Olympiad problems

2013 Paraguay Mathematical Olympiad, 4

Tags: combinatorics

Pedro and Juan are playing the following game: $-$ There are $2$ piles of rocks, with $X$ rocks in one pile and $Y$ rocks in the other pile ($X < 12, Y < 11$). $-$ Each player can draw: -- $1$ rock from one of the piles, or -- $2$ rocks from one of the piles, or -- $1$ rock from each pile, or -- $2$ rock from one pile and $1$ from the other pile. Each player must perform one of these four operations in their turns. The looser is the one who takes the last rock. Pedro plays first and has a winning strategy. What are the three maximum possible values of ($X+Y$)?

2007 Sharygin Geometry Olympiad, 3

Tags: geometry , Triangle

Segments connecting an inner point of a convex non-equilateral n-gon to its vertices divide the n-gon into n equal triangles. What is the least possible n?

1985 Traian Lălescu, 1.2

Tags: series , limits , real analysis , calculus

Calculate $ \sum_{i=2}^{\infty}\frac{i^2-2}{i!} . $

2014 Saudi Arabia IMO TST, 3

Tags: induction , algebra , system of equations , number theory unsolved , number theory

Show that it is possible to write a $n \times n$ array of non-negative numbers (not necessarily distinct) such that the sums of entries on each row and each column are pairwise distinct perfect squares.

2008 District Olympiad, 2

Tags: function , algebra , domain , algebra proposed

Consider the positive reals $ x$, $ y$ and $ z$. Prove that: a) $ \arctan(x) \plus{} \arctan(y) < \frac {\pi}{2}$ iff $ xy < 1$. b) $ \arctan(x) \plus{} \arctan(y) \plus{} \arctan(z) < \pi$ iff $ xyz < x \plus{} y \plus{} z$.

2011 NIMO Problems, 4

Tags: inequalities

Find the number of ordered pairs of integers $(a, b)$ that satisfy the inequality \[ 1 < a < b+2 < 10. \] [i]Proposed by Lewis Chen [/i]

2014 BMT Spring, 20

Tags: algebra , calculus , integration

A certain type of Bessel function has the form $I(x) = \frac{1}{\pi} \int_0^{\pi}e^{x \cos \theta} d\theta$ for all real $x$. Evaluate $\int_0^{\infty} x I(2x) e^{-x^2}dx$.

1996 Bulgaria National Olympiad, 2

Tags: geometry , circumcircle , Bulgaria

The quadrilateral $ABCD$ is inscribed in a circle. The lines $AB$ and $CD$ meet each other in the point $E$, while the diagonals $AC$ and $BD$ in the point $F$. The circumcircles of the triangles $AFD$ and $BFC$ have a second common point, which is denoted by $H$. Prove that $\angle EHF=90^\circ$.

2005 Harvard-MIT Mathematics Tournament, 6

Tags: analytic geometry

Find the sum of the x-coordinates of the distinct points of intersection of the plane curves given by $x^2 = x + y + 4$ and $y^2 = y - 15x + 36$.

2004 Germany Team Selection Test, 2

Tags: linear algebra , matrix , combinatorics , IMO Shortlist

Let $x_1,\ldots, x_n$ and $y_1,\ldots, y_n$ be real numbers. Let $A = (a_{ij})_{1\leq i,j\leq n}$ be the matrix with entries \[a_{ij} = \begin{cases}1,&\text{if }x_i + y_j\geq 0;\\0,&\text{if }x_i + y_j < 0.\end{cases}\] Suppose that $B$ is an $n\times n$ matrix with entries $0$, $1$ such that the sum of the elements in each row and each column of $B$ is equal to the corresponding sum for the matrix $A$. Prove that $A=B$.

2008 Indonesia TST, 3

Tags: geometry

Let $ABCD$ be a convex quadrilateral with $AB$ is not parallel to $CD$ Circle $\Gamma_{1}$ with center $O_1$ passes through $A$ and $B$, and touches segment $CD$ at $P$. Circle $\Gamma_{2}$ with center $O_2$ passes through $C$ and $D$, and touches segment $AB$ at $Q$. Let $E$ and $F$ be the intersection of circles $\Gamma_{1}$ and $\Gamma_{2}$. Prove that $EF$ bisects segment $PQ$ if and only if $BC$ is parallel to $AD$.

2010 ELMO Shortlist, 6

Tags: combinatorics proposed , combinatorics

Hamster is playing a game on an $m \times n$ chessboard. He places a rook anywhere on the board and then moves it around with the restriction that every vertical move must be followed by a horizontal move and every horizontal move must be followed by a vertical move. For what values of $m,n$ is it possible for the rook to visit every square of the chessboard exactly once? A square is only considered visited if the rook was initially placed there or if it ended one of its moves on it. [i]Brian Hamrick.[/i]

Russian TST 2018, P3

Tags: combinatorics

Kirill has $n{}$ identical footballs and two infinite rows of baskets, each numbered with consecutive natural numbers. In one row the baskets are red, in the other they are blue. Kirill puts all the balls into baskets so that the number of balls in the either row of baskets does not increase. Denote by $A{}$ the number of ways to arrange the balls so that the first blue basket contains more balls than any red one, and by $B{}$ the number of arrangements so that the number of some blue basket corresponds with the number of balls in it. Prove that $A = B$.

