Olympiad problems

2019-IMOC, G3

Given a scalene triangle $\vartriangle ABC$ has orthocenter $H$ and circumcircle $\Omega$. The tangent lines passing through $A,B,C$ are $\ell_a,\ell_b,\ell_c$. Suppose that the intersection of $\ell_b$ and $\ell_c$ is $D$. The foots of $H$ on $\ell_a,AD$ are $P,Q$ respectively. Prove that $PQ$ bisects segment $BC$. [img]https://4.bp.blogspot.com/-iiQoxMG8bEs/XnYNK7R8S3I/AAAAAAAALeY/FYvSuF6vQQsofASnXJUgKZ1T9oNnd-02ACK4BGAYYCw/s400/imoc2019g3.png[/img]

2002 Vietnam National Olympiad, 3

Tags: polynomial , algebra

For a positive integer $ n$, consider the equation $ \frac{1}{x\minus{}1}\plus{}\frac{1}{4x\minus{}1}\plus{}\cdots\plus{}\frac{1}{k^2x\minus{}1}\plus{}\cdots\plus{}\frac{1}{n^2x\minus{}1}\equal{}\frac{1}{2}$. (a) Prove that, for every $ n$, this equation has a unique root greater than $ 1$, which is denoted by $ x_n$. (b) Prove that the limit of sequence $ (x_n)$ is $ 4$ as $ n$ approaches infinity.

2001 China Team Selection Test, 3

Tags: algebra , function , calculus , polynomial , minimization

Let $F = \max_{1 \leq x \leq 3} |x^3 - ax^2 - bx - c|$. When $a$, $b$, $c$ run over all the real numbers, find the smallest possible value of $F$.

2011 Iran MO (3rd Round), 3

Tags: polynomial , search , algebra

We define the polynomial $f(x)$ in $\mathbb R[x]$ as follows: $f(x)=x^n+a_{n-2}x^{n-2}+a_{n-3}x^{n-3}+.....+a_1x+a_0$ Prove that there exists an $i$ in the set $\{1,....,n\}$ such that we have $|f(i)|\ge \frac{n!}{\dbinom{n}{i}}$. [i]proposed by Mohammadmahdi Yazdi[/i]

2010 All-Russian Olympiad Regional Round, 11.1

Tags: geometry , right triangle

Each leg of a right triangle is increased by one. Could its hypotenuse increase by more than $\sqrt2$?

1982 IMO Longlists, 21

Tags: inequalities , combinatorics

Al[u][b]l[/b][/u] edges and all diagonals of regular hexagon $A_1A_2A_3A_4A_5A_6$ are colored blue or red such that each triangle $A_jA_kA_m, 1 \leq j < k < m\leq 6$ has at least one red edge. Let $R_k$ be the number of red segments $A_kA_j, (j \neq k)$. Prove the inequality \[\sum_{k=1}^6 (2R_k-7)^2 \leq 54.\]

2021 Kyiv City MO Round 1, 10.4

Tags: inequalities

Positive real numbers $a, b, c$ satisfy $a^2 + b^2 + c^2 + a + b + c = 6$. Prove the following inequality: $$2(\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}) \geq 3 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$$ [i]Proposed by Oleksii Masalitin[/i]

1998 AIME Problems, 5

Tags: trigonometry

Given that $A_k=\frac{k(k-1)}2\cos\frac{k(k-1)\pi}2,$ find $|A_{19}+A_{20}+\cdots+A_{98}|.$

V Soros Olympiad 1998 - 99 (Russia), 11.5

Tags: algebra , inequalities

Find the smallest value of the expression $$(x -y)^2 + (z - u)^2,$$ if $$(x -1)^2 + (y -4)^2 + (z-3)^2 + (u-2)^2 = 1.$$

KoMaL A Problems 2017/2018, A. 722

Tags: combinatorics

The Hawking Space Agency operates $n-1$ space flights between the $n$ habitable planets of the Local Galaxy Cluster. Each flight has a fixed price which is the same in both directions, and we know that using these flights, we can travel from any habitable planet to any habitable planet. In the headquarters of the Agency, there is a clearly visible board on a wall, with a portrait, containing all the pairs of different habitable planets with the total price of the cheapest possible sequence of flights connecting them. Suppose that these prices are precisely $1,2, ... , \binom{n}{2}$ monetary units in some order. prove that $n$ or $n-2$ is a square number.

2008 Princeton University Math Competition, A5/B8

Tags: algebra

Let $H_k =\Sigma_{i=1}^k \frac{1}{i}$ for all positive integers $k$. Find an closed-form expression for $\Sigma_{i=1}^k H_i$ in terms of $n$ and $H_n$.

