Navvye/Olympiad-Problems-CSV huggingface dataset:
2009 Thailand Mathematical Olympiad
A class contains $80$ boys and $80$ girls. On each weekday (Monday to Friday) of the week before final exams, the teacher has $16$ books for the students to borrow, where a book can only be borrowed for one day at a time, and each student can only borrow once during the entire week. Show that there are two days and two books such that one of the following two statements is true:

(i) Both books were not borrowed on both days

(ii) Both books were borrowed on both days, and the four students who borrowed the books on these days are either all boys or all girls.
2009 Thailand Mathematical Olympiad
Find all polynomials of the form $P(x) = (-1)^nx^n + a_1x^{n-1} + a_2x^{n-2} + ...+ a_{n-1}x + a_n$ with the following two properties:

(i) $\{a_1, a_2, . . . , a_n-1, a_n\} =\{0, 1\}$, and

(ii) all roots of $P(x)$ are distinct real numbers
2008 Thailand Mathematical Olympiad
Let $\vartriangle ABC$ be a triangle with $\angle BAC = 90^o$ and $\angle ABC = 60^o$. Point $E$ is chosen on side $BC$ so that $BE : EC = 3 : 2$. Compute $\cos\angle CAE$.
2008 Thailand Mathematical Olympiad
Let $AD$ be the common chord of two equal-sized circles $O_1$ and $O_2$. Let $B$ and $C$ be points on $O_1$ and $O_2$, respectively, so that $D$ lies on the segment $BC$. Assume that $AB = 15, AD = 13$ and $BC = 18$, what is the ratio between the inradii of $\vartriangle ABD$ and $\vartriangle ACD$?
2008 Thailand Mathematical Olympiad
Find all positive real solutions to the equation

$x+\left\lfloor\frac x3\right\rfloor=\left\lfloor\frac{2x}3\right\rfloor+\left\lfloor\frac{3x}5\right\rfloor$
2008 Thailand Mathematical Olympiad
Prove that $$\sqrt{a^2 + b^2 -\sqrt2 ab} +\sqrt{b^2 + c^2 -\sqrt2 bc} \ge \sqrt{a^2 + c^2}$$for all real numbers $a, b, c > 0$
2008 Thailand Mathematical Olympiad
Let $P(x)$ be a polynomial of degree $2008$ with the following property:  all roots of $P$ are real, and for all real $a$, if $P(a) = 0$ then $P(a+ 1) = 1$.  Prove that P must have a repeated root.
2008 Thailand Mathematical Olympiad
Let $f : R \to  R$ be a function satisfying the inequality $|f(x + y) -f(x) - f(y)| < 1$ for all reals $x, y$.

Show that $\left| f\left( \frac{x}{2008 }\right) - \frac{f(x)}{2008} \right|  < 1$ for all real numbers $x$.
2008 Thailand Mathematical Olympiad
Two positive integers $m, n$ satisfy the two equations $m^2 + n^2 = 3789$ and $gcd (m, n) + lcm (m, n) = 633$. Compute $m + n$.
2008 Thailand Mathematical Olympiad
Prove that $2551 \cdot 543^n -2008\cdot 7^n$ is never a perfect square, where $n$ varies over the set of positive integers
2008 Thailand Mathematical Olympiad
Find the number of pairs of sets $(A, B)$ satisfying $A \subseteq B \subseteq \{1, 2, ...,10\}$
2008 Thailand Mathematical Olympiad
On the sides of triangle $\vartriangle ABC$, $17$ points are added, so that there are $20$ points in total (including the vertices of $\vartriangle ABC$.) What is the maximum possible number of (nondegenerate) triangles that can be formed by these points.
2008 Thailand Mathematical Olympiad
Let $P$ be a point outside a circle $\omega$. The tangents from $P$ to $\omega$ are drawn touching $\omega$ at points $A$ and $B$. Let $M$ and $N$ be the midpoints of $AP$ and $AB$, respectively. Line $MN$ is extended to cut $\omega$ at $C$ so that $N$ lies between $M$ and $C$. Line $PC$ intersects $\omega$ again at $D$, and lines $ND$ and $PB$ intersect at $O$. Prove that $MNOP$ is a rhombus.
2008 Thailand Mathematical Olympiad
Find all positive integers $N$ with the following properties:

