Navvye/Olympiad-Problems-CSV huggingface dataset:
2015 Thailand Mathematical Olympiad
Let $\vartriangle ABC$ be a triangle with an obtuse angle $\angle ACB$. The incircle of $\vartriangle ABC$ centered at $I$ is tangent to the sides $AB, BC, CA$ at $D, E, F$ respectively. Lines $AI$ and $BI$ intersect $EF$ at $M$ and $N$ respectively. Let $G$ be the midpoint of $AB$. Show that $M, N, G, D$ lie on a circle.
2015 Thailand Mathematical Olympiad
Let $n$ be an integer greater than $6$.Show that if $n+1$ is a prime number,than

$\left\lceil \frac{(n-1)!}{n(n+1)}\right \rceil$ is $ODD.$
2015 Thailand Mathematical Olympiad
Let $m$ and $n$ be positive integers. Determine the number of ways to fill each cell of an $m \times  n $ table with a number from $\{-2, -1, 1, 2\}$ so that the product of the numbers written in each row and column is $-2$.
2015 Thailand Mathematical Olympiad
Let $A, B, C$ be centers of three circles that are mutually tangent externally, let $r_A, r_B, r_C$ be the radii of the circles, respectively. Let $r$ be the radius of the incircle of $\vartriangle ABC$. Prove that $$r^2  \le \frac19 (r_A^2 + r_B^2+r_C^2)$$and identify, with justification, one case where the equality is attained.
2015 Thailand Mathematical Olympiad
Let $m$ and $n$ be positive integers such that $m - n$ is odd. Show that $(m + 3n)(5m + 7n)$ is not a perfect square.
2015 Thailand Mathematical Olympiad
Determine all functions $f : R \to R$ satisfying $f(f(x) + 2y)= 6x + f(f(y) -x)$ for all real numbers $x,y$
2015 Thailand Mathematical Olympiad
A Boy Scouts camp holds a campfire. The camp has scarfs of m colors with n scarves of each color, and gives each of its $mn$ scouts a scarf, where $m, n \ge 2$ are integers. The camp then divides its scouts into troops by the color of their scarfs.

At the beginning of the campfire, the scouts are seated in a circle so that scouts in the same troop are seated next to each other.

The camp organizer then proceeds to select, round by round, representatives to perform a show, with the following conditions: there must be at least two representatives in each round, they must come from the same troop, and any specific set of representatives can only perform once. (For example, if $\{A, B\}$ has performed, then $\{A, B\}$ cannot perform again, but $\{A, B, C\}$ can still perform.) This process is repeated until all valid sets of representatives have performed.

At this point, the organizers order each scout to hand their scarfs to the scout to the left, and re-group the scouts into troops, again according to their scarf color, and the process above is resumed, until the set of valid sets of representatives is exhausted again. (The sets of representatives after re-grouping must also be distinct from the sets before re-grouping.)

When that happens, the organizers order another re-group, and resumes the process, and this repeats until there can be no further performances. Find, in simple form, the total number of performances that will be performed.
2014 Thailand Mathematical Olympiad
Let $\vartriangle ABC$ be an isosceles triangle with $\angle BAC = 100^o$. Let $D, E$ be points on ray  $\overrightarrow{AB}$ so that $BC = AD = BE$. Show that $BC \cdot  DE = BD \cdot CE$
2014 Thailand Mathematical Olympiad
Find all functions $f : R \to R$ satisfying $f(xy - 1) + f(x)f(y) = 2xy - 1$ for all real numbers $x, y$
2014 Thailand Mathematical Olympiad
Let $M$ and $N$ be positive integers. Pisut walks from point $(0, N)$ to point $(M, 0)$ in steps so that

$\bullet$ each step has unit length and is parallel to either the horizontal or the vertical axis, and

$\bullet$ each point ($x, y)$ on the path has nonnegative coordinates, i.e. $x, y > 0$.

During each step, Pisut measures his distance from the axis parallel to the direction of his step, if after the step he ends up closer from the origin (compared to before the step) he records the distance as a positive number, else he records it as a negative number.

