Navvye/Olympiad-Problems-CSV huggingface dataset:
1995 Taiwan National Olympiad
Let $m_{1},m_{2},...,m_{n}$ be mutually distinct integers. Prove that there exists a $f(x)\in\mathbb{Z}[x]$ of degree $n$ satisfying the following two conditions:

a)$f(m_{i})=-1\forall i=1,2,...,n$.

b)$f(x)$ is irreducible.
1995 Taiwan National Olympiad
Let $P$ be a point on the circumcircle of a triangle $A_{1}A_{2}A_{3}$, and let $H$ be the orthocenter of the triangle. The feet $B_{1},B_{2},B_{3}$ of the perpendiculars from $P$ to $A_{2}A_{3},A_{3}A_{1},A_{1}A_{2}$ lie on a line. Prove that this line bisects the segment $PH$.
1995 Taiwan National Olympiad
Let $a,b,c,d$ are integers such that $(a,b)=(c,d)=1$ and $ad-bc=k>0$. Prove that there are exactly $k$ pairs $(x_{1},x_{2})$ of rational numbers with $0\leq x_{1},x_{2}<1$ for which both $ax_{1}+bx_{2},cx_{1}+dx_{2}$ are integers.
1994 Taiwan National Olympiad
Let $ABCD$ be a quadrilateral with $AD=BC$ and $\widehat{A}+\widehat{B}=120^{0}$. Let us draw equilateral $ACP,DCQ,DBR$ away from $AB$ . Prove that the points $P,Q,R$ are collinear.
1994 Taiwan National Olympiad
Let $a,b,c$ are positive real numbers and $\alpha$ be any real number. Denote $f(\alpha)=abc(a^{\alpha}+b^{\alpha}+c^{\alpha}), g(\alpha)=a^{2+\alpha}(b+c-a)+b^{2+\alpha}(-b+c+a)+c^{2+\alpha}(b-c+a)$. Determine $\min{|f(\alpha)-g(\alpha)|}$ and $\max{|f(\alpha)-g(\alpha)|}$, if they are exists.
1994 Taiwan National Olympiad
Let $a$ be a positive integer such that $5^{1994}-1\mid a$. Prove that the expression of $a$ in base $5$ contains at least $1994$ nonzero digits.
1994 Taiwan National Olympiad
Prove that there are infinitely many positive integers $n$ with the following property: For any $n$ integers $a_{1},a_{2},...,a_{n}$ which form in arithmetic progression, both the mean and the standard deviation of the set $\{a_{1},a_{2},...,a_{n}\}$ are integers.

Remark. The mean and standard deviation of the set $\{x_{1},x_{2},...,x_{n}\}$ are defined by $\overline{x}=\frac{x_{1}+x_{2}+...+x_{n}}{n}$ and $\sqrt{\frac{\sum (x_{i}-\overline{x})^{2}}{n}}$, respectively.
1994 Taiwan National Olympiad
Given $X=\{0,a,b,c\}$, let $M(X)=\{f|f: X\to X\}$ denote the set of all functions from $X$ into itself. An addition table on $X$ is given us follows:

$+$ $0$ $a$ $b$ $c$

$0$ $0$ $a$ $b$ $c$

$a$ $a$ $0$ $c$ $b$

$b$ $b$ $c$ $0$ $a$

$c$ $c$ $b$ $a$ $0$

a)If $S=\{f\in M(X)|f(x+y+x)=f(x)+f(y)+f(x)\forall x,y\in X\}$, find $|S|$.

b)If $I=\{f\in M(X)|f(x+x)=f(x)+f(x)\forall x\in X\}$, find $|I|$.
1994 Taiwan National Olympiad
For $-1\leq x\leq 1$ and $n\in\mathbb N$ define $T_{n}(x)=\frac{1}{2^{n}}[(x+\sqrt{1-x^{2}})^{n}+(x-\sqrt{1-x^{2}})^{n}]$.

