Olympiad problems

2012 CentroAmerican, 3

Let $ABC$ be a triangle with $AB < BC$, and let $E$ and $F$ be points in $AC$ and $AB$ such that $BF = BC = CE$, both on the same halfplane as $A$ with respect to $BC$. Let $G$ be the intersection of $BE$ and $CF$. Let $H$ be a point in the parallel through $G$ to $AC$ such that $HG = AF$ (with $H$ and $C$ in opposite halfplanes with respect to $BG$). Show that $\angle EHG = \frac{\angle BAC}{2}$.

2001 All-Russian Olympiad, 4

Tags: pigeonhole principle , combinatorics unsolved , combinatorics

Consider a convex $2000$-gon, no three of whose diagonals have a common point. Each of its diagonals is colored in one of $999$ colors. Prove that there exists a triangle all of whose sides lie on diagonals of the same color. (Vertices of the triangle need not be vertices of the original polygon.)

2021 USMCA, 23

Tags:

Given real numbers $x, y, z, w$ such that $(x + y + 2z)(x + z + 3w) = 1$, what is the minimum possible value of $x^2 + y^2 + z^2 + w^2$?

2021 CMIMC, 2.1

Tags: algebra , number theory

Find the unique 3 digit number $N=\underline{A}$ $\underline{B}$ $\underline{C},$ whose digits $(A, B, C)$ are all nonzero, with the property that the product $P=\underline{A}$ $\underline{B}$ $\underline{C}$ $\times$ $\underline{A}$ $\underline{B}$ $\times$ $\underline{A}$ is divisible by $1000$. [i]Proposed by Kyle Lee[/i]

2019 Puerto Rico Team Selection Test, 2

Tags: perpendicular , square , geometry

Let $ABCD$ be a square. Let $M$ and $K$ be points on segments $BC$ and $CD$ respectively, such that $MC = KD$. Let $ P$ be the intersection of the segments $MD$ and $BK$. Prove that $AP$ is perpendicular to $MK$.

2001 CentroAmerican, 2

Tags: quadratics , function , arithmetic sequence , absolute value

Let $ a,b$ and $ c$ real numbers such that the equation $ ax^2\plus{}bx\plus{}c\equal{}0$ has two distinct real solutions $ p_1,p_2$ and the equation $ cx^2\plus{}bx\plus{}a\equal{}0$ has two distinct real solutions $ q_1,q_2$. We know that the numbers $ p_1,q_1,p_2,q_2$ in that order, form an arithmetic progression. Show that $ a\plus{}c\equal{}0$.

2010 VTRMC, Problem 2

Tags: number theory

For $n$ a positive integer, define $f_1(n)=n$ and then for $i$ a positive integer, define $f_{i+1}(n)=f_i(n)^{f_i(n)}$. Determine $f_{100}(75)\pmod{17}$. Justify your answer.

2014 Abels Math Contest (Norwegian MO) Final, 4

Tags: number theory , Integer

Find all triples $(a, b, c)$ of positive integers for which $\frac{32a + 3b + 48c}{4abc}$ is also an integer.

1981 AMC 12/AHSME, 9

Tags: geometry , 3D geometry , ratio

In the adjoining figure, $PQ$ is a diagonal of the cube. If $PQ$ has length $a$, then the surface area of the cube is $\text{(A)}\ 2a^2 \qquad \text{(B)}\ 2\sqrt{2}a^2 \qquad \text{(C)}\ 2\sqrt{3}a^2 \qquad \text{(D)}\ 3\sqrt{3}a^2 \qquad \text{(E)}\ 6a^2$

2014 Chile TST Ibero, 3

Tags: algebra , TST , Chile , Iberoamerican

Let $x_0 = 5$ and define the sequence recursively as $x_{n+1} = x_n + \frac{1}{x_n}$. Prove that: \[ 45 < x_{1000} < 45.1. \]

2014 Romania Team Selection Test, 3

Tags: algebra unsolved , algebra , combinatorics

Let $n \in \mathbb{N}$ and $S_{n}$ the set of all permutations of $\{1,2,3,...,n\}$. For every permutation $\sigma \in S_{n}$ denote $I(\sigma) := \{ i: \sigma (i) \le i \}$. Compute the sum $\sum_ {\sigma \in S_{n}} \frac{1}{|I(\sigma )|} \sum_ {i \in I(\sigma)} (i+ \sigma(i))$.

2012 Indonesia TST, 4

Tags: function , number theory unsolved , number theory

Let $\mathbb{N}$ be the set of positive integers. For every $n \in \mathbb{N}$, define $d(n)$ as the number of positive divisors of $n$. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that: a) $d(f(x)) = x$ for all $x \in \mathbb{N}$ b) $f(xy)$ divides $(x-1)y^{xy-1}f(x)$ for all $x,y \in \mathbb{N}$

2019 AMC 8, 23

Tags: AMC 8 , AMC

After Euclid High School's last basketball game, it was determined that $\frac{1}{4}$ of the team's points were scored by Alexa and $\frac{2}{7}$ were scored by Brittany. Chelsea scored 15 points. None of the other 7 team members scored more than 2 points. What was the total number of points scored by the other 7 team members? $\textbf{(A) }10\qquad\textbf{(B) }11\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14$

