Olympiad problems

2013 China Western Mathematical Olympiad, 5

Tags: combinatorics

A nonempty set $A$ is called an [i]$n$-level-good [/i]set if $ A \subseteq \{1,2,3,\ldots,n\}$ and $|A| \le \min_{x\in A} x$ (where $|A|$ denotes the number of elements in $A$ and $\min_{x\in A} x$ denotes the minimum of the elements in $A$). Let $a_n$ be the number of $n$-level-good sets. Prove that for all positive integers $n$ we have $a_{n+2}=a_{n+1}+a_{n}+1$.

2023 All-Russian Olympiad Regional Round, 11.5

Tags: geometry

Given is a triangle $ABC$ with altitude $AH$ and median $AM$. The line $OH$ meets $AM$ at $D$. Let $AB \cap CD=E, AC \cap BD=F$. If $EH$ and $FH$ meet $(ABC)$ at $X, Y$, prove that $BY, CX, AH$ are concurrent.

1986 Tournament Of Towns, (111) 5

Tags: tournament , combinatorics

$20$ football teams take part in a tournament . On the first day all the teams play one match . On the second day all the teams play a further match . Prove that after the second day it is possible to select $10$ teams, so that no two of them have yet played each other. ( S . A . Genkin)

2016 Peru Cono Sur TST, P2

Tags: polygon , geometric inequality , geometry , inequalities , area

Let $\omega$ be a circle. For each $n$, let $A_n$ be the area of a regular $n$-sided polygon circumscribed to $\omega$ and $B_n$ the area of a regular $n$-sided polygon inscribed in $\omega$ . Try that $3A_{2015} + B_{2015}> 4A_{4030}$

MathLinks Contest 2nd, 6.1

Tags: function , algebra , floor function

Determine the parity of the positive integer $N$, where $$N = \lfloor \frac{2002!}{2001 \cdot2003} \rfloor.$$

2014 NIMO Problems, 1

Tags: probability , number theory , expected value , relatively prime

You drop a 7 cm long piece of mechanical pencil lead on the ﬂoor. A bully takes the lead and breaks it at a random point into two pieces. A piece of lead is unusable if it is 2 cm or shorter. If the expected value of the number of usable pieces afterwards is $\frac{m}n$ for relatively prime positive integers $m$ and $n$, compute $100m + n$. [i]Proposed by Aaron Lin[/i]

2020 Estonia Team Selection Test, 1

Tags: arithmetic sequence , arithmetic progression , algebra

Let $a_1, a_2,...$ a sequence of real numbers. For each positive integer $n$, we denote $m_n =\frac{a_1 + a_2 +... + a_n}{n}$. It is known that there exists a real number $c$ such that for any different positive integers $i, j, k$: $(i - j) m_k + (j - k) m_i + (k - i) m_j = c$. Prove that the sequence $a_1, a_2,..$ is arithmetic

1964 All Russian Mathematical Olympiad, 055

Tags: tangential , trapezoid , geometry

Let $ABCD$ be an tangential trapezoid, $E$ is a point of its diagonals intersection, $r_1,r_2,r_3,r_4$ -- the radiuses of the circles inscribed in the triangles $ABE$, $BCE$, $CDE$, $DAE$ respectively. Prove that $$1/(r_1)+1/(r_3) = 1/(r_2)+1/(r_4).$$

2017 Pakistan TST, Problem 3

Tags: algebraic identities , functional equation , algebra

Find all $f:\mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that for all distinct $x,y,z$ $f(x)^2-f(y)f(z)=f(x^y)f(y)f(z)[f(y^z)-f(z^x)]$

2009 Kazakhstan National Olympiad, 2

Tags: perpendicular bisector , geometry

In triangle $ABC$ $AA_1; BB_1; CC_1$-altitudes. Let $I_1$ and $I_2$ be in-centers of triangles $AC_1B_1$ and $CA_1B_1$ respectively. Let in-circle of $ABC$ touch $AC$ in $B_2$. Prove, that quadrilateral $I_1I_2B_1B_2$ inscribed in a circle.

2010 Contests, 3

Tags: combinatorics

Given is the set $M_n=\{0, 1, 2, \ldots, n\}$ of nonnegative integers less than or equal to $n$. A subset $S$ of $M_n$ is called [i]outstanding[/i] if it is non-empty and for every natural number $k\in S$, there exists a $k$-element subset $T_k$ of $S$. Determine the number $a(n)$ of outstanding subsets of $M_n$. [i](41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 3)[/i]

1966 IMO Shortlist, 39

Tags: intersection , circles , geometry , circumcircle

Consider a circle with center $O$ and radius $R,$ and let $A$ and $B$ be two points in the plane of this circle. [b]a.)[/b] Draw a chord $CD$ of the circle such that $CD$ is parallel to $AB,$ and the point of the intersection $P$ of the lines $AC$ and $BD$ lies on the circle. [b]b.)[/b] Show that generally, one gets two possible points $P$ ($P_{1}$ and $P_{2}$) satisfying the condition of the above problem, and compute the distance between these two points, if the lengths $OA=a,$ $OB=b$ and $AB=d$ are given.

