Found problems: 85335
1973 Swedish Mathematical Competition, 4
$p$ is a prime. Find all relatively prime positive integers $m$, $n$ such that
\[
\frac{m}{n}+\frac{1}{p^2}=\frac{m+p}{n+p}
\]
2012 Brazil Team Selection Test, 3
In chess, a king threatens another king if, and only if, they are on neighboring squares, whether horizontally, vertically, or diagonally . Find the greatest amount of kings that can be placed on a $12 \times 12$ board such that each king threatens just another king. Here, we are not considering part colors, that is, consider that the king are all, say, white, and that kings of the same color can threaten each other.
MBMT Team Rounds, 2020.29
The center of circle $\omega_1$ of radius $6$ lies on circle $\omega_2$ of radius $6$. The circles intersect at points $K$ and $W$. Let point $U$ lie on the major arc $\overarc{KW}$ of $\omega_2$, and point $I$ be the center of the largest circle that can be inscribed in $\triangle KWU$. If $KI+WI=11$, find $KI\cdot WI$.
[i]Proposed by Bradley Guo[/i]
2010 Contests, 1
Find all $ 3$-digit numbers such that placing to the right side of the number its successor we get a $ 6$-digit number which is a perfect square.
2016 NIMO Problems, 4
Justine has two fair dice, one with sides labeled $1,2,\ldots, m$ and one with sides labeled $1,2,\ldots, n.$ She rolls both dice once. If $\tfrac{3}{20}$ is the probability that at least one of the numbers showing is at most 3, find the sum of all distinct possible values of $m+n$.
[i]Proposed by Justin Stevens[/i]
2000 Turkey MO (2nd round), 2
A positive real number $a$ and two rays wich intersect at point $A$ are given. Show that all the circles which pass through $A$ and intersect these rays at points $B$ and $C$ where $|AB|+|AC|=a$ have a common point other than $A$.
1999 All-Russian Olympiad, 2
Each rational point on a real line is assigned an integer. Prove that there is a segment such that the sum of the numbers at its endpoints does not exceed twice the number at its midpoint.
1969 IMO Longlists, 33
$(GDR 5)$ Given a ring $G$ in the plane bounded by two concentric circles with radii $R$ and $\frac{R}{2}$, prove that we can cover this region with $8$ disks of radius $\frac{2R}{5}$. (A region is covered if each of its points is inside or on the border of some disk.)
2010 IFYM, Sozopol, 5
Let $A_1 A_2...A_n$ be a convex $n$-gon. What’s the number of $m$-gons with vertices from $A_1,A_2,...,A_n$ such that between each two adjacent vertices of the $m$-gon there are at least $k$ vertices from the $n$-gon?
1965 AMC 12/AHSME, 6
If $ 10^{\log_{10}9} \equal{} 8x \plus{} 5$ then $ x$ equals:
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ \frac {1}{2} \qquad \textbf{(C)}\ \frac {5}{8} \qquad \textbf{(D)}\ \frac {9}{8} \qquad \textbf{(E)}\ \frac {2\log_{10}3 \minus{} 5}{8}$
2013 Saudi Arabia BMO TST, 3
Find the area of the set of points of the plane whose coordinates $(x, y)$ satisfy $x^2 + y^2 \le 4|x| + 4|y|$.
2023 CMI B.Sc. Entrance Exam, 3
Consider the polynomial $p(x) = x^4 + ax^3 + bx^2 + cx + d$.
It is given that $p(x)$ has its only root at $x = r$ i.e $p(r) = 0$.
$\textbf{(a)}$ Show that if $a, b, c, d$ are rational then $r$ is rational.
$\textbf{(b)}$ Show that if $a, b, c, d$ are integers then $r$ is an integer.
[hide=Hint](Hint: Consider the roots of $p'(x)$ )[/hide]
2006 Hungary-Israel Binational, 2
A block of size $ a\times b\times c$ is composed of $ 1\times 1\times 2$ domino blocks. Assuming that each of the three possible directions of domino blocks occurs equally many times, what are the possible values of $ a$, $ b$, $ c$?
2021-IMOC, G7
The incircle of triangle $ABC$ tangents $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. Let the tangents of $E$, $F$ with respect to $\odot(AEF)$ intersect at $P$, and $X$ be a point on $BC$ such that $EF$, $DP$, $AX$ are concurrent. Define $Q$, $Y$ and $R$, $Z$ similarly. Show that $X$, $Y$, $Z$ are collinear.
