Found problems: 85335
2009 China Team Selection Test, 1
Let $ \alpha,\beta$ be real numbers satisfying $ 1 < \alpha < \beta.$ Find the greatest positive integer $ r$ having the following property: each of positive integers is colored by one of $ r$ colors arbitrarily, there always exist two integers $ x,y$ having the same color such that $ \alpha\le \frac {x}{y}\le\beta.$
2024 South Africa National Olympiad, 4
Find all functions $f$ from integers to integers such that
\[ f(m+n) + f(m-n) - 2f(m) = 6mn^2\]
for all integers $m$ and $n$.
2021 Malaysia IMONST 1, 3
There are $10$ girls in a class, all with different heights. They want to form a queue so that no girl stands directly between two girls shorter than her. How many ways are there to form the queue?
2018 Moscow Mathematical Olympiad, 2
In there $2018\times 2018$ square cells colored in white or black. It is known, that exists $10 \times 10$ square with only white cells and $10\times 10$ square with only black cells. For what minimal $d$ always exists square $10\times 10$ such that the number of black and white cells differs by no more than $d$?
Ukrainian TYM Qualifying - geometry, XI.4
Chords $AB$ and $CD$, which do not intersect, are drawn in a circle. On the chord $AB$ or on its extension is taken the point $E$. Using a compass and construct the point $F$ on the arc $AB$ , such that $\frac{PE}{EQ} = \frac{m}{n}$, where $m,n$ are given natural numbers, $P$ is the point of intersection of the chord $AB$ with the chord $FC$, $Q$ is the point of intersection of the chord $AB$ with the chord $FD$. Consider cases where $E\in PQ$ and $E \notin PQ$.
2013 Sharygin Geometry Olympiad, 8
Two fixed circles are given on the plane, one of them lies inside the other one. From a point $C$ moving arbitrarily on the external circle, draw two chords $CA, CB$ of the larger circle such that they tangent to the smalaler one. Find the locus of the incenter of triangle $ABC$.
1988 ITAMO, 7
Given $n \ge 3$ positive integers not exceeding $100$, let $d$ be their greatest common divisor. Show that there exist three of these numbers whose greatest common divisor is also equal to $d$.
MathLinks Contest 5th, 7.1
Prove that the numbers $${{2^n-1} \choose {i}}, i = 0, 1, . . ., 2^{n-1} - 1,$$ have pairwise different residues modulo $2^n$
2000 Bulgaria National Olympiad, 1
Find all polynomials $P(x)$ with real coefficients such that
\[P(x)P(x + 1) = P(x^2), \quad \forall x \in \mathbb R.\]
2018 CMIMC Geometry, 1
Let $ABC$ be a triangle. Point $P$ lies in the interior of $\triangle ABC$ such that $\angle ABP = 20^\circ$ and $\angle ACP = 15^\circ$. Compute $\angle BPC - \angle BAC$.
2009 Portugal MO, 3
Two players play the following game on a circular board with 2009 houses. The two plays put, alternatively, on an empty house, one of three pieces, called [i]explorer (E)[/i], [i]trap (T)[/i] or [i]stone (S)[/i]. A treasure is a sequence of three consecutive filled houses such that the first one (on any direction) has an explorer and the middle one doesn't have a trap. For example, [i]STE[/i] is not a treasure, while [i]TEE[/i] is a treasure. The first player forming a treasure wins. Can any of the players guarantee the victory? And, in affirmative case, who?
2020 LIMIT Category 1, 5
Let $P(x),Q(x)$ be monic polynomials with integer coeeficients. Let $a_n=n!+n$ for all natural numbers $n$. Show that if $\frac{P(a_n)}{Q(a_n)}$ is an integer for all positive integer $n$ then $\frac{P(n)}{Q(n)}$ is an integer for every integer $n\neq0$.
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[i]Hint (given in question): Try applying division algorithm for polynomials [/i]
2018 Rio de Janeiro Mathematical Olympiad, 2
Let $ABC$ be an equilateral triangle with side 3. A circle $C_1$ is tangent to $AB$ and $AC$.
A circle $C_2$, with a radius smaller than the radius of $C_1$, is tangent to $AB$ and $AC$ as well as externally tangent to $C_1$.
Successively, for $n$ positive integer, the circle $C_{n+1}$, with a radius smaller than the radius of $C_n$, is tangent to $AB$ and $AC$ and is externally tangent to $C_n$.
Determine the possible values for the radius of $C_1$ such that 4 circles from this sequence, but not 5, are contained on the interior of the triangle $ABC$.
1994 Turkey MO (2nd round), 5
Find the set of all ordered pairs $(s,t)$ of positive integers such that \[t^{2}+1=s(s+1).\]
1999 Gauss, 14
Which of the following numbers is an odd integer, contains the digit 5, is divisible by 3, and lies between $12^2$ and $13^2$?
