Found problems: 85335
2015 Mexico National Olympiad, 5
Let $I$ be the incenter of an acute-angled triangle $ABC$. Line $AI$ cuts the circumcircle of $BIC$ again at $E$. Let $D$ be the foot of the altitude from $A$ to $BC$, and let $J$ be the reflection of $I$ across $BC$. Show $D$, $J$ and $E$ are collinear.
2009 Saint Petersburg Mathematical Olympiad, 2
$ABCD$ is convex quadrilateral with $AB=CD$. $AC$ and $BD$ intersect in $O$. $X,Y,Z,T$ are midpoints of $BC,AD,AC,BD$. Prove, that circumcenter of $OZT$ lies on $XY$.
1992 Czech And Slovak Olympiad IIIA, 5
The function $f : (0,1) \to R$ is defined by
$f(x) = x$ if $x$ is irrational,
$f(x) = \frac{p+1}{q}$ if $x =\frac{p}{q}$ , where $(p,q) = 1$.
Find the maximum value of $f$ on the interval $(7/8,8/9)$.
2024 Malaysian IMO Training Camp, 3
Given $n$ students in the plane such that the $\frac{n(n-1)}{2}$ distances are pairwise distinct. Each student gives a candy each to the $k$ students closest to him. Given that each student receives the same amount of candies, determine all possible values of $n$ in terms of $k$.
[i]Proposed by Wong Jer Ren[/i]
2024 Princeton University Math Competition, B2
Let $ABCDE$ be a fixed regular pentagon of side length $1.$ An equilateral triangle $\triangle PQR$ with side length $1$ is initially placed with $P$ at $A, Q$ at $B,$ and $R$ outside the pentagon. The triangle then rolls without slipping around the perimeter of pentagon until it comes back to its starting point. The total area covered by any part of the triangle during its journey can be written as $\tfrac{a\sqrt{b}+c\pi}{d}$ for positive integers $a,b,c,d$ with $b$ square-free and $\gcd(a,c,d)=1.$ Find $a+b+c+d.$
2006 Finnish National High School Mathematics Competition, 1
Determine all pairs $(x, y)$ of positive integers for which the equation \[x + y + xy = 2006\] holds.
2021 Polish Junior MO Finals, 2
Point $M$ is the midpoint of the hypotenuse $AB$ of a right angled triangle $ABC$. Points $P$ and $Q$ lie on segments $AM$ and $MB$ respectively and $PQ=CQ$. Prove that $AP\leq 2\cdot MQ$.
2017 India National Olympiad, 2
Suppose $n \ge 0$ is an integer and all the roots of $x^3 +
\alpha x + 4 - ( 2 \times 2016^n) = 0$ are integers. Find all possible values of $\alpha$.
2018 Korea Junior Math Olympiad, 3
Let there be a scalene triangle $ABC$, and denote $M$ by the midpoint of $BC$. The perpendicular bisector of $BC$ meets the circumcircle of $ABC$ at point $P$, on the same side with $A$ with respect to $BC$. Let the incenters of $ABM$ and $AMC$ be $I,J$, respectively. Let $\angle BAC=\alpha$, $\angle ABC=\beta$, $\angle BCA=\gamma$. Find $\angle IPJ$.
2010 India Regional Mathematical Olympiad, 2
Let $P_1(x) = ax^2 - bx - c$, $P_2(x) = bx^2 - cx - a$, $P_3(x) = cx^2 - ax - b$ be three quadratic polynomials. Suppose there exists a real number $\alpha$ such that $P_1(\alpha) = P_2(\alpha) = P_3(\alpha)$. Prove that $a = b = c$.
1971 Bundeswettbewerb Mathematik, 2
The inhabitants of a planet speak a language only using the letters $A$ and $O$. To avoid mistakes, any two words of equal length differ at least on three positions. Show that there are not more than $\frac{2^n}{n+1}$ words with $n$ letters.