2015 All-Russian Olympiad, 3
$110$ teams participate in a volleyball tournament. Every team has played every other team exactly once (there are no ties in volleyball). Turns out that in any set of $55$ teams, there is one which has lost to no more than $4$ of the remaining $54$ teams. Prove that in the entire tournament, there is a team that has lost to no more than $4$ of the remaining $109$ teams.
1990 IMO Longlists, 69
Consider the set of cuboids: the three edges $a, b, c$ from a common vertex satisfy the condition
\[\frac ab = \frac{a^2}{c^5}\]
(i) Prove that there are $100$ pairs of cuboids in this set with equal volumes in each pair.
(ii) For each pair of the above cuboids, find the ratio of the sum of their edges.
2021 Dutch IMO TST, 3
Prove that for every positive integer $n$ there are positive integers $a$ and $b$ exist with $n | 4a^2 + 9b^2 -1$.
2020 Ukrainian Geometry Olympiad - December, 3
In a triangle $ABC$ with an angle $\angle CAB =30^o$ draw median $CD$. If the formed $\vartriangle ACD$ is isosceles, find tan $\angle DCB$.
2021 Olympic Revenge, 4
On a chessboard, Po controls a white queen and plays, in alternate turns, against an invisible black king (there are only those two pieces on the board). The king cannot move to a square where he would be in check, neither capture the queen. Every time the king makes a move, Po receives a message from beyond that tells which direction the king has moved (up, right, up-right, etc). His goal is to make the king unable to make a movement.
Can Po reach his goal with at most $150$ moves, regardless the starting position of the pieces?
2020 Canadian Mathematical Olympiad Qualification, 3
Let $N$ be a positive integer and $A = a_1, a_2, ... , a_N$ be a sequence of real numbers.
Define the sequence $f(A)$ to be
$$f(A) = \left( \frac{a_1 + a_2}{2},\frac{a_2 + a_3}{2}, ...,\frac{a_{N-1} + a_N}{2},\frac{a_N + a_1}{2}\right)$$
and for $k$ a positive integer define $f^k (A)$ to be$ f$ applied to $A$ consecutively $k$ times (i.e. $f(f(... f(A)))$)
Find all sequences $A = (a_1, a_2,..., a_N)$ of integers such that $f^k (A)$ contains only integers for all $k$.
2020 USMCA, 3
Call a polynomial $f$ with positive integer coefficients [i]triangle-compatible[/i] if any three coefficients of $f$ satisfy the triangle inequality. For instance, $3x^3 + 4x^2 + 6x + 5$ is triangle-compatible, but $3x^3 + 3x^2 + 6x + 5$ is not. Given that $f$ is a degree $20$ triangle-compatible polynomial with $-20$ as a root, what is the least possible value of $f(1)$?
2009 Kyrgyzstan National Olympiad, 6
Find all natural $a,b$ such that $\left. {a(a + b) + 1} \right|(a + b)(b + 1) - 1$.
2020 Saint Petersburg Mathematical Olympiad, 5.
Point $I_a$ is the $A$-excircle center of $\triangle ABC$ which is tangent to $BC$ at $X$. Let $A'$ be diametrically opposite point of $A$ with respect to the circumcircle of $\triangle ABC$. On the segments $I_aX, BA'$ and $CA'$ are chosen respectively points $Y,Z$ and $T$ such that $I_aY=BZ=CT=r$ where $r$ is the inradius of $\triangle ABC$.
Prove that the points $X,Y,Z$ and $T$ are concyclic.
2023 India IMO Training Camp, 1
Suppose an acute scalene triangle $ABC$ has incentre $I$ and incircle touching $BC$ at $D$. Let $Z$ be the antipode of $A$ in the circumcircle of $ABC$. Point $L$ is chosen on the internal angle bisector of $\angle BZC$ such that $AL = LI$. Let $M$ be the midpoint of arc $BZC$, and let $V$ be the midpoint of $ID$. Prove that $\angle IML = \angle DVM$
2019 Iran MO (2nd Round), 5
Ali and Naqi are playing a game. At first, they have Polynomial $P(x) = 1+x^{1398}$.
Naqi starts. In each turn one can choice natural number $k \in [0,1398]$ in his trun, and add $x^k$ to the polynomial. For example after 2 moves $P$ can be : $P(x) = x^{1398} + x^{300} + x^{100} +1$. If after Ali's turn, there exist $t \in R$ such that $P(t)<0$ then Ali loses the game. Prove that Ali can play forever somehow he never loses the game!