Olympiad problems

1986 IMO Longlists, 68

Tags: polynomial , algebra

Consider the equation $x^4 + ax^3 + bx^2 + ax + 1 = 0$ with real coefficients $a, b$. Determine the number of distinct real roots and their multiplicities for various values of $a$ and $b$. Display your result graphically in the $(a, b)$ plane.

2015 Turkey EGMO TST, 1

Tags: algebra

$a$ is a real number. Find the all $(x,y)$ real number pairs satisfy;$$y^2=x^3+(a-1)x^2+a^2x$$$$x^2=y^3+(a-1)y^2+a^2y$$

1990 IMO Longlists, 41

Tags: algebra

Let $n$ be an arbitrary positive integer. Calculate $S_n = \sum_{r=0}^n 2^{r-2n} \binom{2n-r}{n}.$

2004 Korea National Olympiad, 1

Tags: function , algebra

For arbitrary real number $x$, the function $f : \mathbb R \to \mathbb R$ satisfies $f(f(x))-x^2+x+3=0$. Show that the function $f$ does not exist.

2018 South East Mathematical Olympiad, 1

Tags: algebra , inequality , inequalities

Assume $c$ is a real number. If there exists $x\in[1,2]$ such that $\max\left\{\left |x+\frac cx\right |, \left |x+\frac cx + 2\right |\right\}\geq 5$, please find the value range of $c$.

2023 Auckland Mathematical Olympiad, 7

Tags: geometry , combinatorial geometry , combinatorics

In a square of area $1$ there are situated $2024$ polygons whose total area is greater than $2023$. Prove that they have a point in common.

2005 Croatia National Olympiad, 1

Tags: modular arithmetic

Find all possible digits $x, y, z$ such that the number $\overline{13xy45z}$ is divisible by $792.$

2009 Iran MO (3rd Round), 4

Tags: incenter , power of a point , radical axis , geometry

4-Point $ P$ is taken on the segment $ BC$ of the scalene triangle $ ABC$ such that $ AP \neq AB,AP \neq AC$.$ l_1,l_2$ are the incenters of triangles $ ABP,ACP$ respectively. circles $ W_1,W_2$ are drawn centered at $ l_1,l_2$ and with radius equal to $ l_1P,l_2P$,respectively. $ W_1,W_2$ intersects at $ P$ and $ Q$. $ W_1$ intersects $ AB$ and $ BC$ at $ Y_1( \mbox{the intersection closer to B})$ and $ X_1$,respectively. $ W_2$ intersects $ AC$ and $ BC$ at $ Y_2(\mbox{the intersection closer to C})$ and $ X_2$,respectively.PROVE THE CONCURRENCY OF $ PQ,X_1Y_1,X_2Y_2$.

1969 IMO Longlists, 64

Tags: inequality , factorial , algebra

$(USS 1)$ Prove that for a natural number $n > 2, (n!)! > n[(n - 1)!]^{n!}.$

1996 Tuymaada Olympiad, 4

Tags: compass , construction , geometry

Given a segment of length $7\sqrt3$ . Is it possible to use only compass to construct a segment of length $\sqrt7$?

2023 China Western Mathematical Olympiad, 2

Tags: combinatorics

In a certain country there are $2023$ islands and $2022$ bridges, such that every bridge connects two different islands and any two islands have at most one bridge in common, and from any island, using bridges one can get to any other island. If in any three islands there is an island with bridges connected to each of the other two islands, call these three islands an "island group". We know that any two "island group"s have at least $1$ common island. What is the minimum number of islands with only $1$ bridge connected to it?

2019 Indonesia MO, 2

Tags: combinatorics , number theory

Given $19$ red boxes and $200$ blue boxes filled with balls. None of which is empty. Suppose that every red boxes have a maximum of $200$ balls and every blue boxes have a maximum of $19$ balls. Suppose that the sum of all balls in the red boxes is less than the sum of all the balls in the blue boxes. Prove that there exists a subset of the red boxes and a subset of the blue boxes such that their sum is the same.

2017 Princeton University Math Competition, 9

Tags: floor function , ceiling function

The set $\{(x, y) \in R^2| \lfloor x + y\rfloor \cdot \lceil x + y\rceil = (\lfloor x\rfloor + \lceil y \rceil ) (\lceil x \rceil + \lfloor y\rfloor), 0 \le x, y \le 100\}$ can be thought of as a collection of line segments in the plane. If the total length of those line segments is $a + b\sqrt{c}$ for $c$ squarefree, find $a + b + c$. ($\lfloor z\rfloor$ is the greatest integer less than or equal to $z$, and $\lceil z \rceil$ is the least integer greater than or equal to $z$, for $z \in R$.)

