problem stringlengths 37 4.98k | answer stringlengths 1 141 | subject stringclasses 8
values | level stringclasses 7
values |
|---|---|---|---|
67. There are three steel pipes, with lengths of 240 cm, 200 cm, and 480 cm, respectively. If they are cut into segments of the same length, the longest each segment can be is $\qquad$ cm. | 40 | Number Theory | olympiads |
1. Find $x$ and $y$ that satisfy the following equation:
$$
(x-y)^{2}+(y-2 \sqrt{x}+2)^{2}=\frac{1}{2}
$$ | 1,\frac{1}{2} | Algebra | olympiads |
Problem 3. Each student's mentor gave them 2 apples, and 19 apples remained in the basket. How many students and how many apples are there, if their total sum is 100? | 27 | Algebra | olympiads |
13. Sarah and Hagar play a game of darts. Let $O_{0}$ be a circle of radius 1. On the $n$th turn, the player whose turn it is throws a dart and hits a point $p_{n}$ randomly selected from the points of $O_{n-1}$. The player then draws the largest circle that is centered at $p_{n}$ and contained in $O_{n-1}$, and calls ... | \frac{6\pi}{7} | Geometry | olympiads |
3. As shown in Figure 1, in $\triangle A B C$, $A B=A C$, points $D, E$ are on sides $A B, A C$ respectively, $D M$ bisects $\angle B D E$, and $E N$ bisects $\angle D E C$. If $\angle D M N=$ $110^{\circ}$, then $\angle D E A=$ ( ).
(A) $40^{\circ}$
(B) $50^{\circ}$
(C) $60^{\circ}$
(D) $70^{\circ}$ | A | Geometry | cn_contest |
2 Find all real solutions of the system of equations
$$
\left\{\begin{array}{l}
5\left(x+\frac{1}{x}\right)=12\left(y+\frac{1}{y}\right)=13\left(z+\frac{1}{z}\right), \\
x y+y z+z x=1
\end{array}\right.
$$
(Supplied by Hua-Wei Zhu) | (\frac{1}{5},\frac{2}{3},1)(-\frac{1}{5},-\frac{2}{3},-1) | Algebra | olympiads |
13.133. A swimming pool has three pipes of different cross-sections for draining water using a uniformly pumping pump. The first and second pipes together, with the third pipe closed, empty a full pool in $a$ minutes; the first and third pipes together, with the second pipe closed, empty a full pool in $b$ minutes; and... | \frac{2}{+-},\frac{2}{+-},\frac{2}{+-} | Algebra | olympiads |
6-161 Let $R$ be the set of all real numbers. Try to find all functions $f: R \rightarrow R$ such that for all $x$ and $y$ in $R$, we have
$$
f\left(x^{2}+f(y)\right)=y+(f(x))^{2} .
$$ | f(x)=x | Algebra | olympiads |
3.339. a) $\cos 36^{\circ}=\frac{\sqrt{5}+1}{4} ;$ b) $\sin 18^{\circ}=\frac{\sqrt{5}-1}{4}$. | \cos36=\frac{\sqrt{5}+1}{4};\sin18=\frac{\sqrt{5}-1}{4} | Algebra | olympiads |
18. Brian chooses an integer, multiplies it by 4 then subtracts 30 . He then multiplies his answer by 2 and finally subtracts 10 . His answer is a two-digit number. What is the largest integer he could choose?
A 10
B 15
C 18
D 20
E 21 | 21 | Algebra | olympiads |
In Sweden, there is allegedly a very deep ground fissure or cave, into which if we drop a stone, we only hear the impact sound after $25 \mathrm{sec}$. How deep is the cave, if we also take into account the speed of sound? | 1867\mathrm{~} | Algebra | olympiads |
3. Given the quadratic function
$$
y=3 a x^{2}+2 b x-(a+b) \text {, }
$$
when $x=0$ and $x=1$, the value of $y$ is positive. Then, when $0<x<1$, the parabola intersects the $x$-axis at $\qquad$ points. | 2 | Algebra | cn_contest |
13. (GDR 1) Find whether among all quadrilaterals whose interiors lie inside a semicircle of radius \( r \) there exists one (or more) with maximal area. If so, determine their shape and area. | \frac{3 \sqrt{3} r^{2}}{4} | Geometry | olympiads_ref |
II. (25 points) Given the quadratic function
$$
y=x^{2}+b x-c
$$
the graph passes through three points
$$
P(1, a), Q(2,3 a)(a \geqslant 3), R\left(x_{0}, y_{0}\right) .