a)Prove that $T_{n}$ is a monic polynomial of degree $n$ in $x$ and that the maximum value of $|T_{n}(x)|$ is $\frac{1}{2^{n-1}}$.

b)Suppose that $p(x)=x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0}\in\mathbb{R}[x]$ is a monic polynomial of degree $n$ such that $p(x)>-\frac{1}{2^{n-1}}$ forall $x$, $-1\leq x\leq 1$. Prove that there exists $x_{0}$, $-1\leq x_{0}\leq 1$ such that $p(x_{0})\geq\frac{1}{2^{n-1}}$.
1993 Taiwan National Olympiad
A sequence $(a_{n})$ of positive integers is given by $a_{n}=[n+\sqrt{n}+\frac{1}{2}]$. Find all of positive integers which belong to the sequence.
1993 Taiwan National Olympiad
Let $E$ and $F$ are distinct points on the diagonal $AC$ of a parallelogram $ABCD$ . Prove that , if there exists a cricle through $E,F$ tangent to rays $BA,BC$ then there also exists a cricle through $E,F$ tangent to rays $DA,DC$.
1993 Taiwan National Olympiad
Find all $ x,y,z\in\mathbb{N}_{0}$ such that $ 7^{x} + 1 = 3^{y} + 5^{z}$.



Alternative formulation: Solve the equation $ 1+7^{x}=3^{y}+5^{z}$ in nonnegative integers $ x$, $ y$, $ z$.
1993 Taiwan National Olympiad
In the Cartesian plane, let $C$ be a unit circle with center at origin $O$. For any point $Q$ in the plane distinct from $O$, define $Q'$ to be the intersection of the ray $OQ$ and the circle $C$. Prove that for any $P\in C$ and any $k\in\mathbb{N}$ there exists a lattice point $Q(x,y)$ with $|x|=k$ or $|y|=k$ such that $PQ'<\frac{1}{2k}$.
1993 Taiwan National Olympiad
Assume $A=\{a_{1},a_{2},...,a_{12}\}$ is a set of positive integers such that for each positive integer $n \leq 2500$  there is a subset $S$ of $A$ whose sum of elements is $n$. If $a_{1}<a_{2}<...<a_{12}$ , what is the smallest possible value of $a_{1}$?
1993 Taiwan National Olympiad
Let $m$ be equal to $1$ or $2$ and $n<10799$ be a positive integer. Determine all such $n$ for which $\sum_{k=1}^{n}\frac{1}{\sin{k}\sin{(k+1)}}=m\frac{\sin{n}}{\sin^{2}{1}}$.
1992 Taiwan National Olympiad
Let $A,B$ be two points on a give circle, and $M$ be the midpoint of one of the arcs $AB$ . Point $C$ is the orthogonal projection of $B$ onto the tangent $l$ to the circle at $A$. The tangent at $M$ to the circle meets $AC,BC$ at $A',B'$ respectively. Prove that if $\hat{BAC}<\frac{\pi}{8}$ then $S_{ABC}<2S_{A'B'C'}$.
1992 Taiwan National Olympiad
Every positive integer can be represented as a sum of one or more consecutive positive integers. For each $n$ , find the number of such  represententation of $n$.
1992 Taiwan National Olympiad
If $x_{1},x_{2},...,x_{n}(n>2)$ are positive real numbers with $x_{1}+x_{2}+...+x_{n}=1$. Prove that $x_{1}^{2}x_{2}+x_{2}^{2}x_{3}+...+x_{n}^{2}x_{1}\leq\frac{4}{27}$.
1992 Taiwan National Olympiad
For a positive integer number $r$, the sequence $a_{1},a_{2},...$ defined by $a_{1}=1$ and $a_{n+1}=\frac{na_{n}+2(n+1)^{2r}}{n+2}\forall n\geq 1$. Prove that each $a_{n}$ is positive integer number, and find $n's$ for which $a_{n}$ is even.
1992 Taiwan National Olympiad
A line through the incenter $I$ of triangle $ABC$, perpendicular to $AI$, intersects $AB$ at $P$ and $AC$ at $Q$. Prove that the circle tangent to $AB$ at $P$ and to $AC$ at $Q$ is also tangent to the circumcircle of triangle $ABC$.
1992 Taiwan National Olympiad
Find the greatest positive integer $A$ with the following property: For every permutation of $\{1001,1002,...,2000\}$ , the sum of some ten consecutive terms is great than or equal to $A$.
2021 Thailand Mathematical Olympiad
Let $\triangle ABC$ be an isosceles triangle such that $AB=AC$. Let $\omega$ be a circle centered at $A$ with a radius strictly less than $AB$. Draw a tangent from $B$ to $\omega$ at $P$, and draw a tangent from $C$ to $\omega$ at $Q$. Suppose that the line $PQ$ intersects the line $BC$ at point $M$. Prove that $M$ is the midpoint of $BC$.
2021 Thailand Mathematical Olympiad
Determine all sequences $a_1,a_2,a_3,\dots$ of positive integers that satisfy the equation