1998 Romania Team Selection Test, 2

Tags: algebra , polynomial , functional equation

Find all positive integers $ k$ for which the following statement is true: If $ F(x)$ is a polynomial with integer coefficients satisfying the condition $ 0 \leq F(c) \leq k$ for each $ c\in \{0,1,\ldots,k \plus{} 1\}$, then $ F(0) \equal{} F(1) \equal{} \ldots \equal{} F(k \plus{} 1)$.

2022 CHMMC Winter (2022-23), 1

Tags: algebra

A wall contains three switches $A,B,C$, each of which powers a light when flipped on. Every $20$ seconds, switch $A$ is turned on and then immediately turned off again. The same occurs for switch $B$ every $21$ seconds and switch $C$ every $22$ seconds. At time $t = 0$, all three switches are simultaneously on. Let $t = T > 0$ be the earliest time that all three switches are once again simultaneously on. Compute the number of times $t > 0$ before $T$ when at least two switches are simultaneously on.

2023 Puerto Rico Team Selection Test, 5

Tags: number theory , algebra

Six fruit baskets contain peaches, apples and pears. The number of peaches in each basket is equal to the total number of apples in the other baskets. The number of apples in each basket is equal to the total number of pears in the other baskets. (a) Find a way to place $31$ fruits in the baskets, satisfying the conditions of the statement. (b) Explain why the total number of fruits must always be multiple of $31$.

1999 Baltic Way, 10

Tags: combinatorics

May the points of a disc of radius $1$ (including its circumference) be partitioned into three subsets in such a way that no subset contains two points separated by a distance $1$?

2016 District Olympiad, 1

Tags: diophantine equation , equation , algebra

Solve in $ \mathbb{N}^2: $ $$ x+y=\sqrt x+\sqrt y+\sqrt{xy} . $$

2012 China Second Round Olympiad, 6

Tags: algebra , inequalities , function

Let $f(x)$ be an odd function on $\mathbb{R}$, such that $f(x)=x^2$ when $x\ge 0$. Knowing that for all $x\in [a,a+2]$, the inequality $f(x+a)\ge 2f(x)$ holds, find the range of real number $a$.

1998 Switzerland Team Selection Test, 9

Tags: algebra , inequalities

If $x$ and $y$ are positive numbers, prove the inequality $\frac{x}{x^4 +y^2 }+\frac{y}{x^2 +y^4} \le \frac{1}{xy}$ .

1991 China National Olympiad, 3

Tags: combinatorics

There are $10$ birds on the ground. For any $5$ of them, there are at least $4$ birds on a circle. Determine the least possible number of birds on the circle with the most birds.

2000 Slovenia National Olympiad, Problem 4

Tags:

Alex and Jack have $1000$ sheets each. Each of them writes the numbers $1,\ldots,2000$ on his sheets in an arbitrary order, with one number on each side of a sheet. The sheets are to be placed on the floor so that one side of each sheet is visible. Prove that they can do so in such a way that each of the numbers from $1$ to $2000$ is visible.

LMT Team Rounds 2021+, 1

Tags: algebra

Kevin writes the multiples of three from $1$ to $100$ on the whiteboard. How many digits does he write?

2020 Nordic, 1

Tags: number theory , arithmetic progression , min , prime

For a positive integer $n$, denote by $g(n)$ the number of strictly ascending triples chosen from the set $\{1, 2, ..., n\}$. Find the least positive integer $n$ such that the following holds:[i] The number $g(n)$ can be written as the product of three different prime numbers which are (not necessarily consecutive) members in an arithmetic progression with common difference $336$.[/i]

2024 Argentina National Math Olympiad Level 3, 5

Tags: geometric inequality , algebra , geometry , inequalities

In triangle $ABC$, let $A'$, $B'$ and $C'$ be points on the sides $BC$, $CA$ and $AB$, respectively, such that$$\frac{BA'}{A'C}=\frac{CB'}{B'A}=\frac{AC'}{C'B}.$$ The line parallel to $B'C'$ passing through $A'$ intersects line $AC$ at $P$ and line $AB$ at $Q$. Prove that$$\frac{PQ}{B'C'} \geqslant 2.$$

2014 Baltic Way, 1

Tags: modular arithmetic , algebra , trigonometry

Show that \[\cos(56^{\circ}) \cdot \cos(2 \cdot 56^{\circ}) \cdot \cos(2^2\cdot 56^{\circ})\cdot . . . \cdot \cos(2^{23}\cdot 56^{\circ}) = \frac{1}{2^{24}} .\]