Found problems: 85335
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!
2020 Vietnam Team Selection Test, 4
Let $n$ be a positive integer. In a $(2n+1)\times (2n+1)$ board, each grid is dyed white or black. In each row and each column, if the number of white grids is smaller than the number of black grids, then we mark all white grids. If the number of white grids is bigger than the number of black grids, then we mark all black grids. Let $a$ be the number of black grids, and $b$ be the number of white grids, $c$ is the number of marked grids.
In this example of $3\times 3$ table, $a=3$, $b=6$, $c=4$. (forget about my watermark)
Proof that no matter how is the dyeing situation in the beginning, there is always $c\geq\frac{1}{2}\min\{a,b\}$.
1979 Miklós Schweitzer, 2
Let $ \Gamma$ be a variety of monoids such that not all monoids of $ \Gamma$ are groups. Prove that if $ A \in \Gamma$ and $ B$ is a submonoid of $ A$, there exist monoids $ S \in \Gamma$ and $ C$ and epimorphisms $ \varphi : S \rightarrow A, \;\varphi_1 : S \rightarrow C$ such that $ ((e)\varphi_1^{\minus{}1})\varphi\equal{}B$ ($ e$ is the identity element of $ C$).
[i]L. Marki[/i]
2020 GQMO, 4
Prove that, for all sufficiently large integers $n$, there exists $n$ numbers $a_1, a_2, \dots, a_n$ satisfying the following three conditions:
[list]
[*] Each number $a_i$ is equal to either $-1, 0$ or $1$.
[*] At least $\frac{2n}{5}$ of the numbers $a_1, a_2, \dots, a_n$ are non-zero.
[*] The sum $\frac{a_1}{1} + \frac{a_2}{2} + \dots + \frac{a_n}{n}$ is $0$.
[/list]
$\textit{Note: Results with 2/5 replaced by a constant } c \textit{ will be awarded points depending on the value of } c$
[i]Proposed by Navneel Singhal, India; Kyle Hess, USA; and Vincent Jugé, France[/i]
2024 Ukraine National Mathematical Olympiad, Problem 8
Find all polynomials $P(x)$ with integer coefficients, such that for each of them there exists a positive integer $N$, such that for any positive integer $n\geq N$, number $P(n)$ is a positive integer and a divisor of $n!$.
[i]Proposed by Mykyta Kharin[/i]
2017 Saudi Arabia JBMO TST, 2
A positive integer $k > 1$ is called nice if for any pair $(m, n)$ of positive integers satisfying the condition $kn + m | km + n$ we have $n | m$.
1. Prove that $5$ is a nice number.
2. Find all the nice numbers.
2016 India Regional Mathematical Olympiad, 4
Prove that $(4\cos^29^o – 3) (4 \cos^227^o– 3) = \tan 9^o$.
2008 Purple Comet Problems, 15
Each of the distinct letters in the following subtraction problem represents a different digit. Find the number represented by the word [b]TEAM[/b]
[size=150][b]
PURPLE
- COMET
________
[color=#FFFFFF].....[/color]TEAM [/b][/size]
2014 ELMO Shortlist, 2
A $2^{2014} + 1$ by $2^{2014} + 1$ grid has some black squares filled. The filled black squares form one or more snakes on the plane, each of whose heads splits at some points but never comes back together. In other words, for every positive integer $n$ greater than $2$, there do not exist pairwise distinct black squares $s_1$, $s_2$, \dots, $s_n$ such that $s_i$ and $s_{i+1}$ share an edge for $i=1,2, \dots, n$ (here $s_{n+1}=s_1$).
What is the maximum possible number of filled black squares?
[i]Proposed by David Yang[/i]
2004 IMO Shortlist, 4
Consider a matrix of size $n\times n$ whose entries are real numbers of absolute value not exceeding $1$. The sum of all entries of the matrix is $0$. Let $n$ be an even positive integer. Determine the least number $C$ such that every such matrix necessarily has a row or a column with the sum of its entries not exceeding $C$ in absolute value.
[i]Proposed by Marcin Kuczma, Poland[/i]
1997 Singapore MO Open, 1
$\vartriangle ABC$ is an equilateral triangle. $L, M$ and $N$ are points on $BC, CA$ and $AB$ respectively. Prove that $MA \cdot AN + NB \cdot BL + LC \cdot CM < BC^2$.
2011 Junior Balkan MO, 2
Find all primes $p$ such that there exist positive integers $x,y$ that satisfy $x(y^2-p)+y(x^2-p)=5p$
2020 IMEO, Problem 2
You are given an odd number $n\ge 3$. For every pair of integers $(i, j)$ with $1\le i \le j \le n$ there is a domino, with $i$ written on one its end and with $j$ written on another (there are $\frac{n(n+1)}{2}$ domino overall). Amin took this dominos and started to put them in a row so that numbers on the adjacent sides of the dominos are equal. He has put $k$ dominos in this way, got bored and went away. After this Anton came to see this $k$ dominos, and he realized that he can't put all the remaining dominos in this row by the rules. For which smallest value of $k$ is this possible?
[i]Oleksii Masalitin[/i]