Olympiad problems

2021 Estonia Team Selection Test, 3

For each prime $p$, construct a graph $G_p$ on $\{1,2,\ldots p\}$, where $m\neq n$ are adjacent if and only if $p$ divides $(m^{2} + 1-n)(n^{2} + 1-m)$. Prove that $G_p$ is disconnected for infinitely many $p$

2016 Saudi Arabia IMO TST, 2

Tags: combinatorics

Given a set of $2^{2016}$ cards with the numbers $1,2, ..., 2^{2016}$ written on them. We divide the set of cards into pairs arbitrarily, from each pair, we keep the card with larger number and discard the other. We now again divide the $2^{2015}$ remaining cards into pairs arbitrarily, from each pair, we keep the card with smaller number and discard the other. We now have $2^{2014}$ cards, and again divide these cards into pairs and keep the larger one in each pair. We keep doing this way, alternating between keeping the larger number and keeping the smaller number in each pair, until we have just one card left. Find all possible values of this final card.

2017 BMT Spring, 11

Tags: combinatorics

Naomi has a class of $100$ students who will compete with each other in five teams. Once the teams are made, each student will shake hands with every other student, except the students in his or her own team. Naomi chooses to partition the students into teams so as to maximize the number of handshakes. How many handshakes will there be?

2019 Korea USCM, 7

Tags: analysis , series , college contests

For a real number $a$ and an integer $n(\geq 2)$, define $$S_n (a) = n^a \sum_{k=1}^{n-1} \frac{1}{k^{2019} (n-k)^{2019}}$$ Find every value of $a$ s.t. sequence $\{S_n(a)\}_{n\geq 2}$ converges to a positive real.

1995 Grosman Memorial Mathematical Olympiad, 3

Tags: combinatorics

Two thieves stole an open chain with $2k$ white beads and $2m$ black beads. They want to share the loot equally, by cutting the chain to pieces in such a way that each one gets $k$ white beads and $m$ black beads. What is the minimal number of cuts that is always sufficient?

2013 Korea National Olympiad, 1

Tags: geometry , circumcircle , geometry proposed

Let $P$ be a point on segment $BC$. $Q, R$ are points on $AC, AB$ such that $PQ \parallel AB $ and $ PR \parallel AC$. $O, O_{1}, O_{2} $ are the circumcenters of triangle $ABC, BPR, PCQ$. The circumcircles of $BPR, PCQ $ meet at point $K (\ne P)$. Prove that $OO_{1} = KO_{2} $.

2014 JBMO Shortlist, 4

Tags: number theory , prime numbers , AZE JBMO TST

Prove that there are not intgers $a$ and $b$ with conditions, i) $16a-9b$ is a prime number. ii) $ab$ is a perfect square. iii) $a+b$ is also perfect square.

2005 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: floor function , number theory unsolved , number theory

Find all numbers $n$ that can be expressed in the form $n=k+2\lfloor\sqrt{k}\rfloor+2$ for some nonnegative integer $k$.

2024 Tuymaada Olympiad, 8

Tags: inequalities , graph theory , graph

A graph $G$ has $n$ vertices ($n>1$). For each edge $e$ let $c(e)$ be the number of vertices of the largest complete subgraph containing $e$. Prove that the inequality (the summation is over all edges of $G$): \[\sum_{e} \frac{c(e)}{c(e)-1}\le \frac{n^2}{2}.\]

2024 Middle European Mathematical Olympiad, 8

Tags: number theory , number theory proposed , Sequence , Divisibility , constant

Let $k$ be a positive integer and $a_1,a_2,\dots$ be an infinite sequence of positive integers such that \[a_ia_{i+1} \mid k-a_i^2\] for all integers $i \ge 1$. Prove that there exists a positive integer $M$ such that $a_n=a_{n+1}$ for all integers $n \ge M$.

2017 Kyiv Mathematical Festival, 5

Tags: Kyiv mathematical festival , geometry , number theory , analytic geometry

A triangle $ABC$ is given on the plane, such that all its vertices have integer coordinates. Does there necessarily exist a straight line which intersects the straight lines $AB,$ $BC,$ and $AC$ at three distinct points with integer coordinates?

Denmark (Mohr) - geometry, 2011.2

Tags: geometry , Regular , octagon , distance

In the octagon below all sides have the length $1$ and all angles are equal. Determine the distance between the corners $A$ and $B$. [img]https://1.bp.blogspot.com/-i6TAFDvcQ8w/XzXCRhnV_kI/AAAAAAAAMVw/rKrQMfPYYJIaCwl8hhdVHdqO4fIn8O7cwCLcBGAsYHQ/s0/2011%2BMogh%2Bp2.png[/img]

2024 Saint Petersburg Mathematical Olympiad, 4

Tags: algebra , number theory , polynomial

Let's consider all possible quadratic trinomials of the form $x^2 + ax + b$, where $a$ and $b$ are positive integers not exceeding some positive integer $N$. Prove that the number of pairs of such trinomials having a common root does not exceed $N^2$.

