Found problems: 85335
TNO 2024 Junior, 6
A box contains 900 cards numbered from 100 to 999. Cards are drawn randomly, one at a time, without replacement, and the sum of their digits is recorded. What is the minimum number of cards that must be drawn to guarantee that at least three of these sums are the same?
2014 Swedish Mathematical Competition, 5
In next year's finals in Schools Mathematics competition, $20$ finalists will participate. The final exam contains six problems. Emil claims that regardless of results, there must be five contestants and two problems such that either all the five contestants solve both problems, or neither of them solve any of the two problems. Is he right?
2015 Switzerland Team Selection Test, 12
Given positive integers $m$ and $n$, prove that there is a positive integer $c$ such that the numbers $cm$ and $cn$ have the same number of occurrences of each non-zero digit when written in base ten.
PEN R Problems, 7
Show that the number $r(n)$ of representations of $n$ as a sum of two squares has $\pi$ as arithmetic mean, that is \[\lim_{n \to \infty}\frac{1}{n}\sum^{n}_{m=1}r(m) = \pi.\]
2012 May Olympiad, 5
There are $27$ boxes located in a row; each contains at least $12$ marbles. The allowed operation is transfer a ball from a box to its neighbor on the right, as long as said neighbor contains more pellets than the box from which the transfer will be made. We will say that a distribution initial of the balls is [i]happy [/i] if it is possible to achieve, by means of a succession of permitted operations, that all the balls are in the same box. Determine what is the smallest total number of marbles with the that you can have a happy initial layout.
2023 Greece Junior Math Olympiad, 4
Find all positive integers $a,b$ with $a>1$ such that, $b$ is a divisor of $a-1$ and $2a+1$ is a divisor of $5b-3$.
2008 Harvard-MIT Mathematics Tournament, 10
Let $ ABC$ be an equilateral triangle with side length 2, and let $ \Gamma$ be a circle with radius $ \frac {1}{2}$ centered at the center of the equilateral triangle. Determine the length of the shortest path that starts somewhere on $ \Gamma$, visits all three sides of $ ABC$, and ends somewhere on $ \Gamma$ (not necessarily at the starting point). Express your answer in the form of $ \sqrt p \minus{} q$, where $ p$ and $ q$ are rational numbers written as reduced fractions.
2001 Brazil Team Selection Test, Problem 1
given that p,q are two polynomials such that each one has at least one root and \[p(1+x+q(x)^2)=q(1+x+p(x)^2)\] then prove that p=q
2018 JBMO Shortlist, G3
Let $\triangle ABC$ and $A'$,$B'$,$C'$ the symmetrics of vertex over opposite sides.The intersection of the circumcircles of $\triangle ABB'$ and $\triangle ACC'$ is $A_1$.$B_1$ and $C_1$ are defined similarly.Prove that lines $AA_1$,$BB_1$ and $CC_1$ are concurent.
Kyiv City MO Seniors 2003+ geometry, 2008.11.4
In the tetrahedron $SABC $ at the height $SH$ the following point $O$ is chosen, such that: $$\angle AOS + \alpha = \angle BOS + \beta = \angle COS + \gamma = 180^o, $$ where $\alpha, \beta, \gamma$ are dihedral angles at the edges $BC, AC, AB $, respectively, at this point $H$ lies inside the base $ABC$. Let ${{A} _ {1}}, \, {{B} _ {1}}, \, {{C} _ {1}} $be the points of intersection of lines and planes: ${{A} _ {1}} = AO \cap SBC $, ${{B} _ {1}} = BO \cap SAC $, ${{C} _ {1}} = CO \cap SBA$ . Prove that if the planes $ABC $ and ${{A} _ {1}} {{B} _ {1}} {{C} _ {1}} $ are parallel, then $SA = SB = SC $.
(Alexey Klurman)
2020 Flanders Math Olympiad, 1
Let $x$ be an angle between $0^o$ and $90^o$ so that
$$\frac{\sin^4 x}{9}+\frac{\cos^4 x}{16 }=\frac{1}{25} .$$
Then what is $\tan x$?
2021 USEMO, 1
Let $n$ be a fixed positive integer and consider an $n\times n$ grid of real numbers. Determine the greatest possible number of cells $c$ in the grid such that the entry in $c$ is both strictly greater than the average of $c$'s column and strictly less than the average of $c$'s row.