$$
If the centroid of $\triangle P Q R$ is on the $y$-axis, find the minimum perimeter of $\triangle P Q R$. | 4 \sqrt{2}+5 \sqrt{5}+\sqrt{37} | Algebra | cn_contest |
Example 4. A discrete random variable $X$ has the following distribution:
| $X$ | 3 | 4 | 5 | 6 | 7 |
| :--- | :--- | :--- | :--- | :--- | :--- |
| $P$ | $p_{1}$ | 0.15 | $p_{3}$ | 0.25 | 0.35 |
Find the probabilities $p_{1}=P(X=3)$ and $p_{3}=P(X=5)$, given that $p_{3}$ is 4 times $p_{1}$. | p_{1}=0.05;p_{3}=0.20 | Algebra | olympiads |
## Aufgabe 17/83
Gesucht sind alle Lösungen $(x ; y ; z)$ in natürlichen Zahlen $x ; y ; z$ des Gleichungssystems:
$$
\begin{aligned}
2 x^{2}-2 y^{2}-3 z+5949 & =0 \\
\lg ^{2} y^{2}+\lg y^{(x-1)(x-y)}+(x-y)^{2} & =0 \\
\lg y^{(y-x)}+x-y & =0
\end{aligned}
$$
| (1;1;1983) | Algebra | olympiads |
9. Given that $m, n$ are positive integers, where $n$ is odd, find the greatest common divisor of $2^{m}+1$ and $2^{n}-1$.
untranslated portion:
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
This part is a note and not part of the problem statement, so it is not translated. | 1 | Number Theory | olympiads |
I1.1 解方程 $\log _{5} a+\log _{3} a=\log _{5} a \cdot \log _{3} a$, 其中 $a>1$ 為實數。 | 15 | Algebra | olympiads |
Let's determine those five-digit square numbers, where the fourth digit - that is, the tens place value - is half as much as the sum of the other four digits. | 11664=108^{2},12996=114^{2},34596=186^{2},53361=231^{2} | Number Theory | olympiads |
5. A natural number is called a "good number" if it is exactly 2007 more than the sum of its digits. Then the sum of all good numbers is $\qquad$ . | 20145 | Number Theory | cn_contest |
Let's determine all solutions of the system of equations $x^{2}+y^{2}=x, 2 x y=y$.
---
Find all solutions of the system of equations $x^{2}+y^{2}=x, 2 x y=y$. | x_{1}=\frac{1}{2},y_{1}=\frac{1}{2};\quadx_{2}=0,y_{2}=0;\quadx_{3}=\frac{1}{2},y_{3}=-\frac{1}{2};\quadx_{4}=1,y_{4}=0 | Algebra | olympiads |
Example 4 Find the value: $\arctan \frac{1}{3}+\arcsin \frac{1}{\sqrt{26}}+\arccos \frac{7}{\sqrt{50}}+\operatorname{arccot} 8$. | \frac{\pi}{4} | Algebra | olympiads |
13) A massless rope passes over a frictionless pulley. Particles of mass $M$ and $M + m$ are suspended from the two different ends of the rope. If $m = 0$, the tension $T$ in the pulley rope is $Mg$. If instead the value m increases to infinity, the value of the tension does which of the following?
A) stays constant
B... | D | Calculus | aops_forum |
1. Add parentheses to the numerical expression $36+144: 9-3 \cdot 2$ so that its value is:
a) 84 ;
b) 14 . | 14 | Algebra | olympiads |
8. Let $A B C$ be a triangle with $A B=A C$ and $\angle B A C=20^{\circ}$. Let $D$ be the point on the side $A B$ such that $\angle B C D=70^{\circ}$. Let $E$ be the point on the side $A C$ such that $\angle C B E=60^{\circ}$. Determine the value of the angle $\angle C D E$.
