Olympiad problems

2022 China Team Selection Test, 2

Two positive real numbers $\alpha, \beta$ satisfies that for any positive integers $k_1,k_2$, it holds that $\lfloor k_1 \alpha \rfloor \neq \lfloor k_2 \beta \rfloor$, where $\lfloor x \rfloor$ denotes the largest integer less than or equal to $x$. Prove that there exist positive integers $m_1,m_2$ such that $\frac{m_1}{\alpha}+\frac{m_2}{\beta}=1$.

1979 AMC 12/AHSME, 10

Tags: asymptote , trapezoid , geometry

If $P_1P_2P_3P_4P_5P_6$ is a regular hexagon whose apothem (distance from the center to midpoint of a side) is $2$, and $Q_i$ is the midpoint of side $P_iP_{i+1}$ for $i=1,2,3,4$, then the area of quadrilateral $Q_1Q_2Q_3Q_4$ is $\textbf{(A) }6\qquad\textbf{(B) }2\sqrt{6}\qquad\textbf{(C) }\frac{8\sqrt{3}}{3}\qquad\textbf{(D) }3\sqrt{3}\qquad\textbf{(E) }4\sqrt{3}$

2010 BAMO, 4

Tags: geometry

Acute triangle $ABC$ has $\angle BAC < 45^\circ$. Point $D$ lies in the interior of triangle $ABC$ so that $BD = CD$ and $\angle BDC = 4 \angle BAC$. Point $E$ is the reflection of $C$ across line $AB$, and point $F$ is the reflection of $B$ across line $AC$. Prove that lines $AD$ and $EF$ are perpendicular.

2003 Silk Road, 3

Tags: algebra

Let $0<a<b<1$ be reals numbers and \[g(x)=\left\{\begin{array}{cc}x+1-a,&\mbox{ if } 0<x<b\\b-a, & \mbox{ if } x=a \\x-a, & \mbox{ if } a<x<b\\1-a ,&\mbox{ if } x=b \\ x-a ,&\mbox{ if } b<x<1 \end{array}\right.\] Give that there exist $n+1$ reals numbers $0<x_0<x_1<...<x_n<1$, for which $g^{[n]}(x_i)=x_i \ (0 \leq i \leq n)$. Prove that there exists a positive integer $N$, such that $g^{[N]}(x)=x$ for all $0<x<1$. ($g^{[n]}(x)= \underbrace{g(g(....(g(x))....))}_{\text{n times}}$) Official solution [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=125&t=365714&p=2011659#p2011659]here[/url]

2018 Estonia Team Selection Test, 5

Tags: geometry , angle chasing , projective geometry , similar triangles

Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.

1983 Spain Mathematical Olympiad, 7

Tags: geometry , 3d geometry , tetrahedron , volume

A regular tetrahedron with an edge of $30$ cm rests on one of its faces. Assuming it is hollow, $2$ liters of water are poured into it. Find the height of the ''upper'' liquid and the area of the ''free'' surface of the water.

2014 IFYM, Sozopol, 3

Tags: combinatorics , clique , graph theory

The graph $G$ with 2014 vertices doesn’t contain any 3-cliques. If the set of the degrees of the vertices of $G$ is $\{1,2,...,k\}$, find the greatest possible value of $k$.

2015 Balkan MO Shortlist, A3

Tags: inequalities , geometric inequality , median

Let a$,b,c$ be sidelengths of a triangle and $m_a,m_b,m_c$ the medians at the corresponding sides. Prove that $$m_a\left(\frac{b}{a}-1\right)\left(\frac{c}{a}-1\right)+ m_b\left(\frac{a}{b}-1\right)\left(\frac{c}{b}-1\right) +m_c\left(\frac{a}{c}-1\right)\left(\frac{b}{c}-1\right)\geq 0.$$ (FYROM)

2021 Harvard-MIT Mathematics Tournament., 9

Tags: algebra

Let $f$ be a monic cubic polynomial satisfying $f(x) + f(-x) = 0$ for all real numbers $x$. For all real numbers $y$, define $g(y)$ to be the number of distinct real solutions $x$ to the equation $f(f(x)) = y$. Suppose that the set of possible values of $g(y)$ over all real numbers $y$ is exactly $\{1, 5, 9\}$. Compute the sum of all possible values of $f(10)$.

2015 European Mathematical Cup, 1

Tags: number theory , combinatorics , pigeonhole principle

$A = \{a, b, c\}$ is a set containing three positive integers. Prove that we can find a set $B \subset A$, $B = \{x, y\}$ such that for all odd positive integers $m, n$ we have $$10\mid x^my^n-x^ny^m.$$ [i]Tomi Dimovski[/i]

2014 PUMaC Individual Finals B, 2

Tags: graph theory , combinatorics

Let $P_1, P_2, \dots, P_n$ be points on the plane. There is an edge between distinct points $P_x, P_y$ if and only if $x \mid y$. Find the largest $n$, such that the graph can be drawn with no crossing edges.

