Olympiad problems

2017 Korea Junior Math Olympiad, 6

Tags: geometry , circumcircle

Let triangle $ABC$ be an acute scalene triangle, and denote $D,E,F$ by the midpoints of $BC,CA,AB$, respectively. Let the circumcircle of $DEF$ be $O_1$, and its center be $N$. Let the circumcircle of $BCN$ be $O_2$. $O_1$ and $O_2$ meet at two points $P, Q$. $O_2$ meets $AB$ at point $K(\neq B)$ and meets $AC$ at point $L(\neq C)$. Show that the three lines $EF,PQ,KL$ are concurrent.

2017 BMT Spring, 5

Tags: combinatorics

You enter an elevator on floor $0$ of a building with some other people, and request to go to floor $10$. In order to be efficient, it doesn’t stop at adjacent floors (so, if it’s at floor $0$, its next stop cannot be floor $ 1$). Given that the elevator will stop at floor $10$, no matter what other floors it stops at, how many combinations of stops are there for the elevator?

2017-2018 SDPC, 5

Tags: algebra

Given positive real numbers $a,b,c$ such that $abc=1$, find the maximum possible value of $$\frac{1}{(4a+4b+c)^3}+\frac{1}{(4b+4c+a)^3}+\frac{1}{(4c+4a+b)^3}.$$

1996 All-Russian Olympiad Regional Round, 10.3

Tags: geometry , trapezoid

Given an angle with vertex $B$. Construct point $M$ as follows. Let us take an arbitrary isosceles trapezoid whose sides lie on the sides of a given angle. Through two opposite ones draw tangents to the vertices of the circle circumscribed around it. Let $M$ denote the point of intersection of these tangents. What figure do all such points $M$ form?

2004 Romania National Olympiad, 3

Tags: function , inequalities , calculus , algebra , logarithm

Let $n>2,n \in \mathbb{N}$ and $a>0,a \in \mathbb{R}$ such that $2^a + \log_2 a = n^2$. Prove that: \[ 2 \cdot \log_2 n>a>2 \cdot \log_2 n -\frac{1}{n} . \] [i]Radu Gologan[/i]

2020 MBMT, 27

Tags:

The perfect square game is played as follows: player 1 says a positive integer, then player 2 says a strictly smaller positive integer, and so on. The game ends when someone says 1; that player wins if and only if the sum of all numbers said is a perfect square. What is the sum of all $n$ such that, if player 1 starts by saying $n$, player 1 has a winning strategy? A winning strategy for player 1 is a rule player 1 can follow to win, regardless of what player 2 does. If player 1 wins, player 2 must lose, and vice versa. Both players play optimally. [i]Proposed by Jacob Stavrianos[/i]

2022 Caucasus Mathematical Olympiad, 5

Tags: combinatorics

Let $S$ be the set of all $5^6$ positive integers, whose decimal representation consists of exactly 6 odd digits. Find the number of solutions $(x,y,z)$ of the equation $x+y=10z$, where $x\in S$, $y\in S$, $z\in S$.

2008 Paraguay Mathematical Olympiad, 2

Tags: parabola , absolute value , conic

Find for which values of $n$, an integer larger than $1$ but smaller than $100$, the following expression has its minimum value: $S = |n-1| + |n-2| + \ldots + |n-100|$

1997 Polish MO Finals, 3

Tags: combinatorics

Given any $n$ points on a unit circle show that at most $\frac{n^2}{3}$ of the segments joining two points have length $> \sqrt{2}$.

2024 Harvard-MIT Mathematics Tournament, 20

Tags: guts

Compute $\sqrt[4]{5508^3+5625^3+5742^3},$ given that it is an integer.

PEN L Problems, 13

Tags: induction , algebra , polynomial , quadratic , difference of squares , special factorizations , linear recurrence

The sequence $\{x_{n}\}_{n \ge 1}$ is defined by \[x_{1}=x_{2}=1, \; x_{n+2}= 14x_{n+1}-x_{n}-4.\] Prove that $x_{n}$ is always a perfect square.

2014 JBMO Shortlist, 2

Tags: diophantine equation , modular arithmetic

Find all triples of primes $(p,q,r)$ satisfying $3p^{4}-5q^{4}-4r^{2}=26$.

2014 Kosovo National Mathematical Olympiad, 1

Tags: algebra

Let $a$ and $b$ be the solutions to $x^2-x+q=0$, find $a^3+b^3+3(a^3b+ab^3)+6(a^3b^2+a^2b^3)$.

