Olympiad problems

2015 AMC 12/AHSME, 12

The parabolas $y=ax^2-2$ and $y=4-bx^2$ intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area $12$. What is $a+b$? $\textbf{(A) }1\qquad\textbf{(B) }1.5\qquad\textbf{(C) }2\qquad\textbf{(D) }2.5\qquad\textbf{(E) }3$

2021 Dutch IMO TST, 1

Tags: number theory , sequence , recurrence relation

The sequence of positive integers $a_0, a_1, a_2, . . .$ is defined by $a_0 = 3$ and $$a_{n+1} - a_n = n(a_n - 1)$$ for all $n \ge 0$. Determine all integers $m \ge 2$ for which $gcd (m, a_n) = 1$ for all $n \ge 0$.

2023 Malaysian IMO Training Camp, 1

Tags: combinatorics

Ivan has a $m \times n$ board, and he color some squares black, so that no three black squares form a L-triomino up to rotations and reflections. What is the maximal number of black squares that Ivan can color? [i]Proposed by Ivan Chan Kai Chin[/i]

2004 AMC 10, 22

Tags: geometry , circumcircle , analytic geometry , inradius , incenter , trigonometry , pythagorean theorem

A triangle with sides of $ 5$, $ 12$, and $ 13$ has both an inscibed and a circumscribed circle. What is the distance between the centers of those circles? $ \textbf{(A)}\ \frac{3\sqrt{5}}{2}\qquad \textbf{(B)}\ \frac{7}{2}\qquad \textbf{(C)}\ \sqrt{15}\qquad \textbf{(D)}\ \frac{\sqrt{65}}{2}\qquad \textbf{(E)}\ \frac{9}{2}$

2019 USMCA, 10

Tags:

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

Tags: geometry

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

Tags: function , algebra , triangle inequality , inequalities

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

Tags: geometry , rectangle

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

Tags: geometry , trapezoid , ratio , similar triangles

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

Tags: divisor , maximum , exponential , number theory

For a positive integer $n$, we dene $D_n$ 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

Tags: floor function , combinatorics

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

Tags: pigeonhole principle , combinatorics

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

Tags: geometry , congruent triangles , pentagon , congruence

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

Tags: geometry

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

Tags: incenter , circumcircle , geometric transformation , reflection , algebra , geometry

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

Tags: circumcircle , geometry

$ABCD$ 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

Tags: function , rational , max , algebra , irrational

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

Tags: combinatorics

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

Tags: geometry

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

Tags: number theory

Determine all pairs $(x, y)$ of positive integers for which the equation \[x + y + xy = 2006\] holds.

2021 Polish Junior MO Finals, 2

Tags: inequalities , geometry , triangle geometry

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

Tags: number theory

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

Tags: geometry , incenter

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

Tags: quadratic , algebra , polynomial

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

Tags: combinatorics

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.