Prove that, after Pisut completes his walk, the sum of the signed distances Pisut measured is zero.
2014 Thailand Mathematical Olympiad
Find $P(x)\in Z[x]$ st : $P(n)|2557^{n}+213.2014$ with any $n\in N^{*}$
2014 Thailand Mathematical Olympiad
Determine the maximal value of $k$ such that the inequality

$$\left(k +\frac{a}{b}\right) \left(k + \frac{b}{c}\right)\left(k + \frac{c}{a}\right)

\le \left( \frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\right) \left(  \frac{b}{a}+ \frac{c}{b}+ \frac{a}{c}\right)$$holds for all positive reals $a, b, c$.
2014 Thailand Mathematical Olympiad
Find all primes $p$ such that $2p^2 - 3p - 1$ is a positive perfect cube
2014 Thailand Mathematical Olympiad
Let $ABCD$ be a convex quadrilateral with shortest side $AB$ and longest side $CD$, and suppose that $AB < CD$. Show that there is a point $E \ne C, D$ on segment $CD$ with the following property:

For all points $P \ne E$ on side $CD$, if we define $O_1$ and $O_2$ to be the circumcenters of $\vartriangle APD$ and $\vartriangle  BPE$ respectively, then the length of $O_1O_2$ does not depend on $P$.
2014 Thailand Mathematical Olympiad
Let $n$ be a positive integer. We want to make up a collection of cards with the following properties:



1. each card has a number of the form $m!$ written on it, where $m$ is a positive integer.



2. for any positive integer  $ t \le n!$, we can select some card(s) from this collection such that the sum of the number(s) on the selected card(s) is $t$.



Determine the smallest possible number of cards needed in this collection.
2013 Thailand Mathematical Olympiad
Find the largest integer that divides $p^4 - 1$ for all primes $p > 4$
2013 Thailand Mathematical Olympiad
Let $\vartriangle ABC$ be a triangle with $\angle  ABC > \angle  BCA \ge 30^o$ . The angle bisectors of $\angle ABC$ and $\angle BCA$ intersect $CA$ and $AB$ at $D$ and $E$ respectively, and $BD$ and $CE$ intersect at $P$. Suppose that $P D = P E$ and the incircle of $\vartriangle ABC$ has unit radius. What is the maximum possible length of $BC$?
2013 Thailand Mathematical Olympiad
Each point on the plane is colored either red or blue. Show that there are three points of the same color that form a triangle with side lengths $1, 2,\sqrt3$.
2013 Thailand Mathematical Olympiad
Determine all monic polynomials $p(x)$ having real coefficients and satisfying the following two conditions:

$\bullet$ $p(x)$ is nonconstant, and all of its roots are distinct reals

$\bullet$ If $a $and $b$ are roots of $p(x)$ then $a + b + ab$ is also a root of $p(x)$.
2013 Thailand Mathematical Olympiad
Find a five-digit positive integer $n$ (in base $10$) such that $n^3 - 1$ is divisible by $2556$ and which minimizes the sum of digits of $n$.
2013 Thailand Mathematical Olympiad
Determine all functions $f$ : $\mathbb R\to\mathbb R$ satisfying $(x^2+y^2)f(xy)=f(x)f(y)f(x^2+y^2)$ $\forall x,y\in\mathbb R$
2013 Thailand Mathematical Olympiad
Let $P_1, ... , P_{2556}$ be distinct points in a regular hexagon $ABCDEF$ with unit side length.

Suppose that no three points in the set $S = \{A, B, C, D, E, F, P_1, ... , P_{2556}\}$ are collinear.

