Olympiad problems

2016 LMT, 11

Tags:

Find all ordered triples $(a,b,c)$ of real numbers such that \[\begin{cases} a+b=c,\\ a^2+b^2=c^2-c-6,\\ a^3+b^3 = c^3-2c^2-5c. \\ \end{cases}\] [i]Proposed by Evan Fang

Kvant 2021, M2666

Tags: number theory

Let $x{}$ and $y{}$ be natural numbers greater than 1. It turns out that $x^2+y^2-1$ is divisible by $x+y-1$. Prove that $x+y-1$ is composite. [i]From the folklore[/i]

There are $n\ge 3$ particles on a circle situated at the vertices of a regular $n$-gon. All these particles move on the circle with the same constant speed. One of the particles moves in the clockwise direction while all others move in the anti-clockwise direction. When particles collide, that is, they are all at the same point, they all reverse the direction of their motion and continue with the same speed as before. Let $s$ be the smallest number of collisions after which all particles return to their original positions. Find $s$. [i]Proposed by N.V. Tejaswi[/i]

2018 India PRMO, 27

Tags: combinatorics

What is the number of ways in which one can color the squares of a $4\times 4$ chessboard with colors red and blue such that each row as well as each column has exactly two red squares and two blue squares?

PEN M Problems, 16

Tags: recursive sequences

Define a sequence $\{a_i\}$ by $a_1=3$ and $a_{i+1}=3^{a_i}$ for $i\geq 1$. Which integers between $00$ and $99$ inclusive occur as the last two digits in the decimal expansion of infinitely many $a_i$?

2018 Auckland Mathematical Olympiad, 2

Tags: number theory , digit

Consider a positive integer, $N = 9 + 99 + 999 + ... +\underbrace{999...9}_{2018}$. How many times does the digit $1$ occur in its decimal representation?

2021 Auckland Mathematical Olympiad, 2

Tags: equilateral , geometry , distance

Given five points inside an equilateral triangle of side length $2$, show that there are two points whose distance from each other is at most $ 1$.

2019 Czech and Slovak Olympiad III A, 2

Tags: constructive geometry , rectangle , geometry

Let be $ABCD$ a rectangle with $|AB|=a\ge b=|BC|$. Find points $P,Q$ on the line $BD$ such that $|AP|=|PQ|=|QC|$. Discuss the solvability with respect to the lengths $a,b$.

2016 JBMO TST - Turkey, 1

Tags: algebra , system of equations

Find all pairs $(x, y)$ of real numbers satisfying the equations \begin{align*} x^2+y&=xy^2 \\ 2x^2y+y^2&=x+y+3xy. \end{align*}

2025 Romania National Olympiad, 2

Tags: complex number , invertible matrix , linear algebra , matrix

Let $n$ be a positive integer, and $a,b$ be two complex numbers such that $a \neq 1$ and $b^k \neq 1$, for any $k \in \{1,2,\dots ,n\}$. The matrices $A,B \in \mathcal{M}_n(\mathbb{C})$ satisfy the relation $BA=a I_n + bAB$. Prove that $A$ and $B$ are invertible.

2009 Harvard-MIT Mathematics Tournament, 8

Tags: algebra , polynomial , vieta , sum of roots

Let $a$, $b$, and $c$ be the $3$ roots of $x^3-x+1=0$. Find $\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}.$

Novosibirsk Oral Geo Oly VIII, 2021.3

Tags: angle , geometry

Find the angle $BCA$ in the quadrilateral of the figure. [img]https://cdn.artofproblemsolving.com/attachments/0/2/974e23be54125cde8610a78254b59685833b5b.png[/img]

1996 Rioplatense Mathematical Olympiad, Level 3, 6

Tags: number theory , greatest common divisor

Find all integers $k$ for which, there is a function $f: N \to Z$ that satisfies: (i) $f(1995) = 1996$ (ii) $f(xy) = f(x) + f(y) + kf(m_{xy})$ for all natural numbers $x, y$,where$ m_{xy}$ denotes the greatest common divisor of the numbers $x, y$. Clarification: $N = \{1,2,3,...\}$ and $Z = \{...-2,-1,0,1,2,...\}$ .

2011 German National Olympiad, 5

Tags: number theory

Prove or disprove: $\exists n\in N$ , s.t. $324 + 455^n$ is prime.

2023 Indonesia TST, G

Tags: geometry

Incircle of triangle $ABC$ tangent to $AB$ and $AC$ on $E$ and $F$ respectively. If $X$ is the midpoint of $EF$, prove $\angle BXC > 90^{\circ}$

1991 Putnam, B1

Tags:

For each integer $n\geq0$, let $S(n)=n-m^2$, where $m$ is the greatest integer with $m^2\leq n$. Define a sequence by $a_0=A$ and $a_{k+1}=a_k+S(a_k)$ for $k\geq0$. For what positive integers $A$ is this sequence eventually constant?

2023 Poland - Second Round, 1

Tags: number theory , divisibility

Find all positive integers $b$ with the following property: there exists positive integers $a,k,l$ such that $a^k + b^l$ and $a^l + b^k$ are divisible by $b^{k+l}$ where $k \neq l$.

1953 Poland - Second Round, 2

Tags: algebra , sum

The board was placed $$ \begin{array}{rcl} 1 & = & 1 \\ 2 + 3 + 4 & = & 1 + 8 \\ 5 + 6 + 7 + 8 + 9 & = & 8 + 27\\ 10 + 11 + 12 + 13 + 14 + 15 + 16 & = & 27 + 64\\ & \ldots & \end{array}$$ Write such a formula for the $ n $-th row of the array that, with the substitutions $ n = 1, 2, 3, 4 $, would give the above four lines of the array and would be true for every natural $ n $.

2008 IMS, 4

Tags: graph theory , combinatorics , spanning tree , square grid

A subset of $ n\times n$ table is called even if it contains even elements of each row and each column. Find the minimum $ k$ such that each subset of this table with $ k$ elements contains an even subset

2006 International Zhautykov Olympiad, 3