Found problems: 85335
2023 Mexico National Olympiad, 4
Let $n \ge 2$ be a positive integer. For every number from $1$ to $n$, there is a card with this number and which is either black or white.
A magician can repeatedly perform the following move: For any two tiles with different number and different colour, he can replace the card with the smaller number by one identical to the other card.
For instance, when $n=5$ and the initial configuration is $(1B, 2B, 3W, 4B,5B)$, the magician can choose $1B, 3W$ on the first move to obtain $(3W, 2B, 3W, 4B, 5B)$ and then $3W, 4B$ on the second move to obtain $(4B, 2B, 3W, 4B, 5B)$.
Determine in terms of $n$ all possible lengths of sequences of moves from any possible initial configuration to any configuration in which no more move is possible.
2007 Mathematics for Its Sake, 1
Consider a trapezium $ ABCD $ in which $ AB\parallel CD. $ Show that
$$ (AC^2+AB^2-BC^2)(BD^2-BC^2+CD^2) =(AC^2-AD^2+CD^2)(BD^2+AB^2-AD^2) . $$
2019 Dutch IMO TST, 4
There are $300$ participants to a mathematics competition. After the competition some of the contestants play some games of chess. Each two contestants play at most one game against each other. There are no three contestants, such that each of them plays against each other. Determine the maximum value of $n$ for which it is possible to satisfy the following conditions at the same time: each contestant plays at most $n$ games of chess, and for each $m$ with $1 \le m \le n$, there is a contestant playing exactly $m$ games of chess.
2016 Latvia National Olympiad, 2
An acute triangle $ABC$ ($AB > AC$) has circumcenter $O$, but $D$ is the midpoint of $BC$. Circle with diameter $AD$ intersects sides $AB$ and $AC$ in $E$ and $F$ respectively. On segment $EF$ pick a point $M$ so that $DM \parallel AO$. Prove that triangles $ABD$ and $FDM$ are similar.
2016 Dutch IMO TST, 1
Prove that for all positive reals $a, b,c$ we have: $a +\sqrt{ab}+ \sqrt[3]{abc}\le \frac43 (a + b + c)$
2009 Moldova National Olympiad, 10.4
Let the isosceles triangle $ABC$ with $| AB | = | AC |$. The point $M$ is the midpoint of the base $[BC]$, the point $N$ is the orthogonal projection of the point $M$ on the line $AC$, and the point $P$ is located on the segment $(MC)$ such that $| MP | = | P C | \sin^2 C$. Prove that the lines $AP$ and $BN$ are perpendicular.
1990 Greece National Olympiad, 3
For which $n$, $ n \in \mathbb{N}$ is the number $1^n+2^n+3^n$ divisible by $7$?
2008 Oral Moscow Geometry Olympiad, 3
Given a quadrilateral $ABCD$. $A ', B', C'$ and $D'$ are the midpoints of the sides $BC, CB, BA$ and $AB$, respectively. It is known that $AA'= CC'$, $BB'= DD'$. Is it true that $ABCD$ is a parallelogram?
(M. Volchkevich)
2024 AMC 10, 21
The numbers, in order, of each row and the numbers, in order, of each column of a $5 \times 5$ array of integers form an arithmetic progression of length $5{.}$ The numbers in positions $(5, 5), \,(2,4),\,(4,3),$ and $(3, 1)$ are $0, 48, 16,$ and $12{,}$ respectively. What number is in position $(1, 2)?$
\[ \begin{bmatrix} . & ? &.&.&. \\ .&.&.&48&.\\ 12&.&.&.&.\\ .&.&16&.&.\\ .&.&.&.&0\end{bmatrix}\]
$\textbf{(A) } 19 \qquad \textbf{(B) } 24 \qquad \textbf{(C) } 29 \qquad \textbf{(D) } 34 \qquad \textbf{(E) } 39$
2023 Middle European Mathematical Olympiad, 6
Let $ABC$ be an acute triangle with $AB < AC$. Let $J$ be the center of the $A$-excircle of $ABC$. Let $D$ be the projection of $J$ on line $BC$. The internal bisectors of angles $BDJ$ and $JDC$ intersectlines $BJ$ and $JC$ at $X$ and $Y$, respectively. Segments $XY$ and $JD$ intersect at $P$. Let $Q$ be the projection of $A$ on line $BC$. Prove that the internal angle bisector of $QAP$ is perpendicular to line $XY$.
[i]Proposed by Dominik Burek, Poland[/i]
2024 Kosovo EGMO Team Selection Test, P1
There are two piles of stones with $1012$ stones each. Ann and Ben play a game. In every move, a player removes two stones from one of the piles and adds one to the other pile. Ann goes first. The first player to remove the last stone in one of the piles wins the game. Which player has a winning strategy and why?
2000 Tuymaada Olympiad, 1
Can the plane be coloured in 2000 colours so that any nondegenerate circle contains points of all 2000 colors?