Show that there is a triangle whose vertices are in $S$ and whose area is less than $\frac{1}{1700}$ .
2013 Thailand Mathematical Olympiad
Let $p(x) = x^{2013} + a_{2012}x^{2012} + a_{2011}x^{2011} +...+ a_1x + a_0$ be a polynomial with real coefficients with roots $- b_{1006}, - b_{1005}, ... , -b_1, 0, b_1, ... , b_{1005}, b_{1006}$, where $b_1, b_2, ... , b_{1006}$ are positive reals with product $1$. Show that $a_3a_{2011} \le 1012036$
2013 Thailand Mathematical Olympiad
Let $ABCD$ be a convex quadrilateral, and let $M$ and$ N$ be midpoints of sides $AB$ and $CD$ respectively. Point $P$ is chosen on $CD$ so that $MP \perp CD$, and point $Q$ is chosen on $AB$ so that $NQ \perp AB$. Show that $AD \parallel BC$ if and only if $\frac{AB}{CD} =\frac{MP}{NQ}$ .
2013 Thailand Mathematical Olympiad
Find all pairs of positive integers $(x, y)$ such that $\frac{xy^3}{x+y}$ is the cube of a prime.
2013 Thailand Mathematical Olympiad
Let $m, n$ be positive integers.

There are $n$ piles of gold coins, so that pile $i$ has $a_i > 0$ coins in it $(i = 1, ..., n)$. Consider the following game:



Step 1. Nadech picks sets $B_1, B_2, ... , B_n$, where each $B_i$ is a nonempty subset of $\{1, 2, . . . , m\}$, and gives them to Yaya.



Step 2. Yaya picks a set $S$ which is also a nonempty subset of $\{1, 2, . . . , m\}$.



Step 3. For each $i = 1, 2, . . . , n$, Nadech wins the coins in pile $i$ if $B_i \cap S$ has an even number of elements, and Yaya wins the coins in pile $i$ if $B_i \cap S$ has an odd number of elements.



Show that, no matter how Nadech picks the sets $B_1, B_2, . . . , B_n$, Yaya can always pick $S$ so that she ends up with more gold coins than Nadech
2013 Thailand Mathematical Olympiad
Let $\omega$ be the incircle of $\vartriangle ABC$, $\omega$  is tangent to sides $BC$ and $AC$ at $D$ and $E$ respectively. The line perpendicular to $BC$ at $D$ intersects $\omega$  again at $P$. Lines $AP$ and $BC$ intersect at $M$. Let $N$ be a point on segment $AC$ so that $AE = CN$. Line $BN$ intersects $\omega$  at $Q$ (closer to $B$) and intersect $AM$ at $R$. Show that the area of $\vartriangle ABR$ is equal to the area of $PQMN$.
2012 Thailand Mathematical Olympiad
Let $\vartriangle ABC$ be a right triangle with $\angle B = 90^o$. Let $P$ be a point on side $BC$, and let $\omega$ be the circle with diameter $CP$. Suppose that $\omega$ intersects $AC $and $AP$ again at $Q$ and $R$, respectively. Show that $CP^2 = AC \cdot CQ - AP \cdot P R$.
2012 Thailand Mathematical Olympiad
Let $a_1, a_2, ..., a_{2012}$ be pairwise distinct integers. Show that the equation $(x -a_1)(x - a_2)...(x - a_{2012}) = (1006!)^2$ has at most one integral solution.
2012 Thailand Mathematical Olympiad
Let $m, n > 1$ be coprime odd integers. Show that

$$\big \lfloor \frac{m^{\phi (n)+1} + n^{\phi (m)+1}}{mn} \rfloor$$is an even integer, where $\phi$ is Euler’s totient function.
2012 Thailand Mathematical Olympiad
Let $ABCD$ be a unit square. Points $E, F, G, H$ are chosen outside $ABCD$ so that $\angle AEB =\angle BF C = \angle CGD = \angle DHA = 90^o$ . Let $O_1, O_2, O_3, O_4$, respectively, be the incenters of $\vartriangle ABE, \vartriangle BCF, \vartriangle CDG, \vartriangle DAH$. Show that the area of $O_1O_2O_3O_4$ is at most $1$.
2012 Thailand Mathematical Olympiad
Determine all functions $f : R \to  R$ satisfying $f(f(x) + xf(y))= 3f(x) + 4xy$ for all real numbers $x,y$.
2012 Thailand Mathematical Olympiad
Let $a, b, m$ be integers such that gcd $(a, b) = 1$ and $5 | ma^2 + b^2$ . Show that there exists an integer $n$ such that $5 | m - n^2$.
2012 Thailand Mathematical Olympiad
$4n$ first grade students at Songkhla Primary School, including $2n$ boys and $2n$ girls, participate in a taekwondo tournament where every pair of students compete against each other exactly once. The tournament is scored as follows:

$\bullet$ In a match between two boys or between two girls, a win is worth $3$ points, a draw $1$ point, and a loss $0$ points.

