Olympiad problems

1983 Swedish Mathematical Competition, 6

Show that the only real solution to \[\left\{ \begin{array}{l} x(x+y)^2 = 9 \\ x(y^3 - x^3) = 7 \\ \end{array} \right. \] is $x = 1$, $y = 2$.

1998 Putnam, 5

Tags:

Let $\mathcal{F}$ be a finite collection of open discs in $\mathbb{R}^2$ whose union contains a set $E\subseteq \mathbb{R}^2$. Show that there is a pairwise disjoint subcollection $D_1,\ldots,D_n$ in $\mathcal{F}$ such that \[E\subseteq\cup_{j=1}^n 3D_j.\] Here, if $D$ is the disc of radius $r$ and center $P$, then $3D$ is the disc of radius $3r$ and center $P$.

2005 Germany Team Selection Test, 1

Tags: number theory , equation , divisor

Let $\tau(n)$ denote the number of positive divisors of the positive integer $n$. Prove that there exist infinitely many positive integers $a$ such that the equation $ \tau(an)=n $ does not have a positive integer solution $n$.

2017 Kazakhstan NMO, Problem 4

Tags: geometry

The acute triangle $ABC$ $(AC> BC)$ is inscribed in a circle with the center at the point $O$, and $CD$ is the diameter of this circle. The point $K$ is on the continuation of the ray $DA$ beyond the point $A$. And the point $L$ is on the segment $BD$ $(DL> LB)$ so that $\angle OKD = \angle BAC$, $\angle OLD = \angle ABC$. Prove that the line $KL$ passes through the midpoint of the segment $AB$.

Kvant 2025, M2828

Tags: algebra , polynomial

Maxim has guessed a polynomial $f(x)$ of degree $n$. Sasha wants to guess it (knowing $n$). During a turn, Sasha can name a certain segment $[a;b]$ and Maxim will give in response the maximum value of $f(x)$ on the segment $[a;b]$. Will Sasha be able to guess $f(x)$ in a finite number of steps? [i]M. Didin[/i]

2004 Bosnia and Herzegovina Team Selection Test, 1

Tags: collinear , touching , geometry , circles

Circle $k$ with center $O$ is touched from inside by two circles in points $S$ and $T,$ respectively. Let those two circles intersect at points $M$ and $N$, such that $N$ is closer to line $ST$. Prove that $OM$ and $MN$ are perpendicular iff $S$, $N$ and $T$ are collinear

1969 IMO Longlists, 30

Tags: number theory , sophie germain identity , composite number

$(GDR 2)^{IMO1}$ Prove that there exist infinitely many natural numbers $a$ with the following property: The number $z = n^4 + a$ is not prime for any natural number $n.$

2014 Canada National Olympiad, 2

Tags: floor function , combinatorics

Let $m$ and $n$ be odd positive integers. Each square of an $m$ by $n$ board is coloured red or blue. A row is said to be red-dominated if there are more red squares than blue squares in the row. A column is said to be blue-dominated if there are more blue squares than red squares in the column. Determine the maximum possible value of the number of red-dominated rows plus the number of blue-dominated columns. Express your answer in terms of $m$ and $n$.

2005 USAMO, 4

Tags: 3d geometry , prism , parallelogram , geometry

Legs $L_1, L_2, L_3, L_4$ of a square table each have length $n$, where $n$ is a positive integer. For how many ordered 4-tuples $(k_1, k_2, k_3, k_4)$ of nonnegative integers can we cut a piece of length $k_i$ from the end of leg $L_i \; (i=1,2,3,4)$ and still have a stable table? (The table is [i]stable[/i] if it can be placed so that all four of the leg ends touch the floor. Note that a cut leg of length 0 is permitted.)

2016 Dutch IMO TST, 1

Tags: algebra , inequalities

Prove that for all positive reals $a, b,c$ we have: $a +\sqrt{ab}+ \sqrt[3]{abc}\le \frac43 (a + b + c)$

1994 Chile National Olympiad, 4

Tags: sphere , combinatorial geometry , combinatorics , circles , 3d geometry , geometry

Consider a box of dimensions $10$ cm $\times 16$ cm $\times 1$ cm. Determine the maximum number of balls of diameter $ 1$ cm that the box can contain.

2002 Chile National Olympiad, 1

Tags: number theory

A Metro ticket, which has six digits, is considered a "lucky number" if its six digits are different and their first three digits add up to the same as the last three (A number such as $026134$ is "lucky number"). Show that the sum of all the "lucky numbers" is divisible by $2002$.