$$(n^2+1)a_{n+1}  - a_n = n^3+n^2+1$$for all positive integers $n$.
2021 Thailand Mathematical Olympiad
Let $a$, $b$, and $c$ be positive real numbers satisfying $ab+bc+ca=abc$. Determine the minimum value of

$$a^abc + b^bca + c^cab.$$
2021 Thailand Mathematical Olympiad
Kan Krao Park is a circular park that has $21$ entrances and a straight line walkway joining each pair of two entrances. No three walkways meet at a single point. Some walkways are paved with bricks, while others are paved with asphalt.



At each intersection of two walkways, excluding the entrances, is planted lotus if the two walkways are paved with the same material, and is planted waterlily if the two walkways are paved with different materials.



Each walkway is decorated with lights if and only if the same type of plant is placed at least $45$ different points along that walkway. Prove that there are at least $11$ walkways decorated with lights and paved with the same material.
2021 Thailand Mathematical Olympiad
Determine all triples $(p,m,k)$ of positive integers such that $p$ is a prime number, $m$ and $k$ are odd integers, and $m^4+4^kp^4$ divides $m^2(m^4-4^kp^4)$.
2021 Thailand Mathematical Olympiad
The cheering team of Ubon Ratchathani University sits on the amphitheater that has $441$ seats arranged into a $21\times 21$ grid. Every seat is occupied by exactly one person, and each person has a blue sign and a yellow sign.



Count the number of ways for each person to raise one sign so that each row and column has an odd number of people raising a blue sign.
2021 Thailand Mathematical Olympiad
Determine all functions $f : \mathbb R^+ \to \mathbb R$ that satisfy the equation

$$f(xy) = f(x)f(y)f(x+y)$$for all positive real numbers $x$ and $y$.
2021 Thailand Mathematical Olympiad
Let $P$ be a point inside an acute triangle $ABC$. Let the lines $BP$ and $CP$ intersect the sides $AC$ and $AB$ at $D$ and $E$, respectively. Let the circles with diameters $BD$ and $CE$ intersect at points $S$ and $T$. Prove that if the points $A$, $S$, and $T$ are colinear, then $P$ lies on a median of $\triangle ABC$.
2021 Thailand Mathematical Olympiad
Let $S$ be a set of positive integers such that if $a$ and $b$ are elements of $S$ such that $a<b$, then $b-a$ divides the least common multiple of $a$ and $b$, and the quotient is an element of $S$. Prove that the cardinality of $S$ is less than or equal to $2$.
2021 Thailand Mathematical Olympiad
Let $d\geq 13$ be an integer, and let $P(x) = a_dx^d + a_{d-1}x^{d-1} + \dots + a_1x+a_0$ be a polynomial of degree $d$ with complex coefficients such that $a_n = a_{d-n}$ for all $n\in\{0,1,\dots,d\}$. Prove that if $P$ has no double roots, then $P$ has two distinct roots $z_1$ and $z_2$ such that $|z_1-z_2|<1$.
2020 Thailand Mathematical Olympiad
Show that $\varphi(2n)\mid n!$ for all positive integer $n$.
2020 Thailand Mathematical Olympiad
There are $63$ houses at the distance of $1, 2, 3, . . . , 63 \text{ km}$ from the north pole, respectively. Santa Clause wants to distribute vaccine to each house. To do so, he will let his assistants, $63$ elfs named $E_1, E_2, . . . , E_{63}$ , deliever the vaccine to each house; each elf will deliever vaccine to exactly one house and never return. Suppose that the elf $E_n$  takes $n$ minutes to travel $1 \text{ km}$ for each $n = 1,2,...,63$ , and that all elfs leave the north pole simultaneously. What is the minimum amount of time to complete the delivery?
2020 Thailand Mathematical Olympiad
Suppose that $f : \mathbb{R}^+\to\mathbb R$ satisfies the equation

