Olympiad problems

2004 Manhattan Mathematical Olympiad, 1

Tags: geometry

Suppose two triangles have equal areas and equal perimeters. Prove that, if a side of one triangle is congruent to a side of the other triangle, then the two triangles are congruent.

2008 Thailand Mathematical Olympiad, 4

Tags: inequalities , algebra

Prove that $$\sqrt{a^2 + b^2 -\sqrt2 ab} +\sqrt{b^2 + c^2 -\sqrt2 bc} \ge \sqrt{a^2 + c^2}$$ for all real numbers $a, b, c > 0$

2012 Turkey Team Selection Test, 2

Tags: inequalities , pythagorean theorem , geometry

In a plane, the six different points $A, B, C, A', B', C'$ are given such that triangles $ABC$ and $A'B'C'$ are congruent, i.e. $AB=A'B', BC=B'C', CA=C'A'.$ Let $G$ be the centroid of $ABC$ and $A_1$ be an intersection point of the circle with diameter $AA'$ and the circle with center $A'$ and passing through $G.$ Define $B_1$ and $C_1$ similarly. Prove that \[ AA_1^2+BB_1^2+CC_1^2 \leq AB^2+BC^2+CA^2 \]

2021 Durer Math Competition Finals, 3

Tags: combinatorics

On the evening of Halloween a group of $n$ kids collected $k$ bars of chocolate of the same type. At the end of the evening they wanted to divide the bars so that everybody gets the same amount of chocolate, and none of the bars is broken into more than two pieces. For which $n$ and $k$ is this possible?

2010 Contests, 3

Tags:

Show that , for any positive integer $n$ , the sum of $8n+4$ consecutive positive integers cannot be a perfect square .

2005 AMC 12/AHSME, 18

Tags: geometry , analytic geometry , graphing lines , slope

Let $ A(2,2)$ and $ B(7,7)$ be points in the plane. Define $ R$ as the region in the first quadrant consisting of those points $ C$ such that $ \triangle ABC$ is an acute triangle. What is the closest integer to the area of the region $ R$? $ \textbf{(A)}\ 25 \qquad \textbf{(B)}\ 39 \qquad \textbf{(C)}\ 51 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 80$

2009 Stanford Mathematics Tournament, 5

Tags: algebra , calculus

Compute $\int_{0}^{\infty} t^5e^{-t}dt$

2008 F = Ma, 16

Tags:

A [i]massless[/i] spring with spring constant $k$ is vertically mounted so that bottom end is firmly attached to the ground, and the top end free. A ball with mass $m$ falls vertically down on the top end of the spring, becoming attached, so that the ball oscillates vertically on the spring. What equation describes the acceleration a of the ball when it is at a height $y$ above the [i]original[/i] position of the top end of the spring? Let down be negative, and neglect air resistance; $g$ is the magnitude of the acceleration of free fall. (a) $a=mv^2/y+g$ (b) $a=mv^2/k-g$ (c) $a=(k/m)y-g$ (d) $a=-(k/m)y+g$ (e) $a=-(k/m)y-g$

PEN H Problems, 26

Tags: diophantine equation , number theory , greatest common divisor

Solve in integers the following equation \[n^{2002}=m(m+n)(m+2n)\cdots(m+2001n).\]

1993 Korea - Final Round, 2

Tags: incenter , geometry

Let be given a triangle $ABC$ with $BC = a, CA = b, AB = c$. Find point $P$ in the plane for which $aAP^{2}+bBP^{2}+cCP^{2}$ is minimum, and compute this minimum.

2013 National Olympiad First Round, 29

Tags: geometry , circumcircle , geometric transformation , reflection

Let $O$ be the circumcenter of triangle $ABC$ with $|AB|=5$, $|BC|=6$, $|AC|=7$. Let $A_1$, $B_1$, $C_1$ be the reflections of $O$ over the lines $BC$, $AC$, $AB$, respectively. What is the distance between $A$ and the circumcenter of triangle $A_1B_1C_1$? $ \textbf{(A)}\ 6 \qquad\textbf{(B)}\ \sqrt {29} \qquad\textbf{(C)}\ \dfrac {19}{2\sqrt 6} \qquad\textbf{(D)}\ \dfrac {35}{4\sqrt 6} \qquad\textbf{(E)}\ \sqrt {\dfrac {35}3} $

2015 IMO Shortlist, G1

Tags: triangle , geometry

Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ = AH$.

Durer Math Competition CD 1st Round - geometry, 2011.C4

Tags: geometry , area

Given a grid rectangle of size $2010 \times 1340$. A grid point is called [i]fair [/i] if the $2$ axis-parallel lines passing through it from the upper left and lower right corners of the large rectangle cut out a rectangle of equal area (such a point is shown in the figure). How many fair grid points lie inside the rectangle? [img]https://cdn.artofproblemsolving.com/attachments/1/b/21d4fb47c94b774994ac1c3aae7690bb98c7ae.png[/img]

2000 Moldova National Olympiad, Problem 3

Tags: number theory

Suppose that $m,n\ge2$ are integers such that $m+n-1$ divides $m^2+n^2-1$. Prove that the number $m+n-1$ is not prime.