(i) $N$ has at least two distinct prime factors, and

(ii) if $d_1 < d_2 < d_3 < d_4$ are the four smallest divisors of $N$ then $N =d_1^2 + d_2 ^2+ d_3 ^2+ d_4^2$
2008 Thailand Mathematical Olympiad
For each positive integer $n$, define $a_n = n(n + 1)$. Prove that

$$n^{1/a_1} + n^{1/a_3} + n^{1/a_5} + ...+ n^{1/a_{2n-1}} \ge  n^{a_{3n+2}/a_{3n+1}}$$.
2008 Thailand Mathematical Olympiad
Let $n$ be a positive integer. Show that

$${2n+1 \choose 1} -{2n+1 \choose 3}2008 + {2n+1 \choose 5}2008^2- ...+(-1)^{2n+1}{2n+1 \choose 2n+1}2008^n $$is not divisible by $19$.
2008 Thailand Mathematical Olympiad
Students in a class consisting of $m$ boys and $n$ girls line up. Over all possible ways of lining up, compute the average number of pairs of two boys or two girls who are next to each other.
2008 Thailand Mathematical Olympiad
Let $f : R^+ \to  R^+$ satisfy $f(xy)^2 = f(x^2)f(y^2)$ for all positive reals $x, y$ with $x^2y^3 > 2008.$

Prove that $f(xy)^2 = f(x^2)f(y^2))$ for all positive reals $x, y$.
2007 Thailand Mathematical Olympiad
In a circle $\odot O$, radius $OA$ is perpendicular to radius $OB$. Chord $AC$ intersects $OB$ at $E$ so that the length of arc $AC$ is one-third the circumference of  $\odot O$. Point $D$ is chosen on $OB$ so that $CD \perp AB$. Suppose that segment $AC$ is $2$ units longer than segment $OD$. What is the length of segment $AC$?
2007 Thailand Mathematical Olympiad
Let $ABCD$ be a cyclic quadrilateral so that arcs $AB$ and $BC$ are equal. Given that $AD = 6, BD = 4$ and $CD = 1$, compute $AB$.
2007 Thailand Mathematical Olympiad
A triangle $\vartriangle ABC$ has $\angle B = 90^o$ . A circle is tangent to $AB$ at $B$ and also tangent to $AC$. Another circle is tangent to the first circle as well as the two sides $AB$ and $AC$. Suppose that $AB =\sqrt3$ and $BC = 3$. What is the radius of the second circle?
2007 Thailand Mathematical Olympiad
A triangle $\vartriangle ABC$ has $AC = 16$ and $BC = 12$. $E$ and $F$ are points on $AC$ and $BC$, respectively, so that $CE = 3CF$. Let $M$ be the midpoint of $AB$, and let lines $EF$ and $CM$ intersect at $G$. Compute the ratio $EG : GF$.
2007 Thailand Mathematical Olympiad
A triangle $\vartriangle ABC$ has $\angle A = 90^o$, and a point $D$ is chosen on $AC$. Point $F$ is the foot of altitude from $A$ to $BC$. Suppose that $BD = DC = CF = 2$. Compute $AC$.
2007 Thailand Mathematical Olympiad
Let $M$ be the midpoint of a given segment $BC$. Point $A$ is chosen to maximize $\angle ABC$ while subject to the condition that $\angle MAC = 20^o$ . What is the ratio $BC/BA$ ?
2007 Thailand Mathematical Olympiad
Let $a, b, c$ be complex numbers such that $a+b+c = 1$, $a^2+b^2+c^2 = 2$ and $a^3+b^3+c^3 = 3$.