$\bullet$ In a math between a boy and a girl, if the boy wins, he receives $2$ points, else he receives $0$ points. If the girl wins, she receives $3$ points, if she draws, she receives $2$ points, and if she loses, she receives $0$ points.

After the tournament, the total score of each student is calculated. Let $P$ be the number of matches ending in a draw, and let $Q$ be the total number of matches. Suppose that the maximum total score is $4n - 1$. Find $P/Q$.
2012 Thailand Mathematical Olympiad
Let $n$ be a positive integer and let $P(x) = x^n + a_{n-1}x^{n-1} +... + a_1x + 1$ be a polynomial with positive real coefficients. Under the assumption that the roots of $P$ are all real, show that $P(x) \ge (x + 1)^n$ for all $x > 0$.
2012 Thailand Mathematical Olympiad
Let $x$ be an irrational number. Show that there are integers $m$ and $n$ such that $\frac{1}{2555}< mx + n <\frac{1}{2012}$
2012 Thailand Mathematical Olympiad
Let $\vartriangle ABC$ be an acute triangle, and let $P$ be the foot of altitude from $C$ to $AB$. Let $\omega$ be the circle with diameter $BC$. The tangents from $A$ to $\omega$  are drawn touching $\omega$  at $D$ and $E$. Lines $AD$ and $AE$ intersect line $BC$ at $M$ and $N$ respectively, so that $B$ lies between $M$ and $C$. Let $CP$ intersect $DE$ at $Q, ME$ intersect $ND$ at $R$, and let $QR$ intersect $BC$ at $S$. Show that $QS$ bisects $\angle DSE$
2012 Thailand Mathematical Olympiad
Let $a, b, c$ be positive integers. Show that if $\frac{a}{b} +\frac{b}{c} +\frac{c}{a}$ is an integer then $abc$ is a perfect cube.
2011 Thailand Mathematical Olympiad
Given a natural number $n$ $\geq 3$. If $p,q$ are primes, such that, $p \mid n!$ and $q \mid (n-1)!-1$. Prove that, $p<q$
2011 Thailand Mathematical Olympiad
Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that $f(2m+2n)=f(m)f(n)$ for all natural numbers $m,n$.
2011 Thailand Mathematical Olympiad
Given a $\Delta ABC$ where $\angle C = 90^{\circ}$, $D$ is a point in the interior of $\Delta ABC$ and lines $AD$ $,$ $BD$ and $CD$ intersect $BC$, $CA$ and $AB$ at points $P$ ,$Q$ and $R$ ,respectively. Let $M$ be the midpoint of $\overline{PQ}$. Prove that, if $\angle BRP$ $ =$ $ \angle PRC$ then $MR=MC$.
2011 Thailand Mathematical Olympiad
There are $900$ students in an International School. There are $59$ international boys and $59$ international girls. The Students are partitioned into $30$ classrooms (each classrooms have equal number of student) and in each of the classrooms, the student will labelled number from $1$ to $30$. The Partition must satisfy at least one follow condition:



Any Two international boys in same classroom can't have consecutive numbers.

For every classroom, the student who is labelled $1$ must be a boy.



