Found problems: 85335
2006 China Western Mathematical Olympiad, 4
Given a positive integer $ n\geq 2$, let $ B_{1}$, $ B_{2}$, ..., $ B_{n}$ denote $ n$ subsets of a set $ X$ such that each $ B_{i}$ contains exactly two elements. Find the minimum value of $ \left|X\right|$ such that for any such choice of subsets $ B_{1}$, $ B_{2}$, ..., $ B_{n}$, there exists a subset $ Y$ of $ X$ such that:
(1) $ \left|Y\right| \equal{} n$;
(2) $ \left|Y \cap B_{i}\right|\leq 1$ for every $ i\in\left\{1,2,...,n\right\}$.
2009 Today's Calculation Of Integral, 430
For a natural number $ n$, let $ a_n\equal{}\int_0^{\frac{\pi}{4}} (\tan x)^{2n}dx$.
Answer the following questions.
(1) Find $ a_1$.
(2) Express $ a_{n\plus{}1}$ in terms of $ a_n$.
(3) Find $ \lim_{n\to\infty} a_n$.
(4) Find $ \lim_{n\to\infty} \sum_{k\equal{}1}^n \frac{(\minus{}1)^{k\plus{}1}}{2k\minus{}1}$.
1970 IMO Longlists, 25
A real function $f$ is defined for $0\le x\le 1$, with its first derivative $f'$ defined for $0\le x\le 1$ and its second derivative $f''$ defined for $0<x<1$. Prove that if $f(0)=f'(0)=f'(1)=f(1)-1 =0$, then there exists a number $0<y<1$ such that $|f''(y)|\ge 4$.
2004 China Girls Math Olympiad, 2
Let $ a, b, c$ be positive reals. Find the smallest value of
\[ \frac {a \plus{} 3c}{a \plus{} 2b \plus{} c} \plus{} \frac {4b}{a \plus{} b \plus{} 2c} \minus{} \frac {8c}{a \plus{} b \plus{} 3c}.
\]
2016 China Girls Math Olympiad, 3
Let $m$ and $n$ are relatively prime integers and $m>1,n>1$. Show that:There are positive integers $a,b,c$ such that $m^a=1+n^bc$ , and $n$ and $c$ are relatively prime.
LMT Theme Rounds, 15
A round robin tournament is held with $2016$ participants. Each player plays each other player once and exactly one game results in a tie. Let $W$ be the sum of the squares of each team's win total and let $L$ be the sum of the squares of each team's loss total. Find the maximum possible value of $W-L$.
[i]Proposed by Matthew Weiss
2014 Sharygin Geometry Olympiad, 4
A square is inscribed into a triangle (one side of the triangle contains two vertices and each of two remaining sides contains one vertex. Prove that the incenter of the triangle lies inside the square.
2010 Flanders Math Olympiad, 2
A parallelogram with an angle of $60^o$ has $a$ as the longest side and a shortest side $b$. Let's take the perpendiculars down from the vertices of the obtuse angles to the longest diagonal, then it is divided into three equal parts. Determine the ratio $\frac{a}{b}$.
2024/2025 TOURNAMENT OF TOWNS, P5
A triangle is constructed on each side of a convex polygon in a manner that the third vertex of each triangle is the meet point of bisectors of the angles adjacent to this side. Prove that these triangles cover all the polygon.
Egor Bakaev
1977 Chisinau City MO, 134
Where is the number $35 351$ in the sequence $1, 8, 22, 43,...$?
2021 All-Russian Olympiad, 4
Given a natural number $n>4$ and $2n+4$ cards numbered with $1, 2, \dots, 2n+4$. On the card with number $m$ a real number $a_m$ is written such that $\lfloor a_{m}\rfloor=m$. Prove that it's possible to choose $4$ cards in such a way that the sum of the numbers on the first two cards differs from the sum of the numbers on the two remaining cards by less than $$\frac{1}{n-\sqrt{\frac{n}{2}}}$$.
2018 CMIMC CS, 3
You are given the existence of an unsorted sequence $a_1,\ldots, a_5$ of five distinct real numbers. The Erdos-Szekeres theorem states that there exists a subsequence of length $3$ which is either strictly increasing or strictly decreasing. You do not have access to the $a_i$, but you do have an oracle which, when given two indexes $1\leq i < j\leq 5$, will tell you whether $a_i < a_j$ or $a_i > a_j$. What is the minimum number of calls to the oracle needed in order to identify an ordered triple of integers $(r,s,t)$ such that $a_r,a_s,a_t$ is one such sequence?