Answer: $\angle C D E=20^{\circ}$. | 20 | Geometry | olympiads |
## Problem Statement
Based on the definition of the derivative, find $f^{\prime}(0)$:
$$
f(x)=\left\{\begin{array}{c}
\ln \left(1-\sin \left(x^{3} \sin \frac{1}{x}\right)\right), x \neq 0 \\
0, x=0
\end{array}\right.
$$ | 0 | Calculus | olympiads |
The integers greater than one are arranged in five columns as follows:
\[\begin{tabular}{c c c c c}\ & 2 & 3 & 4 & 5\\ 9 & 8 & 7 & 6 &\ \\ \ & 10 & 11 & 12 & 13\\ 17 & 16 & 15 & 14 &\ \\ \ & . & . & . & .\\ \end{tabular}\]
(Four consecutive integers appear in each row; in the first, third and other odd numbered rows, t... | B | Number Theory | amc_aime |
32. Given real numbers $a, b, x, y$ satisfy $a+b=x+y=2, a x+b y=5$, then $\left(a^{2}+b^{2}\right) x y+a b\left(x^{2}+y^{2}\right)=$ $\qquad$ | -5 | Algebra | olympiads |
Let $R$ be the region in the Cartesian plane of points $(x,y)$ satisfying $x\geq 0$, $y\geq 0$, and $x+y+\lfloor x\rfloor+\lfloor y\rfloor\leq 5$. Determine the area of $R$. | \frac{9}{2} | Geometry | aops_forum |
8. (10 points) Among the three given phrases “尽心尽力” (exerting oneself to the utmost), “力可拔山” (strength to move mountains), and “山穷水尽” (at the end of one's resources), each Chinese character represents a number between 1 and 8. The same character represents the same number, and different characters represent different n... | 7 | Logic and Puzzles | olympiads |
$4.77 \operatorname{tg} 9^{\circ}+\operatorname{tg} 15^{\circ}-\operatorname{tg} 27^{\circ}-\operatorname{ctg} 27^{\circ}+\operatorname{ctg} 9^{\circ}+\operatorname{ctg} 15^{\circ}=8$. | 8 | Algebra | olympiads |
Let's determine the first term and the common difference of an arithmetic progression, if the sum of the first $n$ terms of this progression is $\frac{n^{2}}{2}$ for all values of $n$. | a_1=\frac{1}{2},1 | Algebra | olympiads |
Let $ A \equal{} \{(a_1,\dots,a_8)|a_i\in\mathbb{N}$ , $ 1\leq a_i\leq i \plus{} 1$ for each $ i \equal{} 1,2\dots,8\}$.A subset $ X\subset A$ is called sparse if for each two distinct elements $ (a_1,\dots,a_8)$,$ (b_1,\dots,b_8)\in X$,there exist at least three indices $ i$,such that $ a_i\neq b_i$.
Find the maxi... | 7! | Combinatorics | aops_forum |
4. Find the quotient and remainder when $(110100111)_{2}$ is divided by $(11101)_{2}$. | (1110)_{2},(10001)_{2} | Number Theory | number_theory |
12. (13 points) In a game activity of a TV entertainment program, each participant needs to complete three tasks: $A$, $B$, and $C$. It is known that the probabilities for participant Jia to complete tasks $A$, $B$, and $C$ are $\frac{3}{4}$, $\frac{3}{4}$, and $\frac{2}{3}$, respectively. Each task is independent of t... | \frac{243}{128} a | Algebra | cn_contest |
[b]6.[/b] Let $f(x)$ be an arbitrary function, differentiable infinitely many times. Then the $n$th derivative of $f(e^{x})$ has the form
$\frac{d^{n}}{dx^{n}}f(e^{x})= \sum_{k=0}^{n} a_{kn}e^{kx}f^{(k)}(e^{x})$ ($n=0,1,2,\dots$).
