Olympiad problems

1972 IMO Shortlist, 12

Prove that from a set of ten distinct two-digit numbers, it is always possible to find two disjoint subsets whose members have the same sum.

1967 All Soviet Union Mathematical Olympiad, 093

Tags: number theory

Given natural number $k$ with a property "if $n$ is divisible by $k$, than the number, obtained from $n$ by reversing the order of its digits is also divisible by $k$". Prove that the $k$ is a divisor of $99$.

2018 PUMaC Team Round, 2

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Let triangle $\triangle{ABC}$ have $AB=90$ and $AC=66$. Suppose that the line $IG$ is perpendicular to side $BC$, where $I$ and $G$ are the incenter and centroid, respectively. Find the length of $BC$.

1958 February Putnam, B4

Tags: geometry , probability , 3d geometry , sphere , calculus

Title is self explanatory. Pick two points on the unit sphere. What is the expected distance between them?

2017 Olympic Revenge, 2

Tags: geometry

Let $\triangle$$ABC$ a triangle with circumcircle $\Gamma$. Suppose there exist points $R$ and $S$ on sides $AB$ and $AC$, respectively, such that $BR=RS=SC$. A tangent line through $A$ to $\Gamma$ meet the line $RS$ at $P$. Let $I$ the incenter of triangle $\triangle$$ARS$. Prove that $PA=PI$

2013 Kazakhstan National Olympiad, 2

Tags: geometry

Let in triangle $ABC$ incircle touches sides $AB,BC,CA$ at $C_1,A_1,B_1$ respectively. Let $\frac {2}{CA_1}=\frac {1}{BC_1}+\frac {1}{AC_1}$ .Prove that if $X$ is intersection of incircle and $CC_1$ then $3CX=CC_1$

2013 Argentina National Olympiad, 3

Tags: digit , number theory , remainder

Find how many are the numbers of $2013$ digits $d_1d_2…d_{2013}$ with odd digits $d_1,d_2,…,d_{2013}$ such that the sum of $1809$ terms $$d_1 \cdot d_2+d_2\cdot d_3+…+d_{1809}\cdot d_{1810}$$ has remainder $1$ when divided by $4$ and the sum of $203$ terms $$d_{1810}\cdot d_{1811}+d_{1811}\cdot d_{1812}+…+d_{2012}\cdot d_{2013}$$ has remainder $1$ when dividing by $4$.

2000 Turkey Team Selection Test, 1

Tags: algebra , polynomial , vieta , number theory

$(a)$ Prove that for every positive integer $n$, the number of ordered pairs $(x, y)$ of integers satisfying $x^2-xy+y^2 = n$ is divisible by $3.$ $(b)$ Find all ordered pairs of integers satisfying $x^2-xy+y^2=727.$

2007 Nicolae Coculescu, 4

Tags: locus , analytic geometry , infimum , geometry

Let $ M $ be a point in the interior of a triangle $ ABC, $ let $ D $ be the intersection of $ AM $ with $ BC, $ let $ E $ be the intersection of $ M $ with AC, let $ F $ be the intersection of $ CM $ with $ AB. $ Knowing that the expression $$ \frac{MA}{MD}\cdot \frac{MB}{ME}\cdot \frac{MC}{MF} $$ is minimized, describe the point $ M. $

1988 China Team Selection Test, 3

Tags: circumcircle , incenter , geometry

In triangle $ABC$, $\angle C = 30^{\circ}$, $O$ and $I$ are the circumcenter and incenter respectively, Points $D \in AC$ and $E \in BC$, such that $AD = BE = AB$. Prove that $OI = DE$ and $OI \bot DE$.

2013 Lusophon Mathematical Olympiad, 3

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An event occurs many years ago. It occurs periodically in $x$ consecutive years, then there is a break of $y$ consecutive years. We know that the event occured in $1964$, $1986$, $1996$, $2008$ and it didn't occur in $1976$, $1993$, $2006$, $2013$. What is the first year in that the event will occur again?

2017 Azerbaijan BMO TST, 3

Tags: geometry

Two circles, $\omega_1$ and $\omega_2$, centered at $O_1$ and $O_2$, respectively, meet at points $A$ and $B$. A line through $B$ meet $\omega_1$ again at $C$, and $\omega_2$ again at $D$. The tangents to $\omega_1$ and $\omega_2$ at $C$ and $D$, respectively, meet at $E$, and the line $AE$ meets the circle $\omega$ through $A, O_1,O_2$ again at $F$. Prove that the length of the segment $EF$ is equal to the diameter of $\omega$.