2024 AMC 12/AHSME, 9

Tags: analytic geometry , number theory

A dartboard is the region B in the coordinate plane consisting of points $(x, y)$ such that $|x| + |y| \le 8$. A target T is the region where $(x^2 + y^2 - 25)^2 \le 49$. A dart is thrown at a random point in B. The probability that the dart lands in T can be expressed as $\frac{m}{n} \pi$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$? $ \textbf{(A) }39 \qquad \textbf{(B) }71 \qquad \textbf{(C) }73 \qquad \textbf{(D) }75 \qquad \textbf{(E) }135 \qquad $

2017 Harvard-MIT Mathematics Tournament, 6

Tags: geometry

Let $ABCD$ be a convex quadrilateral with $AC = 7$ and $BD = 17$. Let $M$, $P$, $N$, $Q$ be the midpoints of sides $AB$, $BC$, $CD$, $DA$ respectively. Compute $MN^2 + PQ^2$ [color = red]The official problem statement does not have the final period.[/color]

2000 Mongolian Mathematical Olympiad, Problem 2

Tags: geometry , circles

Circles $\omega_1,\omega_2,\omega_3$ with centers $O_1,O_2,O_3$, respectively, are externally tangent to each other. The circle $\omega_1$ touches $\omega_2$ at $P_1$ and $\omega_3$ at $P_2$. For any point $A$ on $\omega_1$, $A_1$ denotes the point symmetric to $A$ with respect to $O_1$. Show that the intersection points of $AP_2$ with $\omega_3$, $A_1P_3$ with $\omega_2$, and $AP_3$ with $A_1P_2$ lie on a line.

2013 USAMTS Problems, 3

Tags: geometry , induction , ratio , inequalities , limit , arithmetic sequence

An infinite sequence of positive real numbers $a_1,a_2,a_3,\dots$ is called [i]territorial[/i] if for all positive integers $i,j$ with $i<j$, we have $|a_i-a_j|\ge\tfrac1j$. Can we find a territorial sequence $a_1,a_2,a_3,\dots$ for which there exists a real number $c$ with $a_i<c$ for all $i$?

2007 Baltic Way, 16

Tags: modular arithmetic , relatively prime , number theory

Let $a$ and $b$ be rational numbers such that $s=a+b=a^2+b^2$. Prove that $s$ can be written as a fraction where the denominator is relatively prime to $6$.

1991 Arnold's Trivium, 83

Tags:

Find solutions of the equation $u_t=u_{xxx}+uu_x$ in the form of a traveling wave $u=\varphi(x-ct)$, $\varphi(\pm\infty)=0$.

2016 Bulgaria National Olympiad, Problem 5

Tags: geometry , circumcircle

Let $\triangle {ABC} $ be isosceles triangle with $AC=BC$ . The point $D$ lies on the extension of $AC$ beyond $C$ and is that $AC>CD$. The angular bisector of $ \angle BCD $ intersects $BD$ at point $N$ and let $M$ be the midpoint of $BD$. The tangent at $M$ to the circumcircle of triangle $AMD$ intersects the side $BC$ at point $P$. Prove that points $A,P,M$ and $N$ lie on a circle.

2013 Ukraine Team Selection Test, 8

Tags: geometry , circumcircle , trigonometry , triangle , reflection

Let $ABC$ be a triangle with $AB \neq AC$ and circumcenter $O$. The bisector of $\angle BAC$ intersects $BC$ at $D$. Let $E$ be the reflection of $D$ with respect to the midpoint of $BC$. The lines through $D$ and $E$ perpendicular to $BC$ intersect the lines $AO$ and $AD$ at $X$ and $Y$ respectively. Prove that the quadrilateral $BXCY$ is cyclic.

1999 Harvard-MIT Mathematics Tournament, 8

Tags: algebra , polynomial

If $f(x)$ is a monic quartic polynomial such that $f(-1)=-1$, $f(2)=-4$, $f(-3)=-9$, and $f(4)=-16$, find $f(1)$.

2022 BMT, 9

Tags: geometry

Seven spheres are situated in space such that no three centers are collinear, no four centers are coplanar, and every pair of spheres intersect each other at more than one point. For every pair of spheres, the plane on which the intersection of the two spheres lies in is drawn. What is the least possible number of sets of four planes that intersect in at least one point?

2005 Korea National Olympiad, 5

Tags: geometry

Let $P$ be a point that lies outside of circle $O$. A line passes through $P$ and meets the circle at $A$ and $B$, and another line passes through $P$ and meets the circle at $C$ and $D$. The point $A$ is between $P$ and $B$, $C$ is between $P$ and $D$. Let the intersection of segment $AD$ and $BC$ be $L$ and construct $E$ on ray $(PA$ so that $BL \cdot PE = DL \cdot PD$. Show that $M$ is the midpoint of the segment $DE$, where $M$ is the intersection of lines $PL$ and $DE$.

1984 IMO Longlists, 14

Tags: linear recurrence , algebra , sequence , recurrence relation

Let $c$ be a positive integer. The sequence $\{f_n\}$ is defined as follows: \[f_1 = 1, f_2 = c, f_{n+1} = 2f_n - f_{n-1} + 2 \quad (n \geq 2).\] Show that for each $k \in \mathbb N$ there exists $r \in \mathbb N$ such that $f_kf_{k+1}= f_r.$

2010 IMO Shortlist, 1

Tags: algebra , functional equation , fe

Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds \[ f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \] where $\left\lfloor a\right\rfloor $ is greatest integer not greater than $a.$ [i]Proposed by Pierre Bornsztein, France[/i]