Olympiad problems

2017 Harvard-MIT Mathematics Tournament, 9

Tags: algebra

Jeffrey writes the numbers $1$ and $100000000 = 10^8$ on the blackboard. Every minute, if $x, y$ are on the board, Jeffery replaces them with \[\frac{x + y}{2} \quad \text{and} \quad 2 \left(\frac{1}{x} + \frac{1}{y}\right)^{-1}.\] After $2017$ minutes the two numbers are $a$ and $b$. Find $\min(a, b)$ to the nearest integer.

2019 District Olympiad, 4

Tags: geometry , equilateral , right triangle , isosceles

Consider the isosceles right triangle$ ABC, \angle A = 90^o$, and point $D \in (AB)$ such that $AD = \frac13 AB$. In the half-plane determined by the line $AB$ and point $C$ , consider a point $E$ such that $\angle BDE = 60^o$ and $\angle DBE = 75^o$. Lines $BC$ and $DE$ intersect at point $G$, and the line passing through point $G$ parallel to the line $AC$ intersects the line $BE$ at point $H$. Prove that the triangle $CEH$ is equilateral.

2021 Iran MO (3rd Round), 2

Tags: algebra

If $a, b, c$ and $d$ are complex non-zero numbers such that $$2|a-b|\leq |b|, 2|b-c|\leq |c|, 2|c-d| \leq |d| , 2|d-a|\leq |a|.$$ Prove that $$\frac{7}{2} <\left| \frac{b}{a} + \frac{c}{b} + \frac{d}{c} + \frac{a}{d} \right| .$$

1984 Austrian-Polish Competition, 3

Tags: inequalities

Show that for $n>1$ and any positive real numbers $k,x_{1},x_{2},...,x_{n}$ then \[\frac{f(x_{1}-x_{2})}{x_{1}+x_{2}}+\frac{f(x_{2}-x_{3})}{x_{2}+x_{3}}+...+\frac{f(x_{n}-x_{1})}{x_{n}+x_{1}}\geq \frac{n^2}{2(x_{1}+x_{2}+...+x_{n})}\] Where $f(x)=k^x$. When does equality hold.

2009 Tournament Of Towns, 7

Tags: binomial , combinatorics

Let ${n \choose k}$ be the number of ways that $k$ objects can be chosen (regardless of order) from a set of $n$ objects. Prove that if positive integers k and l are greater than $1$ and less than $n$, then integers ${n \choose k}$ and ${n \choose l}$ have a common divisor greater than $1$.

IV Soros Olympiad 1997 - 98 (Russia), 11.3

Tags: algebra , inequalities , radical

Solve the inequality $$\sqrt{(x-2)^2(x-x^2)}<\sqrt{4x-1-(x^2-3x)^2}$$

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]