Found problems: 85335
2017 Ukraine Team Selection Test, 12
Let $m_1,m_2,...,m_{2013} > 1$ be 2013 pairwise relatively prime positive integers and $A_1,A_2,...,A_{2013}$ be 2013 (possibly empty) sets with $A_i\subseteq \{1,2,...,m_i-1\}$ for $i=1,2,...,2013$. Prove that there is a positive integer $N$ such that
\[ N \le \left( 2\left\lvert A_1 \right\rvert + 1 \right)\left( 2\left\lvert A_2 \right\rvert + 1 \right)\cdots\left( 2\left\lvert A_{2013} \right\rvert + 1 \right) \]
and for each $i = 1, 2, ..., 2013$, there does [i]not[/i] exist $a \in A_i$ such that $m_i$ divides $N-a$.
[i]Proposed by Victor Wang[/i]
India EGMO 2022 TST, 3
Let $I$ be incentre of scalene $\triangle ABC$ and let $L$ be midpoint of arc $BAC$. Let $M$ be midpoint of $BC$ and let the line through $M$ parallel to $AI$ intersect $LI$ at point $P$. Let $Q$ lie on $BC$ such that $PQ\perp LI$. Let $S$ be midpoint of $AM$ and $T$ be midpoint of $LI$. Prove that $IS\perp BC$ if and only if $AQ\perp ST$.
[i]~Mahavir Gandhi[/i]
2008 Iran MO (3rd Round), 2
Prove that there exists infinitely many primes $ p$ such that: \[ 13|p^3\plus{}1\]
2011 QEDMO 9th, 7
Find all functions $f: R\to R$, such that $f(xy + x + y) + f(xy-x-y)=2f (x) + 2f (y)$ for all $x, y \in R$.
2005 Today's Calculation Of Integral, 29
Let $a$ be a real number.
Evaluate
\[\int _{-\pi+a}^{3\pi+a} |x-a-\pi|\sin \left(\frac{x}{2}\right)dx\]
2009 AMC 10, 8
In a certain year the price of gasoline rose by $ 20\%$ during January, fell by $ 20\%$ during February, rose by $ 25\%$ during March, and fell by $ x\%$ during April. The price of gasoline at the end of April was the same as it had been at the beginning of January. To the nearest integer, what is $ x$?
$ \textbf{(A)}\ 12\qquad
\textbf{(B)}\ 17\qquad
\textbf{(C)}\ 20\qquad
\textbf{(D)}\ 25\qquad
\textbf{(E)}\ 35$
2018 Turkey Team Selection Test, 8
For integers $m\geq 3$, $n$ and $x_1,x_2, \ldots , x_m$ if $x_{i+1}-x_i \equiv x_i-x_{i-1} (mod n) $ for every $2\leq i \leq m-1$, we say that the $m$-tuple $(x_1,x_2,\ldots , x_m)$ is an arithmetic sequence in $(mod n)$. Let $p\geq 5$ be a prime number and $1<a<p-1$ be an integer. Let ${a_1,a_2,\ldots , a_k}$ be the set of all possible remainders when positive powers of $a$ are divided by $p$. Show that if a permutation of ${a_1,a_2,\ldots , a_k}$ is an arithmetic sequence in $(mod p)$, then $k=p-1$.
2021 Balkan MO Shortlist, N7
A [i]super-integer[/i] triangle is defined to be a triangle whose lengths of all sides and at least
one height are positive integers. We will deem certain positive integer numbers to be [i]good[/i] with
the condition that if the lengths of two sides of a super-integer triangle are two (not necessarily
different) good numbers, then the length of the remaining side is also a good number. Let $5$ be
a good number. Prove that all integers larger than $2$ are good numbers.
2022 Kosovo Team Selection Test, 2
Find all positive integers $a, b, c$ such that $ab + 1$, $bc + 1$, and $ca + 1$ are all equal to
factorials of some positive integers.
Proposed by [i]Nikola Velov, Macedonia[/i]
1996 Tournament Of Towns, (509) 2
Do there exist three different prime numbers $p$, $q$ and $r$ such that $p^2 + d$ is divisible by $qr$, $q^2 + d$ is divisible by $rp$ and $r^2 + d$ is divisible by $pq$, if
(a) $d = 10$;
(b) $d = 11$?
(V Senderov)
2015 Azerbaijan JBMO TST, 2
Find all non-negative solutions to the equation $2013^x+2014^y=2015^z$
2014 Iran Team Selection Test, 5
if $x,y,z>0$ are postive real numbers such that $x^{2}+y^{2}+z^{2}=x^{2}y^{2}+y^{2}z^{2}+z^{2}x^{2}$
prove that \[((x-y)(y-z)(z-x))^{2}\leq 2((x^{2}-y^{2})^{2}+(y^{2}-z^{2})^{2}+(z^{2}-x^{2})^{2})\]
2018 BMT Spring, Tie 2
Points $A, B, C$ are chosen on the boundary of a circle with center $O$ so that $\angle BAC$ encloses an arc of $120$ degrees. Let $D$ be chosen on $\overline{BA}$ so that $\angle AOD$ is a right angle. Extend $\overline{CD}$ so that it intersects with $O$ again at point $P$. What is the measure of the arc, in degrees, that is enclosed by $\angle ACP$? Please use the $tan^{-1}$ function to express your answer.