2015 Chile TST Ibero, 3

Tags: geometry

Prove that in a scalene acute-angled triangle, the orthocenter, the incenter, and the circumcenter are not collinear.

2007 Bulgarian Autumn Math Competition, Problem 8.2

Tags: geometry , locus of points , equal area

Let $ABCD$ be a convex quadrilateral. Determine all points $M$, which lie inside $ABCD$, such that the areas of $ABCM$ and $AMCD$ are equal.

II Soros Olympiad 1995 - 96 (Russia), 11.2

Tags: geometry , 3d geometry , tetrahedron

Is it possible that the heights of a tetrahedron (that is, a triangular pyramid) would be equal to the numbers $1$, $2$, $3$ and $6$?

2016 IMO Shortlist, A4

Tags: algebra , functional equation

Find all functions $f:(0,\infty)\rightarrow (0,\infty)$ such that for any $x,y\in (0,\infty)$, $$xf(x^2)f(f(y)) + f(yf(x)) = f(xy) \left(f(f(x^2)) + f(f(y^2))\right).$$

2023 Iran Team Selection Test, 5

Tags: algebra , polynomial , function

Find all injective $f:\mathbb{Z}\ge0 \to \mathbb{Z}\ge0 $ that for every natural number $n$ and real numbers $a_0,a_1,...,a_n$ (not everyone equal to $0$), polynomial $\sum_{i=0}^{n}{a_i x^i}$ have real root if and only if $\sum_{i=0}^{n}{a_i x^{f(i)}}$ have real root. [i]Proposed by Hesam Rajabzadeh [/i]

2012 Today's Calculation Of Integral, 840

Tags: calculus , integration , function , trigonometry , algebra , domain , inequalities

Let $x,\ y$ be real numbers. For a function $f(t)=x\sin t+y\cos t$, draw the domain of the points $(x,\ y)$ for which the following inequality holds. \[\left|\int_{-\pi}^{\pi} f(t)\cos t\ dt\right|\leq \int_{-\pi}^{\pi} \{f(t)\}^2dt.\]

2009 USAMTS Problems, 2

Tags:

The ordered pair of four-digit numbers $(2025, 3136)$ has the property that each number in the pair is a perfect square and each digit of the second number is $1$ more than the corresponding digit of the first number. Find, with proof, all ordered pairs of five-digit numbers and ordered pairs of six-digit numbers with the same property: each number in the pair is a perfect square and each digit of the second number is $1$ more than the corresponding digit of the first number.

2014 Contests, 1

Tags: algebra

A sequence $a_0,a_1,a_2,\cdots$ satisfies the conditions $a_0 = 0$ , $a_{n-1}^2 - a_{n-1} = a_n^2 + a_n$ 1) determine the two possible values of $a_1$ . then determine all possible values of $a_2$ . 2)for each $n$, prove that $a_{n+1}=a_n+1$ or $a_{n+1} = -a_n$ 3)Describe the possible values of $a_{1435}$ 4)Prove that the values that you got in (3) are correct

1990 All Soviet Union Mathematical Olympiad, 513

Tags: point , graph , combinatorics , combinatorial geometry , edge

A graph has $30$ points and each point has $6$ edges. Find the total number of triples such that each pair of points is joined or each pair of points is not joined.

2014 Belarusian National Olympiad, 8

Tags: combinatorial geometry , combinatorics , covering

An $n\times n$ square is divided into $n^2$ unit cells. Is it possible to cover this square with some layers of 4-cell figures of the following shape [img]https://cdn.artofproblemsolving.com/attachments/5/7/d42a8011ec4c5c91c337296d8033d412fade5c.png[/img](i.e. each cell of the square must be covered with the same number of these figures) if a) $n=6$? b) $n=7$? (The sides of each figure must coincide with the sides of the cells; the figures may be rotated and turned over, but none of them can go beyond the bounds of the square.)

2009 All-Russian Olympiad, 4

Tags: modular arithmetic , geometry , geometric transformation , rotation , combinatorics

On a circle there are 2009 nonnegative integers not greater than 100. If two numbers sit next to each other, we can increase both of them by 1. We can do this at most $ k$ times. What is the minimum $ k$ so that we can make all the numbers on the circle equal?

2008 District Olympiad, 4

Tags: function , real analysis

Find the values of $a\in [0,\infty)$ for which there exist continuous functions $f:\mathbb{R}\rightarrow \mathbb{R}$, such that $f(f(x))=(x-a)^2,\ (\forall)x\in \mathbb{R}$.