Olympiad problems

2009 Philippine MO, 4

Tags: PMO , 2009 , algebra , Inequality , minimum value

Let $k$ be a positive real number such that $$\frac{1}{k+a} + \frac{1}{k+b} + \frac{1}{k+c} \leq 1$$ for any positive positive real numbers $a$, $b$ and $c$ with $abc = 1$. Find the minimum value of $k$.

2023 Philippine MO, 5

Tags: combinatorics , Strings , Operations , PMO

Silverio is very happy for the 25th year of the PMO. In his jubilation, he ends up writing a finite sequence of As and Gs on a nearby blackboard. He then performs the following operation: if he finds at least one occurrence of the string "AG", he chooses one at random and replaces it with "GAAA". He performs this operation repeatedly until there is no more "AG" string on the blackboard. Show that for any initial sequence of As and Gs, Silverio will eventually be unable to continue doing the operation.

2020 Philippine MO, 4

Tags: PMO , geometry

Let $\triangle ABC$ be an acute triangle with circumcircle $\Gamma$ and $D$ the foot of the altitude from $A$. Suppose that $AD=BC$. Point $M$ is the midpoint of $DC$, and the bisector of $\angle ADC$ meets $AC$ at $N$. Point $P$ lies on $\Gamma$ such that lines $BP$ and $AC$ are parallel. Lines $DN$ and $AM$ meet at $F$, and line $PF$ meets $\Gamma$ again at $Q$. Line $AC$ meets the circumcircle of $\triangle PNQ$ again at $E$. Prove that $\angle DQE = 90^{\circ}$.

2021 Philippine MO, 1

Tags: geometry , PMO

In convex quadrilateral $ABCD$, $\angle CAB = \angle BCD$. $P$ lies on line $BC$ such that $AP = PC$, $Q$ lies on line $AP$ such that $AC$ and $DQ$ are parallel, $R$ is the point of intersection of lines $AB$ and $CD$, and $S$ is the point of intersection of lines $AC$ and $QR$. Line $AD$ meets the circumcircle of $AQS$ again at $T$. Prove that $AB$ and $QT$ are parallel.

2021 Philippine MO, 4

Tags: number theory , algebra , polynomial , PMO

Determine the set of all polynomials $P(x)$ with real coefficients such that the set $\{P(n) | n \in \mathbb{Z}\}$ contains all integers, except possibly finitely many of them.

2023 Philippine MO, 7

Tags: number theory , Fibonacci , PMO

A set of positive integers is said to be [i]pilak[/i] if it can be partitioned into 2 disjoint subsets $F$ and $T$, each with at least $2$ elements, such that the elements of $F$ are consecutive Fibonacci numbers, and the elements of $T$ are consecutive triangular numbers. Find all positive integers $n$ such that the set containing all the positive divisors of $n$ except $n$ itself is pilak.

2009 Philippine MO, 5

Tags: geometry , PMO , 2009 , concurrency , perpendicular bisector

Segments $AC$ and $BD$ intersect at point $P$ such that $PA = PD$ and $PB = PC$. Let $E$ be the foot of the perpendicular from $P$ to the line $CD$. Prove that the line $PE$ and the perpendicular bisectors of the segments $PA$ and $PB$ are concurrent.

2021 Philippine MO, 7

Tags: algebra , number theory , inequalities , Philippines , PMO

Let $a, b, c,$ and $d$ be real numbers such that $a \geq b \geq c \geq d$ and $$a+b+c+d = 13$$ $$a^2+b^2+c^2+d^2=43.$$ Show that $ab \geq 3 + cd$.

2020 Philippine MO, 1

Tags: PMO , combinatorics

A [i]T-tetromino[/i] is formed by adjoining three unit squares to form a $1 \times 3$ rectangle, and adjoining on top of the middle square a fourth unit square. Determine the least number of unit squares that must be removed from a $202 \times 202$ grid so that it can be tiled using T-tetrominoes.

2010 Philippine MO, 1

Tags: number theory , primes , PMO , 2010

Find all primes that can be written both as a sum of two primes and as a difference of two primes.

2020 Philippine MO, 3

Tags: PMO , number theory

Define the sequence $\{a_i\}$ by $a_0=1$, $a_1=4$, and $a_{n+1}=5a_n-a_{n-1}$ for all $n\geq 1$. Show that all terms of the sequence are of the form $c^2+3d^2$ for some integers $c$ and $d$.