Find the value of $a^4 + b^4 + c^4$.
2007 Thailand Mathematical Olympiad
Let $x_1, x_2,... , x_{84}$ be the roots of the equation $x^{84} + 7x - 6 = 0$. Compute $\sum_{k=1}^{84} \frac{x_k}{x_k-1}$.
2007 Thailand Mathematical Olympiad
Let $f : R \to R$ be a function satisfying the equation $f(x^2 + x + 3) + 2f(x^2 - 3x + 5) =6x^2 - 10x + 17$ for all real numbers $x$. What is the value of $f(85)$?
2007 Thailand Mathematical Olympiad
Find the smallest positive integer $n$ such that the equation $\sqrt3 z^{n+1} - z^n - 1 = 0$ has a root on the unit circle.
2007 Thailand Mathematical Olympiad
Compute the number of functions $f : \{1, 2,... , 2550\} \to \{61, 80, 84\}$ such that $\sum_{k=1}^{2550} f(k)$ is divisible by $3$.
2007 Thailand Mathematical Olympiad
An alien with four feet wants to wear four identical socks and four identical shoes, where on each foot a sock must be put on before a shoe. How many ways are there for the alien to wear socks and shoes?
2007 Thailand Mathematical Olympiad
Let $S = \{1, 2,..., 8\}$. How many ways are there to select two disjoint subsets of $S$?
2007 Thailand Mathematical Olympiad
The sum $$\sum_{k=84}^{8000}{k \choose 84}{{8084 - k} \choose 84}$$can be written as a binomial coefficient $a \choose b$ for integers $a, b$. Find a possible pair $(a, b)$
2007 Thailand Mathematical Olympiad
Compute the remainder when $222!^{111} + 111^{222!} + 111!^{222} + 222^{111!}$ is divided by $2007$.
2007 Thailand Mathematical Olympiad
What is the smallest positive integer with $24$ positive divisors?
2007 Thailand Mathematical Olympiad
Compute the product of positive integers $n$ such that $n^2 + 59n + 881$ is a perfect square.
2007 Thailand Mathematical Olympiad
Let $p_k$ be the $k$-th prime number. Find the remainder when $\sum_{k=2}^{2550}p_k^{p_k^4-1}$ is divided by $2550$.
2007 Thailand Mathematical Olympiad
Find all functions $f : R \to  R$ such that the inequality $$\sum_{i=1}^{2549} f(x_i + x_{i+1}) + f (\sum_{i=1}^{2550}x_y) \le \sum_{i=1}^{2550}f(2x_i)$$for all reals $x_1, x_2, . . . , x_{2550}$.
2007 Thailand Mathematical Olympiad
In a dance party there are $n$ girls and $n$ boys, and some $m$ songs are played. Each song is danced to by at least one pair of a boy and a girl, who both receive a malai  each. Prove that for all positive integers $k \le n$, it is possible to select $k$ boys and $n - k$ girls so that the $n$ selected people received at least $m$ malai in total.
2007 Thailand Mathematical Olympiad
Two circles intersect at $X$ and $Y$ . The line through the centers of the circles intersect the first circle at $A$ and $C$, and intersect the second circle at $B$ and $D$ so that $A, B, C, D$ lie in this order. The common chord $XY$ cuts $BC$ at $P$, and a point $O$ is arbitrarily chosen on segment $XP$. Lines $CO$ and $BO$ are extended to intersect the first and second circles at $M$ and $N$, respectively. If lines $AM$ and $DN$ intersect at $Z$, prove that $X, Y$ and $Z$ lie on the same line.
2007 Thailand Mathematical Olympiad
Find all primes $p$ such that $\frac{2^{p-1}-1}{p}$ is a perfect square.
2007 Thailand Mathematical Olympiad
The freshman class of a school consists of $229$ boys and $271$ girls, and is divided into $10$ rooms of $50$ students each, the students in each room are numbered from $1$ to $50$. The physical education teacher wants to select a relay running team consisting of $1$ boy and $3$ girls or $1$ girl and $3$ boys, so that the four students must be two pairs of students with the same number from two rooms. Show that the number of possible teams is odd.
2007 Thailand Mathematical Olympiad
A triangle has perimeter $2s$, inradius $r$, and incenter $I$. If $s_a, s_b$ and $s_c$ are the distances from $I$ to the three vertices, then show that $$\frac34 +\frac{r}{s_a}+\frac{r}{s_b}+\frac{r}{s_c} \le \frac{s^2}{12r^2}$$
2006 Thailand Mathematical Olympiad
Let $O$ be the circumcenter of a triangle $\vartriangle ABC$. It is given that $\angle ABC = 70^o$, $\angle ACB =50^o$. Let the angle bisector of $\angle BAC$ intersect the circumcircle of $\vartriangle ABC$ again at $D$. Compute $\angle ADO$.
2006 Thailand Mathematical Olympiad
Triangle $\vartriangle ABC$ has side lengths $AB = 2$, $CA = 3$ and $BC = 4$. Compute the radius of the circle centered on $BC$ that is tangent to both $AB$ and $AC$.
2006 Thailand Mathematical Olympiad
The three medians of a triangle has lengths $3, 4, 5$. What is the length of the shortest side of this triangle?
2006 Thailand Mathematical Olympiad
Let $P$ be a point outside a circle centered at $O$. From $P$, tangent lines are drawn to the circle, touching the circle at points $A$ and $B$. Ray $\overrightarrow{BO}$ is drawn intersecting the circle again at $C$ and intersecting ray $\overrightarrow{PA}$ at $Q$. If $3QA = 2AP$, what is the value of $\sin \angle CAQ$?
2006 Thailand Mathematical Olympiad
Let $f : Z_{\ge 0} \to Z_{\ge 0}$ satisfy the functional equation $$f(m^2 + n^2) =(f(m) - f(n))^2 + f(2mn)$$for all nonnegative integers $m, n$. If $8f(0) + 9f(1) = 2006$, compute $f(0)$.
2006 Thailand Mathematical Olympiad
A function $f : R \to R$ has $f(1) < 0$, and satisfy the functional equation $$f(\cos (x + y)) = (\cos x)f(\cos y) + 2f(\sin x)f(\sin y)$$for all reals $x, y$. Compute $f \left(\frac{2006}{2549 }\right)$
2006 Thailand Mathematical Olympiad
Let $x, y, z$ be reals summing to $1$ which minimizes $2x^2 + 3y^2 + 4z^2$. Find $x$.
2006 Thailand Mathematical Olympiad
Let $a, b, c$ be the roots of the equation $x^3-9x^2+11x-1 = 0$, and define $s =\sqrt{a}+\sqrt{b}+\sqrt{c}$.