2017 China Team Selection Test, 4

Tags: algebra , polynomial

Find out all the integer pairs $(m,n)$ such that there exist two monic polynomials $P(x)$ and $Q(x)$ ,with $\deg{P}=m$ and $\deg{Q}=n$,satisfy that $$P(Q(t))\not=Q(P(t))$$ holds for any real number $t$.

2011 Dutch IMO TST, 1

Tags: diophantine equation , number theory

Find all pairs $(x, y)$ of integers that satisfy $x^2 + y^2 + 3^3 = 456\sqrt{x - y}$.

1984 IMO Longlists, 13

Tags: diophantine equation , polynomial , equation , quadratic , number theory

Prove: (a) There are infinitely many triples of positive integers $m, n, p$ such that $4mn - m- n = p^2 - 1.$ (b) There are no positive integers $m, n, p$ such that $4mn - m- n = p^2.$

2019 Danube Mathematical Competition, 1

Tags: equation , diophantine equation , algebra

Solve in $ \mathbb{Z}^2 $ the equation: $ x^2\left( 1+x^2 \right) =-1+21^y. $ [i]Lucian Petrescu[/i]

2022 Chile National Olympiad, 4

Tags: geometric inequality , cone , 3d geometry , geometry

In a right circular cone of wood, the radius of the circumference $T$ of the base circle measures $10$ cm, while every point on said circumference is $20$ cm away. from the apex of the cone. A red ant and a termite are located at antipodal points of $T$. A black ant is located at the midpoint of the segment that joins the vertex with the position of the termite. If the red ant moves to the black ant's position by the shortest possible path, how far does it travel?

1961 Putnam, A5

Tags: subset , union of set

Let $\Omega$ be a set of $n$ points, where $n>2$. Let $\Sigma$ be a nonempty subcollection of the $2^n$ subsets of $\Omega$ that is closed with respect to the unions, intersections and complements. If $k$ is the number of elements of $\Sigma,$ what are the possible values of $k?$

2007 May Olympiad, 1

Tags: number theory , algebra

In a year that has $53$ Saturdays, what day of the week is May $12$? Give all chances.

2017 Princeton University Math Competition, A6/B8

Tags:

Jackson begins at $1$ on the number line. At each step, he remains in place with probability $85\%$ and increases his position on the number line by $1$ with probability $15\%$. Let $d_n$ be his position on the number line after $n$ steps, and let $E_n$ be the expected value of $\tfrac{1}{d_n}$. Find the least $n$ such that $\tfrac{1}{E_n} > 2017$.

May Olympiad L2 - geometry, 2018.4

Tags: geometry , parallelogram , distance

In a parallelogram $ABCD$, let $M$ be the point on the $BC$ side such that $MC = 2BM$ and let $N$ be the point of side $CD$ such that $NC = 2DN$. If the distance from point $B$ to the line $AM$ is $3$, calculate the distance from point $N$ to the line $AM$.

2015 Canadian Mathematical Olympiad Qualification, 7

Tags: combinatorics

A $(0_x, 1_y, 2_z)$-string is an infinite ternary string such that: [list] [*] If there is a $0$ in position $i$ then there is a $1$ in position $i + x$, [*] if there is a $1$ in position $j$ then there is a $2$ in position $j + y$, [*] if there is a $2$ in position $k$ then there is a $0$ in position $k + z$. [/list] For how many ordered triples of positive integers $(x, y, z)$ with $x, y, z \leq 100$ does there exist $(0_x, 1_y, 2_z)$-string?

2008 Princeton University Math Competition, B5

Tags: algebra

How many real roots do $x^5 +3x^4 -4x^3 -8x^2 +6x-1$ and $x^5-3x^4 -2x^3 -3x^2 -6x+1$ share?

1989 China Team Selection Test, 1

Tags: geometry

A triangle of sides $\frac{3}{2}, \frac{\sqrt{5}}{2}, \sqrt{2}$ is folded along a variable line perpendicular to the side of $\frac{3}{2}.$ Find the maximum value of the coincident area.

2003 Iran MO (2nd round), 2

Tags: function , geometry , inequalities , parallelogram , triangle inequality , combinatorics

In a village, there are $n$ houses with $n>2$ and all of them are not collinear. We want to generate a water resource in the village. For doing this, point $A$ is [i]better[/i] than point $B$ if the sum of the distances from point $A$ to the houses is less than the sum of the distances from point $B$ to the houses. We call a point [i]ideal[/i] if there doesn’t exist any [i]better[/i] point than it. Prove that there exist at most $1$ [i]ideal[/i] point to generate the resource.

1966 Poland - Second Round, 4

Tags: number theory , perfect square

Prove that if the natural numbers $ a $ and $ b $ satisfy the equation $ a^2+a = 3b^2 $, then the number $ a+1 $ is the square of an integer.