Olympiad problems

Gheorghe Țițeica 2025, P3

Tags: perimeter , geometry

Out of all the nondegenerate triangles with positive integer sides and perimeter $100$, find the one with the smallest area.

2014 Stars Of Mathematics, 2

Tags: number theory

Let $N$ be an arbitrary positive integer. Prove that if, from among any $n$ consecutive integers larger than $N$, one may select $7$ of them, pairwise co-prime, then $n\geq 22$. ([i]Dan Schwarz[/i])

2017-IMOC, A3

Tags: algebra , system of equations

Solve the following system of equations: $$\begin{cases} x^3+y+z=1\\ x+y^3+z=1\\ x+y+z^3=1\end{cases}$$

2021 Israel TST, 3

Tags: geometry , incircle , orthocenter , common tangent

Let $ABC$ be an acute triangle with orthocenter $H$. Prove that there is a line $l$ which is parallel to $BC$ and tangent to the incircles of $ABH$ and $ACH$.

1999 All-Russian Olympiad, 2

Tags: induction , number theory

Find all bounded sequences $(a_n)_{n=1}^\infty$ of natural numbers such that for all $n \ge 3$, \[ a_n = \frac{a_{n-1} + a_{n-2}}{\gcd(a_{n-1}, a_{n-2})}. \]

2000 Korea Junior Math Olympiad, 3

Tags: geometry

Acute triangle $ABC$ is inscribed in circle $O$. $P$ is the foot of altitude from $A$ to $BC$, and $D$ is the intersection of $O$ and line $AP$. $M, N$ are midpoint of $AB, AC$ respectively. $MP$ and $CD$ intersects at $Q$, and $NP$ and $BD$ intersects at $R$. Show that $AD, BQ, CR$ meet at one point if and only if $AB=AC$.

1998 National Olympiad First Round, 3

Tags:

How many ways are there to divide a set with 6 elements into 3 disjoint subsets? $\textbf{(A)}\ 90 \qquad\textbf{(B)}\ 105 \qquad\textbf{(C)}\ 120 \qquad\textbf{(D)}\ 180 \qquad\textbf{(E)}\ 243$

1989 IMO Shortlist, 24

Tags: geometry , 3d geometry , sphere , combinatorics , maximization , geometric inequality

For points $ A_1, \ldots ,A_5$ on the sphere of radius 1, what is the maximum value that $ min_{1 \leq i,j \leq 5} A_iA_j$ can take? Determine all configurations for which this maximum is attained. (Or: determine the diameter of any set $ \{A_1, \ldots ,A_5\}$ for which this maximum is attained.)

2013 Moldova Team Selection Test, 3

Tags: geometry , parallelogram , reflection , moving points , ptolemy sinus lemma

Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ meet at $E$. The extensions of the sides $AD$ and $BC$ beyond $A$ and $B$ meet at $F$. Let $G$ be the point such that $ECGD$ is a parallelogram, and let $H$ be the image of $E$ under reflection in $AD$. Prove that $D,H,F,G$ are concyclic.

2021 Girls in Math at Yale, 9

Tags: college

Ali defines a [i]pronunciation[/i] of any sequence of English letters to be a partition of those letters into substrings such that each substring contains at least one vowel. For example, $\text{A } \vert \text{ THEN } \vert \text{ A}$, $\text{ATH } \vert \text{ E } \vert \text{ NA}$, $\text{ATHENA}$, and $\text{AT } \vert \text{ HEN } \vert \text{ A}$ are all pronunciations of the sequence $\text{ATHENA}$. How many distinct pronunciations does $\text{YALEMATHCOMP}$ have? (Y is not a vowel.) [i]Proposed by Andrew Wu, with significant inspiration from ali cy[/i]

2022 Iberoamerican, 6

Tags: number theory , functional equation

Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$, such that $f(a)f(a+b)-ab$ is a perfect square for all $a, b \in \mathbb{N}$.

2011 Hanoi Open Mathematics Competitions, 7

Tags: number theory , divisible

How many positive integers a less than $100$ such that $4a^2 + 3a + 5$ is divisible by $6$.

2016 Purple Comet Problems, 6

Tags:

Find the number of three-digit positive integers where the digits are three different prime numbers. For example, count 235 but not 553.

