n$ are ... | <think>
Okay, let's see. The problem is about finding all positive integers n (with at least three divisors) such that all their divisors can be arranged in a circle where any two adjacent ones are prime-related. Prime-related means one is a prime multiple of the other. Hmm.
First, let's understand what prime-related ... |
olympiads | 5. On one of two parallel lines, there are 8 points. How many points are on the other line if all the points together determine 640 triangles?
Each task is scored out of 10 points.
The use of a pocket calculator or any reference materials is not allowed. | <think>
Okay, so here's the problem: On one of two parallel lines, there are 8 points. How many points are on the other line if all the points together determine 640 triangles? Hmm, triangles formed by points on two parallel lines. Let me think.
First, I need to recall how triangles are formed with points on two paral... |
olympiads | In a certain sequence of 80 numbers, any term, except for the two endpoints, is equal to the product of its neighboring terms. The product of the first 40 terms of the sequence is 8 and the product of all the terms is also 8. Determine the terms of the sequence. | <think>
Okay, let's tackle this problem step by step. So, we have a sequence of 80 numbers where each term (except the first and last) is equal to the product of its neighboring terms. The product of the first 40 terms is 8, and the product of all 80 terms is also 8. We need to find the terms of the sequence.
Hmm, let... |
olympiads | 2.150. $\frac{2-\sqrt{2}-\sqrt{3}}{2+\sqrt{2}-\sqrt{3}}$. | <think>
Alright, so I have this problem to solve: (2 - √2 - √3)/(2 + √2 - √3). Hmm, I need to simplify this expression. Let me think. I remember from algebra that when you have a denominator with radicals, you often rationalize it by multiplying the numerator and denominator by the conjugate of the denominator. But wai... |
olympiads | Find all solutions to the following system of equations
$$
y=x+\sqrt{x+\sqrt{x+\sqrt{x+\ldots+\sqrt{x+\sqrt{y}}}}}, \quad x+y=6
$$
where there are 1974 square root signs in the equation. | <think>
Okay, so I need to solve this system of equations:
$$
y = x + \sqrt{x + \sqrt{x + \sqrt{x + \ldots + \sqrt{x + \sqrt{y}}}}} \quad \text{(with 1974 square root signs)}
$$
$$
x + y = 6
$$
Hmm, that's a lot of square roots. Let me try to understand what's going on here. The first equation has a nested radical ex... |
olympiads | 10. Given the sequence $\left\{a_{n}\right\}$ satisfies: $a_{1}=\frac{3}{2}$, and $a_{n}=\frac{3 n a_{n-1}}{2 a_{n-1}+n-1}\left(n \geqslant 2, n \in \mathbf{N}_{+}\right)$.
(1) Find the general term formula of the sequence $\left\{a_{n}\right\}$;
(2) For all positive integers $n$, the inequality $a_{1} a_{2} a_{3} \cdo... | <think>
Okay, so I need to solve this problem about a recursive sequence. Let me take it step by step.
First, part (1) is to find the general term formula for the sequence {a_n} where a_1 = 3/2, and for n ≥ 2, a_n = (3n a_{n-1}) / (2 a_{n-1} + n - 1). Hmm. The recursion looks a bit complicated. Let me write down the ... |
olympiads | 3. Circles $\omega_{1}$ and $\omega_{2}$ with centers $O_{1}$ and $O_{2}$ respectively intersect at point $A$. Segment $O_{2} A$ intersects circle $\omega_{1}$ again at point $K$, and segment $O_{1} A$ intersects circle $\omega_{2}$ again at point $L$. The line passing through point $A$ parallel to $K L$ intersects cir... | <think>
Okay, let me try to figure out this geometry problem. So, we have two circles, ω₁ and ω₂, intersecting at point A. Their centers are O₁ and O₂. Then, segment O₂A intersects ω₁ again at K, and segment O₁A intersects ω₂ again at L. There's a line through A parallel to KL that meets ω₁ again at C and ω₂ again at D... |
olympiads | 14. Asymmetric coin. (From 9th grade, 2 points) Billy Bones has two coins - a gold one and a silver one. One of them is symmetric, and the other is not. It is unknown which coin is asymmetric, but it is known that the asymmetric coin lands heads with a probability of \( p = 0.6 \).
Billy Bones tossed the gold coin, an... | <think>
Okay, let's try to figure out this probability problem. Hmm, so we have two coins: gold and silver. One is symmetric (fair) and the other is asymmetric with a probability of landing heads p = 0.6. We don't know which one is which. Billy tossed the gold coin once and got heads. Then he tossed the silver coin twi... |
olympiads | 12. The Musketeers' Journey. The distance between Athos and Aramis, riding on the road, is 20 leagues. In one hour, Athos travels 4 leagues, and Aramis - 5 leagues. What distance will be between them after an hour? | <think>
Okay, so I need to figure out the distance between Athos and Aramis after one hour. Let me start by understanding the problem. The initial distance between them is 20 leagues. Athos is traveling at 4 leagues per hour, and Aramis is going faster at 5 leagues per hour. Hmm, the question is, after one hour, what w... |
olympiads | ## 43. Math Puzzler $12 / 68$
Mr. Dull has nothing to do. He starts counting all the numbers in natural order: $1 +$ $2 + 3 + \cdots$ and so on. The doorbell rings, and Mr. Dull just manages to note down the last intermediate result he found, which is 6328. However, he forgets the last number he added.