From the coefficients $a_{kn}$ compose the sequence of polynomials
$P_{n}(x)= \sum_{k=... | F(t, x) = e^{x(e^t - 1)} | Calculus | aops_forum |
Problem 9.6. At a ball, 29 boys and 15 girls arrived. Some boys danced with some girls (no more than once in each pair). After the ball, each person told their parents how many times they danced. What is the maximum number of different numbers the children could have mentioned? | 29 | Combinatorics | olympiads |
9. Given real numbers $x, y$ satisfy $2^{x}+3^{y}=4^{x}+9^{y}$, try to find the range of $U=8^{x}+27^{y}$. | U\in(1,2] | Algebra | olympiads |
8) Near a water source, there is a cistern with a capacity greater than 30 hectoliters, initially empty. Only two calibrated containers, one of 15 liters and one of 21 liters, are available, with which it is possible to add and remove water from the cistern. Which of the following volumes of water cannot be placed exac... | 5 | Number Theory | olympiads |
A deck of playing cards is laid out on the table (for example, in a row). On top of each card, a card from another deck is placed. Some cards may have matched. Find:
a) the expected value of the number of matches;
b) the variance of the number of matches.
# | 1 | Combinatorics | olympiads |
Find all non-negative integers $x, y$ such that:
$$
2^{x}=y^{2}+y+1
$$ | 0 | Number Theory | olympiads |
In how many ways can the nine digits from 1 to 9 be placed in a $3 \times 3$ grid so that the sums of the rows and the sums of the columns are all equal? | 72 | Combinatorics | olympiads |
8.1. (12 points) In triangle $ABC$, the bisector $BL$ is drawn. Find the area of the triangle if it is known that $AL=2, BL=3\sqrt{10}$, and $CL=3$. | \frac{15\sqrt{15}}{4} | Geometry | olympiads |
Shapovalov A.V.
Thieves Hapok and Glazok are dividing a pile of 100 coins. Hapok grabs a handful of coins from the pile, and Glazok, looking at the handful, decides who of the two will get it. This continues until one of them receives nine handfuls, after which the other takes all the remaining coins (the division may... | 46 | Logic and Puzzles | olympiads |
Including the endpoints, how many points on the line segment joining $(-9,-2)$ and $(6,8)$ have coordinates that are both integers?
(A) 2
(B) 7
(C) 16
(D) 11
(E) 6 | 6 | Number Theory | olympiads |
4. Given in $\triangle A B D$, $A B=m, A D=n$, and an equilateral $\triangle B D C$ is constructed with $D B$ as a side (not overlapping with $\triangle A B D$). When the area of quadrilateral $A B C D$ is maximized, $\angle B A D$ equals ( ).
(A) $60^{\circ}$
(B) $90^{\circ}$
(C) $120^{\circ}$
(D) $150^{\circ}$ | D | Geometry | cn_contest |
Find all functions $f : R \to R$ such that $f(x^2)-f(y^2) \le (f(x)+y) (x-f(y))$ for all $x, y \in R$. | f(x) = x \text{ and } f(x) = -x, \forall x \in \mathbb{R} | Inequalities | aops_forum |
6. Suppose that Ethan has four red chips and two white chips. He selects three chips at random and places them in Urn 1, while the remaining chips are placed in Urn 2. He then lets his brother Josh draw one chip from each urn at random. What is the probabiity that the chips drawn by Josh are both red? | \frac{2}{5} | Combinatorics | olympiads |
203. Find the condition for the compatibility of the equations:
$$
a_{1} x+b_{1} y=c_{1}, \quad a_{2} x+b_{2} y=c_{2}, \quad a_{3} x+b_{3} y=c_{3}
$$ | a_{1}(b_{2}c_{3}-b_{3}c_{2})+a_{2}(b_{3}c_{1}-c_{3}b_{1})+a_{3}(b_{1}c_{2}-b_{2}c_{1})=0 | Algebra | olympiads |
3. Given that $AB$ is a chord of the circle $\odot O$ with radius 1, and the length of $AB$ is the positive root of the equation $x^{2}+x-1=0$. Then the degree of $\angle AOB$ is $\qquad$ . | 36^{\circ} | Geometry | cn_contest |
The last three digits of $1978^{n}$ and $1978^{m}$ are equal. Try to find positive integers $n, m$, such that $m+n$ takes the minimum value. Here $n>m \geqslant 1$.
| 106 | Number Theory | olympiads |
What is the sum of the reciprocals of the roots of the equation
$\frac{2003}{2004}x+1+\frac{1}{x}=0$?