Compute $s^4 -18s^2 - 8s$ .
2006 Thailand Mathematical Olympiad
Compute the largest integer not exceeding $$\frac{2549^3}{2547\cdot 2548} -\frac{2547^3}{2548\cdot 2549}$$
2006 Thailand Mathematical Olympiad
Find the remainder when $26!^{26} + 27!^{27}$ is divided by $29$.
2006 Thailand Mathematical Olympiad
Let $p_n$ be the $n$-th prime number. Find the remainder when $\Pi_{n=1}^{2549} 2006^{p^2_{n-1}}$ is divided by $13$
2006 Thailand Mathematical Olympiad
Let $a_n = 2^{3n-1} + 3^{6n-2} + 5^{6n-3}$. Compute gcd$(a_1, a_2, ... , a_{25})$
2006 Thailand Mathematical Olympiad
Compute the remainder when $\underbrace{\hbox{11...1}}_{\hbox{1862}}$  is divided by $2006$
2006 Thailand Mathematical Olympiad
Find the smallest positive integer $n$ such that $2549 | n^{2545} - 2$.
2006 Thailand Mathematical Olympiad
How many positive integers $n < 2549$ are there such that $x^2 + x - n$ has an integer rooot?
2006 Thailand Mathematical Olympiad
Find the number of triples of sets $(A, B, C)$ such that $A \cup B \cup C = \{1, 2, 3, ... , 2549\}$
2006 Thailand Mathematical Olympiad
Six people, with distinct weights, want to form a triangular position where there are three people in the bottom row, two in the middle row, and one in the top row, and each person in the top two rows must weigh less than both of their supports. How many distinct formations are there?
2006 Thailand Mathematical Olympiad
In May, the traffic police wants to select 10 days to patrol, but no two consecutive days can be selected. How many ways are there for the traffic police to select patrol days?
2006 Thailand Mathematical Olympiad
Show that the product of three consecutive positive integers is never a perfect square.
2006 Thailand Mathematical Olympiad
From a point $P$ outside a circle, two tangents are drawn touching the circle at points $A$ and $C$. Let $B$ be a point on segment $AC$, and let segment $PB$ intersect the circle at point $Q$. The angle bisector of $\angle AQC$ intersects segment $AC$ at $R$. Show that $$\frac{AB}{BC} =\left(\frac{ AR}{RC}\right)^2$$
2006 Thailand Mathematical Olympiad
Let $P(x), Q(x)$ and $R(x)$ be polynomials satisfying the equation $2xP(x^3) + Q(-x -x^3) = (1 + x + x^2)R(x)$.