2021 Philippine MO, 8

Tags: geometry , PMO

In right triangle $ABC$, $\angle ACB = 90^{\circ}$ and $\tan A > \sqrt{2}$. $M$ is the midpoint of $AB$, $P$ is the foot of the altitude from $C$, and $N$ is the midpoint of $CP$. Line $AB$ meets the circumcircle of $CNB$ again at $Q$. $R$ lies on line $BC$ such that $QR$ and $CP$ are parallel, $S$ lies on ray $CA$ past $A$ such that $BR = RS$, and $V$ lies on segment $SP$ such that $AV = VP$. Line $SP$ meets the circumcircle of $CPB$ again at $T$. $W$ lies on ray $VA$ past $A$ such that $2AW = ST$, and $O$ is the circumcenter of $SPM$. Prove that lines $OM$ and $BW$ are perpendicular.

2009 Philippine MO, 1

Tags: algebra , Sequences , PMO , 2009

The sequence ${a_0, a_1, a_2, ...}$ of real numbers satisfies the recursive relation $$n(n+1)a_{n+1}+(n-2)a_{n-1} = n(n-1)a_n$$ for every positive integer $n$, where $a_0 = a_1 = 1$. Calculate the sum $$\frac{a_0}{a_1} + \frac{a_1}{a_2} + ... + \frac{a_{2008}}{a_{2009}}$$.

2021 Philippine MO, 2

Tags: algebra , number theory , PMO

Let $n$ be a positive integer. Show that there exists a one-to-one function $\sigma : \{1,2,...,n\} \to \{1,2,...,n\}$ such that $$\sum_{k=1}^{n} \frac{k}{(k+\sigma(k))^2} < \frac{1}{2}.$$

2023 Philippine MO, 3

Tags: geometry , circumcircle , PMO

In $\triangle ABC$, $AB > AC$. Point $P$ is on line $BC$ such that $AP$ is tangent to its circumcircle. Let $M$ be the midpoint of $AB$, and suppose the circumcircle of $\triangle PMA$ meets line $AC$ again at $N$. Point $Q$ is the reflection of $P$ with respect to the midpoint of segment $BC$. The line through $B$ parallel to $QN$ meets $PN$ at $D$, and the line through $P$ parallel to $DM$ meets the circumcircle of $\triangle PMB$ again at $E$. Show that the lines $PM$, $BE$, and $AC$ are concurrent.

2021 Philippine MO, 3

Tags: functional equation , number theory , PMO

Denote by $\mathbb{Q}^+$ the set of positive rational numbers. A function $f : \mathbb{Q}^+ \to \mathbb{Q}$ satisfies • $f(p) = 1$ for all primes $p$, and • $f(ab) = af(b) + bf(a)$ for all $ a,b \in \mathbb{Q}^+ $. For which positive integers $n$ does the equation $nf(c) = c$ have at least one solution $c$ in $\mathbb{Q}^+$?

2010 Philippine MO, 4

Tags: combinatorics , 2010 , PMO

There are $2008$ blue, $2009$ red and $2010$ yellow chips on a table. At each step, one chooses two chips of different colors, and recolor both of them using the third color. Can all the chips be of the same color after some steps? Prove your answer.

2010 Philippine MO, 3

Tags: function , algebra , functional equation , PMO , 2010

Let $\mathbb{R}^*$ be the set of all real numbers, except $1$. Find all functions $f:\mathbb{R}^* \rightarrow \mathbb{R}$ that satisfy the functional equation $$x+f(x)+2f\left(\frac{x+2009}{x-1}\right)=2010$$.

2023 Philippine MO, 8

Tags: geometry , functional equation , combinatorial geometry , PMO , function , nice

Let $\mathcal{S}$ be the set of all points in the plane. Find all functions $f : \mathcal{S} \rightarrow \mathbb{R}$ such that for all nondegenerate triangles $ABC$ with orthocenter $H$, if $f(A) \leq f(B) \leq f(C)$, then $$f(A) + f(C) = f(B) + f(H).$$

2021 Philippine MO, 6

Tags: combinatorics , PMO

A certain country wishes to interconnect $2021$ cities with flight routes, which are always two-way, in the following manner: • There is a way to travel between any two cities either via a direct flight or via a sequence of connecting flights. • For every pair $(A, B)$ of cities that are connected by a direct flight, there is another city $C$ such that $(A, C)$ and $(B, C)$ are connected by direct flights. Show that at least $3030$ flight routes are needed to satisfy the two requirements.