Olympiad problems

2005 APMO, 2

Let $a, b, c$ be positive real numbers such that $abc=8$. Prove that \[ \frac{a^2}{\sqrt{(1+a^3)(1+b^3)}} +\frac{b^2}{\sqrt{(1+b^3)(1+c^3)}} +\frac{c^2}{\sqrt{(1+c^3)(1+a^3)}} \geq \frac{4}{3} \]

2022 APMO, 2

Tags: geometry , circumcircle , APMO , APMO 2022

Let $ABC$ be a right triangle with $\angle B=90^{\circ}$. Point $D$ lies on the line $CB$ such that $B$ is between $D$ and $C$. Let $E$ be the midpoint of $AD$ and let $F$ be the seconf intersection point of the circumcircle of $\triangle ACD$ and the circumcircle of $\triangle BDE$. Prove that as $D$ varies, the line $EF$ passes through a fixed point.

2018 Brazil Team Selection Test, 3

Tags: combinatorics , APMO

A collection of $n$ squares on the plane is called tri-connected if the following criteria are satisfied: (i) All the squares are congruent. (ii) If two squares have a point $P$ in common, then $P$ is a vertex of each of the squares. (iii) Each square touches exactly three other squares. How many positive integers $n$ are there with $2018\leq n \leq 3018$, such that there exists a collection of $n$ squares that is tri-connected?

2022 APMO, 5

Tags: inequalities , APMO , APMO 2022

Let $a,b,c,d$ be real numbers such that $a^2+b^2+c^2+d^2=1$. Determine the minimum value of $(a-b)(b-c)(c-d)(d-a)$ and determine all values of $(a,b,c,d)$ such that the minimum value is achived.

2016 Brazil Team Selection Test, 1

Tags: geometry , circumcircle , APMO , geometry solved , functional equation , right triangle , reflection

We say that a triangle $ABC$ is great if the following holds: for any point $D$ on the side $BC$, if $P$ and $Q$ are the feet of the perpendiculars from $D$ to the lines $AB$ and $AC$, respectively, then the reflection of $D$ in the line $PQ$ lies on the circumcircle of the triangle $ABC$. Prove that triangle $ABC$ is great if and only if $\angle A = 90^{\circ}$ and $AB = AC$. [i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]

2017 APMO, 4

Tags: number theory , APMO , LTE Lemma , Lifting the Exponent

Call a rational number $r$ [i]powerful[/i] if $r$ can be expressed in the form $\dfrac{p^k}{q}$ for some relatively prime positive integers $p, q$ and some integer $k >1$. Let $a, b, c$ be positive rational numbers such that $abc = 1$. Suppose there exist positive integers $x, y, z$ such that $a^x + b^y + c^z$ is an integer. Prove that $a, b, c$ are all [i]powerful[/i]. [i]Jeck Lim, Singapore[/i]

2019 APMO, 3

Tags: geometry , APMO

Let $ABC$ be a scalene triangle with circumcircle $\Gamma$. Let $M$ be the midpoint of $BC$. A variable point $P$ is selected in the line segment $AM$. The circumcircles of triangles $BPM$ and $CPM$ intersect $\Gamma$ again at points $D$ and $E$, respectively. The lines $DP$ and $EP$ intersect (a second time) the circumcircles to triangles $CPM$ and $BPM$ at $X$ and $Y$, respectively. Prove that as $P$ varies, the circumcircle of $\triangle AXY$ passes through a fixed point $T$ distinct from $A$.

2017 SG Originals, Q4

Tags: number theory , APMO , LTE Lemma , Lifting the Exponent

Call a rational number $r$ [i]powerful[/i] if $r$ can be expressed in the form $\dfrac{p^k}{q}$ for some relatively prime positive integers $p, q$ and some integer $k >1$. Let $a, b, c$ be positive rational numbers such that $abc = 1$. Suppose there exist positive integers $x, y, z$ such that $a^x + b^y + c^z$ is an integer. Prove that $a, b, c$ are all [i]powerful[/i]. [i]Jeck Lim, Singapore[/i]

2023 APMO, 1

Tags: APMO , APMO 2023 , combinatorics , combinatorial geometry

Let $n \geq 5$ be an integer. Consider $n$ squares with side lengths $1, 2, \dots , n$, respectively. The squares are arranged in the plane with their sides parallel to the $x$ and $y$ axes. Suppose that no two squares touch, except possibly at their vertices. Show that it is possible to arrange these squares in a way such that every square touches exactly two other squares.