2000 Saint Petersburg Mathematical Olympiad, 10.1

Tags: algebra , sequances

Sequences $x_1,x_2,\dots,$ and $y_1,y_2,\dots,$ are defined with $x_1=\dfrac{1}{8}$, $y_1=\dfrac{1}{10}$ and $x_{n+1}=x_n+x_n^2$, $y_{n+1}=y_n+y_n^2$. Prove that $x_m\neq y_n$ for all $m,n\in\mathbb{Z}^{+}$. [I]Proposed by A. Golovanov[/i]

1998 Bulgaria National Olympiad, 3

Tags: combinatorics proposed , combinatorics

The sides and diagonals of a regular $n$-gon $R$ are colored in $k$ colors so that: (i) For each color $a$ and any two vertices $A$,$B$ of $R$ , the segment $AB$ is of color $a$ or there is a vertex $C$ such that $AC$ and $BC$ are of color $a$. (ii) The sides of any triangle with vertices at vertices of $R$ are colored in at most two colors. Prove that $k\leq 2$.

2002 Croatia Team Selection Test, 1

Tags: combinatorics , max

In a certain language there are $n$ letters. A sequence of letters is a word, if there are no two equal letters between two other equal letters. Find the number of words of the maximum length.

2024 Thailand TST, 3

Tags: functional equation , IMO Shortlist , Integers

Determine all functions $f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ such that, for all positive integers $a$ and $b$, \[ f^{bf(a)}(a+1)=(a+1)f(b). \]

2013 Saint Petersburg Mathematical Olympiad, 2

Tags: geometry proposed , geometry

in a convex quadrilateral $ABCD$ , $M,N$ are midpoints of $BC,AD$ respectively. If $AM=BN$ and $DM=CN$ then prove that $AC=BD$. S. Berlov

1995 Tournament Of Towns, (445) 1

Tags: number theory

Prove that if $a$, $b$ and $c$ are integers and the sums $$\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \,\,\,\, and \,\,\,\, \frac{a}{c}+\frac{c}{b}+\frac{b}{a}$$ are also integers, then we have $|a| = |v| = |c|$. (A Gribalko)

1982 All Soviet Union Mathematical Olympiad, 345

Tags: table , combinatorics , combinatorial geometry , square table

Given the square table $n\times n$ with $(n-1)$ marked fields. Prove that it is possible to move all the marked fields below the diagonal by moving rows and columns.

1960 AMC 12/AHSME, 24

Tags: logarithms , geometry , 3D geometry , calculus , integration , function , irrational number

If $\log_{2x}216 = x$, where $x$ is real, then $x$ is: $ \textbf{(A)}\ \text{A non-square, non-cube integer} \qquad$ $\textbf{(B)}\ \text{A non-square, non-cube, non-integral rational number} \qquad$ $\textbf{(C)}\ \text{An irrational number} \qquad$ $\textbf{(D)}\ \text{A perfect square}\qquad$ $\textbf{(E)}\ \text{A perfect cube} $

2022 Romania Team Selection Test, 2

Tags: number theory , Sequence , romania , Romanian TST

Fix a nonnegative integer $a_0$ to define a sequence of integers $a_0,a_1,\ldots$ by letting $a_k,k\geq 1$ be the smallest integer (strictly) greater than $a_{k-1}$ making $a_{k-1}+a_k{}$ into a perfect square. Let $S{}$ be the set of positive integers not expressible as the difference of two terms of the sequence $(a_k)_{k\geq 0}.$ Prove that $S$ is finite and determine its size in terms of $a_0.$

1985 Miklós Schweitzer, 1

Tags: college contests , inequalities

[b]1.[/b] Some proper partitions $P_1, \dots , P_n$ of a finite set $S$ (that is, partitions containing at least two parts) are called [i]independent[/i] if no matter how we choose one class from each partition, the intersection of the chosen classes is nonempty. Show that if the inequality $\frac{\left | S \right | }{2} < \left |P_1 \right | \dots \left |P_n \right |\qquad \quad (*)$ holds for some independent partitions, then $P_1, \dots , P_n$ is maximal in the sense that there is no partition $P$ such that $P,P_1, \dots , P_n$ are independent. On the other hand, show that inequality $(*)$ is not necessary for this maximality. ([b]C.20[/b]) [E. Gesztelyi]

2014 Dutch IMO TST, 4

Tags: modular arithmetic , number theory unsolved , number theory

Determine all pairs $(p, q)$ of primes for which $p^{q+1}+q^{p+1}$ is a perfect square.

2008 Korean National Olympiad, 2

Tags: Inequality , algebra

We have $x_i >i$ for all $1 \le i \le n$. Find the minimum value of $\frac{(\sum_{i=1}^n x_i)^2}{\sum_{i=1}^n \sqrt{x^2_i - i^2}}$