Olympiad problems

2025 Harvard-MIT Mathematics Tournament, 2

Mark writes the expression $\sqrt{\underline{abcd}}$ on the board, where $\underline{abcd}$ is a four-digit number and $a \neq 0.$ Derek, a toddler, decides to move the $a,$ changing Mark's expression to $a\sqrt{\underline{bcd}}.$ Surprisingly, these two expressions are equal. Compute the only possible four-digit number $\underline{abcd}.$

2017 IFYM, Sozopol, 4

Tags: number theory

Find all pairs of natural numbers $(a,n)$, $a\geq n \geq 2,$ for which $a^n+a-2$ is a power of $2$.

2012 Centers of Excellency of Suceava, 2

Tags: function , Find all functions , algebra

Find all functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that verify, for any nonzero real number $ x $ the relation $$ xf(x/a)-f(a/x)=b, $$ where $ a\neq 0,b $ are two real numbers. [i]Dan Popescu[/i]

2015 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: algebra , inequation

Solve the inequation: $$5\mid x\mid \leq x(3x+2-2\sqrt{8-2x-x^2})$$

1988 China National Olympiad, 1

Tags: inequalities , function , inequalities unsolved

Let $r_1,r_2,\dots ,r_n$ be real numbers. Given $n$ reals $a_1,a_2,\dots ,a_n$ that are not all equal to $0$, suppose that inequality \[r_1(x_1-a_1)+ r_2(x_2-a_2)+\dots + r_n(x_n-a_n)\leq\sqrt{x_1^2+ x_2^2+\dots + x_n^2}-\sqrt{a_1^2+a_2^2+\dots +a_n^2}\] holds for arbitrary reals $x_1,x_2,\dots ,x_n$. Find the values of $r_1,r_2,\dots ,r_n$.

2015 BAMO, 3

Tags: algebra , Fractions , Compare

Which number is larger, $A$ or $B$, where $$A = \dfrac{1}{2015} (1 + \dfrac12 + \dfrac13 + \cdots + \dfrac{1}{2015})$$ and $$B = \dfrac{1}{2016} (1 + \dfrac12 + \dfrac13 + \cdots + \dfrac{1}{2016}) \text{ ?}$$ Prove your answer is correct.

1979 Dutch Mathematical Olympiad, 2

Tags: number theory , system of equations , System , diophantine

Solve in $N$: $$\begin{cases} a^3=b^3+c^3+12a \\ a^2=5(b+c) \end{cases}$$

2022 IOQM India, 7

Tags: IOQM

Find the number of maps $f: \{1,2,3\} \rightarrow \{1,2,3,4,5\}$ such that $f(i) \le f(j)$ whenever $i < j$.

2014 Contests, 4

Tags: number theory , point , circle , primes

The radius $r$ of a circle with center at the origin is an odd integer. There is a point ($p^m, q^n$) on the circle, with $p,q$ prime numbers and $m,n$ positive integers. Determine $r$.

2011 Postal Coaching, 5

Tags: geometry , inequalities , circumcircle , trigonometry , geometry unsolved

Let $P$ be a point inside a triangle $ABC$ such that \[\angle P AB = \angle P BC = \angle P CA\] Suppose $AP, BP, CP$ meet the circumcircles of triangles $P BC, P CA, P AB$ at $X, Y, Z$ respectively $(\neq P)$ . Prove that \[[XBC] + [Y CA] + [ZAB] \ge 3[ABC]\]

2024 Bulgarian Spring Mathematical Competition, 12.3

Tags: number theory

For a positive integer $n$, denote with $b(n)$ the smallest positive integer $k$, such that there exist integers $a_1, a_2, \ldots, a_k$, satisfying $n=a_1^{33}+a_2^{33}+\ldots+a_k^{33}$. Determine whether the set of positive integers $n$ is finite or infinite, which satisfy: a) $b(n)=12;$ b) $b(n)=12^{12^{12}}.$

2020 LIMIT Category 1, 1

Tags: algebra , polynomial

Find all polynomial $P(x)$ with degree $\leq n$and non negative coefficients such that $$P(x)P(\frac{1}{x})\leq P(1)^2$$ for all positive $x$. Here $n$ is a natuaral number

2008 Indonesia TST, 3

Tags: combinatorics

$10$ people attended a party. For every $3$ people, there exist at least $2$ people who don’t know each other. Prove that there exist $4$ people who don’t know each other.

2023 IFYM, Sozopol, 4

Tags: algebra , functional equation

Find all real numbers $a$ for which there exist functions $f,g: \mathbb{R} \to \mathbb{R}$, where $g$ is strictly increasing, such that $f(1) = 1$, $f(2) = a$, and \[ f(x) - f(y) \leq (x-y)(g(x) - g(y)) \] for all real numbers $x$ and $y$.

