Olympiad problems

2023 Rioplatense Mathematical Olympiad, 1

An integer $n\geq 3$ is [i]poli-pythagorean[/i] if there exist $n$ positive integers pairwise distinct such that we can order these numbers in the vertices of a regular $n$-gon such that the sum of the squares of consecutive vertices is also a perfect square. For instance, $3$ is poli-pythagorean, because if we write $44,117,240$ in the vertices of a triangle we notice: $$44^2+117^2=125^2, 117^2+240^2=267^2, 240^2+44^2=244^2$$ Determine all poli-pythagorean integers.

2018 Sharygin Geometry Olympiad, 20

Tags: geometry

Let the incircle of a nonisosceles triangle $ABC$ touch $AB$, $AC$ and $BC$ at points $D$, $E$ and $F$ respectively. The corresponding excircle touches the side $BC$ at point $N$. Let $T$ be the common point of $AN$ and the incircle, closest to $N$, and $K$ be the common point of $DE$ and $FT$. Prove that $AK||BC$.

2005 Greece Team Selection Test, 2

Tags: geometry , reflection , circumcircle , IMO Shortlist

Let $\Gamma$ be a circle and let $d$ be a line such that $\Gamma$ and $d$ have no common points. Further, let $AB$ be a diameter of the circle $\Gamma$; assume that this diameter $AB$ is perpendicular to the line $d$, and the point $B$ is nearer to the line $d$ than the point $A$. Let $C$ be an arbitrary point on the circle $\Gamma$, different from the points $A$ and $B$. Let $D$ be the point of intersection of the lines $AC$ and $d$. One of the two tangents from the point $D$ to the circle $\Gamma$ touches this circle $\Gamma$ at a point $E$; hereby, we assume that the points $B$ and $E$ lie in the same halfplane with respect to the line $AC$. Denote by $F$ the point of intersection of the lines $BE$ and $d$. Let the line $AF$ intersect the circle $\Gamma$ at a point $G$, different from $A$. Prove that the reflection of the point $G$ in the line $AB$ lies on the line $CF$.

2019 IMO Shortlist, C2

Tags: IMO Shortlist , IMO Shortlist 2019 , combinatorics , induction , Local argument

You are given a set of $n$ blocks, each weighing at least $1$; their total weight is $2n$. Prove that for every real number $r$ with $0 \leq r \leq 2n-2$ you can choose a subset of the blocks whose total weight is at least $r$ but at most $r + 2$.

1996 All-Russian Olympiad Regional Round, 11.8

Tags: combinatorics , periodic

Is there an infinite periodic sequence consisting of the letters $a$ and$ b$, such that if all letters are replaced simultaneously $a$ to $aba$ and letters $b$ to $bba$ does it transform into itself (possibly with a shift)? (A sequence is called periodic if there is such natural number $n$, which for every $i = 1, 2, . . . i$-th member of this sequence is equal to the ($i + n$)- th.)

2024 District Olympiad, P1

Tags: Ring Theory , number theory

Determine the integers $n\geqslant 2$ for which the equation $x^2-\hat{3}\cdot x+\hat{5}=\hat{0}$ has a unique solution in $(\mathbb{Z}_n,+,\cdot).$

2006 Harvard-MIT Mathematics Tournament, 10

Tags: calculus , function , integration , logarithms

Suppose $f$ and $g$ are differentiable functions such that \[xg(f(x))f^\prime(g(x))g^\prime(x)=f(g(x))g^\prime(f(x))f^\prime(x)\] for all real $x$. Moreover, $f$ is nonnegative and $g$ is positive. Furthermore, \[\int_0^a f(g(x))dx=1-\dfrac{e^{-2a}}{2}\] for all reals $a$. Given that $g(f(0))=1$, compute the value of $g(f(4))$.