$\mathrm{(A) \ } -\frac{2004}{2003}\qquad \mathrm{(B) \ } -1\qquad \mathrm{(C) \ } \frac{2003}{2004}\qquad \mathrm{(D) \ } 1\qquad \mathrm{(E) \ } \frac{2004}{2003}$ | -1 | Algebra | amc_aime |
6.21. Find the derivative $y_{x}^{\prime}(x)$ of the function $f(x, y(x))=$ $=\ln y(x)+\cot x^{2}-2 x=0$, given implicitly. | y_{x}^{\}(x)=y(x)\cdot(2x/(\sin^{2}x^{2})+2) | Calculus | olympiads |
1B. Lena bought a package of blue and red paper leaves, in which the ratio of the number of blue to the number of red leaves is $2: 7$. Every day, Lena uses 1 blue and 3 red leaves. One day, after taking 3 red leaves and using the last blue leaf in the package, 15 red leaves remained. How many leaves were there in the ... | 135 | Algebra | olympiads |
On a farm - where there are more horses than ducks - the number of cows is one third of the total number of horses and ducks. The sum of the heads and legs of ducks and horses is 100. How many cows are on the farm? | 8 | Algebra | olympiads |
Quadrilateral $A B C D$ has $\angle B C D=\angle D A B=90^{\circ}$. The perimeter of $A B C D$ is 224 and its area is 2205. One side of $A B C D$ has length 7. The remaining three sides have integer lengths. The sum of the squares of the side lengths of $A B C D$ is $S$. What is the integer formed by the rightmost two ... | 60 | Geometry | olympiads |
5. Given complex numbers $z_{1}, z_{2}, z_{3}$ such that $\frac{z_{1}}{z_{2}}$ is a pure imaginary number, and
$$
\left|z_{1}\right|=\left|z_{2}\right|=1,\left|z_{1}+z_{2}+z_{3}\right|=1 \text {. }
$$
Then the minimum value of $\left|z_{3}\right|$ is $\qquad$ . | \sqrt{2}-1 | Algebra | olympiads |
12.127. The development of the lateral surface of a cylinder is a rectangle, the diagonals of which intersect at an angle $\alpha$. The length of the diagonal is $d$. Find the lateral surface area of the cylinder. | \frac{1}{2}^{2}\sin\alpha | Geometry | olympiads |
Each of two boxes contains both black and white marbles, and the total number of marbles in the two boxes is $25.$ One marble is taken out of each box randomly. The [probability](https://artofproblemsolving.com/wiki/index.php/Probability) that both marbles are black is $27/50,$ and the probability that both marbles are... | 26 | Combinatorics | amc_aime |
3. The number n is the product of three different prime numbers. If we increase the two smaller ones by 1 and leave the largest one unchanged, the product of the three will increase by 915. Determine the number n. | 2013 | Number Theory | olympiads |
$17 \cdot 49$ In $\triangle A B C$, $B D$ is a median, $C F$ intersects $B D$ at $E$, and $B E=E D$, point $F$ is on $A B$, if $B F=5$, then $B A$ equals
(A) 10.
(B) 12.
(C) 15.
(D) 20.
(E) None of the above.
