Olympiad problems

2022 Austrian MO National Competition, 4

Decide whether for every polynomial $P$ of degree at least $1$, there exist infinitely many primes that divide $P(n)$ for at least one positive integer $n$. [i](Walther Janous)[/i]

1990 Vietnam National Olympiad, 1

Tags: trigonometry , algebra

The sequence $ (x_n)$, $ n\in\mathbb{N}^*$ is defined by $ |x_1|<1$, and for all $ n \ge 1$, \[ x_{n\plus{}1} \equal{}\frac{\minus{}x_n \plus{}\sqrt{3\minus{}3x_n^2}}{2}\] (a) Find the necessary and sufficient condition for $ x_1$ so that each $ x_n > 0$. (b) Is this sequence periodic? And why?

2020 JBMO Shortlist, 3

Tags: shortlist , algebra , inequality , minimum

Find all triples of positive real numbers $(a, b, c)$ so that the expression $M = \frac{(a + b)(b + c)(a + b + c)}{abc}$ gets its least value.

2020-IMOC, A2

Tags: functional equation , function , algebra

Find all function $f:\mathbb{R}^+$ $\rightarrow \mathbb{R}^+$ such that: $f(f(x) + y)f(x) = f(xy + 1) \forall x, y \in \mathbb{R}^+$

2003 Miklós Schweitzer, 2

Tags: linear algebra , matrix , vector

Let $p$ be a prime and let $M$ be an $n\times m$ matrix with integer entries such that $Mv\not\equiv 0\pmod{p}$ for any column vector $v\neq 0$ whose entries are $0$ are $1$. Show that there exists a row vector $x$ with integer entries such that no entry of $xM$ is $0\pmod{p}$. (translated by L. Erdős)

1988 IMO Longlists, 60

Tags: number theory

Given integers $a_1, \ldots, a_{10},$ prove that there exist a non-zero sequence $\{x_1, \ldots, x_{10}\}$ such that all $x_i$ belong to $\{-1,0,1\}$ and the number $\sum^{10}_{i=1} x_i \cdot a_i$ is divisible by 1001.

2007 Estonia National Olympiad, 4

Tags: limit , number theory

Find all pairs $ (m, n)$ of positive integers such that $ m^n \minus{} n^m \equal{} 3$.

2020 Centroamerican and Caribbean Math Olympiad, 5

Tags: algebra , polynomial , inequality , inequalities , n-variable inequality

Let $P(x)$ be a polynomial with real non-negative coefficients. Let $k$ be a positive integer and $x_1, x_2, \dots, x_k$ positive real numbers such that $x_1x_2\cdots x_k=1$. Prove that $$P(x_1)+P(x_2)+\cdots+P(x_k)\geq kP(1).$$

2005 Croatia National Olympiad, 1

Tags: function , number theory

Find all positive integer solutions of the equation $k!l! = k!+l!+m!.$

2016 CCA Math Bonanza, T5

Tags:

How many permutations of the word ``ACADEMY'' have that there exist two vowels that are separated by an odd distance? For example, the X and Y in XAY are separated by an even distance, while the X and Y in XABY are separated by an odd distance. Note: the vowels are A, E, I, O, and U. Y is [b]NOT[/b] a vowel. [i]2016 CCA Math Bonanza Team #5[/i]

1900 Eotvos Mathematical Competition, 3

Tags: algebra

A cliff is $300$ meters high. Consider two free-falling raindrops such that the second one leaves the top of the cliff when the first one has already fallen $0.001$ millimeters. What is the distance between the drops at the moment the first hits the ground? (Compute the answer to within $0.1$ mm. Neglect air resistance, etc.)

2007 AIME Problems, 5

Tags: geometry , rectangle , floor function , function , 3d geometry

The graph of the equation $9x+223y=2007$ is drawn on graph paper with each square representing one unit in each direction. How many of the $1$ by $1$ graph paper squares have interiors lying entirely below the graph and entirely in the first quadrant?

2016 Greece Team Selection Test, 3

Tags: functional equation

Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\] holds for all $x,y\in\mathbb{Z}$.

1996 Singapore MO Open, 2

Tags: geometry , angle , square

In the following figure, $ABCD$ is a square of unit length and $P, Q$ are points on $AD$ and $AB$ respectively. Find $\angle PCQ$ if $|AP| + |AQ| + |PQ| = 2$. [img]https://cdn.artofproblemsolving.com/attachments/2/c/2f40db978c1d3fcbc0161f874b5cbec926058e.png[/img]

2017 German National Olympiad, 6

Tags: diophantine equation , number theory

Prove that there exist infinitely many positive integers $m$ such that there exist $m$ consecutive perfect squares with sum $m^3$. Specify one solution with $m>1$.