2001 Estonia National Olympiad, 1

Tags: number theory , digit , integer

John had to solve a math problem in the class. While cleaning the blackboard, he accidentally erased a part of his problem as well: the text that remained on board was $37 \cdot(72 + 3x) = 14**45$, where $*$ marks an erased digit. Show that John can still solve his problem, knowing that $x$ is an integer

1910 Eotvos Mathematical Competition, 3

Tags: geometry , angle bisector

The lengths of sides $CB$ and $CA$ of $\vartriangle ABC$ are $a$ and $b$, and the angle between them is $\gamma = 120^o$. Express the length of the bisector of $\gamma$ in terms of $a$ and $b$.

2018 Stanford Mathematics Tournament, 2

Tags: combinatorics , game , game strategy

Consider a game played on the integers in the closed interval $[1, n]$. The game begins with some tokens placed in $[1, n]$. At each turn, tokens are added or removed from$ [1, n]$ using the following rule: For each integer $k \in [1, n]$, if exactly one of $k - 1$ and $k + 1$ has a token, place a token at $k$ for the next turn, otherwise leave k blank for the next turn. We call a position [i]static [/i] if no changes to the interval occur after one turn. For instance, the trivial position with no tokens is static because no tokens are added or removed after a turn (because there are no tokens). Find all non-trivial static positions.

2018 Brazil National Olympiad, 1

Tags: inequality , triangle inequality , inequalities , geometry

We say that a polygon $P$ is [i]inscribed[/i] in another polygon $Q$ when all vertices of $P$ belong to perimeter of $Q$. We also say in this case that $Q$ is [i]circumscribed[/i] to $P$. Given a triangle $T$, let $l$ be the maximum value of the side of a square inscribed in $T$ and $L$ be the minimum value of the side of a square circumscribed to $T$. Prove that for every triangle $T$ the inequality $L/l \ge 2$ holds and find all the triangles $T$ for which the equality occurs.

2019 Bosnia and Herzegovina Junior BMO TST, 1

Tags: algebra

Let $x,y,z$ be real numbers ( $x \ne y$, $y\ne z$, $x\ne z$) different from $0$. If $\frac{x^2-yz}{x(1-yz)}=\frac{y^2-xz}{y(1-xz)}$, prove that the following relation holds: $$x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$$

2022 Balkan MO Shortlist, C5

Tags:

Given is a cube of side length $2021$. In how many different ways is it possible to add somewhere on the boundary of this cube a $1\times 1\times 1$ cube in such a way that the new shape can be filled in with $1\times 1\times k$ shapes, for some natural number $k$, $k\geq 2$?

2013 Kazakhstan National Olympiad, 1

Tags: inequalities , algebra

Find maximum value of $|a^2-bc+1|+|b^2-ac+1|+|c^2-ba+1|$ when $a,b,c$ are reals in $[-2;2]$.

2016 Tournament Of Towns, 1

Tags: combinatorics

$100$ children stand in a line each having $100$ candies. In one move, one of them may take some of their candies and distribute them to a non-empty set of the remaining children. After what least number of moves can it happen that no two children have the same number of candies? [i](N. Chernyatevya)[/i] (Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.[/url])

1961 IMO Shortlist, 2

Tags: inequalities , trigonometry , geometry , area of a triangle , heron's formula

Let $ a$, $ b$, $ c$ be the sides of a triangle, and $ S$ its area. Prove: \[ a^{2} \plus{} b^{2} \plus{} c^{2}\geq 4S \sqrt {3} \] In what case does equality hold?

1996 IberoAmerican, 1

Tags: induction , number theory

Given a natural number $n \geq 2$, consider all the fractions of the form $\frac{1}{ab}$, where $a$ and $b$ are natural numbers, relative primes and such that: $a < b \leq n$, $a+b>n$. Show that for each $n$, the sum of all this fractions are $\frac12$.

2017 Bundeswettbewerb Mathematik, 1

Tags: sequence , combinatorics

For which integers $n \geq 4$ is the following procedure possible? Remove one number of the integers $1,2,3,\dots,n+1$ and arrange them in a sequence $a_1,a_2,\dots,a_n$ such that of the $n$ numbers \[ |a_1-a_2|,|a_2-a_3|,\dots,|a_{n-1}-a_n|,|a_n-a_1| \] no two are equal.

2014 AIME Problems, 7

Tags: sum , trigonometry , logarithm

Let $f(x) = (x^2+3x+2)^{\cos(\pi x)}$. Find the sum of all positive integers $n$ for which \[\left| \sum_{k=1}^n \log_{10} f(k) \right| = 1.\]