$$f(a+b+c+d) = f(a)+f(b)+f(c)+f(d)$$for all $a,b,c,d$ that are the four sides of some tangential quadrilateral. Show that $f(x+y)=f(x)+f(y)$ for all $x,y\in\mathbb{R}^+$.
2020 Thailand Mathematical Olympiad
Let $\triangle ABC$ be a triangle with altitudes $AD,BE,CF$. Let the lines $AD$ and $EF$ meet at $P$, let the tangent to the circumcircle of $\triangle ADC$ at $D$ meet the line $AB$ at $X$, and let the tangent to the circumcircle of $\triangle ADB$ at $D$ meet the line $AC$ at $Y$. Prove that the line $XY$ passes through the midpoint of $DP$.
2020 Thailand Mathematical Olympiad
You have an $n\times n$ grid and want to remove all edges of the grid by the sequence of the following moves. In each move, you can select a cell and remove exactly three edges surrounding that cell; in particular, that cell must have at least three remaining edges for the operation to be valid. For which positive integers $n$ is this possible?
2020 Thailand Mathematical Olympiad
Let the incircle of an acute triangle $\triangle ABC$ touches $BC,CA$, and $AB$ at points $D,E$, and $F$, respectively. Place point $K$ on the side $AB$ so that $DF$ bisects $\angle ADK$, and place point $L$ on the side $AB$ so that $EF$ bisects $\angle BEL$.



Prove that $\triangle ALE\sim\triangle AEB$.

Prove that $FK=FL$.
2020 Thailand Mathematical Olympiad
Determine all functions $f:\mathbb{R}\to\mathbb{Z}$ satisfying the inequality $(f(x))^2+(f(y))^2 \leq 2f(xy)$ for all reals $x,y$.
2020 Thailand Mathematical Olympiad
For all positive real numbers $a,b,c$ with $a+b+c=3$, prove the inequality

$$\frac{a^6}{c^2+2b^3} + \frac{b^6}{a^2+2c^3} + \frac{c^6}{b^2+2a^3} \geq 1.$$
2020 Thailand Mathematical Olympiad
Let $n,k$ be positive integers such that $n>k$. There is a square-shaped plot of land, which is divided into $n\times n$ grid so that each cell has the same size. The land needs to be plowed by $k$ tractors; each tractor will begin on the lower-left corner cell and keep moving to the cell sharing a common side until it reaches the upper-right corner cell. In addition, each tractor can only move in two directions: up and right. Determine the minimum possible number of unplowed cells.
2020 Thailand Mathematical Olympiad
Determine all polynomials $P(x)$ with integer coefficients which satisfies $P(n)\mid n!+2$ for all postive integer $n$.
2019 Thailand Mathematical Olympiad
Let $ABCDE$ be a convex pentagon with $\angle AEB=\angle BDC=90^o$ and line $AC$ bisects $\angle BAE$ and $\angle DCB$ internally. The circumcircle of $ABE$ intersects line $AC$ again at $P$.

