Found problems: 85335
2018 Israel National Olympiad, 1
$n$ people sit in a circle. Each of them is either a liar (always lies) or a truthteller (always tells the truth). Every person knows exactly who speaks the truth and who lies. In their turn, each person says 'the person two seats to my left is a truthteller'. It is known that there's at least one liar and at least one truthteller in the circle.
[list=a]
[*] Is it possible that $n=2017?$
[*] Is it possible that $n=5778?$
[/list]
1974 Chisinau City MO, 75
Through point $P$, which lies on one of the sides of the triangle $ABC$, draw a line dividing its area in half.
2000 AIME Problems, 2
Let $u$ and $v$ be integers satisfying $0<v<u.$ Let $A=(u,v),$ let $B$ be the reflection of $A$ across the line $y=x,$ let $C$ be the reflection of $B$ across the y-axis, let $D$ be the reflection of $C$ across the x-axis, and let $E$ be the reflection of $D$ across the y-axis. The area of pentagon $ABCDE$ is 451. Find $u+v.$
1999 Tournament Of Towns, 4
(a) On each of the $1 \times 1$ squares of the top row of an $8 \times 8$ chessboard there is a black pawn, and on each of the $1 \times 1$ squares of the bottom row of this chessboard there is a white pawn. On each move one can shift any pawn vertically or horizontally to any adjacent empty $1 \times 1$ square. What is the smallest number of moves that are needed to move all white pawns to the top row and all black pawns to the bottom one?
(b) The same question for a $7 \times 7$ board.
(A Shapovalov_
2018 SIMO, Bonus
Simon plays a game on an $n\times n$ grid of cells. Initially, each cell is filled with an integer. Every minute, Simon picks a cell satisfying the following:
[list]
[*] The magnitude of the integer in the chosen cell is less than $n^{n^n}$
[*] The sum of all the integers in the neighboring cells (sharing one side with the chosen cell) is non-zero
[/list]
Simon then adds each integer in a neighboring cell to the chosen cell.
Show that Simon will eventually not be able to make any valid moves.
2004 Estonia National Olympiad, 4
In the beginning, number $1$ has been written to point $(0,0)$ and $0$ has been written to any other point of integral coordinates. After every second, all numbers are replaced with the sum of the numbers in four neighbouring points at the previous second. Find the sum of numbers in all points of integral coordinates after $n$ seconds.
2006 Germany Team Selection Test, 3
Suppose that $ a_1$, $ a_2$, $ \ldots$, $ a_n$ are integers such that $ n\mid a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n$.
Prove that there exist two permutations $ \left(b_1,b_2,\ldots,b_n\right)$ and $ \left(c_1,c_2,\ldots,c_n\right)$ of $ \left(1,2,\ldots,n\right)$ such that for each integer $ i$ with $ 1\leq i\leq n$, we have
\[ n\mid a_i \minus{} b_i \minus{} c_i
\]
[i]Proposed by Ricky Liu & Zuming Feng, USA[/i]
2003 Turkey Team Selection Test, 4
Find the least
a. positive real number
b. positive integer
$t$ such that the equation $(x^2+y^2)^2 + 2tx(x^2 + y^2) = t^2y^2$ has a solution where $x,y$ are positive integers.
2021 Romania National Olympiad, 3
Given is an positive integer $a>2$
a) Prove that there exists positive integer $n$ different from $1$, which is not a prime, such that $a^n=1(mod n)$
b) Prove that if $p$ is the smallest positive integer, different from $1$, such that $a^p=1(mod p)$, then $p$ is a prime.
c) There does not exist positive integer $n$, different from $1$, such that $2^n=1(mod n)$
2016 ASDAN Math Tournament, 7
Let $x$, $y$, and $z$ be real numbers satisfying the equations
\begin{align*}
4x+2yz-6z+9xz^2&=4\\
xyz&=1.
\end{align*}
Find all possible values of $x+y+z$.
2021 HMNT, 4
The sum of the digits of the time $19$ minutes ago is two less than the sum of the digits of the time right now. Find the sum of the digits of the time in $19$ minutes. (Here, we use a standard $12$-hour clock of the form $hh:mm$.)
1963 Polish MO Finals, 3
From a given triangle, cut out the rectangle with the largest area.
2007 Tournament Of Towns, 1
(from The Good Soldier Svejk) Senior military doctor Bautze exposed $abccc$ malingerers among $aabbb$ draftees who claimed not to be fit for the military service. He managed to expose all but one draftees. (He would for sure expose this one too, if the lucky guy was not taken by a stroke at the very moment when the doctor yelled at him "Turn around !. . . ") How many malingerers were exposed by the vigilant doctor?
Each digit substitutes a letter. The same digits substitute the same letters, while distinct digits substitute distinct letters.
