Olympiad problems

1993 Mexico National Olympiad, 2

Find all numbers between $100$ and $999$ which equal the sum of the cubes of their digits.

2020 Purple Comet Problems, 10

Tags: number theory

Given that $a, b$, and $c$ are distinct positive integers such that $a \cdot b \cdot c = 2020$, the minimum possible positive value of $\frac{1}{a}-\frac{1}{b}-\frac{1}{c}$, is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

2005 Junior Balkan Team Selection Tests - Romania, 14

Tags: trigonometry , geometry , inequalities

Let $a,b,c$ be three positive real numbers with $a+b+c=3$. Prove that \[ (3-2a)(3-2b)(3-2c) \leq a^2b^2c^2 . \] [i]Robert Szasz[/i]

2015 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory

Prove that number $1$ can be represented as a sum of a finite number $n$ of real numbers, less than $1,$ not necessarily distinct, which contain in their decimal representation only the digits $0$ and/or $7.$ Which is the least possible number $n$?

2021 Durer Math Competition Finals, 15

Tags: combinatorics

King Albrecht founded a family. In the family everyone has exactly $ 8$ children. The only, but really important rule is that among the grandchildren of any person at most $x$ can be named Bela. (None of Albrecht’s children is called Bela.) For which $x$ is it possible that after a certain time each newborn in the family has at least one direct ancestor in the Royal family called Bela. No two of Albrecht’s descendants (including himself) have a common child.

2001 Baltic Way, 16

Tags: function , number theory , induction

Let $f$ be a real-valued function defined on the positive integers satisfying the following condition: For all $n>1$ there exists a prime divisor $p$ of $n$ such that $f(n)=f\left(\frac{n}{p}\right)-f(p)$. Given that $f(2001)=1$, what is the value of $f(2002)$?

2014 India National Olympiad, 5

Tags: geometric transformation , circumcircle , trigonometry , geometry , perpendicular bisector , reflection

In a acute-angled triangle $ABC$, a point $D$ lies on the segment $BC$. Let $O_1,O_2$ denote the circumcentres of triangles $ABD$ and $ACD$ respectively. Prove that the line joining the circumcentre of triangle $ABC$ and the orthocentre of triangle $O_1O_2D$ is parallel to $BC$.

1988 AMC 12/AHSME, 6

Tags: rectangle , rhombus , trapezoid , geometry , parallelogram

A figure is an equiangular parallelogram if and only if it is a $ \textbf{(A)}\ \text{rectangle}\qquad\textbf{(B)}\ \text{regular polygon}\qquad\textbf{(C)}\ \text{rhombus}\qquad\textbf{(D)}\ \text{square}\qquad\textbf{(E)}\ \text{trapezoid} $

1992 Poland - First Round, 3

Tags: symmetry

Given is a hexagon $ABCDEF$ with a center of symmetry. The lines $AB$ and $EF$ meet at the point $A'$, the lines $BC$ and $AF$ meet at the point $B'$, and the lines $AB$ and $CD$ meet at the point $C'$. Prove that $AB \cdot BC \cdot CD = AA' \cdot BB' \cdot CC'$.

2003 National Olympiad First Round, 9

Tags: inradius , geometry

How many integer triangles are there with inradius $1$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{Infinite} $

2009 Hanoi Open Mathematics Competitions, 10

Tags: geometric inequality , inequalities , geometry

Prove that $d^2+(a-b)^2<c^2$ ,where $d$ is diameter of the inscribed circle of $\vartriangle ABC$

2023 Canadian Junior Mathematical Olympiad, 4

Tags: combinatorics

There are 20 students in a high school class, and each student has exactly three close friends in the class. Five of the students have bought tickets to an upcoming concert. If any student sees that at least two of their close friends have bought tickets, then they will buy a ticket too. Is it possible that the entire class buys tickets to the concert? (Assume that friendship is mutual; if student $A$ is close friends with student $B$, then $B$ is close friends with $A$.)

2005 IMO Shortlist, 5

Tags: modular arithmetic , invariant , combinatorics

There are $ n$ markers, each with one side white and the other side black. In the beginning, these $ n$ markers are aligned in a row so that their white sides are all up. In each step, if possible, we choose a marker whose white side is up (but not one of the outermost markers), remove it, and reverse the closest marker to the left of it and also reverse the closest marker to the right of it. Prove that, by a finite sequence of such steps, one can achieve a state with only two markers remaining if and only if $ n \minus{} 1$ is not divisible by $ 3$. [i]Proposed by Dusan Dukic, Serbia[/i]

2008 Romania National Olympiad, 3

Tags: prime number , number theory , inequalities

Let $ p,q,r$ be 3 prime numbers such that $ 5\leq p <q<r$. Knowing that $ 2p^2\minus{}r^2 \geq 49$ and $ 2q^2\minus{}r^2\leq 193$, find $ p,q,r$.