1954 Putnam, B7

Tags: limit , exponential

Let $a>0$. Show that $$ \lim_{n \to \infty} \sum_{s=1}^{n} \left( \frac{a+s}{n} \right)^{n}$$ lies between $e^a$ and $e^{a+1}.$

V Soros Olympiad 1998 - 99 (Russia), 11.2

Tags: algebra , inequalities

Find the greatest value of $C$ for which, for any $x, y, z,u$, and such that for $0\le x\le y \le z\le u$, holds the inequality $$(x + y +z + u)^2 \ge Cyz .$$

2000 Harvard-MIT Mathematics Tournament, 1

Tags: inequalities , absolute value

How many integers $x$ satisfy $|x|+5<7$ and $|x-3|>2$?

2022 Iran MO (3rd Round), 2

Tags: combinatorics , rectangle , parity

$m\times n$ grid is tiled by mosaics $2\times2$ and $1\times3$ (horizontal and vertical). Prove that the number of ways to choose a $1\times2$ rectangle (horizontal and vertical) such that one of its cells is tiled by $2\times2$ mosaic and the other cell is tiled by $1\times3$ mosaic [horizontal and vertical] is an even number.

2004 APMO, 3

Tags: combinatorics

Let a set $S$ of 2004 points in the plane be given, no three of which are collinear. Let ${\cal L}$ denote the set of all lines (extended indefinitely in both directions) determined by pairs of points from the set. Show that it is possible to colour the points of $S$ with at most two colours, such that for any points $p,q$ of $S$, the number of lines in ${\cal L}$ which separate $p$ from $q$ is odd if and only if $p$ and $q$ have the same colour. Note: A line $\ell$ separates two points $p$ and $q$ if $p$ and $q$ lie on opposite sides of $\ell$ with neither point on $\ell$.

2008 Princeton University Math Competition, B1

Tags: algebra

Find all pairs of positive real numbers $(a,b)$ such that $\frac{n-2}{a} \leq \left\lfloor bn \right\rfloor < \frac{n-1}{a}$ for all positive integes $n$.

2016 Saudi Arabia Pre-TST, 1.1

Tags: algebra , inequalities

Let $x, y, z$ be positive real numbers satisfy the condition $x^2 +y^2 + z^2 = 2(x y + yz + z x)$. Prove that $x + y + z + \frac{1}{2x yz} \ge 4$

2016 Indonesia MO, 3

Tags: combinatorics , balls

There are $5$ boxes arranged in a circle. At first, there is one a box containing one ball, while the other boxes are empty. At each step, we can do one of the following two operations: i. select one box that is not empty, remove one ball from the box and add one ball into both boxes next to the box, ii. select an empty box next to a non-empty box, from the box the non-empty one moves one ball to the empty box. Is it possible, that after a few steps, obtained conditions where each box contains exactly $17^{5^{2016}}$ balls?

2023 Romanian Master of Mathematics, 2

Tags: combinatorics , combinatorial geometry

Fix an integer $n \geq 3$. Let $\mathcal{S}$ be a set of $n$ points in the plane, no three of which are collinear. Given different points $A,B,C$ in $\mathcal{S}$, the triangle $ABC$ is [i]nice[/i] for $AB$ if $[ABC] \leq [ABX]$ for all $X$ in $\mathcal{S}$ different from $A$ and $B$. (Note that for a segment $AB$ there could be several nice triangles). A triangle is [i] beautiful [/i] if its vertices are all in $\mathcal{S}$ and is nice for at least two of its sides. Prove that there are at least $\frac{1}{2}(n-1)$ beautiful triangles.

2012 National Olympiad First Round, 22

Tags: symmetry

How many integer pairs $(m,n)$ are there satisfying $4mn(m+n-1)=(m^2+1)(n^2+1)$? $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 1$

2008 Flanders Math Olympiad, 2

Tags: number theory , perfect square

Let $a, b$ and $c$ be integers such that $a+b+c = 0$. Prove that $\frac12(a^4 +b^4 +c^4)$ is a perfect square.

2008 F = Ma, 1

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A bird flying in a straight line, initially at $\text{10 m/s}$, uniformly increases its speed to $\text{18 m/s}$ while covering a distance of $\text{40 m}$. What is the magnitude of the acceleration of the bird? (a) $\text{0.1 m/s}^2$ (b) $\text{0.2 m/s}^2$ (c) $\text{2.0 m/s}^2$ (d) $\text{2.8 m/s}^2$ (e) $\text{5.6 m/s}^2$

2011 HMNT, 4

Tags: number theory

Determine the remainder when $$2^{\frac {1 \cdot 2}{2}} + 2^{\frac {2 \cdot 3}{2}}+ ...+ 2^{\frac {2011 \cdot 2012}{2}}$$ is divided by $7$.