2023 BMT, 9

Tags: algebra

The boxes in the expression below are filled with the numbers $3$, $4$, $5$, $6$, $7$, and $8$, so that each number is used exactly once. What is the least possible value of the expression? $$\square \times \square +\square \times \square -\square \times \square$$

2014 India IMO Training Camp, 3

Tags: circumcircle , isosceles triangle , angle chasing , inversion , geometry

Let $ABC$ be a triangle with $\angle B > \angle C$. Let $P$ and $Q$ be two different points on line $AC$ such that $\angle PBA = \angle QBA = \angle ACB $ and $A$ is located between $P$ and $C$. Suppose that there exists an interior point $D$ of segment $BQ$ for which $PD=PB$. Let the ray $AD$ intersect the circle $ABC$ at $R \neq A$. Prove that $QB = QR$.

2018 Romania Team Selection Tests, 2

Tags: geometry , mixtilinear incircle , incenter , circumcircle

Let $ABC$ be a triangle, let $I$ be its incenter, let $\Omega$ be its circumcircle, and let $\omega$ be the $A$- mixtilinear incircle. Let $D,E$ and $T$ be the intersections of $\omega$ and $AB,AC$ and $\Omega$, respectively, let the line $IT$ cross $\omega$ again at $P$, and let lines $PD$ and $PE$ cross the line $BC$ at $M$ and $N$ respectively. Prove that points $D,E,M,N$ are concyclic. What is the center of this circle?

2020-21 IOQM India, 10

Tags: statistics , median , mean , algebra

Five students take a test on which any integer score from $0$ to $100$ inclusive is possible. What is the largest possible difference between the median and the mean of the scores? [i](The median of a set of scores is the middlemost score when the data is arranged in increasing order. It is exactly the middle score when there are an odd number of scores and it is the average of the two middle scores when there are an even number of scores.)[/i]

2005 Federal Math Competition of S&M, Problem 1

Tags: power mean , divisibility , algebra , number theory

Let $a$ and $b$ be positive integers and $K=\sqrt{\frac{a^2+b^2}2}$, $A=\frac{a+b}2$. If $\frac KA$ is a positive integer, prove that $a=b$.

2018 JBMO Shortlist, NT4

Tags: number theory

Prove that there exist infinitely many positive integers $n$ such that $\frac{4^n+2^n+1}{n^2+n+1}$ is a positive integer.

2024 AIME, 15

Tags: 3d geometry , prism , inequalities , geometry

Let $\mathcal{B}$ be the set of rectangular boxes that have volume $23$ and surface area $54$. Suppose $r$ is the least possible radius of a sphere that can fit any element of $\mathcal{B}$ inside it. Then $r^{2}$ can be expressed as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

2025 Sharygin Geometry Olympiad, 18

Tags: geometry

Let $ABCD$ be a quadrilateral such that the excircles $\omega_{1}$ and $\omega_{2}$ of triangles $ABC$ and $BCD$ touching their sides $AB$ and $BD$ respectively touch the extension of $BC$ at the same point $P$. The segment $AD$ meets $\omega_{2}$ at point $Q$, and the line $AD$ meets $\omega_{1}$ at $R$ and $S$. Prove that one of angles $RPQ$ and $SPQ$ is right Proposed by: I.Kukharchuk

2020 USA EGMO Team Selection Test, 6

Tags: algebra , polynomial , integer polynomial , periodic sequence , modular arithmetic

Find the largest integer $N \in \{1, 2, \ldots , 2019 \}$ such that there exists a polynomial $P(x)$ with integer coefficients satisfying the following property: for each positive integer $k$, $P^k(0)$ is divisible by $2020$ if and only if $k$ is divisible by $N$. Here $P^k$ means $P$ applied $k$ times, so $P^1(0)=P(0), P^2(0)=P(P(0)),$ etc.

2020 MBMT, 3

Tags:

Square $ABCD$ has a side length of 1. Point $E$ lies on the interior of $ABCD$, and is on the line $\overleftrightarrow{AC}$ such that the length of $\overline{AE}$ is 1. Find the shortest distance from point $E$ to a side of square $ABCD$. [i]Proposed by Chris Tong[/i]

2022 Moldova Team Selection Test, 7

Tags: number theory

Let $f:\mathbb{N} \rightarrow \mathbb{N},$ $f(n)=n^2-69n+2250$ be a function. Find the prime number $p$, for which the sum of the digits of the number $f(p^2+32)$ is as small as possible.