Found problems: 85335
1978 Germany Team Selection Test, 4
Let $B$ be a set of $k$ sequences each having $n$ terms equal to $1$ or $-1$. The product of two such sequences $(a_1, a_2, \ldots , a_n)$ and $(b_1, b_2, \ldots , b_n)$ is defined as $(a_1b_1, a_2b_2, \ldots , a_nb_n)$. Prove that there exists a sequence $(c_1, c_2, \ldots , c_n)$ such that the intersection of $B$ and the set containing all sequences from $B$ multiplied by $(c_1, c_2, \ldots , c_n)$ contains at most $\frac{k^2}{2^n}$ sequences.
2010 Contests, 2
In an acute-angled triangle $ABC$, $M$ is the midpoint of side $BC$, and $D, E$ and $F$ the feet of the altitudes from $A, B$ and $C$, respectively. Let $H$ be the orthocenter of $\Delta ABC$, $S$ the midpoint of $AH$, and $G$ the intersection of $FE$ and $AH$. If $N$ is the intersection of the median $AM$ and the circumcircle of $\Delta BCH$, prove that $\angle HMA = \angle GNS$.
[i]Proposed by Marko Djikic[/i]
2002 Tournament Of Towns, 7
Some domino pieces are placed in a chain according to standard rules. In each move, we may remove a sub-chain with equal numbers at its ends, turn the whole sub-chain around, and put it back in the same place. Prove that for every two legal chains formed from the same pieces and having the same numbers at their ends, we can transform one to another in a finite sequence of moves.
1995 AMC 12/AHSME, 21
Two nonadjacent vertices of a rectangle are $(4,3)$ and $(-4,-3)$, and the coordinates of the other two vertices are integers. The number of such rectangles is
$\textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 5$
2014 Junior Balkan Team Selection Tests - Romania, 3
Let $ABC$ be an acute triangle and $D \in (BC) , E \in (AD)$ be mobile points. The circumcircle of triangle $CDE$ meets the median from $C$ of the triangle $ABC$ at $F$ Prove that the circumcenter of triangle $AEF$ lies on a fixed line.
1992 India National Olympiad, 3
Find the remainder when $19^{92}$ is divided by 92.
1995 Baltic Way, 2
Let $a$ and $k$ be positive integers such that $a^2+k$ divides $(a-1)a(a+1)$. Prove that $k\ge a$.
2015 HMMT Geometry, 4
Let $ABCD$ be a cyclic quadrilateral with $AB=3$, $BC=2$, $CD=2$, $DA=4$. Let lines perpendicular to $\overline{BC}$ from $B$ and $C$ meet $\overline{AD}$ at $B'$ and $C'$, respectively. Let lines perpendicular to $\overline{BC}$ from $A$ and $D$ meet $\overline{AD}$ at $A'$ and $D'$, respectively. Compute the ratio $\frac{[BCC'B']}{[DAA'D']}$, where $[\overline{\omega}]$ denotes the area of figure $\overline{\omega}$.
1995 Tournament Of Towns, (442) 2
Three grasshoppers $A$, $B$ and $C$ are placed on a line. Grasshopper $B$ sits at the midpoint between $A$ and $C$. Every second, one of the grasshoppers jumps over one of the others to the symmetrical point on the other side (if $X$ jumps over $Y$ to point $X'$, then $XY $= $YX'$). After several jumps it so happened that they returned to the three initial points (but maybe in different order). Prove that in this case $B$ returns to his initial middle position.
(AK Kovaldzhy)
1998 Switzerland Team Selection Test, 2
Find all nonnegative integer solutions $(x,y,z)$ of the equation
$\frac{1}{x+2}+\frac{1}{y+2}=\frac{1}{2} +\frac{1}{z+2}$
2016 Azerbaijan BMO TST, 2
There are $100$ students who praticipate at exam.Also there are $25$ members of jury.Each student is checked by one jury.Known that every student likes $10$ jury
$a)$ Prove that we can select $7$ jury such that any student likes at least one jury.
$b)$ Prove that we can make this every student will be checked by the jury that he likes and every jury will check at most $10$ students.
1965 Swedish Mathematical Competition, 1
The feet of the altitudes in the triangle $ABC$ are $A', B', C'$. Find the angles of $A'B'C'$ in terms of the angles $A, B, C$. Show that the largest angle in $A'B'C'$ is at least as big as the largest angle in $ABC$. When is it equal?
2020 Switzerland - Final Round, 8
Let $u_1, u_2, \dots, u_{2019}$ be real numbers satisfying \[u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.\] Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that
\[
a b \leqslant-\frac{1}{2019}.
\]
2021 AMC 10 Spring, 10
An inverted cone with base radius $12 \text{ cm}$ and height $18 \text{ cm}$ is full of water. The water is poured into a tall cylinder whose horizontal base has a radius of $24 \text{ cm}$. What is the height in centimeters of the water in the cylinder?
$\textbf{(A) }1.5 \qquad \textbf{(B) }3 \qquad \textbf{(C) }4 \qquad \textbf{(D) }4.5 \qquad \textbf{(E) }6$
2017 Bundeswettbewerb Mathematik, 4
The sequence $a_0,a_1,a_2,\dots$ is recursively defined by \[ a_0 = 1 \quad \text{and} \quad a_n = a_{n-1} \cdot \left(4-\frac{2}{n} \right) \quad \text{for } n \geq 1. \] Prove for each integer $n \geq 1$:
(a) The number $a_n$ is a positive integer.