Can you help h... | <think>
Okay, so Mr. Dull was adding numbers starting from 1, like 1 + 2 + 3 + 4 + ... and so on. Then the doorbell rang, and he wrote down the last sum he had, which is 6328. But he forgot what the last number he added was. My job is to figure out what that last number is. Hmm, let's see.
First, I remember that the s... |
olympiads | 3. Insert arithmetic operation signs and parentheses into the expression consisting of three numbers, $\frac{1}{8} \ldots \frac{1}{9} \ldots \frac{1}{28}$, so that the result of the calculations is equal to $\frac{1}{2016}$.
ANSWER: $\frac{1}{8} \times \frac{1}{9} \times \frac{1}{28}$ or $\left(\frac{1}{8}-\frac{1}{9}... | <think>
Okay, so I need to figure out how to insert arithmetic operations and parentheses between 1/8, 1/9, and 1/28 to get 1/2016. Hmm, let's start by recalling the possible operations: addition, subtraction, multiplication, division. Maybe parentheses to change the order of operations. Let's see.
First, let's comput... |
aops_forum | Let $p$ and $q$ be two given positive integers. A set of $p+q$ real numbers $a_1<a_2<\cdots <a_{p+q}$ is said to be balanced iff $a_1,\ldots,a_p$ were an arithmetic progression with common difference $q$ and $a_p,\ldots,a_{p+q}$ where an arithmetic progression with common difference $p$. Find the maximum possible numbe... | <think>
Okay, so I need to solve this problem about balanced sets. Let me try to understand what's given and what's being asked.
First, we have two positive integers p and q. A set of p + q real numbers, arranged in increasing order as a₁ < a₂ < ... < a_{p+q}, is called balanced if two conditions are satisfied:
1. Th... |
olympiads | 3. Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=\frac{7}{6}$, and for any positive integer $n$, the quadratic equation $a_{n} x^{2}-$ $a_{n+1} x+1=0$ has two roots $\alpha_{n}, \beta_{n}$ that satisfy $6 \alpha_{n}-2 \alpha_{n} \beta_{n}+6 \beta_{n}=3$. Then the general term formula for $\left\{a_{n}\righ... | <think>
Okay, let's try to tackle this problem step by step. So we have a sequence {a_n} where a₁ = 7/6, and for each positive integer n, the quadratic equation a_n x² - a_{n+1} x + 1 = 0 has two roots α_n and β_n. These roots satisfy the equation 6α_n - 2α_n β_n + 6β_n = 3. We need to find a general formula for a_n.
... |
olympiads | 1. When $2 x^{2}$ was added to the quadratic trinomial $f(x)$, its maximum value increased by 10, and when $5 x^{2}$ was subtracted from it, its maximum value decreased by $\frac{15}{2}$. How will the maximum value of $f(x)$ change if $3 x^{2}$ is added to it? | <think>
Okay, let's see. I need to solve this problem about a quadratic trinomial f(x). The problem says that when 2x² is added to f(x), its maximum value increases by 10. Then, when 5x² is subtracted from f(x), the maximum value decreases by 15/2. The question is asking how the maximum value of f(x) will change if 3x²... |
olympiads | 3. Two numbers x and y satisfy the equation $280 x^{2}-61 x y+3 y^{2}-13=0$ and are the fourth and ninth terms, respectively, of a decreasing arithmetic progression consisting of integers. Find the common difference of this progression. | <think>
Okay, so I need to find the common difference of a decreasing arithmetic progression where the fourth and ninth terms are x and y, respectively. These numbers satisfy the equation 280x² - 61xy + 3y² - 13 = 0. Hmm, let me break this down step by step.
First, let's recall that in an arithmetic progression (AP), ... |
aops_forum | Let $a_1=2021$ and for $n \ge 1$ let $a_{n+1}=\sqrt{4+a_n}$. Then $a_5$ can be written as $$\sqrt{\frac{m+\sqrt{n}}{2}}+\sqrt{\frac{m-\sqrt{n}}{2}},$$ where $m$ and $n$ are positive integers. Find $10m+n$. | <think>
Okay, so I need to find the value of 10m + n where m and n are positive integers such that a5 can be expressed in that square root form they mentioned. The sequence starts with a1 = 2021, and each subsequent term is the square root of 4 plus the previous term. Let me break this down step by step.
First, let me... |
olympiads | Suppose that for the positive integer $n$, $2^{n}+1$ is prime. What remainder can this prime give when divided by $240$? | <think>
Okay, let me try to figure out this problem. So, the question is: Suppose that for a positive integer \( n \), \( 2^{n} + 1 \) is prime. What remainder can this prime give when divided by 240?
First, I need to recall some properties of primes of the form \( 2^n + 1 \). I remember that primes like this are call... |
olympiads |
4. There are four basket-ball players $A, B, C, D$. Initially, the ball is with $A$. The ball is always passed from one person to a different person. In how many ways can the ball come back to $A$ after seven passes? (For example $A \rightarrow C \rightarrow B \rightarrow D \rightarrow A \rightarrow B \rightarrow C \r... | <think>
Okay, so there's this basketball passing problem. Let me try to figure it out step by step.
We have four players: A, B, C, D. The ball starts with A. Each time, the ball is passed to a different person. After seven passes, we need to find the number of ways the ball can come back to A.