$\textbf{(A)}\ 105 \qquad \textbf{(B)}\ 147 \qquad \textbf{(C)}\ 156 \qquad \textbf{(D)}\ 165 \qquad \textbf{(E)}\ 175$
2010 Romanian Master of Mathematics, 2
For each positive integer $n$, find the largest real number $C_n$ with the following property. Given any $n$ real-valued functions $f_1(x), f_2(x), \cdots, f_n(x)$ defined on the closed interval $0 \le x \le 1$, one can find numbers $x_1, x_2, \cdots x_n$, such that $0 \le x_i \le 1$ satisfying
\[|f_1(x_1)+f_2(x_2)+\cdots f_n(x_n)-x_1x_2\cdots x_n| \ge C_n\]
[i]Marko Radovanović, Serbia[/i]
2021 China National Olympiad, 2
Let $m>1$ be an integer. Find the smallest positive integer $n$, such that for any integers $a_1,a_2,\ldots ,a_n; b_1,b_2,\ldots ,b_n$ there exists integers $x_1,x_2,\ldots ,x_n$ satisfying the following two conditions:
i) There exists $i\in \{1,2,\ldots ,n\}$ such that $x_i$ and $m$ are coprime
ii) $\sum^n_{i=1} a_ix_i \equiv \sum^n_{i=1} b_ix_i \equiv 0 \pmod m$
2022 Austrian MO National Competition, 5
Let $ABC$ be an isosceles triangle with base $AB$. We choose a point $P$ inside the triangle on altitude through $C$. The circle with diameter $CP$ intersects the straight line through $B$ and $P$ again at the point $D_P$ and the Straight through $A$ and $C$ one more time at point $E_P$. Prove that there is a point $F$ such that for any choice of $P$ the points $D_P , E_P$ and $F$ lie on a straight line.
[i](Walther Janous)[/i]
2021 MOAA, 2
On Andover's campus, Graves Hall is $60$ meters west of George Washington Hall, and George Washington Hall is $80$ meters north of Paresky Commons. Jessica wants to walk from Graves Hall to Paresky Commons. If she first walks straight from Graves Hall to George Washington Hall and then walks straight from George Washington Hall to Paresky Commons, it takes her $8$ minutes and $45$ seconds while walking at a constant speed. If she walks with the same speed directly from Graves Hall to Paresky Commons, how much time does she save, in seconds?
[i]Proposed by Nathan Xiong[/i]
2019 USA TSTST, 7
Let $f: \mathbb Z\to \{1, 2, \dots, 10^{100}\}$ be a function satisfying
$$\gcd(f(x), f(y)) = \gcd(f(x), x-y)$$
for all integers $x$ and $y$. Show that there exist positive integers $m$ and $n$ such that $f(x) = \gcd(m+x, n)$ for all integers $x$.
[i]Ankan Bhattacharya[/i]
2017 China Team Selection Test, 2
In $\varDelta{ABC}$,the excircle of $A$ is tangent to segment $BC$,line $AB$ and $AC$ at $E,D,F$ respectively.$EZ$ is the diameter of the circle.$B_1$ and $C_1$ are on $DF$, and $BB_1\perp{BC}$,$CC_1\perp{BC}$.Line $ZB_1,ZC_1$ intersect $BC$ at $X,Y$ respectively.Line $EZ$ and line $DF$ intersect at $H$,$ZK$ is perpendicular to $FD$ at $K$.If $H$ is the orthocenter of $\varDelta{XYZ}$,prove that:$H,K,X,Y$ are concyclic.
1975 AMC 12/AHSME, 14
If the $ whatsis$ is $ so$ when the $ whosis$ is $ is$ and the $ so$ and $ so$ is $ is \cdot so$, what is the $ whosis \cdot whatsis$ when the $ whosis$ is $ so$, the $ so$ and $ so$ is $ so \cdot so$ and the $ is$ is two ($ whatsis$, $ whosis$, $ is$ and $ so$ are variables taking positive values)?
$ \textbf{(A)}\ whosis \cdot is \cdot so \qquad
\textbf{(B)}\ whosis \qquad
\textbf{(C)}\ is \qquad
\textbf{(D)}\ so \qquad
\textbf{(E)}\ so \text{ and } so$
2019 IMEO, 5
Find all pairs of positive integers $(s, t)$, so that for any two different positive integers $a$ and $b$ there exists some positive integer $n$, for which $$a^s + b^t | a^n + b^{n+1}.$$
[i]Proposed by Oleksii Masalitin (Ukraine)[/i]
2021-2022 OMMC, 14
The corners of a $2$-dimensional room in the shape of an isosceles right triangle are labeled $A$, $B$, $C$ where $AB = BC$. Walls $BC$ and $CA$ are mirrors. A laser is shot from $A$, hits off of each of the mirrors once and lands at a point $X$ on $AB$. Let $Y$ be the point where the laser hits off $AC$. If $\tfrac{AB}{AX} = 64$, $\tfrac{CA}{AY} = \tfrac pq$ for coprime positive integers $p$, $q$. Find $p + q$.
[i]Proposed by Sid Doppalapudi[/i]
2005 Greece Junior Math Olympiad, 2
If $f(n)=\frac{2n+1+\sqrt{n(n+1)}}{\sqrt{n+1}+\sqrt{n}}$ for all positive integers $n$, evaluate
(a) $f(1)$,
(b) the sum $A=f(1)+f(2)+...+f(400)$.