(10th American High School Mathematics Examination, 1959) | 15 | Geometry | olympiads |
Find all monotonic positive functions $f(x)$ defined on the positive reals such that $f(xy) f\left( \frac{f(y)}{x}\right) = 1$ for all $x, y$. | f(x) = \frac{1}{x} | Other | aops_forum |
1. A triangle has three sides of integer lengths, with the longest side being 11. The number of such triangles is ( ).
A. 32
B. 34
C. 36
D. 40 | 36 | Geometry | olympiads |
4. Inside triangle $ABC$, where $\angle C=70^{\circ}, \angle B=80^{\circ}$, a point $M$ is taken such that triangle $CMB$ is equilateral. Find the angles $MAB$ and $MAC$. | \angleMAB=20,\angleMAC=10 | Geometry | olympiads |
A hollow glass sphere with uniform wall thickness, which is empty inside, has an outer diameter of $16 \mathrm{~cm}$ and floats in water such that $\frac{3}{8}$ of its surface remains dry. What is the wall thickness if the specific gravity of the glass is $s=2.523$? | 0.8\mathrm{~} | Geometry | olympiads |
Let $f:\mathbb{R}\to\mathbb{R}$ be a bijective function. Does there always exist an infinite number of functions $g:\mathbb{R}\to\mathbb{R}$ such that $f(g(x))=g(f(x))$ for all $x\in\mathbb{R}$?
[i]Proposed by Daniel Liu[/i] | \text{YES} | Logic and Puzzles | aops_forum |
$4 \cdot 210$ On the same route, there are four people: the first person is in a car, the second person is on a motorcycle, the third person is on a moped, and the fourth person is on a bicycle. The speeds of the vehicles are constant. The person in the car catches up with the person on the moped at 12 o'clock, meets t... | 15:20 | Logic and Puzzles | olympiads |
3. Given
$$
\sin \alpha+\sqrt{3} \sin \beta=1, \cos \alpha+\sqrt{3} \cos \beta=\sqrt{3} \text {. }
$$
Then the value of $\cos (\alpha-\beta)$ is $\qquad$ . | 0 | Algebra | cn_contest |
9.160. $0.008^{x}+5^{1-3 x}+0.04^{\frac{3}{2}(x+1)}<30.04$. | x\in(-\frac{1}{3};\infty) | Inequalities | olympiads |
3. Let $f(x)=x^{2}+3 x+2$, and $S=\{0,1,2, \cdots, 100\}$. If $a \in S$ has $f(a)$ divisible by 6, then the number of such $a$ is $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 67 | Combinatorics | olympiads |
20. (6 points) Observe the calculation patterns of the following equations:
First row $1+2+3=6$
Second row $3+5+7=15$
Third row $5+8+11=24$
$\qquad$
The equation in the twelfth row is $\qquad$ | 23+35+47=105 | Algebra | olympiads |
17. Initially, there were some gold coins and empty boxes to hold the gold coins. If each box is filled with 9 gold coins, there will be two empty boxes left; if each box is filled with 6 gold coins, there will be 3 gold coins left. Therefore, there are ( ) gold coins.
(A) 9
(B) 27
(C) 45
(D) 63
(E) 81 | C | Number Theory | cn_contest |
8. (5 points) The sum of two numbers is 363. When the larger number is divided by the smaller number, the quotient is 16 with a remainder of 6. The larger of the two numbers is $\qquad$.
| 342 | Algebra | olympiads |
2. Let $k, n$ be natural numbers. For the statement “the number $(n-1)(n+1)$ is divisible by $k$,” Adam concluded that either the number $n-1$ or the number $n+1$ is divisible by $k$. Determine all natural numbers $k$ for which Adam's reasoning is correct for every natural $n$.
| k=4,\quadk=p^{},\quadk=2p^{} | Number Theory | olympiads |
6.9. \( f(x)=4 x \ln x+5 \frac{e^{x}}{\cos x} \). | \lnx+1-5\cdot\frac{e^x\cdot\cosx+e^x\cdot\sinx}{\cos^2x} | Calculus | olympiads |
15. Random Vectors. There are $n$ random vectors of the form $\left(y_{1}, y_{2}, y_{3}\right)$, where exactly one random coordinate is equal to 1, and the others are 0. They are added together. The resulting random vector is $\vec{a}$ with coordinates $\left(Y_{1}, Y_{2}, Y_{3}\right)$.
a) (from 9th grade. 2 points).... | \frac{2n+n^{2}}{3} | Algebra | olympiads |
G7.3-G7.4 Chicken eggs cost $\$ 0.50$ each, duck eggs cost $\$ 0.60$ each and goose eggs cost $\$ 0.90$ each. A man sold $x$ chicken eggs, $y$ duck eggs, $z$ goose eggs and received $\$ 60$. If $x, y, z$ are all positive numbers with $x+y+z=100$ and two of the values $x, y, z$ are equal,
G7.3 find $x$.