2015 India PRMO, 18
$18.$ A subset $B$ of the set of first $100$ positive integers has the property that no two elements of $B$ sum to $125.$ What is the maximum possible number of elements in $B ?$
2011 Stars Of Mathematics, 1
Let $ABC$ be an acute-angled triangle with $AB \neq BC$, $M$ the midpoint of $AC$, $N$ the point where the median $BM$ meets again the circumcircle of $\triangle ABC$, $H$ the orthocentre of $\triangle ABC$, $D$ the point on the circumcircle for which $\angle BDH = 90^{\circ}$, and $K$ the point that makes $ANCK$ a parallelogram.
Prove the lines $AC$, $KH$, $BD$ are concurrent.
(I. Nagel)
2016 CCA Math Bonanza, L3.3
Triangle $ABC$ has side length $AB = 5$, $BC = 12$, and $CA = 13$. Circle $\Gamma$ is centered at point $X$ exterior to triangle $ABC$ and passes through points $A$ and $C$. Let $D$ be the second intersection of $\Gamma$ with segment $\overline{BC}$. If $\angle BDA = \angle CAB$, the radius of $\Gamma$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]2016 CCA Math Bonanza Lightning #3.3[/i]
2015 Purple Comet Problems, 23
Larry and Diane start $100$ miles apart along a straight road. Starting at the same time, Larry and Diane
drive their cars toward each other. Diane drives at a constant rate of 30 miles per hour. To make it
interesting, at the beginning of each 10 mile stretch, if the two drivers have not met, Larry flips a fair coin.
If the coin comes up heads, Larry drives the next 10 miles at 20 miles per hour. If the coin comes up tails,
Larry drives the next 10 miles at 60 miles per hour. Larry and Diane stop driving when they meet. The expected number of times that Larry flips the coin is $\frac{m}{n}$ , where m and n are relatively prime positive integers. Find $m + n.$
2006 Czech-Polish-Slovak Match, 2
There are $n$ children around a round table. Erika is the oldest among them and she has $n$ candies, while no other child has any candy. Erika decided to distribute the candies according to the following rules. In every round, she chooses a child with at least two candies and the chosen child sends a candy to each of his/her two neighbors. (So in the first round Erika must choose herself). For which $n \ge 3$ is it possible to end the distribution after a finite number of rounds with every child having exactly one candy?
2013 ELMO Shortlist, 3
Find all $f:\mathbb{R}\to\mathbb{R}$ such that for all $x,y\in\mathbb{R}$, $f(x)+f(y) = f(x+y)$ and $f(x^{2013}) = f(x)^{2013}$.
[i]Proposed by Calvin Deng[/i]
2020 MBMT, 31
Consider the infinite sequence $\{a_i\}$ that extends the pattern
\[1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, \dots\]
Formally, $a_i = i-T(i)$ for all $i \geq 1$, where $T(i)$ represents the largest triangular number less than $i$ (triangle numbers are integers of the form $\frac{k(k+1)}2$ for some nonnegative integer $k$). Find the number of indices $i$ such that $a_i = a_{i + 2020}$.
[i]Proposed by Gabriel Wu[/i]
2015 Germany Team Selection Test, 2
Let $ABC$ be an acute triangle with the circumcircle $k$ and incenter $I$. The perpendicular through $I$ in $CI$ intersects segment $[BC]$ in $U$ and $k$ in $V$. In particular $V$ and $A$ are on different sides of $BC$. The parallel line through $U$ to $AI$ intersects $AV$ in $X$.
Prove: If $XI$ and $AI$ are perpendicular to each other, then $XI$ intersects segment $[AC]$ in its midpoint $M$.
[i](Notation: $[\cdot]$ denotes the line segment.)[/i]
1967 IMO Shortlist, 6
A line $l$ is drawn through the intersection point $H$ of altitudes of acute-angle triangles. Prove that symmetric images $l_a, l_b, l_c$ of $l$ with respect to the sides $BC,CA,AB$ have one point in common, which lies on the circumcircle of $ABC.$
2007 Canada National Olympiad, 1
What is the maximum number of non-overlapping $ 2\times 1$ dominoes that can be placed on a $ 8\times 9$ checkerboard if six of them are placed as shown? Each domino must be placed horizontally or vertically so as to cover two adjacent squares of the board.
2020 USAMTS Problems, 3:
Given a nonconstant polynomial with real coefficients $f(x),$ let $S(f)$ denote the sum of its roots. Let p and q be nonconstant polynomials with real coefficients such that $S(p) = 7,$ $S(q) = 9,$ and $S(p-q)= 11$. Find, with proof, all possible values for $S(p + q)$.
1997 Bundeswettbewerb Mathematik, 3
A semicircle with diameter $AB = 2r$ is divided into two sectors by an arbitrary radius. To each of the sectors a circle is inscribed. These two circles touch A$B$ at $S$ and $T$. Show that $ST \ge 2r(\sqrt{2}-1)$.
1990 AIME Problems, 3
Let $ P_1$ be a regular $ r$-gon and $ P_2$ be a regular $ s$-gon $ (r\geq s\geq 3)$ such that each interior angle of $ P_1$ is $ \frac {59}{58}$ as large as each interior angle of $ P_2$. What's the largest possible value of $ s$?