2023 APMO, 2

Tags: number theory , APMO , APMO 2023

Find all integers $n$ satisfying $n \geq 2$ and $\dfrac{\sigma(n)}{p(n)-1} = n$, in which $\sigma(n)$ denotes the sum of all positive divisors of $n$, and $p(n)$ denotes the largest prime divisor of $n$.

2017 APMO, 3

Tags: combinatorics , APMO

Let $A(n)$ denote the number of sequences $a_1\ge a_2\ge\cdots{}\ge a_k$ of positive integers for which $a_1+\cdots{}+a_k = n$ and each $a_i +1$ is a power of two $(i = 1,2,\cdots{},k)$. Let $B(n)$ denote the number of sequences $b_1\ge b_2\ge \cdots{}\ge b_m$ of positive integers for which $b_1+\cdots{}+b_m =n$ and each inequality $b_j\ge 2b_{j+1}$ holds $(j=1,2,\cdots{}, m-1)$. Prove that $A(n) = B(n)$ for every positive integer $n$. [i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]

2022 APMO, 1

Tags: number theory , APMO , APMO 2022 , divisible , multiple , Hi

Find all pairs $(a,b)$ of positive integers such that $a^3$ is multiple of $b^2$ and $b-1$ is multiple of $a-1$.

2021 APMO, 1

Tags: algebra , APMO , APMO 2021

Prove that for each real number $r>2$, there are exactly two or three positive real numbers $x$ satisfying the equation $x^2=r\lfloor x \rfloor$.

PEN H Problems, 21

Tags: quadratics , LaTeX , Diophantine Equations , APMO

Prove that the equation \[6(6a^{2}+3b^{2}+c^{2}) = 5n^{2}\] has no solutions in integers except $a=b=c=n=0$.

2020 Brazil Team Selection Test, 4

Tags: algebra , polynomial , APMO

Let $\mathbb{Z}$ denote the set of all integers. Find all polynomials $P(x)$ with integer coefficients that satisfy the following property: For any infinite sequence $a_1$, $a_2$, $\dotsc$ of integers in which each integer in $\mathbb{Z}$ appears exactly once, there exist indices $i < j$ and an integer $k$ such that $a_i +a_{i+1} +\dotsb +a_j = P(k)$.

2013 Brazil Team Selection Test, 4

Tags: algebra , polynomial , combinatorics , APMO

Let $a$ and $b$ be positive integers, and let $A$ and $B$ be finite sets of integers satisfying (i) $A$ and $B$ are disjoint; (ii) if an integer $i$ belongs to either to $A$ or to $B$, then either $i+a$ belongs to $A$ or $i-b$ belongs to $B$. Prove that $a\left\lvert A \right\rvert = b \left\lvert B \right\rvert$. (Here $\left\lvert X \right\rvert$ denotes the number of elements in the set $X$.)

2018 APMO, 3

Tags: combinatorics , APMO

A collection of $n$ squares on the plane is called tri-connected if the following criteria are satisfied: (i) All the squares are congruent. (ii) If two squares have a point $P$ in common, then $P$ is a vertex of each of the squares. (iii) Each square touches exactly three other squares. How many positive integers $n$ are there with $2018\leq n \leq 3018$, such that there exists a collection of $n$ squares that is tri-connected?

2021 APMO, 3

Tags: geometry , circumcircle , APMO

Let $ABCD$ be a cyclic convex quadrilateral and $\Gamma$ be its circumcircle. Let $E$ be the intersection of the diagonals of $AC$ and $BD$. Let $L$ be the center of the circle tangent to sides $AB$, $BC$, and $CD$, and let $M$ be the midpoint of the arc $BC$ of $\Gamma$ not containing $A$ and $D$. Prove that the excenter of triangle $BCE$ opposite $E$ lies on the line $LM$.