Olympiad problems

2006 MOP Homework, 2

Tags:

Determine all unordered triples $(x,y,z)$ of integers for which the number $\sqrt{\frac{2005}{x+y}}+\sqrt{\frac{2005}{y+z}}+\sqrt{\frac{2005}{z+x}}$ is an integer.

2019 India IMO Training Camp, P2

Tags: function , algebra

Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.

2007 Moldova Team Selection Test, 4

Tags: ratio , combinatorial geometry , geometry

Consider five points in the plane, no three collinear. The convex hull of this points has area $S$. Prove that there exist three points of them that form a triangle with area at most $\frac{5-\sqrt 5}{10}S$

Indonesia MO Shortlist - geometry, g1.1

Tags: circumcircle , geometric transformation , rotation , geometry

Given triangle $ ABC$. Points $ D,E,F$ outside triangle $ ABC$ are chosen such that triangles $ ABD$, $ BCE$, and $ CAF$ are equilateral triangles. Prove that cicumcircles of these three triangles are concurrent.

2015 AMC 10, 25

Tags: probability , integration , geometry , trigonometry , calculus , rectangle , geometric probability

Let $S$ be a square of side length $1$. Two points are chosen independently at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\tfrac12$ is $\tfrac{a-b\pi}c$, where $a$, $b$, and $c$ are positive integers and $\gcd(a,b,c)=1$. What is $a+b+c$? $\textbf{(A) }59\qquad\textbf{(B) }60\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$

2008 AMC 12/AHSME, 2

Tags:

What is the reciprocal of $ \frac{1}{2}\plus{}\frac{2}{3}$? $ \textbf{(A)}\ \frac{6}{7} \qquad \textbf{(B)}\ \frac{7}{6} \qquad \textbf{(C)}\ \frac{5}{3} \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ \frac{7}{2}$

2017 Regional Olympiad of Mexico West, 3

Tags: combinatorics

In a building there are $119$ inhabitants who live in $120$ apartments (several inhabitants can live in the same apartment). We call an apartment [i]overcrowded [/i] if $15$ or more people live in it. Every day in some overcrowded apartment (if there is one) its inhabitants have a fight and yes they all go to live in a different apartment (which may or may not be already inhabited). Should you always terminate this process?

2016 Stars of Mathematics, 1

Tags: algebra , number theory

Determine all positive integers $ k,n $ for which $ 2^k+10n^2+n^4 $ is a perfect square. [i]Japan EGMO 2016 Shortlist[/i]

1962 All-Soviet Union Olympiad, 6

Tags: geometry

Given the lengths $AB$ and $BC$ and the fact that the medians to those two sides are perpendicular, construct the triangle $ABC$.

1975 Dutch Mathematical Olympiad, 4

Tags: geometry , analytic geometry , lattice points , equilateral

Given is a rectangular plane coordinate system. (a) Prove that it is impossible to find an equilateral triangle whose vertices have integer coordinates. (b) In the plane the vertices $A, B$ and $C$ lie with integer coordinates in such a way that $AB = AC$. Prove that $\frac{d(A,BC)}{BC}$ is rational.

2008 Balkan MO, 4

Tags: inequalities , polynomial , modular arithmetic , induction , number theory , relatively prime , algebra

Let $ c$ be a positive integer. The sequence $ a_1,a_2,\ldots$ is defined as follows $ a_1\equal{}c$, $ a_{n\plus{}1}\equal{}a_n^2\plus{}a_n\plus{}c^3$ for all positive integers $ n$. Find all $ c$ so that there are integers $ k\ge1$ and $ m\ge2$ so that $ a_k^2\plus{}c^3$ is the $ m$th power of some integer.