Hmm, okay. So each pa... |
olympiads | 10.074. One of the two parallel lines is tangent to a circle of radius $R$ at point $A$, while the other intersects this circle at points $B$ and $C$. Express the area of triangle $A B C$ as a function of the distance $x$ between the lines. | <think>
Okay, let's see. The problem is about two parallel lines, one tangent to a circle with radius R at point A, and the other line intersects the circle at points B and C. We need to find the area of triangle ABC as a function of the distance x between the two lines. Hmm, okay.
First, I need to visualize this. The... |
olympiads | 90. Even or odd product
$$
(7 a+b-2 c+1)(3 a-5 b+4 c+10)
$$
where the numbers $a, b, c$ are integers | <think>
Okay, let's see. The problem is asking whether the product (7a + b - 2c + 1)(3a - 5b + 4c + 10) is even or odd when a, b, c are integers. Hmm, so I need to determine if this product will always be even, always be odd, or if it can be either depending on the values of a, b, c. But the way the question is phrased... |
olympiads | 1. A professor from the Department of Mathematical Modeling at FEFU last academic year gave 6480 twos, thereby overachieving the commitments taken at the beginning of the year. In the next academic year, the number of professors increased by 3, and each of them started giving more twos. As a result, a new record was se... | <think>
Alright, so I need to figure out how many professors there were initially in the Department of Mathematical Modeling at FEFU. Let me try to break down the problem step by step.
First, the information given:
1. Last academic year, a professor (or professors?) gave 6480 twos. It mentions "a professor" but then ... |
olympiads | 3. On each of the lines $x=2$ and $x=15$, there are 400 points with ordinates $1,2,3, \ldots, 400$. In how many ways can three points be chosen from the 800 marked points so that they form the vertices of a right triangle? | <think>
Okay, so I have this problem where there are two vertical lines, x=2 and x=15, each with 400 points having y-coordinates from 1 to 400. I need to find the number of ways to choose three points such that they form a right triangle. Hmm, right triangles... right triangles have one 90-degree angle. So, I remember ... |
olympiads | 11.2. Several married couples came to the New Year's Eve party, each of whom had from 1 to 10 children. Santa Claus chose one child, one mother, and one father from three different families and took them for a ride in his sleigh. It turned out that he had exactly 3630 ways to choose the required trio of people. How man... | <think>
Okay, let's try to tackle this problem. So, we have several married couples at a party, each with 1 to 10 children. Santa picks one child, one mother, and one father from three different families. The number of ways he can do this is exactly 3630. We need to find the total number of children that could be at th... |
olympiads | 4. On the coordinate plane, squares are considered, all vertices of which have integer non-negative coordinates, and the center is located at the point $(60 ; 45)$. Find the number of such squares. | <think>
Okay, so I need to figure out how many squares there are with integer coordinates for all their vertices and their center at (60, 45). Hmm, let's start by recalling what defines the center of a square. The center of a square is the midpoint of its diagonals. So, if we have a square with vertices (x1, y1), (x2, ... |
olympiads | 10. $n>10$ teams take part in a soccer tournament. Each team plays every other team exactly once. A win gives two points, a draw one point and a defeat no points. After the tournament, it turns out that each team has won exactly half of its points in the games against the 10 worst teams (in particular, each of these 10... | <think>
Okay, let me try to work through this problem step by step. So, we have a soccer tournament with n teams where n is greater than 10. Each team plays every other team exactly once. The scoring is standard: 2 points for a win, 1 for a draw, 0 for a loss. After the tournament, it turns out that each team has won e... |
olympiads | 3. In the number $2 * 0 * 1 * 6 * 0 * 2 *$, each of the 6 asterisks needs to be replaced with any of the digits $1,2,3,4,5,6,7,8,9$ (digits can repeat) so that the resulting 12-digit number is divisible by 18. In how many ways can this be done? | <think>
Okay, so I need to figure out how many ways I can replace the six asterisks in the number 2*0*1*6*0*2* with the digits 1 through 9 (allowing repeats) such that the resulting 12-digit number is divisible by 18. Hmm, let's break this down step by step.
First, I remember that a number divisible by 18 must be divi... |
olympiads | 7. Given a complex number $z$ satisfying $|z|=1$. Then the maximum value of $\left|z^{3}-3 z-2\right|$ is $\qquad$ .
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | <think>
Okay, so I need to find the maximum value of |z³ - 3z - 2| where z is a complex number with |z| = 1. Hmm, let's start by understanding what the problem is asking. Since |z| = 1, z lies on the unit circle in the complex plane. That means we can represent z as e^(iθ) where θ is the argument of z. Maybe writing z ... |
olympiads | ## Task A-4.5.
In a circle, a finite number of real numbers are arranged. Each number is colored red, white, or blue. Each red number is twice as small as the sum of its two neighboring numbers, each white number is equal to the sum of its two neighboring numbers, and each blue number is twice as large as the sum of i... | <think>
Okay, let's see. I need to find the ratio of the sum of all white numbers (b) to the sum of all blue numbers (p) in a circle where each number is red, white, or blue. The rules are:
- Red number is half the sum of its two neighbors.
- White number is equal to the sum of its two neighbors.
- Blue number is twic... |
cn_contest | 3. If $\left|3^{x}-\log _{3} x\right|=3^{x}+\log _{3} x=k$, then the value of $k$ is ( ).
(A) 3,
(B) 1.
(C) 0.