MOAA Gunga Bowls, 2023.8
Let $ABCD$ be a parallelogram with area 160. Let diagonals $AC$ and $BD$ intersect at $E$. Point $P$ is on $\overline{AE}$ such that $EC = 4EP$. If line $DP$ intersects $AB$ at $F$, find the area of $BFPC$.
[i]Proposed by Andy Xu[/i]
2014 Contests, 4
We are given a circle $c(O,R)$ and two points $A,B$ so that $R<AB<2R$.The circle $c_1 (A,r)$ ($0<r<R$) crosses the circle $c$ at C,D ($C$ belongs to the short arc $AB$).From $B$ we consider the tangent lines $BE,BF$ to the circle $c_1$ ,in such way that $E$ lays out of the circle $c$.If $M\equiv EC\cap DF$ show that the quadrilateral $BCFM$ is cyclic.
2011 Canadian Students Math Olympiad, 2
For a fixed positive integer $k$, prove that there exist infinitely many primes $p$ such that there is an integer $w$, where $w^2-1$ is not divisible by $p$, and the order of $w$ in modulus $p$ is the same as the order of $w$ in modulus $p^k$.
[i]Author: James Rickards[/i]
2012 Grand Duchy of Lithuania, 4
Let $m$ be a positive integer. Find all bounded sequences of integers $a_1, a_2, a_3,... $for which $a_n + a_{n+1} + a_{n+m }= 0$ for all $n \in N$.
2010 All-Russian Olympiad Regional Round, 10.8
Let's call it a [i] staircase of height [/i]$n$, a figure consisting from all square cells $n\times n$ lying no higher diagonals (the figure shows a [i]staircase of height [/i] $4$ ). In how many different ways can a [i]staircase of height[/i] $n$ can be divided into several rectangles whose sides go along the grid lines, but the areas are different in pairs?
[img]https://cdn.artofproblemsolving.com/attachments/f/0/f66d7e9ada0978e8403fbbd8989dc1b201f2cd.png[/img]
1990 Vietnam National Olympiad, 2
Suppose $ f(x)\equal{}a_0x^n\plus{}a_1x^{n\minus{}1}\plus{}\ldots\plus{}a_{n\minus{}1}x\plus{}a_n$ ($ a_0\neq 0$) is a polynomial with real coefficients satisfying $ f(x)f(2x^2) \equal{} f(2x^3 \plus{} x)$ for all $ x \in\mathbb{R}$. Prove that $ f(x)$ has no real roots.
2013 Online Math Open Problems, 18
Determine the absolute value of the sum \[ \lfloor 2013\sin{0^\circ} \rfloor + \lfloor 2013\sin{1^\circ} \rfloor + \cdots + \lfloor 2013\sin{359^\circ} \rfloor, \] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.
(You may use the fact that $\sin{n^\circ}$ is irrational for positive integers $n$ not divisible by $30$.)
[i]Ray Li[/i]
2009 IMO Shortlist, 5
Let $P(x)$ be a non-constant polynomial with integer coefficients. Prove that there is no function $T$ from the set of integers into the set of integers such that the number of integers $x$ with $T^n(x)=x$ is equal to $P(n)$ for every $n\geq 1$, where $T^n$ denotes the $n$-fold application of $T$.
[i]Proposed by Jozsef Pelikan, Hungary[/i]
2019 BMT Spring, 13
Two circles $O_1$ and $O_2$ intersect at points $A$ and $B$. Lines $\overline{AC}$ and $\overline{BD}$ are drawn such that $C$ is on $O_1$ and $D$ is on $O_2$ and $\overline{AC} \perp \overline{AB}$ and $\overline{BD} \perp \overline{AB}$. If minor arc $AB= 45$ degrees relative to $O_1$ and minor arc $AB= 60$ degrees relative to $O_2$ and the radius of $O_2 = 10$, the area of quadrilateral $CADB$ can be expressed in simplest form as $a + b\sqrt{k} + c\sqrt{\ell}$. Compute $a + b + c + k +\ell$.
2002 Tournament Of Towns, 2
All the species of plants existing in Russia are catalogued (numbered by integers from $2$ to $2000$ ; one after another, without omissions or repetitions). For any pair of species the gcd of their catalogue numbers was calculated and recorded but the catalogue numbers themselves were lost. Is it possible to restore the catalogue numbers from the data in hand?
2017 Hanoi Open Mathematics Competitions, 5
Write $2017$ following numbers on the blackboard: $-\frac{1008}{1008}, -\frac{1007}{1008}, ..., -\frac{1}{1008}, 0,\frac{1}{1008},\frac{2}{1008}, ... ,\frac{1007}{1008},\frac{1008}{1008}$ .
One processes some steps as: erase two arbitrary numbers $x, y$ on the blackboard and then write on it the number $x + 7xy + y$. After $2016$ steps, there is only one number. The last one on the blackboard is
(A): $-\frac{1}{1008}$ (B): $0$ (C): $\frac{1}{1008}$ (D): $-\frac{144}{1008}$ (E): None of the above
2000 South africa National Olympiad, 3
Let $c \geq 1$ be an integer, and define the sequence $a_1,\ a_2,\ a_3,\ \dots$ by \[ \begin{aligned} a_1 & = 2, \\ a_{n + 1} & = ca_n + \sqrt{\left(c^2 - 1\right)\left(a_n^2 - 4\right)}\textrm{ for }n = 1,2,3,\dots\ . \end{aligned} \] Prove that $a_n$ is an integer for all $n$.