Olympiad problems

2013 JBMO Shortlist, 1

Tags: combinatorics

Find the maximum number of different integers that can be selected from the set $ \{1,2,...,2013\}$ so that no two exist that their difference equals to $17$.

1971 IMO Longlists, 34

Tags: algebra , sequence , recurrence relation , linear recurrence

Let $T_k = k - 1$ for $k = 1, 2, 3,4$ and \[T_{2k-1} = T_{2k-2} + 2^{k-2}, T_{2k} = T_{2k-5} + 2^k \qquad (k \geq 3).\] Show that for all $k$, \[1 + T_{2n-1} = \left[ \frac{12}{7}2^{n-1} \right] \quad \text{and} \quad 1 + T_{2n} = \left[ \frac{17}{7}2^{n-1} \right],\] where $[x]$ denotes the greatest integer not exceeding $x.$

LMT Accuracy Rounds, 2022 S7

Tags: combinatorics

A teacher wishes to separate her $12$ students into groups. Yesterday, the teacher put the students into $4$ groups of $3$. Today, the teacher decides to put the students into $4$ groups of $3$ again. However, she doesn’t want any pair of students to be in the same group on both days. Find how many ways she could formthe groups today.

2014 BMT Spring, 5

Tags: limit , real analysis

Determine $$\lim_{x\to\infty}\frac{\sqrt{x+2014}}{\sqrt x+\sqrt{x+2014}}$$

1966 AMC 12/AHSME, 15

Tags: inequalities

If $x-y>x$ and $x+y<y$, then $\text{(A)} \ y<x \qquad \text{(B)} \ x<y \qquad \text{(C)} \ x<y<0 \qquad \text{(D)} \ x<0,y<0$ $\text{(E)} \ x<0,y>0$

2014 Sharygin Geometry Olympiad, 6

Tags: tangent , geometry , tangent circle

Two circles $k_1$ and $k_2$ with centers $O_1$ and $O_2$ are tangent to each other externally at point $O$. Points $X$ and $Y$ on $k_1$ and $k_2$ respectively are such that rays $O_1X$ and $O_2Y$ are parallel and codirectional. Prove that two tangents from $X$ to $k_2$ and two tangents from $Y$ to $k_1$ touch the same circle passing through $O$. (V. Yasinsky)

2018 IMO, 6

Tags: geometry , symmedian , isogonal conjugate

A convex quadrilateral $ABCD$ satisfies $AB\cdot CD = BC\cdot DA$. Point $X$ lies inside $ABCD$ so that \[\angle{XAB} = \angle{XCD}\quad\,\,\text{and}\quad\,\,\angle{XBC} = \angle{XDA}.\] Prove that $\angle{BXA} + \angle{DXC} = 180^\circ$. [i]Proposed by Tomasz Ciesla, Poland[/i]

2021 Belarusian National Olympiad, 9.5

Tags: number theory

Prove that for some positive integer $n$ there exist positive integers $a$,$b$ and $c$ such that $a^2-n=xy$, $b^2-n=yz$ and $c^2-n=xz$ where $x,y$ and $z$ - some pairwise different positive integers.

2010 Dutch IMO TST, 4

Tags: geometry , cyclic quadrilateral , equal angles , midpoint

Let $ABCD$ be a cyclic quadrilateral satisfying $\angle ABD = \angle DBC$. Let $E$ be the intersection of the diagonals $AC$ and $BD$. Let $M$ be the midpoint of $AE$, and $N$ be the midpoint of $DC$. Show that $MBCN$ is a cyclic quadrilateral.

2010 Contests, 4

Tags: polynomial , inequalities , algebra

Let $P(x)=ax^3+bx^2+cx+d$ be a polynomial with real coefficients such that \[\min\{d,b+d\}> \max\{|{c}|,|{a+c}|\}\] Prove that $P(x)$ do not have a real root in $[-1,1]$.

1990 China Team Selection Test, 2

Tags: geometry

Finitely many polygons are placed in the plane. If for any two polygons of them, there exists a line through origin $O$ that cuts them both, then these polygons are called "properly placed". Find the least $m \in \mathbb{N}$, such that for any group of properly placed polygons, $m$ lines can drawn through $O$ and every polygon is cut by at least one of these $m$ lines.

2018 Harvard-MIT Mathematics Tournament, 1

Tags: probability

Four standard six-sided dice are rolled. Find the probability that, for each pair of dice, the product of the two numbers rolled on those dice is a multiple of 4.