1998 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 1
Let $ a \geq b$ be real number such that $ a^2\plus{}b^2 \equal{} 31$ and $ ab \equal{} 3.$ Then $ a\minus{}b$ equals
$ \text{(A)}\ 5 \qquad \text{(B)}\ \frac{31}{6} \qquad \text{(C)}\ 2 \sqrt{6} \qquad \text{(D)}\ \frac{5}{6} \sqrt{31} \qquad \text{(E)}\ \frac{5}{6} \sqrt{37}$
2023 Assara - South Russian Girl's MO, 6
Aunt Raya has $14$ wheels of cheese. She found out that out of any $6$ wheels, she could choose $4$ and put them on the scales so that the scales came into balance. Aunt Raya wants to give Daud Kazbekovich two of these $14$ wheels , and divide the rest equally (by weight) between Pavel and Kirill. Prove that she can make her wish come true.
the 16th XMO, 3
$m$ is an integer satisfying $m \ge 2024$ , $p$ is the smallest prime factor of $m$ , for an arithmetic sequence $\{a_n\}$ of positive numbers with the common difference $m$ satisfying : for any integer $1 \le i \le \frac{p}{2} $ , there doesn’t exist an integer $x , y \le \max \{a_1 , m\}$ such that $a_i=xy$ Try to proof that there exists a positive real number $c$ such that for any $ 1\le i \le j \le n $ , $gcd(a_i , a_j ) = c \times gcd(i , j)$
1998 Canada National Olympiad, 2
Find all real numbers $x$ such that: \[ x = \sqrt{ x - \frac{1}{x} } + \sqrt{ 1 - \frac{1}{x} } \]
1982 IMO Longlists, 49
Simplify
\[\sum_{k=0}^n \frac{(2n)!}{(k!)^2((n-k)!)^2}.\]
1977 USAMO, 4
Prove that if the opposite sides of a skew (non-planar) quadrilateral are congruent, then the line joining the midpoints of the two diagonals is perpendicular to these diagonals, and conversely, if the line joining the midpoints of the two diagonals of a skew quadrilateral is perpendicular to these diagonals, then the opposite sides of the quadrilateral are congruent.
2022 Bundeswettbewerb Mathematik, 3
In an acute triangle $ABC$ with $AC<BC$, lines $m_a$ and $m_b$ are the perpendicular bisectors of sides $BC$ and
$AC$, respectively. Further, let $M_c$ be the midpoint of side $AB$. The Median $CM_c$ intersects $m_a$ in point $S_a$ and $m_b$ in point $S_b$; the lines $AS_b$ und $BS_a$ intersect in point $K$.
Prove: $\angle ACM_c = \angle KCB$.
2023 China Second Round, 3
Find the smallest positive integer ${k}$ with the following properties $:{}{}{}{}{}$If each positive integer is arbitrarily colored red or blue${}{}{},$
there may be ${}{}{}{}9$ distinct red positive integers $x_1,x_2,\cdots ,x_9,$ satisfying
$$x_1+x_2+\cdots +x_8<x_9,$$
or there are $10{}{}{}{}{}{}$ distinct blue positive integers $y_1,y_2,\cdots ,y_{10}$ satisfiying
$${y_1+y_2+\cdots +y_9<y_{10}}.$$
2020 MIG, 16
Two $1$ inch by $1$ inch squares are cutout from opposite corners of a $7$ inch by $5$ inch piece of paper to form an octagon. What is the distance, in inches, between the two dotted points, both of which lie on corners of the octagon?
[asy]
size(120);
draw((0,1)--(0,5)--(6,5));
draw((1,0)--(7,0)--(7,4));
draw((0,1)--(0,0)--(1,0),dashed);
draw((6,5)--(7,5)--(7,4),dashed);
draw((0,1)--(1,1)--(1,0));
draw((6,5)--(6,4)--(7,4));
draw((1,1)--(6,4),dashed);
dot((1,1),linewidth(5));
dot((6,4),linewidth(5));
label("$?$",(1,1)--(6,4),N);
[/asy]
$\textbf{(A) }5\qquad\textbf{(B) }\sqrt{34}\qquad\textbf{(C) }5\sqrt2\qquad\textbf{(D) }8\qquad\textbf{(E) }\sqrt{74}$
2020 AMC 10, 1
What is the value of $$1-(-2)-3-(-4)-5-(-6)?$$
$\textbf{(A) } -20 \qquad\textbf{(B) } -3 \qquad\textbf{(C) } 3 \qquad\textbf{(D) } 5 \qquad\textbf{(E) } 21$
1984 All Soviet Union Mathematical Olympiad, 393
Given three circles $c_1,c_2,c_3$ with $r_1,r_2,r_3$ radiuses, $r_1 > r_2, r_1 > r_3$. Each lies outside of two others. The A point -- an intersection of the outer common tangents to $c_1$ and $c_2$ -- is outside $c_3$. The $B$ point -- an intersection of the outer common tangents to $c_1$ and $c_3$ -- is outside $c_2$. Two pairs of tangents -- from $A$ to $c_3$ and from $B$ to $c_2$ -- are drawn. Prove that the quadrangle, they make, is circumscribed around some circle and find its radius.
Indonesia MO Shortlist - geometry, g4
Given an isosceles triangle $ABC$ with $AB = AC$, suppose $D$ is the midpoint of the $AC$. The circumcircle of the $DBC$ triangle intersects the altitude from $A$ at point $E$ inside the triangle $ABC$, and the circumcircle of the triangle $AEB$ cuts the side $BD$ at point $F$. If $CF$ cuts $AE$ at point $G$, prove that $AE = EG$.
2022 Germany Team Selection Test, 3
A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes a colouring of the cells with finitely many colours. The rabbit then secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly moves to an adjacent cell that it has not visited before (two cells are adjacent if they share an edge). The hunter wins if after some finite time either:[list][*]the rabbit cannot move; or
[*]the hunter can determine the cell in which the rabbit started.[/list]Decide whether there exists a winning strategy for the hunter.
[i]Proposed by Aron Thomas[/i]