Show that $x - 1$ divides $P(x) - Q(x)$.
2006 Thailand Mathematical Olympiad
In a classroom, $28$ students are divided into $4$ groups of $7$, and in each group the students are labeled $1, 2,..., 7$ in some order. Show that no matter how the labels are assigned, there must be four students of the same gender  who come from two groups and share the same two labels.
2006 Thailand Mathematical Olympiad
Show that there are coprime positive integers $m$ and $n$ such that $2549 | (25 \cdot  49)^m + 25^n - 2 \cdot  49^n$
2006 Thailand Mathematical Olympiad
Let $a, b, c$ be positive reals. Show that $$1 +\frac{3}{ab + bc + ca}\ge \frac{6}{a + b + c}$$
2005 Thailand Mathematical Olympiad
Let $ABCD$ be a trapezoid inscribed in a unit circle with diameter $AB$. If $DC = 4AD$, compute $AD$.
2005 Thailand Mathematical Olympiad
Let $\vartriangle ABC$ be an acute triangle, and let $A'$ and $B'$ be the feet of altitudes from $A$ to $BC$ and from $B$ to $CA$, respectively; the altitudes intersect at $H$. If $BH$ is equal to the circumradius of $\vartriangle ABC$, find $\frac{A'B}{AB}$ .
2005 Thailand Mathematical Olympiad
Triangle $\vartriangle ABC$ is isosceles with $AB = AC$ and $\angle ABC = 2\angle BAC$. Compute $\frac{AB}{BC}$ .
2005 Thailand Mathematical Olympiad
Triangle $\vartriangle ABC$ is inscribed in the circle with diameter $BC$. If $AB = 3$, $AC = 4$, and $O$ is the incenter of $\vartriangle ABC$, then find $BO \cdot OC$.
2005 Thailand Mathematical Olympiad
A die is thrown six times. How many ways are there for the six rolls to sum to $21$?
2005 Thailand Mathematical Olympiad
Find the number of positive integer solutions to the equation $(x_1+x_2+x_3)^2(y_1+y_2) = 2548$.
2005 Thailand Mathematical Olympiad
How many ways are there to express $2548$ as a sum of at least two positive integers, where two sums that differ in order are considered different?
2005 Thailand Mathematical Olympiad
For each subset $T$ of $S = \{1, 2, ... , 7\}$, the result $r(T)$ of T is computed as follows: the elements of $T$ are written, largest to smallest, and alternating signs $(+, -)$ starting with $+$ are put in front of each number. The value of the resulting expression is$ r(T)$. (For example, for $T =\{2, 4, 7\}$, we have $r(T) = +7 - 4 + 2 = 5$.) Compute the sum of $r(T)$ as $T$ ranges over all subsets of $S$.
2005 Thailand Mathematical Olympiad
Compute gcd $\left( \frac{135^{90}-45^{90}}{90^2} , 90^2 \right)$
2005 Thailand Mathematical Olympiad
What is the remainder when $\sum_{k=1}^{2005}k^{2005\cdot 2^{2005}}$ is divided by $2^{2005}$?
2005 Thailand Mathematical Olympiad
Find the smallest positive integer $x$ such that $2^{254}$ divides $x^{2005} + 1$.
2005 Thailand Mathematical Olympiad
Find the number of even integers n such that $0 \le n \le 100$ and $5 | n^2 \cdot  2^{{2n}^2}+ 1$.
2005 Thailand Mathematical Olympiad
Find all odd integers $k$ for which there exists a positive integer $m$ satisfying the equation

