Olympiad problems

2024 CMIMC Algebra and Number Theory, 5

Tags: algebra

Let \[f(x)=(x+1)^{6}+(x-1)^{5}+(x+1)^{4}+(x-1)^3+(x+1)^2+(x-1)^1+1.\] Find the remainder when $\sum_{j=-126}^{126}jf(j)$ is divided by 1000. [i]Proposed by Hari Desikan[/i]

2017 Romania EGMO TST, P3

Tags: algebra , function , functional equation

Determine all functions $f:\mathbb R\to\mathbb R$ such that \[f(xy-1)+f(x)f(y)=2xy-1,\]for any real numbers $x{}$ and $y{}.$

2023 Bangladesh Mathematical Olympiad, P1

Tags: factorial , number theory , diophantine equation

Find all possible non-negative integer solution ($x,$ $y$) of the following equation- $$x!+2^y=z!$$ Note: $x!=x\cdot(x-1)!$ and $0!=1$. For example, $5!=5\times4\times3\times2\times1=120$.

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 $\left. {{{40}^n} \cdot n!} \right|(5n)!$