Cono Sur Shortlist - geometry, 2003.G7.3

Tags: geometry

Let $ABC$ be an acute triangle such that $\angle{B}=60$. The circle with diameter $AC$ intersects the internal angle bisectors of $A$ and $C$ at the points $M$ and $N$, respectively $(M\neq{A},$ $N\neq{C})$. The internal bisector of $\angle{B}$ intersects $MN$ and $AC$ at the points $R$ and $S$, respectively. Prove that $BR\leq{RS}$.

1954 AMC 12/AHSME, 47

Tags:

At the midpoint of line segment $ AB$ which is $ p$ units long, a perpendicular $ MR$ is erected with length $ q$ units. An arc is described from $ R$ with a radius equal to $ \frac{1}{2}AB$, meeting $ AB$ at $ T$. Then $ AT$ and $ TB$ are the roots of: $ \textbf{(A)}\ x^2\plus{}px\plus{}q^2\equal{}0 \\ \textbf{(B)}\ x^2\minus{}px\plus{}q^2\equal{}0 \\ \textbf{(C)}\ x^2\plus{}px\minus{}q^2\equal{}0 \\ \textbf{(D)}\ x^2\minus{}px\minus{}q^2\equal{}0 \\ \textbf{(E)}\ x^2\minus{}px\plus{}q\equal{}0$

2018 Macedonia National Olympiad, Problem 2

Tags: absolute value , combinatorics

Let $n$ be a natural number and $C$ a non-negative real number. Determine the number of sequences of real numbers $1, x_{2}, ..., x_{n}, 1$ such that the absolute value of the difference between any two adjacent terms is equal to $C$.

2015 Canada National Olympiad, 4

Tags: geometry , circumcircle , national olympiads

Let $ABC$ be an acute-angled triangle with circumcenter $O$. Let $I$ be a circle with center on the altitude from $A$ in $ABC$, passing through vertex $A$ and points $P$ and $Q$ on sides $AB$ and $AC$. Assume that \[BP\cdot CQ = AP\cdot AQ.\] Prove that $I$ is tangent to the circumcircle of triangle $BOC$.

2020 Saint Petersburg Mathematical Olympiad, 6.

Tags: number theory

The sequence $a_n$ is given as $$a_1=1, a_2=2 \;\;\; \text{and} \;\;\;\; a_{n+2}=a_n(a_{n+1}+1) \quad \forall n\geq 1$$ Prove that $a_{a_n}$ is divisible by $(a_n)^n$ for $n\geq 100$.

2010 Contests, 4

Tags: geometry

Point $O$ is chosen in a triangle $ABC$ such that ${d_a},{d_b},{d_c}$ are distance from point $O$ to sides $BC,CA,AB$, respectively. Find position of point $O$ so that product ${d_a} \cdot {d_b} \cdot {d_c}$ becomes maximum.

1935 Moscow Mathematical Olympiad, 003

Tags: pyramid , geometry , 3d geometry , angle

The base of a pyramid is an isosceles triangle with the vertex angle $\alpha$. The pyramid’s lateral edges are at angle $\phi$ to the base. Find the dihedral angle $\theta$ at the edge connecting the pyramid’s vertex to that of angle $\alpha$.

2018 China Western Mathematical Olympiad, 8

Tags: combinatorics

Let $n,k$ be positive integers, satisfying $n$ is even, $k\geq 2$ and $n>4k.$ There are $n$ points on the circumference of a circle. If the endpoints of $\frac{n}{2}$ chords in a circle that do not intersect with each other are exactly the $n$ points, we call these chords a matching.Determine the maximum of integer $m,$ such that for any matching, there exists $k$ consecutive points, satisfying all the endpoints of at least $m$ chords are in the $k$ points.

1989 IMO Longlists, 36

Tags: combinatorics

Connecting the vertices of a regular $ n$-gon we obtain a closed (not necessarily convex) $ n$-gon. Show that if $ n$ is even, then there are two parallel segments among the connecting segments and if $ n$ is odd then there cannot be exactly two parallel segments.

2007 Balkan MO, 3

Tags: algebra , polynomial , floor function , induction , number theory

Find all positive integers $n$ such that there exist a permutation $\sigma$ on the set $\{1,2,3, \ldots, n\}$ for which \[\sqrt{\sigma(1)+\sqrt{\sigma(2)+\sqrt{\ldots+\sqrt{\sigma(n-1)+\sqrt{\sigma(n)}}}}}\] is a rational number.