2013 NIMO Problems, 6
Given a regular dodecagon (a convex polygon with 12 congruent sides and angles) with area 1, there are two possible ways to dissect this polygon into 12 equilateral triangles and 6 squares. Let $T_1$ denote the union of all triangles in the first dissection, and $S_1$ the union of all squares. Define $T_2$ and $S_2$ similarly for the second dissection. Let $S$ and $T$ denote the areas of $S_1 \cap S_2$ and $T_1 \cap T_2$, respectively. If $\frac{S}{T} = \frac{a+b\sqrt{3}}{c}$ where $a$ and $b$ are integers, $c$ is a positive integer, and $\gcd(a,c)=1$, compute $10000a+100b+c$.
[i]Proposed by Lewis Chen[/i]
2017 India IMO Training Camp, 3
Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\Omega$ with $AC \perp BD$. Let $P=AC \cap BD$ and $W,X,Y,Z$ be the projections of $P$ on the lines $AB, BC, CD, DA$ respectively. Let $E,F,G,H$ be the mid-points of sides $AB, BC, CD, DA$ respectively.
(a) Prove that $E,F,G,H,W,X,Y,Z$ are concyclic.
(b) If $R$ is the radius of $\Omega$ and $d$ is the distance between its centre and $P$, then find the radius of the circle in (a) in terms of $R$ and $d$.
2017 Kosovo National Mathematical Olympiad, 1
Find all ordered pairs $(a,b)$, of natural numbers, where $1<a,b\leq 100$, such that
$\frac{1}{\log_{a}{10}}+\frac{1}{\log_{b}{10}}$ is a natural number.
1961 Polish MO Finals, 2
Prove that if $ a + b = 1 $, then $$
a^5 + b^5 \geq \frac{1}{16}$$
2012 Stars of Mathematics, 4
Consider a set $X$ with $|X| = n\geq 1$ elements. A family $\mathcal{F}$ of distinct subsets of $X$ is said to have property $\mathcal{P}$ if there exist $A,B \in \mathcal{F}$ so that $A\subset B$ and $|B\setminus A| = 1$.
i) Determine the least value $m$, so that any family $\mathcal{F}$ with $|\mathcal{F}| > m$ has property $\mathcal{P}$.
ii) Describe all families $\mathcal{F}$ with $|\mathcal{F}| = m$, and not having property $\mathcal{P}$.
([i]Dan Schwarz[/i])
2017 Thailand Mathematical Olympiad, 8
Let $a, b, c$ be side lengths of a right triangle. Determine the minimum possible value of $\frac{a^3 + b^3 + c^3}{abc}$.
2010 Stanford Mathematics Tournament, 8
Let $P(x)$ be a polynomial of degree $n$ such that $P(x)=3^k$ for $0\le k \le n$. Find $P(n+1)$
2007 Switzerland - Final Round, 9
Find all pairs $(a, b)$ of natural numbers such that $$\frac{a^3 + 1}{2ab^2 + 1}$$ is an integer.
1966 IMO Shortlist, 53
Prove that in every convex hexagon of area $S$ one can draw a diagonal that cuts off a triangle of area not exceeding $\frac{1}{6}S.$
1967 Dutch Mathematical Olympiad, 2
Consider arithmetic sequences where all terms are natural numbers. If the first term of such a sequence is $1$, prove that that sequence contains infinitely many terms that are the cube of a natural number. Give an example of such a sequence in which no term is the cube of a natural number and show the correctness of this example.
2019 Saudi Arabia JBMO TST, 1
Let $n$ be positive integer. Given is a grid $nxn$. Some cells of the grid are colored in green, so that no two green squares share a common side. Is it possible, however the green cells are colored, to place $n$ rooks, so that none of the rooks is on green cell, and no two rooks attack each other, if
a) n=19
b) n=20
2006 Bundeswettbewerb Mathematik, 2
Find all functions $f: Q^{+}\rightarrow R$ such that
$f(x)+f(y)+2xyf(xy)=\frac{f(xy)}{f(x+y)}$ for all $x,y\in Q^{+}$
2012 JBMO ShortLists, 7
Find all $a , b , c \in \mathbb{N}$ for which \[1997^a+15^b=2012^c\]
2010 ISI B.Stat Entrance Exam, 8
Take $r$ such that $1\le r\le n$, and consider all subsets of $r$ elements of the set $\{1,2,\ldots,n\}$. Each subset has a smallest element. Let $F(n,r)$ be the arithmetic mean of these smallest elements. Prove that: \[ F(n,r)={n+1\over r+1}. \]