$k + (k + 5) + (k + 10) + ... + (k + 5(m - 1)) = 1372$.
2005 Thailand Mathematical Olympiad
A function $f : Z \to Z$ is given so that $f(m + n) = f(m) + f(n) + 2mn - 2548$ for all positive integers $m, n$. Given that $f(2548) = -2548$, find the value of $f(2)$.
2005 Thailand Mathematical Olympiad
A function $f : R \to R$ satisfy the functional equation $f(x + 2y) + 2f(y - 2x) = 3x -4y + 6$ for all reals $x, y$. Compute $f(2548)$.
2005 Thailand Mathematical Olympiad
Compute the sum of roots of $(2 - x)^{2005} + x^{2005} = 0$.
2005 Thailand Mathematical Olympiad
For $a, b \ge 0$ we define $a *  b = \frac{a+b+1}{ab+12}$ . Compute $0*(1*(2*(... (2003*(2004*2005))...)))$.
2005 Thailand Mathematical Olympiad
Compute the sum $$\sum_{k=0}^{1273}\frac{1}{1 + tan^{2548}\left(\frac{k\pi}{2548}\right)}$$
2005 Thailand Mathematical Olympiad
Let $P(x)$ be a monic polynomial of degree $4$ such that for $k = 1, 2, 3$, the remainder when $P(x)$ is divided by $x - k$ is equal to $k$. Find the value of $P(4) + P(0)$.
2005 Thailand Mathematical Olympiad
Let $a, b, c, d > 0$ satisfy $36a + 4b + 4c + 3d = 25$. What is the maximum possible value of $ab^{1/2}c^{1/3}d^{1/4}$ ?
2005 Thailand Mathematical Olympiad
Compute the minimum value of $cos(a-b) + cos(b-c) + cos(c-a)$ as $a,b,c$  ranges over the real numbers.
2005 Thailand Mathematical Olympiad
A point $A$ is chosen outside a circle with diameter $BC$ so that $\vartriangle ABC$ is acute. Segments $AB$ and $AC$ intersect the circle at $D$ and $E$, respectively, and $CD$ intersects $BE$ at $F$. Line $AF$ intersects the circle again at $G$ and intersects $BC$ at $H$. Prove that $AH \cdot F H = GH^2$.

.
2005 Thailand Mathematical Olympiad
Let $S $be a set of three distinct integers. Show that there are $a, b \in  S$ such that $a \ne b$ and $10 | a^3b - ab^3$.
2005 Thailand Mathematical Olympiad
Does there exist a function $f : Z^+ \to Z^+$ such that $f(f(n)) = 2n$ for all positive integers $n$? Justify your answer, and if the answer is yes, give an explicit construction.
2005 Thailand Mathematical Olympiad
Let $O_1$ be the center of a semicircle $\omega_1$ with diameter $AB$ and let $O_2$ be the center of a circle $\omega_2$ inscribed in $\omega_1$ and which is tangent to $AB$ at $O_1$. Let $O_3$ be a point on $AB$ that is the center of a semicircle $\omega_3$ which is tangent to both $\omega_1$ and $\omega_2$. Let $P$ be the intersection of the line through $O_3$ perpendicular to $AB$ and the line through $O_2$ parallel to $AB$. Show that $P$ is the center of a circle $\Gamma$ tangent to all of $\omega_1, \omega_2$ and $\omega_3$.
2005 Thailand Mathematical Olympiad
Let $a, b, c$ be distinct real numbers. Prove that