(D) Cannot be determined. | <think>
Okay, so I need to solve this problem where we have the equation |3^x - log₃x| = 3^x + log₃x = k. And we need to find the value of k from the options given. Let me try to parse this step by step.
First, the problem states that the absolute value of (3^x minus log base 3 of x) is equal to both (3^x plus log bas... |
olympiads | 4. An $8 \times 6$ grid is placed in the first quadrant with its edges along the axes, as shown. A total of 32 of the squares in the grid are shaded. A line is drawn through $(0,0)$ and $(8, c)$ cutting the shaded region into two equal areas. What is the value of $c$ ? | <think>
Alright, so I have this problem here about an 8x6 grid placed in the first quadrant with edges along the axes. They say that 32 of the squares are shaded, and we need to find a value c such that a line drawn through (0,0) and (8,c) divides the shaded region into two equal areas. Hmm, okay. Let me try to visuali... |
olympiads | 2.119 The number of values of $x$ that satisfy the equation $\frac{2 x^{2}-10 x}{x^{2}-5 x}=x-3$ is
(A) 0.
(B) 1.
(C) 2.
(D) 3.
(E) An integer greater than 3.
(17th American High School Mathematics Examination, 1966) | <think>
Okay, so I have this equation to solve: (2x² - 10x)/(x² - 5x) = x - 3. And the question is asking how many values of x satisfy this equation, with the options being 0, 1, 2, 3, or more than 3. Hmm. Let me start by trying to simplify the equation step by step.
First, I notice that both the numerator and the den... |
olympiads | Exercise 10. Find all triplets ( $\mathbf{p , q , r} \mathbf{)}$ ) of prime numbers such that the 3 differences
$$
|p-q|,|q-r|,|r-p|
$$
are also prime numbers. | <think>
Okay, so I need to find all triplets of prime numbers (p, q, r) such that the absolute differences between each pair, |p - q|, |q - r|, and |r - p|, are also prime numbers. Hmm, primes... differences... Let me think through this step by step.
First, primes are numbers greater than 1 that have no divisors other... |
olympiads | The base of the triangle is 20; the medians drawn to the lateral sides are 18 and 24. Find the area of the triangle.
# | <think>
Okay, so I need to find the area of a triangle where the base is 20 units, and the medians to the two lateral (I guess that means the other two) sides are 18 and 24 units. Hmm, let's try to visualize this. The triangle has a base of 20, and there are two medians, each drawn to the other two sides. Medians are t... |
olympiads | 6. Klárka had a three-digit number written on a piece of paper. When she correctly multiplied it by nine, she got a four-digit number that started with the same digit as the original number, the middle two digits were the same, and the last digit was the sum of the digits of the original number. What four-digit number ... | <think>
Okay, so Klárka has a three-digit number, and when she multiplies it by 9, she gets a four-digit number. Let me break down the problem step by step.
First, let's note the conditions given:
1. The original number is a three-digit number.
2. Multiplying it by 9 gives a four-digit number.
3. The four-digit resu... |
olympiads | From a survey at our school, it was found that
- all children who like mathematics solve the Mathematical Olympiad,
- $90\%$ of children who do not like mathematics do not solve the Mathematical Olympiad,
- $46\%$ of children solve the Mathematical Olympiad.
What percentage of children in our school like mathematics?... | <think>
Okay, so I need to figure out the percentage of children in the school who like mathematics based on the given survey results. Let me start by breaking down the information provided.
First, all children who like mathematics solve the Mathematical Olympiad. That means if a child likes math, they definitely part... |
olympiads | 1B. Calculate the product
$$
P=\left(1+\frac{1}{2}\right)\left(1+\frac{1}{2^{2}}\right)\left(1+\frac{1}{2^{4}}\right)\left(1+\frac{1}{2^{8}}\right) \cdot \ldots \cdot\left(1+\frac{1}{2^{2^{n}}}\right)
$$ | <think>
Okay, so I need to calculate the product P, which is a product of terms like (1 + 1/2^{2^k}) from k=0 to k=n. Let me write that down again to make sure I have it right:
P = (1 + 1/2)(1 + 1/2²)(1 + 1/2⁴)(1 + 1/2⁸)...(1 + 1/2^{2^n})
Each term in the product is of the form (1 + 1/2^{2^k}) where k starts at 0 and... |
olympiads | 9. Vremyankin and Puteykin simultaneously set out from Utrenneye to Vechernoye. The first of them walked half of the time spent on the journey at a speed of $5 \mathrm{km} / \mathrm{h}$, and then at a speed of $4 \mathrm{km} /$ h. The second, however, walked the first half of the distance at a speed of 4 km/h, and then... | <think>
Okay, let's see. So there's this problem where two people, Vremyankin and Puteykin, start walking from Utrenneye to Vechernoye at the same time. The question is who arrives first. Hmm. Both have different strategies for their speeds. Let me try to break this down.
