Found problems: 85335
2019 USMCA, 10
Let $a, b$ be positive real numbers with $a>b$. Compute the minimum possible value of the expression
\[\frac{a^2b - ab^2 + 8}{ab - b^2}.\]
2022 Pan-American Girls' Math Olympiad, 3
Let $ABC$ be an acute triangle with $AB< AC$. Denote by $P$ and $Q$ points on the segment $BC$ such that $\angle BAP = \angle CAQ < \frac{\angle BAC}{2}$. $B_1$ is a point on segment $AC$. $BB_1$ intersects $AP$ and $AQ$ at $P_1$ and $Q_1$, respectively. The angle bisectors of $\angle BAC$ and $\angle CBB_1$ intersect at $M$. If $PQ_1\perp AC$ and $QP_1\perp AB$, prove that $AQ_1MPB$ is cyclic.
2016 IFYM, Sozopol, 1
Find all functions $f: \mathbb{R}^+\rightarrow \mathbb{R}^+$ with the following property: $a,b,$ and $c$ are lengths of sides of a triangle, if and only if $f(a),f(b),$ and $f(c)$ are lengths of sides of a triangle.
1970 AMC 12/AHSME, 12
A circle with radius $r$ is tangent to sides $AB$, $AD$, and $CD$ of rectangle $ABCD$ and passes through the midpoint of diagonal $AC$.The area of the rectangle in terms of $r$, is
$\textbf{(A) }4r^2\qquad\textbf{(B) }6r^2\qquad\textbf{(C) }8r^2\qquad\textbf{(D) }12r^2\qquad \textbf{(E) }20r^2$
2005 Dutch Mathematical Olympiad, 4
Let $ABCD$ be a quadrilateral with $AB \parallel CD$, $AB > CD$. Prove that the line passing through $AC \cap BD$ and $AD \cap BC$ passes through the midpoints of $AB$ and $CD$.
2015 Dutch IMO TST, 5
For a positive integer $n$, we de ne $D_n$ as the largest integer that is a divisor of $a^n + (a + 1)^n + (a + 2)^n$ for all positive integers $a$.
1. Show that for all positive integers $n$, the number $D_n$ is of the form $3^k$ with $k \ge 0$ an integer.
2. Show that for all integers $k \ge 0$ there exists a positive integer n such that $D_n = 3^k$.
2002 Irish Math Olympiad, 1
A $ 3 \times n$ grid is filled as follows. The first row consists of the numbers from $ 1$ to $ n$ arranged in ascending order. The second row is a cyclic shift of the top row: $ i,i\plus{}1,...,n,1,2,...,i\minus{}1$ for some $ i$. The third row has the numbers $ 1$ to $ n$ in some order so that in each of the $ n$ columns, the sum of the three numbers is the same. For which values of $ n$ is it possible to fill the grid in this way? For all such $ n$, determine the number of different ways of filling the grid.
2013 Olympic Revenge, 1
Let $n$ to be a positive integer. A family $\wp$ of intervals $[i, j]$ with $0 \le i < j \le n$ and $i$, $j$ integers is considered [i]happy[/i] if, for any $I_1 = [i_1, j_1] \in \wp$ and $I_2 = [i_2, j_2] \in \wp$ such that $I_1 \subset I_2$, we have $i_1 = i_2$ [b]or[/b] $j_1 = j_2$.
Determine the maximum number of elements of a [i]happy[/i] family.
2023 Novosibirsk Oral Olympiad in Geometry, 4
Inside the convex pentagon $ABCDE$, a point $O$ was chosen, and it turned out that all five triangles $AOB$, $BOC$, $COD$, $DOE$ and $EOA$ are congrunet to each other. Prove that these triangles are isosceles or right-angled.
1999 IMO Shortlist, 5
Let $ABC$ be a triangle, $\Omega$ its incircle and $\Omega_{a}, \Omega_{b}, \Omega_{c}$ three circles orthogonal to $\Omega$ passing through $(B,C),(A,C)$ and $(A,B)$ respectively. The circles $\Omega_{a}$ and $\Omega_{b}$ meet again in $C'$; in the same way we obtain the points $B'$ and $A'$. Prove that the radius of the circumcircle of $A'B'C'$ is half the radius of $\Omega$.
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 $$
a^5 + b^5 \geq \frac{1}{16}$$