Prove that there are $2$ classrooms, each of which has $2$ international boys with their labels difference equal.
2011 Thailand Mathematical Olympiad
Find all $n$ such that $$n = d (n) ^ 4$$

Where $d (n)$ is the number of divisors of $n$, for example $n = 2 \cdot 3\cdot 5\implies  d (n) = 2 \cdot 2\cdot 2$.
2011 Thailand Mathematical Olympiad
For any $0\leq x_1,x_2,\ldots,x_{2011} \leq 1$, Find the maximum value of \begin{align*} \sum_{k=1}^{2011}(x_k-m)^2 \end{align*}where $m$ is the arithmetic mean of $x_1,x_2,\ldots,x_{2011}$.
2011 Thailand Mathematical Olympiad
Let $a,b,c,d \in \mathbb{R}^+$ and suppose that all roots of the equation

\begin{align*} x^5-ax^4+bx^3-cx^2+dx=1 \end{align*}are real. Prove

\begin{align*} \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} \le \frac{3}{5} \end{align*}
2011 Thailand Mathematical Olympiad
Given $\Delta ABC$ and its centroid $G$, If line $AC$ is tangent to $\odot (ABG)$. Prove that, \begin{align*} AB+BC \leq 2AC \end{align*}
2011 Thailand Mathematical Olympiad
Prove that, for all $n \in \mathbb{N}$ \begin{align*} \frac{1}{1}+\frac{1}{3}+\frac{1}{5}+\ldots+\frac{1}{2n+1} \not\in \mathbb{Z} \end{align*}
2011 Thailand Mathematical Olympiad
Does there exists a function $f : \mathbb{N} \longrightarrow \mathbb{N}$

\begin{align*} f \left( m+ f(n) \right) = f(m) +f(n) + f(n+1) \end{align*}for all $m,n \in \mathbb{N}$ ?
2011 Thailand Mathematical Olympiad
In $\Delta ABC$, Let the Incircle touch $\overline{BC}, \overline{CA}, \overline{AB}$ at $X,Y,Z$. Let $I_A,I_B,I_C$ be $A$,$B$,$C-$excenters, respectively. Prove that Incenter of $\Delta ABC$, orthocenter of $\Delta XYZ$ and centroid of $\Delta I_AI_BI_C$ are collinear.
2011 Thailand Mathematical Olympiad
$7662$ chairs are placed in a circle around the city of Chiang Mai. They are also marked with a label for either $1$st, $2$nd, or $3$rd grade students, so that there are $2554$ chairs labeled with each label. The following situations happen, in order



$2554$ students each from the $1$st, $2$nd, and $3$rd grades are given a ball as follows: $1$st grade students receive footballs, $2$nd grade students receive basketballs, and $3$rd grade students receive volleyballs.

The students go sit in chairs labeled for their grade

The students simultaneously send their balls to the student to their left, and this happens some positive number of times.