2007 Princeton University Math Competition, 8

Tags: geometry , analytic geometry , quadratic , calculus , integration , quadratic formula , algebra

For how many rational numbers $p$ is the area of the triangle formed by the intercepts and vertex of $f(x) = -x^2+4px-p+1$ an integer?

2000 Tuymaada Olympiad, 8

Tags: combinatorics , graph theory

There are $2000$ cities in the country, each of which has exactly three roads to other cities. Prove that you can close $1000$ roads, so that there is not a single closed route in the country, consisting of an odd number of roads.

2023 All-Russian Olympiad, 8

Tags: inequalities , algebra

Given is a real number $a \in (0,1)$ and positive reals $x_0, x_1, \ldots, x_n$ such that $\sum x_i=n+a$ and $\sum \frac{1}{x_i}=n+\frac{1}{a}$. Find the minimal value of $\sum x_i^2$.

LMT Team Rounds 2021+, 8

Tags: algebra

Let $x, y$, and $z$ be positive reals that satisfy the system $$\begin{cases} x^2 + x y + y^2 = 10 \\ x^2 + xz + z^2 = 20 \\ y^2 + yz + z^2 = 30\end{cases}$$ Find $x y + yz + xz$.

2024 Korea Summer Program Practice Test, 5

Tags: combinatorics

Call a set $\{a,b,c,d\}$ [i]epic[/i] if for any four different positive integers $a, b, c, d$, there is a unique way to select three of them to form the sides of a triangle. Find all positive integers $n$ such that $\{1, 2, \ldots, 4n\}$ can be partitioned into $n$ disjoint epic sets.

2019 Polish Junior MO Finals, 2.

Tags: geometry , trapezoid

Let $ABCD$ be the isosceles trapezium with bases $AB$ and $CD$, such that $AC = BC$. The point $M$ is the midpoint of side $AD$. Prove that $\sphericalangle ACM = \sphericalangle CBD$.

1984 Vietnam National Olympiad, 3

Tags: trigonometry , incenter , geometry

Consider a trihedral angle $Sxyz$ with $\angle xSz = \alpha, \angle xSy = \beta$ and $\angle ySz =\gamma$. Let $A,B,C$ denote the dihedral angles at edges $y, z, x$ respectively. $(a)$ Prove that $\frac{\sin\alpha}{\sin A}=\frac{\sin\beta}{\sin B}=\frac{\sin\gamma}{\sin C}$ $(b)$ Show that $\alpha + \beta = 180^{\circ}$ if and only if $\angle A + \angle B = 180^{\circ}.$ $(c)$ Assume that $\alpha=\beta =\gamma = 90^{\circ}$. Let $O \in Sz$ be a fixed point such that $SO = a$ and let $M,N$ be variable points on $x, y$ respectively. Prove that $\angle SOM +\angle SON +\angle MON$ is constant and find the locus of the incenter of $OSMN$.

2001 USA Team Selection Test, 8

Tags: number theory

Find all pairs of nonnegative integers $(m,n)$ such that \[(m+n-5)^2=9mn.\]

2009 Kyrgyzstan National Olympiad, 5

Tags: induction , floor function , number theory

Prove for all natural $n$ that $\left. {{{40}^n} \cdot n!} \right|(5n)!$

Ukrainian TYM Qualifying - geometry, 2013.17

Tags: geometry , 3d geometry , tetrahedron , concurrency

Through the point of intersection of the medians of each of the faces a tetrahedron is drawn perpendicular to this face. Prove that all these four lines intersect at one point if and only if the four lines containing the heights of this tetrahedron intersect at one point .

Kvant 2024, M2825

Tags: geometry

At the same time, three beetles with identical speeds began to crawl along the heights of an acute-angled non-isosceles triangle from its vertices. At some point, it turned out that the first and second beetles were on a circle inscribed in a triangle. Prove that at this moment the third beetle is also on this circle. [i]A. Kuznetsov[/i]

2004 China Second Round Olympiad, 3

Tags: relatively prime , number theory

For integer $n\ge 4$, find the minimal integer $f(n)$, such that for any positive integer $m$, in any subset with $f(n)$ elements of the set ${m, m+1, \ldots, m+n+1}$ there are at least $3$ relatively prime elements.