[i](1 point)[/i]
2016 Postal Coaching, 2
Determine all functions $f : \mathbb R \to \mathbb R$ such that $$f(f(x)- f(y)) = f(f(x)) - 2x^2f(y) + f\left(y^2\right),$$ for all reals $x, y$.
2017 India PRMO, 21
Attached below:
2016 Singapore MO Open, 5
A total of $731$ objects are put into $n$ nonempty bags where $n$ is a positive integer. These bags can be distributed into $17$ red boxes and also into $43$ blue boxes so that each red and each blue box contain $43$ and $17$ objects, respectively. Find the minimum value of $n$.
1984 AMC 12/AHSME, 2
If $x,y$ and $y - \frac{1}{x}$ are not 0, then \[\frac{x - \frac{1}{y}}{y - \frac{1}{x}}\] equals
$\textbf{(A) }1\qquad\textbf{(B) } \frac{x}{y}\qquad\textbf{(C) }\frac{y}{x}\qquad\textbf{(D) }\frac{x}{y} - \frac{y}{x}\qquad\textbf{(E) } xy - \frac{1}{xy}$
2024 Putnam, A3
Let $S$ be the set of bijections
\[
T\colon\{1,\,2,\,3\}\times\{1,\,2,\,\ldots,\,2024\}\to\{1,\,2,\,\ldots,\,6072\}
\]
such that $T(1,\,j)<T(2,\,j)<T(3,\,j)$ for all $j\in\{1,\,2,\,\ldots,\,2024\}$ and $T(i,\,j)<T(i,\,j+1)$ for all $i\in\{1,\,2,\,3\}$ and $j\in\{1,\,2,\,\ldots,\,2023\}$. Do there exist $a$ and $c$ in $\{1,\,2,\,3\}$ and $b$ and $d$ in $\{1,\,2,\,\ldots,\,2024\}$ such that the fraction of elements $T$ in $S$ for which $T(a,\,b)<T(c,\,d)$ is at least $1/3$ and at most $2/3$.
2002 AIME Problems, 13
In triangle $ ABC$ the medians $ \overline{AD}$ and $ \overline{CE}$ have lengths 18 and 27, respectively, and $ AB \equal{} 24$. Extend $ \overline{CE}$ to intersect the circumcircle of $ ABC$ at $ F$. The area of triangle $ AFB$ is $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.
2005 Miklós Schweitzer, 9
prove that if $r_n$ is a rational function whose numerator and denominator have at most degrees $n$, then $$||r_n||_{1/2}+\left\|\frac{1}{r_n}\right\|_2\geq\frac{1}{2^{n-1}}$$ where $||\cdot||_a$ denotes the supremum over a circle of radius $a$ around the origin.
2024 Belarusian National Olympiad, 10.4
A parallelogram $ABCD$ is given. The incircle of triangle $ABC$ with center $I$ touches $AB,BC,CA$ at $R,P,Q$. Ray $DI$ intersects segment $AB$ at $S$. It turned out that $\angle DPR=90$
Prove that the circle with diameter $AS$ is tangent to the circumcircle of triangle $DPQ$
[i]M. Zorka[/i]
2014 AMC 12/AHSME, 25
What is the sum of all positive real solutions $x$ to the equation \[2\cos(2x)\left(\cos(2x)-\cos\left(\frac{2014\pi^2}{x}\right)\right)=\cos(4x)-1?\]
$\textbf{(A) }\pi\qquad
\textbf{(B) }810\pi\qquad
\textbf{(C) }1008\pi\qquad
\textbf{(D) }1080\pi\qquad
\textbf{(E) }1800\pi\qquad$
1982 IMO Longlists, 43
[b](a)[/b] What is the maximal number of acute angles in a convex polygon?
[b](b)[/b] Consider $m$ points in the interior of a convex $n$-gon. The $n$-gon is partitioned into triangles whose vertices are among the $n + m$ given points (the vertices of the $n$-gon and the given points). Each of the $m$ points in the interior is a vertex of at least one triangle. Find the number of triangles obtained.
2021 Peru MO (ONEM), 2
The numbers $1$ to $25$ will be written in a table $5 \times 5$. First, Ana chooses $k$ of these numbers($1$ to $25$), and write in some cells. Then, Enrique writes the remaining numbers with the following goal: The product of the numbers in some column/row is a perfect square.
[b]a)[/b] Prove that if $k=5$, Ana can [b]avoid[/b] Enrique to reach his goal.
[b]b)[/b] Prove that if $k=4$, Enrique can reach his goal.
1982 All Soviet Union Mathematical Olympiad, 342
What minimal number of numbers from the set $\{1,2,...,1982\}$ should be deleted to provide the property:
[i]none of the remained numbers equals to the product of two other remained numbers[/i]?