2018 IFYM, Sozopol, 1

Tags: set , inequality , algebra

$A = \{a_1, a_2, . . . , a_k\}$ is a set of positive integers for which the sum of some (we can have only one number too) different numbers from the set is equal to a different number i.e. there $2^k - 1$ different sums of different numbers from $A$. Prove that the following inequality holds: $\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_k}<2$

VII Soros Olympiad 2000 - 01, 9.4

Tags: algebra

The distance between cities $A$ and $B$ is $30$ km. A bus departed from $A$, which makes a stop every $5$ km for $2$ minutes. The bus moves between stops at a speed of $80$ km / h. Simultaneously with the departure of the bus from $A$, a cyclist leaves $B$ to meet it, traveling at a speed of $27$ km / h. How far from $A$ will the cyclist meet the bus?

2009 Stars Of Mathematics, 4

Tags: algebra , polynomial , number theory

Determine all non-constant polynomials $ f\in \mathbb{Z}[X]$ with the property that there exists $ k\in\mathbb{N}^*$ such that for any prime number $ p$, $ f(p)$ has at most $ k$ distinct prime divisors.

2012 Rioplatense Mathematical Olympiad, Level 3, 5

Tags: number theory , divisor

Let $a \ge 2$ and $n \ge 3$ be integers . Prove that one of the numbers $a^n+ 1 , a^{n + 1}+ 1 , ... , a^{2 n-2}+ 1$ does not share any odd divisor greater than $1$ with any of the other numbers.

2020 Purple Comet Problems, 16

Tags: inequalities

Find the maximum possible value of $$\left( \frac{a^3}{b^2c}+\frac{b^3}{c^2a}+\frac{c^3}{a^2b} \right)^2$$ where $a, b$, and $c$ are nonzero real numbers satisfying $$a \sqrt[3]{\frac{a}{b}}+b\sqrt[3]{\frac{b}{c}}+c\sqrt[3]{\frac{c}{a}}=0$$

2010 VJIMC, Problem 2

Tags: linear algebra , matrix

If $A,B\in M_2(C)$ such that $AB-BA=B^2$ then prove that \[AB=BA\]

2020 Jozsef Wildt International Math Competition, W14

Tags: fibonacci , number theory

Let $\{F_n\}_{n\ge1}$ be the Fibonacci sequence defined by $F_1=F_2=1$ and for all $n\ge3$, $F_n=F_{n-1}+F_{n-2}$. Prove that among the first $10000000000000002$ terms of the sequence there is one term that ends up with $8$ zeroes. [i]Proposed by José Luis Díaz-Barrero[/i]

1987 China Team Selection Test, 1

Tags: geometry , rectangle , ratio , analytic geometry , symmetry , hyperbola , conic

Given a convex figure in the Cartesian plane that is symmetric with respect of both axis, we construct a rectangle $A$ inside it with maximum area (over all posible rectangles). Then we enlarge it with center in the center of the rectangle and ratio lamda such that is covers the convex figure. Find the smallest lamda such that it works for all convex figures.

1965 German National Olympiad, 5

Tags: digit , number theory

Determine all triples of nonzero decimal digits $(x,y,z)$ for which the equality $\sqrt{ \underbrace{xxx...x}_{2n}- \underbrace{yy...y}_{n}}= \underbrace{zzz...z}_{n}$ holds for at least two different natural numbers $n$.

2008 Cuba MO, 8

Tags: geometry , rectangle , locus

Let $ABC$ an acute-angle triangle. Let $R$ be a rectangle with vertices in the edges of $ABC$. Let $O$ be the center of $R$. a) Find the locus of all the points $O$. b) Decide if there is a point that is the center of three of these rectangles.

2016 CMIMC, 5

Tags: geometry

Let $\mathcal{P}$ be a parallelepiped with side lengths $x$, $y$, and $z$. Suppose that the four space diagonals of $\mathcal{P}$ have lengths $15$, $17$, $21$, and $23$. Compute $x^2+y^2+z^2$.

1971 All Soviet Union Mathematical Olympiad, 158

Tags: combinatorics , minimum

A switch has two inputs $1, 2$ and two outputs $1, 2$. It either connects $1$ to $1$ and $2$ to $2$, or $1$ to $2$ and $2$ to 1. If you have three inputs $1, 2, 3$ and three outputs $1, 2, 3$, then you can use three switches, the first across $1$ and $2$, then the second across $2$ and $3$, and finally the third across $1$ and $2$. It is easy to check that this allows the output to be any permutation of the inputs and that at least three switches are required to achieve this. What is the minimum number of switches required for $4$ inputs, so that by suitably setting the switches the output can be any permutation of the inputs?

2024 Indonesia MO, 5

Tags: integer , combinatorics , number theory

Each integer is colored with exactly one of the following colors: red, blue, or orange, and all three colors are used in the coloring. The coloring also satisfies the following properties: 1. The sum of a red number and an orange number results in a blue-colored number, 2. The sum of an orange and blue number results in an orange-colored number; 3. The sum of a blue number and a red number results in a red-colored number. (a) Prove that $0$ and $1$ must have distinct colors. (b) Determine all possible colorings of the integers which also satisfy the properties stated above.