Olympiad problems

2012 IFYM, Sozopol, 8

Tags: combinatorics

On a chess tournament two teams $A$ and $B$ are playing between each other and each consists of $n$ participants. It was noticed that however they arranged them in pairs, there was at least one pair that already played a match. Prove that there can be chosen $a$ chess players from $A$ and $b$ chess players from $B$ so that $a+b>n$ and each from the first chosen group has played a match earlier with each from the second group.

2011 Pre-Preparation Course Examination, 7

Tags: combinatorics

prove or disprove: in a connected graph $G$, every three longest paths have a vertex in common.

2010 Contests, 1

Tags: geometry

Let $CD$ be a chord of a circle $\Gamma_1$ and $AB$ a diameter of $\Gamma_1$ perpendicular to $CD$ at $N$ with $AN > NB$. A circle $\Gamma_2$ centered at $C$ with radius $CN$ intersects $\Gamma_1$ at points $P$ and $Q$. The line $PQ$ intersects $CD$ at $M$ and $AC$ at $K$; and the extension of $NK$ meets $\Gamma_2$ at $L$. Prove that $PQ$ is perpendicular to $AL$

2004 Thailand Mathematical Olympiad, 3

Tags: algebra , polynomial , cubic

Let $u, v, w$ be the roots of $x^3 -5x^2 + 4x-3 = 0$. Find a cubic polynomial having $u^3, v^3, w^3$ as roots.

2020 Paraguay Mathematical Olympiad, 3

Tags: geometry , area

In triangle $ABC$, side $AC$ is $8$ cm. Two segments are drawn parallel to $AC$ that have their ends on $AB$ and $BC$ and that divide the triangle into three parts of equal area. What is the length of the parallel segment closest to $AC$?

2025 India National Olympiad, P2

Tags: combinatorics , casework

Let $n\ge 2$ be a positive integer. The integers $1,2,\cdots,n$ are written on a board. In a move, Alice can pick two integers written on the board $a\neq b$ such that $a+b$ is an even number, erase both $a$ and $b$ from the board and write the number $\frac{a+b}{2}$ on the board instead. Find all $n$ for which Alice can make a sequence of moves so that she ends up with only one number remaining on the board. [b]Note.[/b] When $n=3$, Alice changes $(1,2,3)$ to $(2,2)$ and can't make any further moves. [i]Proposed by Rohan Goyal[/i]

1947 Moscow Mathematical Olympiad, 127

Tags: equal circles , circles , geometry , circumcircle

Point $O$ is the intersection point of the heights of an acute triangle $\vartriangle ABC$. Prove that the three circles which pass: a) through $O, A, B$, b) through $O, B, C$, and c) through $O, C, A$, are equal

2009 Sharygin Geometry Olympiad, 22

Tags: search , quadratic , trigonometry , circumcircle , 3d geometry , prism , geometry

Construct a quadrilateral which is inscribed and circumscribed, given the radii of the respective circles and the angle between the diagonals of quadrilateral.

2011 China Second Round Olympiad, 10

Tags: integration , algebra

A sequence $a_n$ satisfies $a_1 =2t-3$ ($t \ne 1,-1$), and $a_{n+1}=\dfrac{(2t^{n+1}-3)a_n+2(t-1)t^n-1}{a_n+2t^n-1}$. [list] [b][i]i)[/i][/b] Find $a_n$, [b][i]ii)[/i][/b] If $t>0$, compare $a_{n+1}$ with $a_n$.[/list]

2008 Princeton University Math Competition, 2

Tags: algebra

Find $\log_2 3 * \log_3 4 * \log_4 5 * ... * \log_{62} 63 * \log_{63} 64$ .

1992 Putnam, B6

Tags: linear algebra , matrix

Let $M$ be a set of real $n \times n$ matrices such that i) $I_{n} \in M$, where $I_n$ is the identity matrix. ii) If $A\in M$ and $B\in M$, then either $AB\in M$ or $-AB\in M$, but not both iii) If $A\in M$ and $B \in M$, then either $AB=BA$ or $AB=-BA$. iv) If $A\in M$ and $A \ne I_n$, there is at least one $B\in M$ such that $AB=-BA$. Prove that $M$ contains at most $n^2 $ matrices.

