Found problems: 85335
2014 BAMO, 1
The four bottom corners of a cube are colored red, green, blue, and purple. How many ways are there to color the top four corners of the cube so that every face has four different colored corners? Prove that your answer is correct.
2018 Cyprus IMO TST, 1
Determine all integers $n \geq 2$ for which the number $11111$ in base $n$ is a perfect square.
2014 Argentina National Olympiad Level 2, 4
There is a number written in each square of a $13\times13$ board such that any two numbers in squares with a common side differ by exactly $1$. Each of the numbers $2$ and $24$ is written twice. How many times is the number $13$ written? Find all possibilities.
2012 Danube Mathematical Competition, 2
Let $ABC$ be an acute triangle and let $A_1$, $B_1$, $C_1$ be points on the sides $BC, CA$ and $AB$, respectively. Show that the triangles $ABC$ and $A_1B_1C_1$ are similar ($\angle A = \angle A_1, \angle B = \angle B_1,\angle C = \angle C_1$) if and only if the orthocentre of the triangle $A_1B_1C_1$ and the circumcentre of the triangle $ABC$ coincide.
2001 May Olympiad, 5
On the board are written the natural numbers from $1$ to $2001$ inclusive. You have to delete some numbers so that among those that remain undeleted it is impossible to choose two different numbers such that the result of their multiplication is equal to one of the numbers that remain undeleted. What is the minimum number of numbers that must be deleted? For that amount, present an example showing which numbers are erased. Justify why, if fewer numbers are deleted, the desired property is not obtained.
2014 ASDAN Math Tournament, 1
Consider a square of side length $1$ and erect equilateral triangles of side length $1$ on all four sides of the square such that one triangle lies inside the square and the remaining three lie outside. Going clockwise around the square, let $A$, $B$, $C$, $D$ be the circumcenters of the four equilateral triangles. Compute the area of $ABCD$.
2022 Saudi Arabia BMO + EGMO TST, 2.4
Consider the function $f : R^+ \to R^+$ and satisfying
$$f(x + 2y + f(x + y)) = f(2x) + f(3y), \,\, \forall \,\, x, y > 0.$$
1. Find all functions $f(x)$ that satisfy the given condition.
2. Suppose that $f(4\sin^4x)f(4\cos^4x) \ge f^2(1)$ for all $x \in \left(0\frac{\pi}{2}\right) $. Find the minimum value of $f(2022)$.
2014 Sharygin Geometry Olympiad, 11
Points $K, L, M$ and $N$ lying on the sides $AB, BC, CD$ and $DA$ of a square $ABCD$ are vertices of another square. Lines $DK$ and $N M$ meet at point $E$, and lines $KC$ and $LM$ meet at point $F$ . Prove that $EF\parallel AB$.
2014 ASDAN Math Tournament, 7
$f(x)$ is a quartic polynomial with a leading coefficient $1$ where $f(2)=4$, $f(3)=9$, $f(4)=16$, and $f(5)=25$. Compute $f(8)$.
2021 LMT Spring, A9
Find the sum of all positive integers $n$ such that $7<n < 100$ and $1573_{n}$ has $6$ factors when written in base $10$.
[i]Proposed by Aidan Duncan[/i]
2006 Italy TST, 3
Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that for all integers $m,n$,
\[f(m - n + f(n)) = f(m) + f(n).\]
2021 Swedish Mathematical Competition, 5
Let $ n$ be a positive integer congruent to $1$ modulo $4$. Xantippa has a bag of $n + 1$ balls numbered from $ 0$ to $n$. She draws a ball (randomly, equally distributed) from the bag and reads its number: $k$, say. She keeps the ball and then picks up another $k$ balls from the bag (randomly, equally distributed, without repossession). Finally, she adds up the numbers of all the $k + 1$ balls she picked up. What is the probability that the sum will be odd?
1986 Polish MO Finals, 6
$ABC$ is a triangle. The feet of the perpendiculars from $B$ and $C$ to the angle bisector at $A$ are $K, L$ respectively. $N$ is the midpoint of $BC$, and $AM$ is an altitude. Show that $K,L,N,M$ are concyclic.
2007 Korea National Olympiad, 1
For all positive reals $ a$, $ b$, and $ c$, what is the value of positive constant $ k$ satisfies the following inequality?