First, let's parse the information. Vremyankin... |
olympiads | ## Problem Statement
Calculate the volumes of solids formed by rotating figures bounded by the graphs of functions. The axis of rotation is $O y$.
$$
y=\arcsin \frac{x}{5}, y=\arcsin x, y=\frac{\pi}{2}
$$ | <think>
Okay, so I need to find the volume of the solid formed by rotating the region bounded by the curves y = arcsin(x/5), y = arcsin(x), and y = π/2 around the y-axis (Oy). Hmm, let me start by visualizing the region. Rotating around the y-axis suggests that maybe using the method of cylindrical shells or the washer... |
olympiads_ref | Determine all pairs of positive integers $(x, y)$ for which
$$
x^{3}+y^{3}=4\left(x^{2} y+x y^{2}-5\right) .
$$ | <think>
Okay, let's try to tackle this problem: find all pairs of positive integers (x, y) such that x³ + y³ = 4(x²y + xy² - 5). Hmm, first off, this equation involves cubes and quadratic terms. Maybe I can manipulate it a bit to make it easier to handle. Let me write it down again:
x³ + y³ = 4(x²y + xy² - 5)
Hmm, th... |
olympiads | 7.4. In triangle $A B C$, the angles $A$ and $C$ at the base are $20^{\circ}$ and $40^{\circ}$ respectively. It is known that $A C - A B = 5$ (cm). Find the length of the angle bisector of angle $B$. | <think>
Okay, so I need to find the length of the angle bisector of angle B in triangle ABC. The given information is that angles at A and C are 20 degrees and 40 degrees respectively, and the difference between sides AC and AB is 5 cm. Let me start by visualizing the triangle. Let me label the triangle properly: verte... |
olympiads | ## Problem Statement
Calculate the indefinite integral:
$$
\int \frac{x^{3}+6 x^{2}+13 x+6}{(x-2)(x+2)^{3}} d x
$$ | <think>
Okay, so I need to integrate the rational function (x³ + 6x² + 13x + 6) divided by (x - 2)(x + 2)³. Hmm, rational functions. The first thing that comes to mind is partial fraction decomposition. Let me recall how that works.
Partial fractions are used when you can express the numerator as a sum of simpler fra... |
olympiads | 2.2.1. A covered football field of rectangular shape with a length of 90 m and a width of 60 m is being designed, which should be illuminated by four spotlights, each hanging at some point on the ceiling. Each spotlight illuminates a circle, the radius of which is equal to the height at which the spotlight is hanging. ... | <think>
Okay, so I need to figure out the minimum ceiling height for a football field that's 90 meters long and 60 meters wide. There are four spotlights, each hanging from the ceiling, and each spotlight illuminates a circle with a radius equal to the height it's hanging at. The ceiling height has to be a multiple of ... |
olympiads | 2. (2 points) In the quarry, there was a pile of 20160000 sand grains. A truck carried away from the quarry a quantity of sand grains per trip, which was some power of 8 (including possibly $8^{0}=1$). Could it have carried away the entire pile of sand in exactly 1000 trips? | <think>
Okay, so I have this problem here about a pile of sand grains in a quarry. The total number of sand grains is 20,160,000. A truck is supposed to carry away this entire pile in exactly 1000 trips. The catch is that each time the truck carries a quantity of sand grains that is a power of 8. That includes 8^0, whi... |
olympiads | 69. This year, the sum of Dan Dan's, her father's, and her mother's ages is 100 years. If 6 years ago, her mother's age was 4 times Dan Dan's age, and 13 years ago, her father's age was 15 times Dan Dan's age, then Dan Dan is $\qquad$ years old this year. | <think>
Okay, so I need to figure out Dan Dan's current age based on the information given. Let me start by breaking down the problem step by step.
First, the problem states that this year, the sum of Dan Dan's, her father's, and her mother's ages is 100 years. So, if I let Dan Dan's current age be D, her father's ag... |
olympiads | In a circle with a radius of 1.4, determine the distance from the center to the chord if it intercepts an arc of $120^{\circ}$.
# | <think>
Okay, so I need to find the distance from the center of a circle to a chord. The circle has a radius of 1.4, and the chord intercepts an arc of 120 degrees. Hmm, let me think. I remember that when a chord is drawn in a circle, the perpendicular distance from the center to the chord can be found using some trigo... |
olympiads | Yasinsky V.
On the plane, there are $n(n>2)$ points, no three of which lie on the same line. In how many different ways can this set of points be divided into two non-empty subsets such that the convex hulls of these subsets do not intersect? | <think>
Okay, so I have this problem here about dividing a set of n points on a plane into two non-empty subsets such that their convex hulls don't intersect. No three points are colinear, which probably simplifies things a bit. Hmm, let me try to think through this step by step.
First, let me recall what a convex hul... |
cn_contest | 11. (20 points) Let the sequence of rational numbers $\left\{a_{n}\right\}$ be defined as follows:
$a_{k}=\frac{x_{k}}{y_{k}}$, where $x_{1}=y_{1}=1$, and
if $y_{k}=1$, then $x_{k+1}=1, y_{k+1}=x_{k}+1$;
if $y_{k} \neq 1$, then $x_{k+1}=x_{k}+1, y_{k+1}=y_{k}-1$.
How many terms in the first 2011 terms of this sequence ... | <think>
Alright, let's try to figure out this problem. So, we have a sequence of rational numbers defined as a_k = x_k / y_k. The starting point is x₁ = y₁ = 1, so the first term is 1/1, which is 1. Then, there are rules for generating the next terms based on whether y_k is 1 or not.
Let me restate the rules to make ... |
olympiads | 123. How many right-angled triangles are there with sides of integer lengths, if one of the legs of these triangles is equal to 15? | <think>
Okay, so I need to find how many right-angled triangles with integer side lengths have one of the legs equal to 15. Hmm, right-angled triangles, so Pythagorean triples where one leg is 15. Let me think.