2018 Saudi Arabia IMO TST, 2

Tags: geometry , circumcircle , right angle , angle bisector

Let $ABC$ be an acute-angled triangle inscribed in circle $(O)$. Let $G$ be a point on the small arc $AC$ of $(O)$ and $(K)$ be a circle passing through $A$ and $G$. Bisector of $\angle BAC$ cuts $(K)$ again at $P$. The point $E$ is chosen on $(K)$ such that $AE$ is parallel to $BC$. The line $PK$ meets the perpendicular bisector of $BC$ at $F$. Prove that $\angle EGF = 90^o$.

1991 Kurschak Competition, 3

Tags: combinatorics unsolved , combinatorics

Consider $998$ red points on the plane with no three collinear. We select $k$ blue points in such a way that inside each triangle whose vertices are red points, there is a blue point as well. Find the smallest $k$ for which the described selection of blue points is possible for any configuration of $998$ red points.

2007 Stanford Mathematics Tournament, 6

Tags: probability , Stanford , college

Team Stanford has a $ \frac{1}{3}$ chance of winning any given math contest. If Stanford competes in 4 contests this quarter, what is the probability that the team will win at least once?

2024 IMO, 6

Tags: algebra , IMO

Let $\mathbb{Q}$ be the set of rational numbers. A function $f: \mathbb{Q} \to \mathbb{Q}$ is called aquaesulian if the following property holds: for every $x,y \in \mathbb{Q}$, \[ f(x+f(y)) = f(x) + y \quad \text{or} \quad f(f(x)+y) = x + f(y). \] Show that there exists an integer $c$ such that for any aquaesulian function $f$ there are at most $c$ different rational numbers of the form $f(r) + f(-r)$ for some rational number $r$, and find the smallest possible value of $c$.

2022 Stars of Mathematics, 2

Tags: geometry

Let $ABCD$ be a convex quadrilateral and $P$ be a point in its interior, such that $\angle APB+\angle CPD=\angle BPC+\angle DPA$, $\angle PAD+\angle PCD=\angle PAB+\angle PCB$ and $\angle PDC+ \angle PBC= \angle PDA+\angle PBA$. Prove that the quadrilateral is circumscribed.

1986 IMO Longlists, 44

Tags: geometry , incenter , circumcircle , collinearity , IMO Shortlist

The circle inscribed in a triangle $ABC$ touches the sides $BC,CA,AB$ in $D,E, F$, respectively, and $X, Y,Z$ are the midpoints of $EF, FD,DE$, respectively. Prove that the centers of the inscribed circle and of the circles around $XYZ$ and $ABC$ are collinear.

2020 LMT Spring, 13

Tags:

In the game of Flow, a path is drawn through a $3\times3$ grid of squares obeying the following rules: i A path is continuous with no breaks (it can be drawn without lifting a pencil). ii A path that spans multiple squares can only be drawn between colored squares that share a side. iii A path cannot go through a square more than once. Compute the number of ways to color a positive number of squares on the grid such that a valid path can be drawn. An example of one such coloring and a valid path is shown below. [Insert Diagram] [i]Proposed by Alex Li[/i]

2021 AMC 10 Spring, 20

Tags: AMC , AMC 10 , AMC 10 A , 2021 AMC 10A , probs

In how many ways can the sequence $1,2,3,4,5$ be arranged so that no three consecutive terms are increasing and no three consecutive terms are decreasing? $\textbf{(A) } 10 \qquad \textbf{(B) } 18 \qquad \textbf{(C) } 24 \qquad \textbf{(D) } 32 \qquad \textbf{(E) } 44$

2024 ELMO Shortlist, A1

Tags: algebra

Let $m,n,a_1,a_2,\dots,a_n$ be positive integers and $r$ be a real number. Prove that the equation \[\lfloor a_1x\rfloor+\lfloor a_2x\rfloor+\cdots+\lfloor a_nx\rfloor=sx+r\] has exactly $ms$ solutions in $x$, where $s=a_1+a_2+\cdots+a_n+\frac1m$. [i]Linus Tang[/i]

2018 IMO Shortlist, A2

Tags: IMO , algebra , Sequence , system of equations , imo 2018 , IMO Shortlist , Hi