(a) Show that $P$ is the circumcenter of $BDE$.

(b) Show that $A, C, D, E$ are concyclic.
2019 Thailand Mathematical Olympiad
Let $a,b$ be two different positive integers. Suppose that $a,b$ are relatively prime. Prove that $\dfrac{2a(a^2+b^2)}{a^2-b^2}$ is not an integer.
2019 Thailand Mathematical Olympiad
Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that $f(x+yf(x)+y^2) = f(x)+2y$ for every $x,y\in\mathbb{R}^+$.
2019 Thailand Mathematical Olympiad
A rabbit initially stands at the position $0$, and repeatedly jumps on the real line. In each jump, the rabbit can jump to any position corresponds to an integer but it cannot stand still. Let $N(a)$ be the number of ways to jump with a total distance of $2019$ and stop at the position $a$. Determine all integers $a$ such that $N(a)$ is odd.
2019 Thailand Mathematical Olympiad
Let $a,b,c$ be positive reals such that $abc=1$. Prove the inequality

$$\frac{4a-1}{(2b+1)^2} + \frac{4b-1}{(2c+1)^2} + \frac{4c-1}{(2a+1)^2}\geqslant 1.$$
2019 Thailand Mathematical Olympiad
Determine all function $f:\mathbb{R}\to\mathbb{R}$ such that $xf(y)+yf(x)\leqslant xy$ for all $x,y\in\mathbb{R}$.
2019 Thailand Mathematical Olympiad
Let $A=\{-2562,-2561,...,2561,2562\}$. Prove that for any bijection (1-1, onto function) $f:A\to A$,

$$\sum_{k=1}^{2562}\left\lvert f(k)-f(-k)\right\rvert\text{ is maximized if and only if } f(k)f(-k)<0\text{ for any } k=1,2,...,2562.$$
2019 Thailand Mathematical Olympiad
Let $ABC$ be a triangle such that $AB\ne AC$ and $\omega$ be  the circumcircle of this triangle.

Let $I$ be the center of the inscribed circle of $ABC$ which touches $BC$ at $D$.

Let the circle with diameter $AI$ meets $\omega$ again at $K$.

If the line $AI$ intersects $\omega$ again at $M$, show that $K, D, M$ are collinear.
2019 Thailand Mathematical Olympiad
A chaisri figure is a triangle which the three vertices are vertices of a regular $2019$-gon. Two different chaisri figure may be formed by different regular $2019$-gon.

A thubkaew figure is a convex polygon which can be dissected into multiple chaisri figure where each vertex of a dissected chaisri figure does not necessarily lie on the border of the convex polygon.



Determine the maximum number of vertices that a thubkaew figure may have.
2019 Thailand Mathematical Olympiad
Prove that there are infinitely many positive odd integer $n$ such that $n!+1$ is composite number.
2018 Thailand Mathematical Olympiad
In $\vartriangle ABC$, the incircle is tangent to the sides $BC, CA, AB$ at $D, E, F$ respectively. Let $P$ and $Q$ be the midpoints of $DF$ and $DE$ respectively. Lines $P C$ and $DE$ intersect at $R$, and lines $BQ$ and$ DF$ intersect at $S$. Prove that

a) Points $B, C, P, Q$ lie on a circle.

b) Points $P, Q, R, S$ lie on a circle.
2018 Thailand Mathematical Olympiad
Show that there are no functions $f : R \to  R$ satisfying $f(x + f(y)) = f(x) + y^2$ for all real numbers $x$ and $y$
2018 Thailand Mathematical Olympiad
Karakade has three flash drives of each of the six capacities $1, 2, 4, 8, 16, 32$ gigabytes. She gives each of her $6$ servants three flash drives of different capacities. Prove that either there are two capacities where each servant has at most one of the two capacities, or all servants have flash drives with different sums of capacities.
2018 Thailand Mathematical Olympiad
Let $a, b, c$ be nonzero real numbers such that $a + b + c = 0$. Determine the maximum  possible value of