First, let me recall the Pythagorean theorem: for a right-angled triangle with legs a and b, and hypotenuse ... |
olympiads | 5. Find all values of $a$, for each of which the system
$$
\left\{\begin{array}{l}
|y|+|y-x| \leqslant a-|x-1| \\
(y-4)(y+3) \geqslant(4-x)(3+x)
\end{array}\right.
$$
has exactly two solutions. | <think>
Okay, so I need to find all values of a for which the given system of inequalities has exactly two solutions. Let me first try to understand the problem step by step.
The system is:
1. |y| + |y - x| ≤ a - |x - 1|
2. (y - 4)(y + 3) ≥ (4 - x)(3 + x)
I need to visualize these inequalities and find where their i... |
olympiads | 11. Given the function $f(x)=-x^{2}+x+m+2$, if the solution set of the inequality $f(x) \geqslant|x|$ contains exactly one integer, then the range of the real number $m$ is $\qquad$ . | <think>
Okay, let's tackle this problem step by step. The function given is f(x) = -x² + x + m + 2. We need to find the range of real number m such that the solution set of the inequality f(x) ≥ |x| contains exactly one integer. Hmm, okay. Let's parse this carefully.
First, the inequality is f(x) ≥ |x|. So, substituti... |
olympiads | 1. Calculate the sum
$$
S=\frac{1-i}{\sqrt{2}}+\left(\frac{1-i}{\sqrt{2}}\right)^{2}+\left(\frac{1-i}{\sqrt{2}}\right)^{3}+\ldots+\left(\frac{1-i}{\sqrt{2}}\right)^{1987}
$$
where \( i \) is the imaginary unit. | <think>
Okay, so I need to calculate this sum S, which is a geometric series starting from the first term (1 - i)/√2 up to the 1987th power. Hmm, geometric series, right? Let me recall the formula for the sum of a geometric series. If I have a common ratio r, then the sum of the first n terms is S_n = a1*(1 - r^n)/(1 -... |
olympiads | 14. (3 points) $A$ and $B$ bought the same number of sheets of letter paper. $A$ put 1 sheet of paper in each envelope, and after using all the envelopes, 40 sheets of paper were left; $B$ put 3 sheets of paper in each envelope, and after using all the sheets of paper, 40 envelopes were left. They both bought $\qquad$ ... | <think>
Okay, so here's the problem again. Both A and B bought the same number of sheets of letter paper. A used 1 sheet per envelope and after using all the envelopes, had 40 sheets left. B used 3 sheets per envelope and after using all the sheets, had 40 envelopes left. We need to find how many sheets they each bough... |
olympiads | ## Problem Statement
Find the cosine of the angle between vectors $\overrightarrow{A B}$ and $\overrightarrow{A C}$.
$A(-1 ; 2 ;-3), B(0 ; 1 ;-2), C(-3 ; 4 ;-5)$ | <think>
Okay, so I need to find the cosine of the angle between vectors AB and AC. The points given are A(-1, 2, -3), B(0, 1, -2), and C(-3, 4, -5). Hmm, let me recall how to approach this.
First, I think I need to find the vectors AB and AC. To get vector AB, I subtract the coordinates of point A from point B, right?... |
olympiads | 20. A barcode of the type shown in the two examples is composed of alternate strips of black and white, where the leftmost and rightmost strips are always black. Each strip (of either colour) has a width of 1 or 2 . The total width of the barcode is 12 . The barcodes are always read from left to right. How many distinc... | <think>
Alright, let's try to figure out how many distinct barcodes there are with the given conditions. The problem states that the barcode is made up of alternating black and white strips, starting and ending with black. Each strip can be 1 or 2 units wide, and the total width has to be 12. Hmm, okay. Let me break th... |
olympiads | The number of edges of a convex polyhedron is 99. What is the maximum number of edges that a plane, not passing through its vertices, can intersect?
# | <think>
Okay, so the problem is: A convex polyhedron has 99 edges. What's the maximum number of edges that a plane, not passing through its vertices, can intersect? Hmm, interesting. Let me try to figure this out step by step.
First, I remember that in a convex polyhedron, any plane that intersects the polyhedron will... |
olympiads | Problem 12.4. The point $K$ on the edge $A B$ of the cube $A B C D A_{1} B_{1} C_{1} D_{1}$ is such that the angle between the line $A_{1} B$ and the plane $\left(B_{1} C K\right)$ is equal to $60^{\circ}$. Find $\tan \alpha$, where $\alpha$ is the angle between the planes $\left(B_{1} C K\right)$ and $(A B C)$.
Oleg ... | <think>
Okay, let me try to figure out this geometry problem. So, we have a cube named ABCD A₁B₁C₁D₁. The point K is on the edge AB. The angle between the line A₁B and the plane (B₁CK) is 60 degrees. We need to find tan α, where α is the angle between the planes (B₁CK) and (ABC).