G7.4 find $y$. | 60 | Algebra | olympiads |
4. An isosceles triangle has legs of length $a$ and a base of length $b(a$ $>b)$, and another isosceles triangle has legs of length $b$ and a base of length $a$. If the vertex angles of the two isosceles triangles are supplementary, then $\frac{a^{2}+b^{2}}{a^{2}-b^{2}}=$ | \sqrt{3} | Geometry | cn_contest |
25. [13] In convex quadrilateral $A B C D$ with $A B=11$ and $C D=13$, there is a point $P$ for which $\triangle A D P$ and $\triangle B C P$ are congruent equilateral triangles. Compute the side length of these triangles. | 7 | Geometry | olympiads |
Derek and Julia are two of 64 players at a casual basketball tournament. The players split up into 8 teams of 8 players at random. Each team then randomly selects 2 captains among their players. What is the probability that both Derek and Julia are captains? | \frac{5}{84} | Combinatorics | aops_forum |
A function $f$ is defined for all real numbers and satisfies \[f(2 + x) = f(2 - x)\qquad\text{and}\qquad f(7 + x) = f(7 - x)\] for all real $x$. If $x = 0$ is a root of $f(x) = 0$, what is the least number of roots $f(x) = 0$ must have in the interval $-1000 \le x \le 1000$? | 401 | Logic and Puzzles | aops_forum |
9. (14 points) Let positive real numbers $x, y, z$ satisfy $xyz=1$. Try to find the maximum value of $f(x, y, z)=(1-yz+z)(1-xz+x)(1-xy+y)$ and the values of $x, y, z$ at that time. | 1 | Algebra | olympiads |
Quadratic trinomial $f(x)$ is allowed to be replaced by one of the trinomials $x^2f(1+\frac{1}{x})$ or $(x-1)^2f(\frac{1}{x-1})$. With the use of these operations, is it possible to go from $x^2+4x+3$ to $x^2+10x+9$? | \text{isn't possible to go from}\ x^2+4x+3\ \text{to}\ x^2+10x+9 | Logic and Puzzles | aops_forum |
32. [15] Pirate ships Somy and Lia are having a tough time. At the end of the year, they are both one pillage short of the minimum required for maintaining membership in the Pirate Guild, so they decide to pillage each other to bring their counts up. Somy by tradition only pillages $28 \cdot 3^{k}$ coins for integers $... | 2 | Number Theory | olympiads |
## Task B-1.6.
Determine the value of the real parameter $a$ for which each of the equations
$$
2 a-1=\frac{3-3 a}{x-1} \quad \text { and } \quad a^{2}(2 x-4)-1=a(4-5 x)-2 x
$$
has a unique solution and their solutions are equal. | -1 | Algebra | olympiads |
Determine all solutions $(x, y) \in \mathbf{N}^{2}$ of the equation:
$$
x(x+1)=4 y(y+1)
$$ | (x,y)=(0,0) | Number Theory | olympiads |
## Task 10/89
At least two of the four prime numbers $p_{1} ; p_{2} ; p_{3} ; p_{4}$ with $p_{1}<p_{2} ; p_{3} ; p_{4}$ are to be determined, for which $P=p_{1}^{p_{2}}+p_{3}^{2 p_{4}}$ can be a prime number. | 761 | Number Theory | olympiads |
An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example, 3, 5, 7, 9 is an arithmetic sequence with four terms.