[i]Proposed by Holden Mui[/i]
2010 Bundeswettbewerb Mathematik, 4
Find all numbers that can be expressed in exactly $2010$ different ways as the sum of powers of two with non-negative exponents, each power appearing as a summand at most three times. A sum can also be made from just one summand.
2018 IMO Shortlist, A7
Find the maximal value of
\[S = \sqrt[3]{\frac{a}{b+7}} + \sqrt[3]{\frac{b}{c+7}} + \sqrt[3]{\frac{c}{d+7}} + \sqrt[3]{\frac{d}{a+7}},\]
where $a$, $b$, $c$, $d$ are nonnegative real numbers which satisfy $a+b+c+d = 100$.
[i]Proposed by Evan Chen, Taiwan[/i]
1980 Tournament Of Towns, (006) 3
We are given $30$ non-zero vectors in $3$ dimensional space.
Prove that among these there are two such that the angle between them is less than $45^o$.
2003 BAMO, 4
An integer $n > 1$ has the following property: for every (positive) divisor $d$ of $n, d + 1$ is a divisor of $n + 1$.
Prove that $n$ is prime.
2006 Korea Junior Math Olympiad, 8
De ne the set $F$ as the following: $F = \{(a_1,a_2,... , a_{2006}) : \forall i = 1, 2,..., 2006, a_i \in \{-1,1\}\}$
Prove that there exists a subset of $F$, called $S$ which satis es the following:
$|S| = 2006$
and for all $(a_1,a_2,... , a_{2006})\in F$ there exists $(b_1,b_2,... , b_{2006}) \in S$, such that $\Sigma_{i=1} ^{2006}a_ib_i = 0$.
2000 Italy TST, 3
Given positive numbers $a_1$ and $b_1$, consider the sequences defined by
\[a_{n+1}=a_n+\frac{1}{b_n},\quad b_{n+1}=b_n+\frac{1}{a_n}\quad (n \ge 1)\]
Prove that $a_{25}+b_{25} \geq 10\sqrt{2}$.
2017 HMNT, 5
Ashwin the frog is traveling on the $xy$-plane in a series of $2^{2017} -1$ steps, starting at the origin. At the $n^{th}$ step, if $n$ is odd, then Ashwin jumps one unit to the right. If $n$ is even, then Ashwin jumps $m$ units up, where $m$ is the greatest integer such that $2^m$ divides $n$. If Ashwin begins at the origin, what is the area of the polygon bounded by Ashwin’s path, the line $x = 2^{2016}$, and the $x$-axis?
2005 AMC 8, 20
Alice and Bob play a game involving a circle whose circumference is divided by 12 equally-spaced points. The points are numbered clockwise, from 1 to 12. Both start on point 12. Alice moves clockwise and Bob, counterclockwise.
In a turn of the game, Alice moves 5 points clockwise and Bob moves 9 points counterclockwise. The game ends when they stop on the same point. How many turns will this take?
$ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 24 $
1968 IMO Shortlist, 22
Find all natural numbers $n$ the product of whose decimal digits is $n^2-10n-22$.
2018 MOAA, 1
In $\vartriangle ABC$, $AB = 3$, $BC = 5$, and $CA = 6$. Points $D$ and $E$ are chosen such that $ACDE$ is a square which does not overlap with $\vartriangle ABC$. The length of $BD$ can be expressed in the form $\sqrt{m + n\sqrt{p}}$, where $m$, $n$, and $p$ are positive integers and $p$ is not divisible by the square of a prime. Determine the value of $m + n + p$.
2021 AMC 12/AHSME Spring, 6
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?
$\textbf{(A) }1.5 \qquad \textbf{(B) }3 \qquad \textbf{(C) }4 \qquad \textbf{(D) }4.5 \qquad \textbf{(E) }6$
1980 IMO Longlists, 10
Two circles $C_{1}$ and $C_{2}$ are (externally or internally) tangent at a point $P$. The straight line $D$ is tangent at $A$ to one of the circles and cuts the other circle at the points $B$ and $C$. Prove that the straight line $PA$ is an interior or exterior bisector of the angle $\angle BPC$.
2024 Harvard-MIT Mathematics Tournament, 3
Let $ABC$ be a scalene triangle and $M$ be the midpoint of $BC$. Let $X$ be the point such that $CX \parallel AB$ and $\angle AMX = 90^{\circ}.$ Prove that $AM$ bisects $\angle BAX$.