First, let me visualize the cube. Let... |
olympiads | 5. Determine the unknown number $x$ and explain.
| 9 | 22 | 13 | 16 |
| :---: | :---: | :---: | :---: |
| 144 | 176 | 143 | 192 |
| 16 | 8 | 11 | $x$ |
Tasks worth 10 points | <think>
Okay, let's see here. I need to figure out the value of x in this table. There are three rows: the first row has numbers 9, 22, 13, 16; the second row has 144, 176, 143, 192; and the third row has 16, 8, 11, x. My job is to find x. Hmm, let me try to find a pattern or relationship between the numbers in each co... |
olympiads |
N2. Find the maximum number of natural numbers $x_{1}, x_{2}, \ldots, x_{m}$ satisfying the conditions:
a) No $x_{i}-x_{j}, 1 \leq i<j \leq m$ is divisible by 11 ; and
b) The sum $x_{2} x_{3} \ldots x_{m}+x_{1} x_{3} \ldots x_{m}+\cdots+x_{1} x_{2} \ldots x_{m-1}$ is divisible by 11 .
| <think>
Okay, so I need to find the maximum number of natural numbers \( x_1, x_2, \ldots, x_m \) that satisfy two conditions:
a) No difference \( x_i - x_j \) for \( 1 \leq i < j \leq m \) is divisible by 11.
b) The sum \( S = x_2 x_3 \ldots x_m + x_1 x_3 \ldots x_m + \cdots + x_1 x_2 \ldots x_{m-1} \) is divisible ... |
olympiads | 1. (16 points) The dividend is five times larger than the divisor, and the divisor is four times larger than the quotient. Find the dividend.
# | <think>
Okay, so I need to solve this problem where the dividend is five times larger than the divisor, and the divisor is four times larger than the quotient. Hmm, let's break this down step by step.
First, let's recall some basic division terminology. In a division equation: Dividend ÷ Divisor = Quotient. So, if we... |
aops_forum | Find the maximal value of the following expression, if $a,b,c$ are nonnegative and $a+b+c=1$.
\[ \frac{1}{a^2 -4a+9} + \frac {1}{b^2 -4b+9} + \frac{1}{c^2 -4c+9} \] | <think>
Alright, so I need to find the maximum value of the expression \(\frac{1}{a^2 -4a +9} + \frac{1}{b^2 -4b +9} + \frac{1}{c^2 -4c +9}\) where \(a\), \(b\), and \(c\) are nonnegative real numbers that add up to 1. Hmm, okay. Let me think step by step.
First, since the variables are nonnegative and their sum is 1,... |
olympiads | 8. Given the sequence $\left\{a_{n}\right\}$ with the first term being 2, and satisfying $6 S_{n}=3 a_{n+1}+4^{n}-1$. Then the maximum value of $S_{n}$ is | <think>
Okay, let's try to figure out this problem. We have a sequence {a_n} where the first term is 2. The condition given is 6 S_n = 3 a_{n+1} + 4^n - 1. We need to find the maximum value of S_n, which is the sum of the first n terms of the sequence.
First, let me recall that S_n is the sum of the first n terms, so ... |
olympiads | 5. Find the number of lattice points in the plane region (excluding the boundary) bounded by the parabola $x^{2}=2 y$, the $x$-axis, and the line $x=21$.
| <think>
Okay, let me try to figure out how to solve this problem. So, the question is asking for the number of lattice points inside the region bounded by the parabola \(x^2 = 2y\), the x-axis, and the line \(x = 21\). And we need to exclude the boundary points. Hmm, lattice points are points with integer coordinates, ... |
olympiads_ref | 11. (IND $\left.3^{\prime}\right)^{\mathrm{IMO}}$ Given a circle with two chords $A B, C D$ that meet at $E$, let $M$ be a point of chord $A B$ other than $E$. Draw the circle through $D, E$, and $M$. The tangent line to the circle $D E M$ at $E$ meets the lines $B C, A C$ at $F, G$, respectively. Given $\frac{A M}{A B... | <think>
Okay, let's try to tackle this geometry problem. Hmm, so we have a circle with two chords AB and CD intersecting at point E. Then there's a point M on chord AB (not at E). We need to draw a circle through D, E, and M. The tangent to this circle at E intersects lines BC and AC at points F and G respectively. We’... |
olympiads | 35. The fish soup prepared by the tourists on the seashore after a successful fishing trip was good, but due to lack of culinary experience, they put too little salt in it, so they had to add salt at the table. The next time they made the same amount of fish soup, they put in twice as much salt as the first time, but e... | <think>
Okay, let's see. The problem is about tourists making fish soup twice. Both times, they didn't put enough salt the first time, so they had to add salt at the table. The second time, they used twice as much salt as the first time, but still had to add salt, though only half as much as the first time. We need to ... |
olympiads | 6.23 The sum of the first three terms of an increasing arithmetic progression is 21. If 1 is subtracted from the first two terms of this progression, and 2 is added to the third term, the resulting three numbers will form a geometric progression. Find the sum of the first eight terms of the geometric progression. | <think>
Okay, let's see. I have this problem about an arithmetic progression and a geometric progression. Hmm, let me read it again to make sure I understand.
