Olympiad problems

2020 Centroamerican and Caribbean Math Olympiad, 5

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.

2008 Oral Moscow Geometry Olympiad, 5

Tags: midpoint , geometry , orthocenter , construction

Reconstruct an acute-angled triangle given the orthocenter and midpoints of two sides. (A. Zaslavsky)

2019 Jozsef Wildt International Math Competition, W. 70

Tags: trigonometry , inequalities

If $x \in \left(0,\frac{\pi}{2}\right)$ then$$\left(\frac{\sin \left(\frac{\pi}{2}\sin x\right)}{\sin x}\right)^2+\left(\frac{\sin \left(\frac{\pi}{2}\cos x\right)}{\cos x}\right)^2\geq 3$$

2007 Today's Calculation Of Integral, 176

Tags: calculus , integration , trigonometry , limit , induction , symmetry , calculus computations

Let $f_{n}(x)=\sum_{k=1}^{n}\frac{\sin kx}{\sqrt{k(k+1)}}.$ Find $\lim_{n\to\infty}\int_{0}^{2\pi}\{f_{n}(x)\}^{2}dx.$

2017 ASDAN Math Tournament, 12

Tags:

Anna has a magical compass which can point only in four directions: North, East, South, West. Initially, the compass points North. After each minute, the compass can either turn left, turn right, or stay at its current orientation, with each action occurring with equal probability. What is the probability that the compass points South after $6$ minutes?

1994 AIME Problems, 12

Tags: inequalities , ratio

A fenced, rectangular field measures 24 meters by 52 meters. An agricultural researcher has 1994 meters of fence that can be used for internal fencing to partition the field into congruent, square test plots. The entire field must be partitioned, and the sides of the squares must be parallel to the edges of the field. What is the largest number of square test plots into which the field can be partitioned using all or some of the 1994 meters of fence?

2015 Oral Moscow Geometry Olympiad, 1

Tags: diagonal , geometry , trapezoid , equal angles , congruent

Two trapezoid angles and diagonals are respectively equal. Is it true that such are the trapezoid equal?

2006 Sharygin Geometry Olympiad, 4

Tags: square , geometry , regular , pentagon

a) Given two squares $ABCD$ and $DEFG$, with point $E$ lying on the segment $CD$, and points$ F,G$ outside the square $ABCD$. Find the angle between lines $AE$ and $BF$. b) Two regular pentagons $OKLMN$ and $OPRST$ are given, and the point $P$ lies on the segment $ON$, and the points $R, S, T$ are outside the pentagon $OKLMN$. Find the angle between straight lines $KP$ and $MS$.