2014 JHMMC 7 Contest, 19

Tags: JHMMC

Lev and Alex are racing on a number line. Alex is much faster, so he goes to sleep until Lev reaches $100$. Lev runs at $5$ integers per minute and Alex runs at $7$ integers per minute (in the same direction). How many minutes from the START of the race will it take Alex to catch up to Lev (who is still running after Alex wakes up)?

2020 LIMIT Category 2, 10

Tags: limit , trigonometry , geometry

In a triangle $\triangle XYZ$, $\tan(x)\tan(z)=2$, $\tan(y)\tan(z)=18$. Then what is $\tan^2(z)$?

2006 Czech and Slovak Olympiad III A, 1

Tags: number theory proposed , number theory

Define a sequence of positive integers $\{a_n\}$ through the recursive formula: $a_{n+1}=a_n+b_n(n\ge 1)$,where $b_n$ is obtained by rearranging the digits of $a_n$ (in decimal representation) in reverse order (for example,if $a_1=250$,then $b_1=52,a_2=302$,and so on). Can $a_7$ be a prime?

2013 Iran MO (3rd Round), 6

Tags: combinatorics unsolved , combinatorics

Planet Tarator is a planet in the Yoghurty way galaxy. This planet has a shape of convex $1392$-hedron. On earth we don't have any other information about sides of planet tarator. We have discovered that each side of the planet is a country, and has it's own currency. Each two neighbour countries have their own constant exchange rate, regardless of other exchange rates. Anybody who travels on land and crosses the border must change all his money to the currency of the destination country, and there's no other way to change the money. Incredibly, a person's money may change after crossing some borders and getting back to the point he started, but it's guaranteed that crossing a border and then coming back doesn't change the money. On a research project a group of tourists were chosen and given same amount of money to travel around the Tarator planet and come back to the point they started. They always travel on land and their path is a nonplanar polygon which doesn't intersect itself. What is the maximum number of tourists that may have a pairwise different final amount of money? [b]Note 1:[/b] Tourists spend no money during travel! [b]Note 2:[/b] The only constant of the problem is 1392, the number of the sides. The exchange rates and the way the sides are arranged are unknown. Answer must be a constant number, regardless of the variables. [b]Note 3:[/b] The maximum must be among all possible polyhedras. Time allowed for this problem was 90 minutes.

1994 Tournament Of Towns, (437) 3

Tags: geometry , median , angles

The median $AD$ of triangle $ABC$ intersects its inscribed circle (with center $O$) at the points $X$ and $Y$. Find the angle $XOY$ if $AC = AB + AD$. (A Fedotov)

2003 Korea Junior Math Olympiad, 1

Tags: number theory , Squares , Integer

Show that for any non-negative integer $n$, the number $2^{2n+1}$ cannot be expressed as a sum of four non-zero square numbers.

2019 Saint Petersburg Mathematical Olympiad, 4

Tags: number theory , reciprocal sum , Sum , Divisors , Irreducible

Olya wrote fractions of the form $1 / n$ on cards, where $n$ is all possible divisors the numbers $6^{100}$ (including the unit and the number itself). These cards she laid out in some order. After that, she wrote down the number on the first card, then the sum of the numbers on the first and second cards, then the sum of the numbers on the first three cards, etc., finally, the sum of the numbers on all the cards. Every amount Olya recorded on the board in the form of irreducible fraction. What is the least different denominators could be on the numbers on the board?

1941 Moscow Mathematical Olympiad, 079

Tags: algebra , equation , absolute value

Solve the equation: $|x + 1| - |x| + 3|x - 1| - 2|x - 2| = x + 2$.

2009 Switzerland - Final Round, 9

Tags: inequalities , Functional inequality , functional , algebra

Find all injective functions $f : N\to N$ such that holds for all natural numbers $n$: $$f(f(n)) \le \frac{f(n) + n}{2}$$

1989 ITAMO, 4

Tags: geometry , cyclic quadrilateral , Concyclic

Points $A,M,B,C,D$ are given on a circle in this order such that $A$ and $B$ are equidistant from $M$. Lines $MD$ and $AC$ intersect at $E$ and lines $MC$ and $BD$ intersect at $F$. Prove that the quadrilateral $CDEF$ is inscridable in a circle.

2008 Bulgarian Autumn Math Competition, Problem 8.4

Tags: Geoemtry , combinatorics , Obtuse triangle

Let $M$ be a set of $99$ different rays with a common end point in a plane. It's known that two of those rays form an obtuse angle, which has no other rays of $M$ inside in. What is the maximum number of obtuse angles formed by two rays in $M$?

1972 Bundeswettbewerb Mathematik, 4

Tags: combinatorics proposed , combinatorics

$p>2$ persons participate at a chess tournament, two players play at most one game against each other. After $n$ games were made, no more game is running and in every subset of three players, we can find at least two that havem't played against each other. Show that $n \leq \frac{p^{2}}4$.