A labelling of the chairs is called lin-ping if it is possible for all $1$st, $2$nd, and $3$rd grade students to now hold volleyballs, footballs, and basketballs respectively. Compute the number of lin-ping labellings
2010 Thailand Mathematical Olympiad
Find the number of ways to distribute $11$ balls into $5$ boxes with different sizes, so that each box receives at least one ball, and the total number of balls in the largest and smallest boxes is more than the total number of balls in the remaining boxes.
2010 Thailand Mathematical Olympiad
Let $\vartriangle ABC$ be an isosceles triangle with $AB = AC$. A circle passing through $B$ and $C$ intersects sides $AB$ and $AC$ at $D$ and $E$ respectively. A point $F$ on this circle is chosen so that $EF\perp  BC$. If $BC = x$, $CF = y$, and $BF = z$, find the length of $DF$ in terms of $x, y, z$.
2010 Thailand Mathematical Olympiad
Show that there are infinitely many positive integers n such that $2\underbrace{555...55}_{n}3$  is divisible by $2553$.
2010 Thailand Mathematical Olympiad
Let $\vartriangle ABC$ be an equilateral triangle, and let $M$ and $N$ be points on $AB$ and $AC$, respectively, so that $AN = BM$ and $3MB = AB$. Lines $CM$ and $BN$ intersect at $O$. Find $\angle AOB$.
2010 Thailand Mathematical Olympiad
In a round-robin table tennis tournament between $2010$ athletes, where each match ends with a winner and a loser, let $a_1,... , a_{2010}$ denote the number of wins of each athlete, and let $b_1, .., b_{2010}$ denote the number of losses of each athlete. Show that $a^2_1+a^2_2+...+a^2_{2010} =b^2_1 + b^2_2 + ... + b^2_{2010}$.
2010 Thailand Mathematical Olympiad
Let $f : R \to R$ be a function satisfying the functional equation $f(3x + y) + f(3x-y) = f(x + y) + f(x - y) + 16f(x)$ for all reals $x, y$. Show that $f$ is even, that is, $f(-x) = f(x)$ for all reals $x$
2010 Thailand Mathematical Olympiad
Let $a, b, c$ be positive reals. Show that $\frac{a^5}{bc^2} + \frac{b^5}{ca^2} + \frac{c^5}{ab^2} \ge a^2 + b^2 + c^2.$
2010 Thailand Mathematical Olympiad
Define the modulo $2553$ distance $d(x, y)$ between two integers $x, y$ to be the smallest nonnegative integer $d$ equivalent to either $x - y$ or $y - x$ modulo $2553$. Show that, given a set S of integers such that $|S| \ge 70$, there must be $m, n \in S$ with $d(m, n) \le 36$.
2010 Thailand Mathematical Olympiad
Let $a, b, c$ be real numbers so that all roots of the equation $2x^5 + 5x^4 + 5x^3 + ax^2 + bx + c = 0$ are real. Find the smallest real root of the equation above.
2010 Thailand Mathematical Olympiad
Find all primes $p$ such that ${100 \choose p} + 7$ is divisible by $p$.
2010 Thailand Mathematical Olympiad
Show that, for every positive integer $x$, there is a positive integer $y\in \{2, 5, 13\}$ such that $xy - 1$ is not a perfect square.
2010 Thailand Mathematical Olympiad
The Ministry of Education selects $2010$ students from $5$ regions of Thailand to participate in a debate tournament, where each pair of students will debate in one of the three topics: politics, economics, and societal problems. Show that there are $3$ students who were born in the same month, come from the same region, are of the same gender , and whose pairwise debates are on the same topic.
2010 Thailand Mathematical Olympiad
Let $\vartriangle ABC$ be a scalene triangle with $AB < BC < CA$. Let $D$ be the projection of $A$ onto the angle bisector of $\angle ABC$, and let $E$ be the projection of $A$ onto the angle bisector of $\angle ACB$. The line $DE$ cuts sides $AB$ and AC at $M$ and $N$, respectively. Prove that $$\frac{AB+AC}{BC} =\frac{DE}{MN} + 1$$
2010 Thailand Mathematical Olympiad
For $i = 1, 2$ let $\vartriangle A_iB_iC_i$ be a triangle with side lengths $a_i, b_i, c_i$ and altitude lengths $p_i, q_i, r_i$.  Define $a_3 =\sqrt{a_1^2 + a_2^2}, b_3 =\sqrt{b_1^2 + b_2^2}$ , and $c_3 =\sqrt{c_1^2 + c_2^2}$.