1991 Arnold's Trivium, 59

Tags: trigonometry

Investigate the existence and uniqueness of the solution of the problem $yu_x = xu_y, u|_{x=1} =\cos y$ in a neighbourhood of the point $(1, y_0)$.

2002 AMC 10, 2

Tags: function

For the nonzero numbers $ a$, $ b$, $ c$, define \[(a,b,c)\equal{}\frac{a}{b}\plus{}\frac{b}{c}\plus{}\frac{c}{a}.\] Find $ (2,12,9)$. $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8$

2004 Denmark MO - Mohr Contest, 5

Tags: combinatorics , combinatorial geometry , tiles , tiling

Determine for which natural numbers $n$ you can cover a $2n \times 2n$ chessboard with non-overlapping $L$ pieces. An $L$ piece covers four spaces and has appearance like the letter $L$. The piece may be rotated and mirrored at will.

2006 All-Russian Olympiad Regional Round, 10.5

Tags: algebra , inequalities , trigonometry

Prove that for every $x$ such that $\sin x \ne 0$, there is such natural $n$, which $$ | \sin nx| \ge \frac{\sqrt3}{2}.$$

1992 Poland - First Round, 4

Tags: function , functional equation , algebra

Determine all functions $f: R \longrightarrow R$ such that $f(x+y)-f(x-y)=f(x)*f(y)$ for $x,y \in R$

2005 Today's Calculation Of Integral, 41

Tags: calculus , integration , trigonometry , derivative , geometry , calculus computations

Evaluate \[\int_0^a \sqrt{2ax-x^2}\ dx \ (a>0)\]

1998 Belarus Team Selection Test, 2

Tags: sequence , algebra

For any sequence of real numbers $(a_n), n \in N$, define a new sequence $(b_n)$ as $b_n =a_{n+2}+sa_{n+1}+ta_{n}$, where $s,t$ are given real numbers. Find all ordered pairs $(s,t)$ satisfying the following property: any sequence $(a_n)$ converges as soon as the sequence $(b_n)$ converges.

2018 Iran MO (3rd Round), 3

Tags: algebra , complex number , geometry

A)Let $x,y$ be two complex numbers on the unit circle so that: $\frac{\pi }{3} \le \arg (x)-\arg (y) \le \frac{5 \pi }{3}$ Prove that for any $z \in \mathbb{C}$ we have: $|z|+|z-x|+|z-y| \ge |zx-y|$ B)Let $x,y$ be two complex numbers so that: $\frac{\pi }{3} \le \arg (x)-\arg (y) \le \frac{2 \pi }{3}$ Prove that for any $z \in \mathbb{C}$ we have: $|z|+|z-y|+|z-x| \ge | \frac{\sqrt{3}}{2} x +(y-\frac{x}{2})i|$

2002 China Team Selection Test, 2

Tags: function , algebra

Given an integer $k$. $f(n)$ is defined on negative integer set and its values are integers. $f(n)$ satisfies \[ f(n)f(n+1)=(f(n)+n-k)^2, \] for $n=-2,-3,\cdots$. Find an expression of $f(n)$.

2021 Puerto Rico Team Selection Test, 4

Tags: digit , multiple , divisible , number theory

How many numbers $\overline{abcd}$ with different digits satisfy the following property: if we replace the largest digit with the digit $1$ results in a multiple of $30$?

2000 All-Russian Olympiad Regional Round, 10.4

Tags: combinatorics , tiles , combinatorial geometry

For what smallest $n$ can a $n \times n$ square be cut into squares $40 \times 40$ and $49 \times 49$ so that squares of both types are present?

2005 Germany Team Selection Test, 2

Tags: inequalities

Let n be a positive integer, and let $a_1$, $a_2$, ..., $a_n$, $b_1$, $b_2$, ..., $b_n$ be positive real numbers such that $a_1\geq a_2\geq ...\geq a_n$ and $b_1\geq a_1$, $b_1b_2\geq a_1a_2$, $b_1b_2b_3\geq a_1a_2a_3$, ..., $b_1b_2...b_n\geq a_1a_2...a_n$. Prove that $b_1+b_2+...+b_n\geq a_1+a_2+...+a_n$.

2006 Baltic Way, 11

Tags: geometry

The altitudes of a triangle are $12$, $15$, and $20$. What is the area of this triangle?

2003 Federal Competition For Advanced Students, Part 1, 1

Tags: number theory

Find all triples of prime numbers $(p, q, r)$ such that $p^q + p^r$ is a perfect square.