2011 Today's Calculation Of Integral, 746

Tags: calculus , integration , floor function , inequalities , calculus computations

Prove the following inequality. \[n^ne^{-n+1}\leq n!\leq \frac 14(n+1)^{n+1}e^{-n+1}.\]

2010 Saudi Arabia Pre-TST, 4.1

Tags: diophantine , number theory , System , system of equations

Find all triples $(a, b, c)$ of positive integers for which $$\begin{cases} a + bc=2010 \\ b + ca = 250\end{cases}$$

Geometry Mathley 2011-12, 8.4

Tags: concurrent circles , circles , tangent circles , orthocenter

Let $ABC$ a triangle inscribed in a circle $(O)$ with orthocenter $H$. Two lines $d_1$ and $d_2$ are mutually perpendicular at $H$. Let $d_1$ meet $BC,CA,AB$ at $X_1, Y_1,Z_1$ respectively. Let $A_1B_1C_1$ be a triangle formed by the line through $X_1$ perpendicular to $BC$, the line through $Y_1$ perpendicular to CA, the line through $Z_1$ perpendicular perpendicular to $AB$. Triangle $A_2B_2C_2$ is defined in the same manner. Prove that the circumcircles of triangles $A_1B_1C_1$ and $A_2B_2C_2$ touch each other at a point on $(O)$. Nguyễn Văn Linh

2018-2019 SDML (High School), 2

Tags: 2018-2019 Round 2 , SDML High School

When a positive integer $N$ is divided by $60$, the remainder is $49$. When $N$ is divided by $15$, the remainder is $ \mathrm{(A) \ } 0 \qquad \mathrm{(B) \ } 3 \qquad \mathrm {(C) \ } 4 \qquad \mathrm{(D) \ } 5 \qquad \mathrm{(E) \ } 8$

2006 Princeton University Math Competition, 10

Tags:

If $x,y,z$ are real numbers and \begin{align*} 2x+y+z\leq66\\ x+2y+z\leq60\\ x+y+2z\leq70\\ x+2y+3z\leq110\\ 3x+y+2z\leq98\\ 2x+3y+z\leq89\\ \end{align*} What is the maximum possible value of $x+y+z$?

1987 All Soviet Union Mathematical Olympiad, 452

Tags: algebra , inequalities

The positive numbers $a,b,c,A,B,C$ satisfy a condition $$a + A = b + B = c + C = k$$ Prove that $$aB + bC + cA \le k^2$$

2009 Oral Moscow Geometry Olympiad, 6

Tags: geometry , incenter , circles

Fixed two circles $w_1$ and $w_2$, $\ell$ one of their external tangent and $m$ one of their internal tangent . On the line $m$, a point $X$ is chosen, and on the line $\ell$, points $Y$ and $Z$ are constructed so that $XY$ and $XZ$ touch $w_1$ and $w_2$, respectively, and the triangle $XYZ$ contains circles $w_1$ and $w_2$. Prove that the centers of the circles inscribed in triangles $XYZ$ lie on one line. (P. Kozhevnikov)

2003 India IMO Training Camp, 8

Tags: geometry , inradius , geometry proposed

Let $ABC$ be a triangle, and let $r, r_1, r_2, r_3$ denoted its inradius and the exradii opposite the vertices $A,B,C$, respectively. Suppose $a>r_1, b>r_2, c>r_3$. Prove that (a) triangle $ABC$ is acute, (b) $a+b+c>r+r_1+r_2+r_3$.

2015 Romania Team Selection Tests, 3

Tags: arithmetic sequence , Sets , Product , algebra

Given a positive real number $t$ , determine the sets $A$ of real numbers containing $t$ , for which there exists a set $B$ of real numbers depending on $A$ , $|B| \geq 4$ , such that the elements of the set $AB =\{ ab \mid a\in A , b \in B \}$ form a finite arithmetic progression .

2008 Balkan MO Shortlist, C4

Tags:

An array $n \times n$ is given, consisting of $n^2$ unit squares. A pawn is placed arbitrarily on a unit square. A [i]move[/i] of the pawn means a jump from a square of the $k$th column to any square of the $k$th row. Show that there exists a sequence of $n^2$ moves of the pawn so that all the unit squares of the array are visited once and, in the end, the pawn returns to the original position.

1983 Tournament Of Towns, (042) O5

Tags: combinatorial geometry , combinatorics , projection , regular polygon

A point is chosen inside a regular $k$-gon in such a way that its orthogonal projections on to the sides all meet the respective sides at interior points. These points divide the sides into $2k$ segments. Let these segments be enumerated consecutively by the numbers $1,2, 3, ... ,2k$. Prove that the sum of the lengths of the segments having even numbers equals the sum of the segments having odd numbers. (A Andjans, Riga)