(b) Each prime $p$ with $n < p \leq 2n$ is a divisor of $a_n$.
(c) If $n$ is a prime, then $a_n-2$ is divisible by $n$.
2014 AMC 8, 16
The "Middle School Eight" basketball conference has $8$ teams. Every season, each team plays every other conference team twice (home and away), and each team also plays $4$ games against non-conference opponents. What is the total number of games in a season involving the "Middle School Eight" teams?
$\textbf{(A) }60\qquad\textbf{(B) }88\qquad\textbf{(C) }96\qquad\textbf{(D) }144\qquad \textbf{(E) }160$
2009 Mid-Michigan MO, 10-12
[b]p1.[/b] Compute the sum of sharp angles at all five nodes of the star below.
( [url=http://www.math.msu.edu/~mshapiro/NewOlympiad/Olymp2009/10_12_2009.pdf]figure missing[/url] )
[b]p2.[/b] Arrange the integers from $1$ to $15$ in a row so that the sum of any two consecutive numbers is a perfect square. In how many ways this can be done?
[b]p3.[/b] Prove that if $p$ and $q$ are prime numbers which are greater than $3$ then $p^2 -q^2$ is divisible by $ 24$.
[b]p4.[/b] A city in a country is called Large Northern if comparing to any other city of the country it is either larger or farther to the North (or both). Similarly, a city is called Small Southern. We know that in the country all cities are Large Northern city. Show that all the cities in this country are simultaneously Small Southern.
[b]p5.[/b] You have four tall and thin glasses of cylindrical form. Place on the flat table these four glasses in such a way that all distances between any pair of centers of the glasses' bottoms are equal.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2017 Purple Comet Problems, 14
Let a and b be positive integers such that $a + ab = 1443$ and $ab + b = 1444$. Find $10a + b$.
2024 AMC 12/AHSME, 10
Let $\alpha$ be the radian measure of the smallest angle in a $3{-}4{-}5$ right triangle. Let $\beta$ be the radian measure of the smallest angle in a $7{-}24{-}25$ right triangle. In terms of $\alpha$, what is $\beta$?
$
\textbf{(A) }\frac{\alpha}{3}\qquad
\textbf{(B) }\alpha - \frac{\pi}{8}\qquad
\textbf{(C) }\frac{\pi}{2} - 2\alpha \qquad
\textbf{(D) }\frac{\alpha}{2}\qquad
\textbf{(E) }\pi - 4\alpha\qquad
$
2005 iTest, 3
Carrie, Miranda, Charlotte, and Samantha are sitting at a table with $5$ numbered chairs (numbered $1$ through $5$). One chair is left open for Big, should he decide to join the four for lunch. In how many distinct ways can the four women occupy the table?
1966 IMO Longlists, 42
Given a finite sequence of integers $a_{1},$ $a_{2},$ $...,$ $a_{n}$ for $n\geq 2.$ Show that there exists a subsequence $a_{k_{1}},$ $a_{k_{2}},$ $...,$ $a_{k_{m}},$ where $1\leq k_{1}\leq k_{2}\leq...\leq k_{m}\leq n,$ such that the number $a_{k_{1}}^{2}+a_{k_{2}}^{2}+...+a_{k_{m}}^{2}$ is divisible by
$n.$
[b]Note by Darij:[/b] Of course, the $1\leq k_{1}\leq k_{2}\leq ...\leq k_{m}\leq n$ should be understood as $1\leq k_{1}<k_{2}<...<k_{m}\leq n;$ else, we could take $m=n$ and $k_{1}=k_{2}=...=k_{m},$ so that the number $a_{k_{1}}^{2}+a_{k_{2}}^{2}+...+a_{k_{m}}^{2}=n^{2}a_{k_{1}}^{2}$ will surely be divisible by $n.$
2021 CMIMC, 1.6
Alice and Bob each flip $20$ fair coins. Given that Alice flipped at least as many heads as Bob, what is the expected number of heads that Alice flipped?
[i]Proposed by Adam Bertelli[/i]
2017 European Mathematical Cup, 3
Let $ABC$ be a scalene triangle and let its incircle touch sides $BC$, $CA$ and $AB$ at points $D$, $E$ and
$F$ respectively. Let line $AD$ intersect this incircle at point $X$. Point $M$ is chosen on the line $FX$ so that the
quadrilateral $AFEM$ is cyclic. Let lines $AM$ and $DE$ intersect at point $L$ and let $Q$ be the midpoint of segment
$AE$. Point $T$ is given on the line $LQ$ such that the quadrilateral $ALDT$ is cyclic. Let $S$ be a point such that
the quadrilateral $TFSA$ is a parallelogram, and let $N$ be the second point of intersection of the circumcircle of
triangle $ASX$ and the line $TS$. Prove that the circumcircles of triangles $TAN$ and $LSA$ are tangent to each
other.
Kvant 2023, M2765
We have 101 coins and a two-pan scale. In one weighing, we can compare the weights of two coins. What is the smallest number of weighings required in order to decide whether there exist 51 coins which all have the same weight?
2008 German National Olympiad, 1
Find all real numbers $ x$ such that \[ \sqrt{x\plus{}1}\plus{}\sqrt{x\plus{}3} \equal{} \sqrt{2x\minus{}1}\plus{}\sqrt{2x\plus{}1}.\]