A geometric sequence is a sequence in which each term after the first is obtained by multiplying the previous term by a con... | \frac{3}{2},\frac{1}{2},\frac{1+\sqrt{5}}{2} | Algebra | olympiads |
13.113. A material particle entered the pipe through an opening, and 6.8 minutes later, a second particle entered the same opening. Upon entering the pipe, each particle immediately began linear motion along the pipe: the first particle moved uniformly at a speed of 5 m/min, while the second particle covered 3 m in the... | 17 | Algebra | olympiads |
18.96 The perimeter of an equilateral triangle is $1989 \mathrm{~cm}$ larger than that of a square, the side length of the triangle is $d \mathrm{~cm}$ larger than that of the square, and the perimeter of the square is greater than 0. Then the number of positive integers that $d$ cannot take is
(A) 0.
(B) 9.
(C) 221.
(... | 663 | Geometry | olympiads |
A3. The number line between 0 and 2 is divided into 7 equal parts. Which number does point $A$ represent?
(A) $\frac{3}{14}$
(B) $\frac{6}{14}$
(C) $\frac{3}{7}$
(D) $\frac{6}{7}$
(E) 1 | \frac{6}{7} | Geometry | olympiads |
The circle constructed on side $A C$ of triangle $A B C$ as a diameter passes through the midpoint of side $B C$ and intersects side $A B$ at point $D$ such that $A D=\frac{1}{3} A B$. Find the area of triangle $A B C$, if $A C=1$.
# | \frac{\sqrt{2}}{3} | Geometry | olympiads |
## Task 4
From class 1a of the Ernst-Thälmann-Oberschule, 6 students participate in a Pioneer sports festival, from class 1b 7 students, and from class 1c 5 students.
How many students from all three first classes participate in the Pioneer sports festival? | 18 | Other | olympiads |
## 163. The Court.
In the language of the tribe inhabiting the island located not far from the island of zombies, the words "bal" and "da" mean "yes" and "no," but words that sound the same do not necessarily have the same meaning. Some of the island's inhabitants answer questions with "bal" and "da," while others, br... | notguilty | Logic and Puzzles | olympiads |
5-5. Along a straight alley, 400 lamps are placed at equal intervals, numbered in order from 1 to 400. At the same time, from different ends of the alley, Alla and Boris started walking towards each other at different constant speeds (Alla from the first lamp, Boris from the four hundredth). When Alla was at the 55th l... | 163 | Algebra | olympiads |
Problem 6. Master Li Si Qing makes fans. Each fan consists of 6 sectors, painted on both sides in red and blue (see fig.). Moreover, if one side of a sector is painted red, the opposite side is painted blue and vice versa. Any two fans made by the master differ in coloring (if one coloring can be transformed into anoth... | 36 | Combinatorics | olympiads |
Example 4 (2003 National High School Mathematics Competition) If $x \in\left[-\frac{5}{12} \pi,-\frac{\pi}{3}\right]$, then what is the maximum value of $y=\tan \left(x+\frac{2}{3} \pi\right)-$ $\tan \left(x+\frac{\pi}{6}\right)+\cos \left(x+\frac{\pi}{6}\right)$? | \frac{11}{6}\sqrt{3} | Algebra | olympiads |
11. In the Cartesian coordinate system, the ellipse $\Gamma: \frac{x^{2}}{4}+\frac{y^{2}}{3}=1$, point $P$ is inside the ellipse $\Gamma$ and moves along the line $y=x$. Points $K, L$ are on $\Gamma$, such that $\overrightarrow{P K}, \overrightarrow{P L}$ are in the positive directions of the $x$-axis and $y$-axis, res... | \frac{5\sqrt{3}}{3} | Geometry | olympiads |
82. The Cart. "Three men," said Crackham, "Atkins, Brown, and Cranby, decided to go on a short trip. They have a distance of 40 km to cover. Atkins walks at a speed of 1 km/h, Brown at 2 km/h, and Cranby, with his cart pulled by a donkey, travels at 8 km/h. For some time, Cranby carries Atkins, then drops him off to wa... | 10\frac{5}{41} | Logic and Puzzles | olympiads |
A triangle, two of whose sides are 3 and 4 , is inscribed in a circle. Find the minimal possible radius of the circle. | 2 | Geometry | olympiads |
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