1995 Tournament Of Towns, (466) 4

Tags: geometry , angle bisector , equal segments

From the vertex $A$ of a triangle $ABC$, three segments are drawn: the bisectors $AM$ and $AN$ of its interior and exterior angles and the tangent $AK$ to the circumscribed circle of the triangle (the points $M$, $K$ and $N$ lie on the line $BC$). Prove that $MK = KN$. (I Sharygin)

2014 Saint Petersburg Mathematical Olympiad, 3

Tags: number theory , combinatorics , algebra , inequalities

$N$ in natural. There are natural numbers from $N^3$ to $N^3+N$ on the board. $a$ numbers was colored in red, $b$ numbers was colored in blue. Sum of red numbers in divisible by sum of blue numbers. Prove, that $b|a$

2020 Nigerian MO round 3, #3

Tags: combinatorics , analytic geometry

given any 3 distinct points $X,Y,Z$on the integer coordinates of the x-axis,the following operation is allowed:A point say $X$ is reflected over another point say $Y$. Note that after each operation only one among three points is moved. we perform these operations till 2 out of the 3 points coincide. let $N=N(X,Y,Z)$ denote the minimum number of operations before we are forced to stop.(this could happen in different ways). show that there are at most $2^N$coordinates that point $X$ could end up if we are forced to stop after $N$operations

2017 India Regional Mathematical Olympiad, 2

Tags: modular arithmetic , number theory

Show that the equation $a^3+(a+1)^3+\ldots+(a+6)^3=b^4+(b+1)^4$ has no solutions in integers $a,b$.

MOAA Gunga Bowls, 2021.21

Tags: MOAA 2021 , Gunga

King William is located at $(1, 1)$ on the coordinate plane. Every day, he chooses one of the eight lattice points closest to him and moves to one of them with equal probability. When he exits the region bounded by the $x, y$ axes and $x+y = 4$, he stops moving and remains there forever. Given that after an arbitrarily large amount of time he must exit the region, the probability he ends up on $x+y = 4$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by Andrew Wen[/i]

2024 ELMO Shortlist, N8

Tags: Elmo , number theory

Let $d(n)$ be the number of divisors of a nonnegative integer $n$ (we set $d(0)=0$). Find all positive integers $d$ such that there exists a two-variable polynomial $P(x,y)$ of degree $d$ with integer coefficients such that: [list] [*] for any positive integer $y$, there are infinitely many positive integers $x$ such that $\gcd(x,y)=1$ and $d(|P(x,y)|) \mid x$, and [*] for any positive integer $x$, there are infinitely many positive integers $y$ such that $\gcd(x,y)=1$ and $d(|P(x,y)|) \mid y$. [/list] [i]Allen Wang[/i]

2005 Sharygin Geometry Olympiad, 10.1

Tags: geometry , parallelogram , interior , convex quadrilateral

A convex quadrangle without parallel sides is given. For each triple of its vertices, a point is constructed that supplements this triple to a parallelogram, one of the diagonals of which coincides with the diagonal of the quadrangle. Prove that of the four points constructed, exactly one lies inside the original quadrangle.

2008 iTest Tournament of Champions, 4

Tags: iTest Tournament of Champions

Euclid places a morsel of food at the point $(0,0)$ and an ant at the point $(1,2)$. Every second, the ant walks one unit in one of the four coordinate directions. However, whenever the ant moves to $(x,\pm 3)$, Euclid's malicious brother Mobius picks it up and puts it at $(-x,\mp 2)$, and whenever it moves to $(\pm 2,y)$, his cousin Klein puts it at $(\mp 1,y)$. If $p$ and $q$ are relatively prime positive integers such that $\tfrac pq$ is the expected number of steps the ant takes before reaching the food, find $p+q$.

2008 India Regional Mathematical Olympiad, 4

Tags: number theory , Divisibility , Natural Numbers

Determine all the natural numbers $n$ such that $21$ divides $2^{2^{n}}+2^n+1.$

2018-IMOC, N2

Tags: number theory , least common multiple , greatest common divisor , fe , functional equation

Find all functions $f:\mathbb N\to\mathbb N$ satisfying $$\operatorname{lcm}(f(x),y)\gcd(f(x),f(y))=f(x)f(f(y))$$ for all $x,y\in\mathbb N$.

2011 JBMO Shortlist, 1

Tags: inequalities , inequalities unsolved

Let $a,b,c$ be positive real numbers such that $abc = 1$. Prove that: $\displaystyle\prod(a^5+a^4+a^3+a^2+a+1)\geq 8(a^2+a+1)(b^2+b+1)(c^2+c+1)$

2002 China Western Mathematical Olympiad, 3

Tags: induction , number theory proposed , number theory

Assume that $ \alpha$ and $ \beta$ are two roots of the equation: $ x^2\minus{}x\minus{}1\equal{}0$. Let $ a_n\equal{}\frac{\alpha^n\minus{}\beta^n}{\alpha\minus{}\beta}$, $ n\equal{}1, 2, \cdots$. (1) Prove that for any positive integer $ n$, we have $ a_{n\plus{}2}\equal{}a_{n\plus{}1}\plus{}a_n$. (2) Find all positive integers $ a$ and $ b$, $ a<b$, satisfying $ b \mid a_n\minus{}2na^n$ for any positive integer $ n$.