Find all integers $n \geq 3$ for which there exist real numbers $a_1, a_2, \dots a_{n + 2}$ satisfying $a_{n + 1} = a_1$, $a_{n + 2} = a_2$ and $$a_ia_{i + 1} + 1 = a_{i + 2},$$ for $i = 1, 2, \dots, n$. [i]Proposed by Patrik Bak, Slovakia[/i]

1990 IMO Longlists, 1

Tags: geometry , circumcircle , parallelogram , IMO Shortlist , IMO Longlist

In triangle $ABC, O$ is the circumcenter, $H$ is the orthocenter. Construct the circumcircles of triangles $CHB, CHA$ and $AHB$, and let their centers be $A_1, B_1, C_1$, respectively. Prove that triangles $ABC$ and $A_1B_1C_1$ are congruent, and their nine-point circles coincide.

2018 Federal Competition For Advanced Students, P2, 2

Tags: geometric inequality , geometry , cyclic quadrilateral

Let $A, B, C$ and $D$ be four different points lying on a common circle in this order. Assume that the line segment $AB$ is the (only) longest side of the inscribed quadrilateral $ABCD$. Prove that the inequality $AB + BD > AC + CD$ holds. [i](Proposed by Karl Czakler)[/i]

Gheorghe Țițeica 2024, P4

Tags: Olympiad Combinatorics

For a set $M$ of $n\geq 3$ points in the plane we define a [i]path[/i] to be a polyline $A_1A_2\dots A_n$, where $M=\{A_1,A_2,\dots ,A_n\}$ and define its length to be $A_1A_2+A_2A_3+\dots +A_{n-1}A_n$. We call $M$ [i]path-unique[/i] if any two distinct paths have different lengths and [i]segment-unique[/i] if any two nondegenerate segments with their ends among the points in $M$ have different lengths. Determine the positive integers $n\geq 3$ such that: a) any segment-unique set $M$ of $n$ points in the plane is path-unique; b) any path-unique set $M$ of $n$ points in the plane is segment-unique. [i]Cristi Săvescu[/i]

2015 Geolympiad Spring, 2

Tags: Geolympiad Spring 2015

Let $ABC$ be a triangle and $w$ its incircle. $w$ touches $BC,CA$ at $A_1,B_1$ respectively. The second intersection of $AA_1$ and $w$ is $A_2$, similarly define $B_2$. Then $AB,A_1B_1,A_2B_2$ concur at a point $C_3$.

2012 Saint Petersburg Mathematical Olympiad, 4

Tags: number theory

Some notzero reals numbers are placed around circle. For every two neighbour numbers $a,b$ it true, that $a+b$ and $\frac{1}{a}+\frac{1}{b}$ are integer. Prove that there are not more than $4$ different numbers.

2024 Spain Mathematical Olympiad, 6

Tags: number theory , Spain , modular arithmetic , modulo class , floor function

Let $a$, $b$ and $n$ be positive integers, satisfying that $bn$ divides $an-a+1$. Let $\alpha=a/b$. Prove that, when the numbers $\lfloor\alpha\rfloor,\lfloor2\alpha\rfloor,\dots,\lfloor(n-1)\alpha\rfloor$ are divided by $n$, the residues are $1,2,\dots,n-1$, in some order.

1965 Putnam, A5

Tags: Putnam

In how many ways can the integers from $1$ to $n$ be ordered subject to the condition that, except for the first integer on the left, every integer differs by $1$ from some integer to the left of it?

2014 Online Math Open Problems, 4

Tags: Online Math Open , geometry , parallelogram

A crazy physicist has discovered a new particle called an emon. He starts with two emons in the plane, situated a distance $1$ from each other. He also has a crazy machine which can take any two emons and create a third one in the plane such that the three emons lie at the vertices of an equilateral triangle. After he has five total emons, let $P$ be the product of the $\binom 52 = 10$ distances between the $10$ pairs of emons. Find the greatest possible value of $P^2$. [i]Proposed by Yang Liu[/i]