The problem says: The sum of the first three terms of an increasing arithmetic progression is 21. If 1 is subtracted from the first two terms of this progressio... |
olympiads | 10. (10 points) As shown in the figure, in triangle $ABC$, $AD=2BD$, $AD=EC$, $BC=18$, the area of triangle $AFC$ is equal to the area of quadrilateral $DBEF$. What is the length of $AB$? | <think>
Alright, so I have this geometry problem here. Let me try to visualize it first. The triangle is ABC. There's a point D on AB such that AD is twice BD, right? So AD = 2BD. Then, AD is also equal to EC. Hmm, EC must be a segment on BC or maybe somewhere else? Wait, the problem says "as shown in the figure," but ... |
aops_forum | Sally is thinking of a positive four-digit integer. When she divides it by any one-digit integer greater than $1$, the remainder is $1$. How many possible values are there for Sally's four-digit number? | <think>
Okay, so Sally has a four-digit number, right? Let's call her number N. The problem says that when she divides N by any one-digit integer greater than 1, the remainder is 1. Hmm, so the one-digit integers greater than 1 are 2, 3, 4, 5, 6, 7, 8, 9. That means N leaves a remainder of 1 when divided by each of the... |
olympiads | 18. (12 points) Given vectors $\boldsymbol{m}=(\sin A, \cos A)$, $\boldsymbol{n}=(\cos B, \sin B)$, $\boldsymbol{m} \cdot \boldsymbol{n}=\sin 2 C$, and $\angle A$, $\angle B$, $\angle C$ are the angles opposite to sides $a$, $b$, $c$ of $\triangle ABC$ respectively.
(1) Find the size of $\angle C$;
(2) If $\sin A$, $\s... | <think>
Okay, so I have this problem here with vectors and trigonometry related to a triangle. Let's take it step by step.
First, part (1) asks to find the size of angle C. The given information is vectors m = (sin A, cos A) and n = (cos B, sin B), and their dot product m · n equals sin 2C. Also, angles A, B, C are th... |
aops_forum | A circle $\omega$ and a point $P$ not lying on it are given. Let $ABC$ be an arbitrary equilateral triangle inscribed into $\omega$ and $A', B', C'$ be the projections of $P$ to $BC, CA, AB$. Find the locus of centroids of triangles $A' B'C'$. | <think>
Okay, so there's this problem about a circle ω and a point P not on it. Then, for any equilateral triangle ABC inscribed in ω, we need to find the locus of the centroids of triangles A'B'C', where A', B', C' are the projections (orthogonal projections, I assume) of P onto the sides BC, CA, AB respectively. Hmm,... |
olympiads | 1. Solve the equation $x^{\log _{5}(0.008 x)}=\frac{125}{x^{5}}$. | <think>
Alright, so I have this equation to solve: \( x^{\log _{5}(0.008 x)} = \frac{125}{x^{5}} \). Hmm, looks a bit intimidating at first glance, but let's take it step by step.
First, I need to recall some logarithmic and exponential properties. Maybe taking logarithms on both sides could simplify things? That usu... |
olympiads | 2. Given real numbers $a, b, x, y$ satisfy the system of equations:
$$
\left\{\begin{array}{l}
a x+b y=3, \\
a x^{2}+b y^{2}=7, \\
a x^{3}+b y^{3}=16, \\
a x^{4}+b y^{4}=42 .
\end{array}\right.
$$
Find the value of $a x^{5}+b y^{5}$. | <think>
Okay, so I need to find the value of \( a x^5 + b y^5 \) given the system of equations:
\[
\begin{cases}
a x + b y = 3, \\
a x^2 + b y^2 = 7, \\
a x^3 + b y^3 = 16, \\
a x^4 + b y^4 = 42.
\end{cases}
\]
Hmm, this seems like a sequence where each term is built from the previous ones. Maybe there's a recurrence... |
olympiads | 1. Which whole numbers from 1 to 80000 (inclusive) are there more of, and by how many: those containing only even digits or those containing only odd digits? | <think>
Okay, so I need to figure out whether there are more numbers from 1 to 80,000 that have only even digits or only odd digits, and by how much. Hmm, let's start by understanding the problem.
First, the numbers range from 1 to 80,000 inclusive. That's 80,000 numbers. But we have to consider two categories: number... |
cn_contest | 4. From the numbers $1,2,3,4,5$, select several (at least 2) without repetition and find their sum, the number of different sums that can be obtained is ( ) .
(A) 26
(B) 17
(C) 13
(D) 12 | <think>
Okay, let's see. I need to figure out how many different sums can be obtained by selecting several numbers from 1, 2, 3, 4, 5, with at least 2 numbers each time and no repetition. The options are given, so maybe after calculating I can check which one it matches. Hmm, let's start step by step.
First, the numbe... |
olympiads | [ Properties of medians. The centroid of a triangle. ] [ The ratio of the areas of triangles with a common angle ]
Given triangle $A B C$, the area of which is 2. On the medians $A K, B L$ and $C N$ of triangle $A B C$, points $P, Q$ and $R$ are taken respectively such that $A P: P K=1, B Q: Q L=1: 2, C R: R N=5: 4$. ... | <think>
Okay, let's see. The problem is about finding the area of triangle PQR, where P, Q, R are points on the medians of triangle ABC. The original triangle ABC has an area of 2. The ratios given are AP:PK = 1:1, BQ:QL = 1:2, and CR:RN = 5:4. Hmm, medians and centroids... I remember that the centroid divides each med... |
olympiads | 1. Let $\triangle A B C$ have interior angles $\angle A, \angle B, \angle C$ with opposite sides of lengths $a, b, c$ respectively. If $c=1$, and the area of $\triangle A B C$ is equal to $\frac{a^{2}+b^{2}-1}{4}$, then the maximum value of the area of $\triangle A B C$ is $\qquad$ | <think>
Okay, so I have this problem here about triangle ABC. Let me read it again. It says that in triangle ABC, the sides opposite angles A, B, C are a, b, c respectively. Given that c = 1, and the area of the triangle is equal to (a² + b² - 1)/4. I need to find the maximum value of the area. Hmm, interesting. Let me... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.