2020 Dutch IMO TST, 2

Tags: fixed point , fixed , geometry , circumcircle , reflection , angle bisector

Given is a triangle $ABC$ with its circumscribed circle and $| AC | <| AB |$. On the short arc $AC$, there is a variable point $D\ne A$. Let $E$ be the reflection of $A$ wrt the inner bisector of $\angle BDC$. Prove that the line $DE$ passes through a fixed point, regardless of point $D$.

2018 Hanoi Open Mathematics Competitions, 11

Tags: divisible , number theory

Find all positive integers $k$ such that there exists a positive integer $n$, for which $2^n + 11$ is divisible by $2^k - 1$.

2016 ISI Entrance Examination, 3

Tags: polynomial

If $P(x)=x^n+a_1x^{n-1}+...+a_{n-1}$ be a polynomial with real coefficients and $a_1^2<a_2$ then prove that not all roots of $P(x)$ are real.

1994 BMO TST – Romania, 3:

Tags: algebra , combinatorics

Let $M_1, M_2, . . ., M_{11}$ be $5-$element sets such that $M_i \cap M_j \neq {\O}$ for all $i, j \in \{1, . . ., 11\}$. Determine the minimum possible value of the greatest number of the given sets that have nonempty intersection.

2000 Mediterranean Mathematics Olympiad, 4

Tags: geometry , vector

Let $P,Q,R,S$ be the midpoints of the sides $BC,CD,DA,AB$ of a convex quadrilateral, respectively. Prove that \[4(AP^2+BQ^2+CR^2+DS^2)\le 5(AB^2+BC^2+CD^2+DA^2)\]

2015 Postal Coaching, Problem 5

Tags: combinatorics , analytic geometry

Let $S$ be a set of in $3-$ space such that each of the points in $S$ has integer coordinates $(x,y,z)$ with $1 \le x,y,z \le n $. Suppose the pairwise distances between these points are all distinct. Prove that $$|S| \le min \{(n+2)\sqrt{\frac{n}{3}},n\sqrt{6} \}$$

2018 PUMaC Algebra B, 8

Tags: algebra

Let $a, b, c$ be non-zero real numbers that satisfy $\frac{1}{abc} + \frac{1}{a} + \frac{1}{c} = \frac{1}{b}$. The expression $\frac{4}{a^2 + 1} + \frac{4}{b^2 + 1} + \frac{7}{c^2 + 1}$ has a maximum value $M$. Find the sum of the numerator and denominator of the reduced form of $M$.

2011 Saudi Arabia Pre-TST, 4.4

Tags: equal segments , geometry , orthocenter

In a triangle $ABC$, let $O$ be the circumcenter, $H$ the orthocenter, and $M$ the midpoint of the segment $AH$. The perpendicular at $M$ onto $OM$ intersects lines $AB$ and $AC$ at $P$ and $Q$, respectively. Prove that $MP = MQ$.

1999 Baltic Way, 2

Tags: 3d geometry , geometry , number theory

Determine all positive integers $n$ with the property that the third root of $n$ is obtained by removing its last three decimal digits.

2013 Costa Rica - Final Round, 3

Tags: right triangle , concyclic , geometry

Let $ABC$ be a triangle, right-angled at point $ A$ and with $AB>AC$. The tangent through $ A$ of the circumcircle $G$ of $ABC$ cuts $BC$ at $D$. $E$ is the reflection of $ A$ over line $BC$. $X$ is the foot of the perpendicular from $ A$ over $BE$. $Y$ is the midpoint of $AX$, $Z$ is the intersection of $BY$ and $G$ other than $ B$, and $F$ is the intersection of $AE$ and $BC$. Prove $D, Z, F, E$ are concyclic.

1974 All Soviet Union Mathematical Olympiad, 191

Tags: diagonal , hexagon , convex , geometry

a) Each of the side of the convex hexagon is longer than $1$. Does it necessary have a diagonal longer than $2$? b) Each of the main diagonals of the convex hexagon is longer than $2$. Does it necessary have a side longer than $1$?