Prove that $a_3, b_3, c_3$ are side lengths of a triangle, and if $p_3, q_3, r_3$ are the lengths of altitudes of this triangle, then $p_3^2 \ge p_1^2 +p_2^2$, $q_3^2 \ge q_1^2 +q_2^2$ , and $r_3^2 \ge r_1^2 +r_2^2$
2010 Thailand Mathematical Olympiad
Determine all functions $f : R \times R \to  R$ satisfying the equation $f(x - t, y) + f(x + t, y) + f(x, y - t) + f(x, y + t) = 2010$ for all real numbers $x, y$ and for all nonzero $t$
2010 Thailand Mathematical Olympiad
Show that no triples of primes $p, q, r$ satisfy $p > r, q > r$, and $pq | r^p + r^q$
2009 Thailand Mathematical Olympiad
Let $S \subset Z^+$ be a set of positive integers with the following property: for any $a, b  \in  S$, if $a \ne b$ then $a + b$ is a perfect square. Given that $2009 \in S$ and $2087  \in  S$, what is the maximum number of elements in $S$?
2009 Thailand Mathematical Olympiad
Let $k$ and $n$ be positive integers with $k < n$. Find the number of subsets of $\{1, 2, . . . , n\}$ such that the difference between the largest and smallest elements in the subset is $k$.
2009 Thailand Mathematical Olympiad
Teeradet is a student in a class with $19$ people. He and his classmates form clubs, so that each club must have at least one student, and each student can be in more than one club. Suppose that any two clubs differ by at least one student, and all clubs Teeradet is in have an odd number of students. What is the maximum possible number of clubs?
2009 Thailand Mathematical Olympiad
In triangle $\vartriangle ABC$, $D$ is the midpoint of $BC$. Points $E$ and $F$ are chosen on side $AC$ so that $AF = F E = EC$. Let $AD$ intersect $BE$ and $BF$ and $G$ and $H$, respectively. Find the ratio of the areas of $\vartriangle BGH$ and $\vartriangle ABC$.
2009 Thailand Mathematical Olympiad
Determine all functions $f : R\to  R$ satisfying:  $$f(xy + 2x + 2y - 1) = f(x)f(y) + f(y) + x -2$$for all real numbers $x, y$.
2009 Thailand Mathematical Olympiad
Let $\vartriangle ABC$ be a triangle with $AB > AC$, its incircle is tangent to $BC$ at $D$. Let $DE$ be a diameter of the incircle, and let $F$ be the intersection between line $AE$ and side $BC$. Find the ratio between the areas of $\vartriangle DEF$ and $\vartriangle ABC$ in terms of the three side lengths of$\vartriangle ABC$.
2009 Thailand Mathematical Olympiad
Let $a, b, c$ be real numbers, and define $S_n = a^n + b^n + c^n$ for positive integers $n$. Suppose that $S_1, S_2, S_3$ are integers satisfying $6 | 5S_1 - 3S_2 - 2S_3$. Show that $S_n$ is an integer for all positive integers $n$.
2009 Thailand Mathematical Olympiad
Let $a, b, c$ be side lengths of a triangle, and define $s =\frac{a+b+c}{2}$.  Prove that

$$\frac{2a(2a-s)}{b + c}+\frac{2b(2b - s)}{c + a}+\frac{2c(2c - s)}{a + b}\ge  s.$$
2009 Thailand Mathematical Olympiad
In triangle $\vartriangle ABC, D$ and $E$ are midpoints of the sides $BC$ and $AC$, respectively. Lines $AD$ and $BE$ are drawn intersecting at $P$. It turns out that $\angle CAD = 15^o$ and $\angle APB = 60^o$. What is the value of $AB/BC$ ?
2009 Thailand Mathematical Olympiad
Let $p > 5$ be a prime. Suppose that $$\frac{1}{2^2} + \frac{1}{4^2}+ \frac{1}{6^2}+ ...+ \frac{1}{(p -1)^2} =\frac{a}{b}$$where $a/b$ is a fraction in lowest terms. Show that $p | a$.
2009 Thailand Mathematical Olympiad
Let $a$ and $b$ be integers and $p$ a prime. For each positive integer k, define$ A_k = \{n \in Z^+ |p^k$ divides $a^n - b^n\}$. Show that if $A_1$ is nonempty then $A_k$ is nonempty for all positive integers $k$
2009 Thailand Mathematical Olympiad
Is there an injective function $f : Z^+ \to  Q$ satisfying the equation $f(xy) = f(x) + f(y)$ for all positive integers $x$ and $y$?
2009 Thailand Mathematical Olympiad
Let $ABCD$ be a convex quadrilateral with the property that $MA \cdot  MC + MA  \cdot  CD = MB  \cdot  MD$, where $M$ is the intersection of the diagonals $AC$ and $BD$. The angle bisector of $\angle ACD$ is drawn intersecting ray $\overrightarrow{BA}$ at $K$. Prove that $BC = DK$ if and only if $AB \parallel CD$.
2009 Thailand Mathematical Olympiad
Let $k$ be a positive integer. Show that there are infinitely many positive integer solutions $(m, n)$ to

$(m - n)^2 = kmn + m + n$.