$ \frac{a}{c\plus{}kb}\plus{}\frac{b}{a\plus{}kc}\plus{}\frac{c}{b\plus{}ka}\geq\frac{1}{2007}$ .
2009 Singapore Junior Math Olympiad, 1
In $\vartriangle ABC, \angle A= 2 \angle B$. Let $a,b,c$ be the lengths of its sides $BC,CA,AB$, respectively. Prove that $a^2 = b(b + c)$.
2008 Cuba MO, 1
Given a polynomial of degree $2$, $p(x) = ax^2 +bx+c$ define the function $$S(p) = (a -b)^2 + (b - c)^2 + (c - a)^2.$$ Determine the real number$ r$such that, for any polynomial $p(x)$ of degree $2$ with real roots, holds $S(p) \ge ra^2$
2009 Junior Balkan Team Selection Tests - Moldova, 6
Prove that there are no pairs of nonnegative integers $(x,y)$ that satisfy the equality $$x^3-y^3=x-y+2^{x-y}.$$
2023 Saint Petersburg Mathematical Olympiad, 6
There are several gentlemen in the meeting of the Diogenes Club, some of which are friends with each other (friendship is mutual). Let's name a participant unsociable if he has exactly one friend among those present at the meeting. By the club rules, the only friend of any unsociable member can leave the meeting (gentlemen leave the meeting one at a time). The purpose of the meeting is to achieve a situation in which that there are no friends left among the participants. Prove that if the goal is achievable, then the number of participants remaining at the meeting does not depend on who left and in what order.
1985 Putnam, A1
Determine, with proof, the number of ordered triples $\left(A_{1}, A_{2}, A_{3}\right)$ of sets which have the property that
(i) $A_{1} \cup A_{2} \cup A_{3}=\{1,2,3,4,5,6,7,8,9,10\},$ and
(ii) $A_{1} \cap A_{2} \cap A_{3}=\emptyset.$
Express your answer in the form $2^{a} 3^{b} 5^{c} 7^{d},$ where $a, b, c, d$ are nonnegative integers.
2010 QEDMO 7th, 7
Let $ABC$ be a triangle. Let $x_1$ and $x_2$ be two congruent circles, which touch each other and the segment $BC$, and which both lie within triangle $ABC$, and for which it also holds that $x_1$ touches the segment $CA$, and that $x_2$ is the segment $AB$. Let $X$ be the contact point of these two circles $x_1$ and $x_2$. Let $y_1$ and $y_2$ two congruent circles that touch each other and the segment $CA$, and both within of triangle $ABC$, and for which it also holds that $y_1$ touches the segment $AB$, and that $y_2$ the segment $BC$. Let $Y$ be the contact point of these two circles $y_1$ and $y_2$. Let $z_1$ and $z_2$ be two congruent circles that touch each other and the segment $AB$, and both within triangle $ABC$, and for which it also holds that $z_1$ touches the segment $BC$, and that $z_2$ the segment $CA$. Let $Z$ be the contact point of these two circles $z_1$ and $z_2$. Prove that the straight lines $AX, BY$ and $CZ$ intersect at a point.
2001 Federal Math Competition of S&M, Problem 3
Let $k$ be a positive integer and $N_k$ be the number of sequences of length $2001$, all members of which are elements of the set $\{0,1,2,\ldots,2k+1\}$, and the number of zeroes among these is odd. Find the greatest power of $2$ which divides $N_k$.
1967 Polish MO Finals, 3
There are 100 persons in a hall, everyone knowing at least 67 of the others. Prove that there always exist four of them who know each other
1986 Austrian-Polish Competition, 4
Find all triples (m,n,N) of positive integers numbers m,n and N such that
$m^N-n^N=2^{100}$ with N>1
2012 QEDMO 11th, 8
Prove that there are $2012$ points in the plane, none of which are three on one straight line and in pairs have integer distances .
2008 Abels Math Contest (Norwegian MO) Final, 4b
A point $D$ lies on the side $BC$ , and a point $E$ on the side $AC$ , of the triangle $ABC$ , and $BD$ and $AE$ have the same length. The line through the centres of the circumscribed circles of the triangles $ADC$ and $BEC$ crosses $AC$ in $K$ and $BC$ in $L$. Show that $KC$ and $LC$ have the same length.