Olympiad problems

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.

2007 Hong Kong TST, 3

Tags: trigonometry , inequalities

[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=107262]IMO 2007 HKTST 1[/url] Problem 3 Let $A$, $B$ and $C$ be real numbers such that (i) $\sin A \cos B+|\cos A \sin B|=\sin A |\cos A|+|\sin B|\cos B$, (ii) $\tan C$ and $\cot C$ are defined. Find the minimum value of $(\tan C-\sin A)^{2}+(\cot C-\cos B)^{2}$.

2023 Bulgarian Autumn Math Competition, 9.4

Tags: number theory

Let $p, q$ be coprime integers, such that $|\frac{p} {q}| \leq 1$. For which $p, q$, there exist even integers $b_1, b_2, \ldots, b_n$, such that $$\frac{p} {q}=\frac{1}{b_1+\frac{1}{b_2+\frac{1}{b_3+\ldots}}}? $$

2019 Latvia Baltic Way TST, 2

Tags: function , functional equation , algebra

Let $\mathbb R$ be set of real numbers. Determine all functions $f:\mathbb R\to \mathbb R$ such that $$f(y^2 - f(x)) = yf(x)^2+f(x^2y+y)$$ holds for all real numbers $x; y$

2021 German National Olympiad, 3

Tags: digit , sum of cubes , combinatorics , algebra

For a fixed $k$ with $4 \le k \le 9$ consider the set of all positive integers with $k$ decimal digits such that each of the digits from $1$ to $k$ occurs exactly once. Show that it is possible to partition this set into two disjoint subsets such that the sum of the cubes of the numbers in the first set is equal to the sum of the cubes in the second set.

2012 CHMMC Spring, 10

Tags: combinatorics , geometry

A convex polygon in the Cartesian plane has all of its vertices on integer coordinates. One of the sides of the polygon is $AB$ where $A = (0, 0)$ and $B = (51, 51)$, and the interior angles at $A$ and $B$ are both at most $45$ degrees. Assuming no $180$ degree angles, what is the maximum number of vertices this polygon can have?