$$\left(\frac{2a - b}{a -b} \right)^2+\left(\frac{2b - c}{b - c}  \right)^2+\left(\frac{2c - a}{c - a} \right)^2 \ge 5$$
2004 Thailand Mathematical Olympiad
Given that $\cos 4A =\frac13$ and $-\frac{\pi}{4} \le A \le \frac{\pi}{4}$ , find the value of $\cos^8 A - \sin^8 A$.
2004 Thailand Mathematical Olympiad
Let a and b be real numbers such that $a^6 - 3a^2b^4 = 3$ and $b^6 - 3a^4b^2 = 3\sqrt2$. What is the value of $a^4 + b^4$ ?
2004 Thailand Mathematical Olympiad
Let $u, v, w$ be the roots of $x^3 -5x^2 + 4x-3 = 0$. Find a cubic polynomial having $u^3, v^3, w^3$ as roots.
2004 Thailand Mathematical Olympiad
Find all real solutions $x$ to the equation $$x =\sqrt{x -\frac{1}{x}} +\sqrt{1 -\frac{1}{x}}$$
2004 Thailand Mathematical Olympiad
Let $n$ be a given positive integer. Find the solution set of the equation $\sum_{k=1}^{2n} \sqrt{x^2 -2kx + k^2} =|2nx - n - 2n^2|$
2004 Thailand Mathematical Olympiad
Let $f(x) = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$. Find the remainder when $f(x^7)$ is divided by $f(x)$.
2004 Thailand Mathematical Olympiad
Let f be a function such that $f(0) = 0, f(1) = 1$, and $f(n) = 2f(n-1)- f(n- 2) + (-1)^n(2n - 4)$ for all integers $n \ge 2$. Find f(n) in terms of $n$.
2004 Thailand Mathematical Olympiad
Let $f : R \to R$ satisfy $f(x + f(y)) = 2x + 4y + 2547$ for all reals $x, y$. Compute $f(0)$.
2004 Thailand Mathematical Olympiad
Compute the sum $$\sum_{k=0}^{n}\frac{(2n)!}{k!^2(n-k)!^2}.$$
2004 Thailand Mathematical Olympiad
Find the number of ways to select three distinct numbers from ${1, 2, . . . , 3n}$ with a sum divisible by $3$.
2004 Thailand Mathematical Olympiad
Find the number of positive integer solutions to $(x_1 + x_2 + x_3)(y_1 + y_2 + y_3 + y_4) = 91$
2004 Thailand Mathematical Olympiad
Let $n$ be a positive integer and define $A_n = \{1, 2, ..., n\}$. How many functions $f : A_n \to A_n$ are there such that for all $x, y \in A_n$, if $x < y$ then $f(x) \ge f(y)$?
2004 Thailand Mathematical Olympiad
Compute the remainder when $29^{30 }+ 31^{28} + 28! \cdot 30!$ is divided by $29 \cdot 31$.
2004 Thailand Mathematical Olympiad
Compute gcd$(5^{2547} - 1, 5^{2004} - 1)$.
2004 Thailand Mathematical Olympiad
Find the largest positive integer $n \le 2004$ such that $3^{3n+3} - 27$ is divisible by $169$.
2004 Thailand Mathematical Olympiad
What are last three digits of $2^{2^{2004}}$ ?
2004 Thailand Mathematical Olympiad
Compute the remainder when $1^{2547} + 2^{2547} +...+ 2547^{2547}$ is divided by $2$.
2004 Thailand Mathematical Olympiad
Find positive reals $a, b, c$ which maximizes the value of $abc$ subject to the constraint that $b(a^2 + 2) + c(a + 2) = 12$.
2004 Thailand Mathematical Olympiad
Find positive reals $a, b, c$ which maximizes the value of $a+ 2b+ 3c$ subject to the constraint that $9a^2 + 4b^2 + c^2 = 91$
2004 Thailand Mathematical Olympiad
Two pillars of height $a$ and $b$ are erected perpendicular to the ground. On each pillar, a straight cable is placed connecting the top of the pillar to the base of the other pillar; the two lines of cable intersect at a point above ground. What is the height of this point?
2004 Thailand Mathematical Olympiad
The ratio between the circumradius and the inradius of a given triangle is $7 : 2$. If the length of two sides of the triangle are $3$ and $7$, and the length of the remaining side is also an integer, what is the length of the remaining side?
2004 Thailand Mathematical Olympiad
A $\vartriangle ABC$ is given with $\angle A = 70^o$. The angle bisectors of $\vartriangle ABC$ intersect at $I$. Suppose that  $CA + AI=BC$. Find, with proof, the value of $\angle B$.
2004 Thailand Mathematical Olympiad
Let $f : Q \to  Q$ be a function satisfying the equation $f(x + y) = f(x) + f(y) + 2547$ for all rational numbers $x, y$. If $f(2004) = 2547$, find $f(2547)$.
2004 Thailand Mathematical Olympiad
$18$ students with pairwise distinct heights line up. Ideally, the teacher wants the students to be ordered by height so that the tallest student is in the back of the line. However, it turns out that this is not the case, so when the teacher sees two consecutive students where the taller of the two is in front, the two students are swapped. It turns out that $150$ swaps must be made before the students are lined up in the correct order. How many possible starting orders are there?
2004 Thailand Mathematical Olympiad
Let $ABCD$ be a convex quadrilateral. Prove that area $(ABCD) \le \frac{AB^2 + BC^2 + CD^2 + DA^2}{4}$
2004 Thailand Mathematical Olympiad
Find all primes $p$ such that $p^2 + 2543$ has at most $16$ divisors.
2004 Thailand Mathematical Olympiad
Let $a, b, c > 0$ satisfy $a + b + c \ge \frac{1}{a} +\frac{1}{b} +\frac{1}{c}$. Prove that $a^3 + b^3 + c^3 \ge a + b + c$