2011 Morocco National Olympiad, 3

Tags: linear algebra , matrix , algebra

Solve in $\mathbb{R}^{3}$ the following system \[\left\{\begin{matrix} \sqrt{x^{2}-y}=z-1\\ \sqrt{y^{2}-z}=x-1\\ \sqrt{z^{2}-x}=y-1 \end{matrix}\right.\]

2000 Mexico National Olympiad, 2

Tags: pascal triangle , number theory

A triangle of numbers is constructed as follows. The first row consists of the numbers from $1$ to $2000$ in increasing order, and under any two consecutive numbers their sum is written. (See the example corresponding to $5$ instead of $2000$ below.) What is the number in the lowermost row? 1 2 3 4 5 3 5 7 9 8 12 16 20 28 4

2024 LMT Fall, 6

Tags: guts

Let $P$ be a point in rectangle $ABCD$ such that the area of $PAB$ is $20$ and the area of $PCD$ is $24$. Find the area of $ABCD$.

2020 LIMIT Category 2, 13

Tags: sum of digits , limit

For every $n \in N $, let $d(n)$ denote the sum of digits of $n$. It is easy to see that the sequence $d(n), d(d(n))$, $d(d(d(n))), ... $ will eventually become a constant integer between $1$ and $9$ (both inclusive). This number is called the digital root of $n$ . Denote it by $b(n)$. Then for how many natural numbers $k<1000 , \lim_{n \to \infty} b(k^n)$ exists.

2014 Turkey Junior National Olympiad, 4

Tags: circumcircle , trapezoid , geometry

$ABC$ is an acute triangle with orthocenter $H$. Points $D$ and $E$ lie on segment $BC$. Circumcircle of $\triangle BHC$ instersects with segments $AD$,$AE$ at $P$ and $Q$, respectively. Prove that if $BD^2+CD^2=2DP\cdot DA$ and $BE^2+CE^2=2EQ\cdot EA$, then $BP=CQ$.

2001 All-Russian Olympiad Regional Round, 10.1

Tags: algebra , geometry , combinatorial geometry

The lengths of the sides of the polygon are $a_1$, $a_2$,. $..$ ,$a_n$. The square trinomial $f(x)$ is such that $f(a_1) = f(a_2 +...+ a_n)$. Prove that if $A$ is the sum of the lengths of several sides of a polygon, $B$ is the sum of the lengths of its remaining sides, then $f(A) = f(B)$.

2011 Lusophon Mathematical Olympiad, 2

Tags: geometry , rectangle , symmetry , geometric transformation , parallelogram , reflection , rotation

Consider two circles, tangent at $T$, both inscribed in a rectangle of height $2$ and width $4$. A point $E$ moves counterclockwise around the circle on the left, and a point $D$ moves clockwise around the circle on the right. $E$ and $D$ start moving at the same time; $E$ starts at $T$, and $D$ starts at $A$, where $A$ is the point where the circle on the right intersects the top side of the rectangle. Both points move with the same speed. Find the locus of the midpoints of the segments joining $E$ and $D$.

2020 New Zealand MO, 2

Tags: midpoint , square , geometry

Let $ABCD$ be a square and let $X$ be any point on side $BC$ between $B$ and $C$. Let $Y$ be the point on line $CD$ such that $BX = YD$ and $D$ is between $C$ and $Y$ . Prove that the midpoint of $XY$ lies on diagonal $BD$.

2006 Abels Math Contest (Norwegian MO), 3

Tags: integer , number theory , diophantine , rational

(a) Let $a$ and $b$ be rational numbers such that line $y = ax + b$ intersects the circle $x^2 + y^2 = 5$ at two different points. Show that if one of the intersections has two rational coordinates, so does the other intersection. (b) Show that there are infinitely many triples ($k, n, m$) that are such that $k^2 + n^2 = 5m^2$, where $k, n$ and $m$ are integers, and not all three have any in common prime factor.

1991 Austrian-Polish Competition, 3

Tags: midpoint , equal segments , combinatorial geometry , combinatorics , geometry

Given two distinct points $A_1,A_2$ in the plane, determine all possible positions of a point $A_3$ with the following property: There exists an array of (not necessarily distinct) points $P_1,P_2,...,P_n$ for some $n \ge 3$ such that the segments $P_1P_2,P_2P_3,...,P_nP_1$ have equal lengths and their midpoints are $A_1, A_2, A_3, A_1, A_2, A_3, ...$ in this order.