1982 IMO Longlists, 9

Tags: function , floor function , number theory

Given any two real numbers $\alpha$ and $\beta , 0 \leq \alpha < \beta \leq 1$, prove that there exists a natural number $m$ such that \[\alpha < \frac{\phi(m)}{m} < \beta.\]

2011 Mathcenter Contest + Longlist, 10

Tags: algebra , inequalities

Let $p,q,r\in R $ with $pqr=1$. Prove that $$\left(\frac{1}{1-p}\right)^2+\left(\frac{1}{1-q}\right)^2+\left(\frac{1}{1-r}\right)^2\ge 1$$ [i](Real Matrik)[/i]

2019 JBMO Shortlist, C1

Tags: combinatorics

Let $S$ be a set of $100$ positive integer numbers having the following property: “Among every four numbers of $S$, there is a number which divides each of the other three or there is a number which is equal to the sum of the other three.” Prove that the set $S$ contains a number which divides all other $99$ numbers of $S$. [i]Proposed by Tajikistan[/i]

2016 AMC 8, 15

Tags:

What is the largest power of 2 that is a divisor of $13^4-11^4$? $\textbf{(A) } 8\qquad\textbf{(B) } 16\qquad\textbf{(C) } 32\qquad\textbf{(D) } 64\qquad \textbf{(E) } 128$

PEN A Problems, 25

Tags: number theory , least common multiple , floor function , modular arithmetic , function , inequalities , logarithm

Show that ${2n \choose n} \; \vert \; \text{lcm}(1,2, \cdots, 2n)$ for all positive integers $n$.

2007 Bulgaria Team Selection Test, 3

Tags: inequalities

Let $n\geq 2$ is positive integer. Find the best constant $C(n)$ such that \[\sum_{i=1}^{n}x_{i}\geq C(n)\sum_{1\leq j<i\leq n}(2x_{i}x_{j}+\sqrt{x_{i}x_{j}})\] is true for all real numbers $x_{i}\in(0,1),i=1,...,n$ for which $(1-x_{i})(1-x_{j})\geq\frac{1}{4},1\leq j<i \leq n.$

2006 Stanford Mathematics Tournament, 10

Tags: ratio , geometric sequence

Evaluate: $\sum^{\infty}_{k=1} \tfrac{k}{a^{k-1}}$ for all $|a|<1$.

2024 OMpD, 1

Tags: algebra , number theory

Let $O, M, P$ and $D$ be distinct digits from each other, and different from zero, such that $O < M < P < D$, and the following equation is true: \[ \overline{\text{OMPD}} \times \left( \overline{\text{OM}} - \overline{\text{D}} \right) = \overline{\text{MDDMP}} - \overline{\text{OM}} \] (a) Using estimates, explain why it is impossible for the value of $O$ to be greater than or equal to $3$. (b) Explain why $O$ cannot be equal to $1$. (c) Is it possible for $M$ to be greater than or equal to $5$? Justify. (d) Determine the values of $M$, $P$, and $D$.

2013 Romanian Masters In Mathematics, 3

Tags: geometry , geometric transformation , reflection , combinatorics

A token is placed at each vertex of a regular $2n$-gon. A [i]move[/i] consists in choosing an edge of the $2n$-gon and swapping the two tokens placed at the endpoints of that edge. After a finite number of moves have been performed, it turns out that every two tokens have been swapped exactly once. Prove that some edge has never been chosen.

2010 Paraguay Mathematical Olympiad, 4

Tags:

Find all 4-digit numbers $\overline{abcd}$ that are multiples of $11$, such that the 2-digit number $\overline{ac}$ is a multiple of $7$ and $a + b + c + d = d^2$.

2009 Germany Team Selection Test, 1

Tags: algebra , number theory , equation

Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations \[ a^n + pb = b^n + pc = c^n + pa\] then $a = b = c$. [i]Proposed by Angelo Di Pasquale, Australia[/i]

2019 BMT Spring, Tie 1

Tags: algebra

Compute the maximum real value of $a$ for which there is an integer $b$ such that $\frac{ab^2}{a+2b} = 2019$. Compute the maximum possible value of $a$.

1995 Baltic Way, 1

Tags: algebra , system of equations , number theory

Find all triples $(x,y,z)$ of positive integers satisfying the system of equations \[\begin{cases} x^2=2(y+z)\\ x^6=y^6+z^6+31(y^2+z^2)\end{cases}\]

2014 Contests, 1

Tags: limit , induction , logarithm

Let $\{a_n\}_{n\geq 1}$ be a sequence of real numbers which satisfies the following relation: \[a_{n+1}=10^n a_n^2\] (a) Prove that if $a_1$ is small enough, then $\displaystyle\lim_{n\to\infty} a_n =0$. (b) Find all possible values of $a_1\in \mathbb{R}$, $a_1\geq 0$, such that $\displaystyle\lim_{n\to\infty} a_n =0$.