$\frac{a^2b^2c^2}{ (a^2 + ab + b^2)(b^2 + bc + c^2)(c^2 + ca + a^2)}$

.
2018 Thailand Mathematical Olympiad
Let a, b be positive integers such that $5 \nmid a, b$ and $5^5 \mid a^5+b^5$. What is the minimum possible value of $a + b$?
2018 Thailand Mathematical Olympiad
Let $A$ be the set of all triples $(x, y, z)$ of positive integers satisfying $2x^2 + 3y^3 = 4z^4$ .

a) Show that if $(x, y, z) \in A$ then $6$ divides all of $x, y, z$.

b) Show that $A$ is an infinite set.
2018 Thailand Mathematical Olympiad
We color each number in the set $S = \{1, 2, ..., 61\}$ with one of $25$ given colors, where it is not necessary that every color gets used. Let $m$ be the number of non-empty subsets of $S$ such that every number in the subset has the same color. What is the minimum possible value of $m$?
2018 Thailand Mathematical Olympiad
There are $2n + 1$ tickets, each with a unique positive integer as the ticket number. It is known that the sum of all ticket numbers is more than $2330$, but the sum of any $n$ ticket numbers is at most $1165$. What is the maximum value of $n$?
2018 Thailand Mathematical Olympiad
In  $\vartriangle ABC$ the incircle is tangent to $AB$ at $D$. Let $P$ be a point on $BC$ different from $B$ and $C$, and let $K$ and $L$ be incenters of  $\vartriangle ABP$ and  $\vartriangle ACP$ respectively. Suppose that the circumcircle of $\vartriangle KP L$ cuts $AP$ again at $Q$. Prove that $AD = AQ$.
2018 Thailand Mathematical Olympiad
Let $a,b,c$ be non-zero real numbers.Prove that if function $f,g:\mathbb{R}\to\mathbb{R}$ satisfy $af(x+y)+bf(x-y)=cf(x)+g(y)$ for all real number $x,y$ that $y>2018$ then there exists a function $h:\mathbb{R}\to\mathbb{R}$ such that $f(x+y)+f(x-y)=2f(x)+h(y)$ for all real number $x,y$.
2017 Thailand Mathematical Olympiad
Let $p$ be a prime. Show that $\sqrt[3]{p} +\sqrt[3]{p^5} $  is irrational.
2017 Thailand Mathematical Olympiad
A cyclic quadrilateral $ABCD$ has circumcenter $O$, its diagonals $AC$ and $BD$ intersect at $G$. Let $P, Q, R, S$ be the circumcenters of $\vartriangle AGB, \vartriangle BGC, \vartriangle CGD, \vartriangle DGA$ respectively. Lines $P R$ and $QS$ intersect at $M$. Show that $M$ is the midpoint of $OG$.
2017 Thailand Mathematical Olympiad
Determine all functions $f : R \to  R$ satisfying $f(f(x) - y)  \le  xf(x) + f(y)$ for all real numbers $x, y$.
2017 Thailand Mathematical Olympiad
In a math competition, $14$ schools participate, each sending $14$ students. The students are separated into $14$ groups of $14$ so that no two students from the same school are in the same group. The tournament organizers noted that, from the competitors, exactly $15$ have participated in the competition before. The organizers want to select two representatives, with the conditions that they must be former participants, must come from different schools, and must also be in different groups. It turns out that there are $ n$ ways to do this. What is the minimum possible value of $n$?
2017 Thailand Mathematical Olympiad
Does there exist $2017$ consecutive positive integers, none of which could be written as $a^2 + b^2$ for some integers $a, b$? Justify your answer.
2017 Thailand Mathematical Olympiad
In an acute triangle $\vartriangle ABC$, $D$ is the foot of altitude from $A$ to $BC$. Suppose that $AD = CD$, and define $N$ as the intersection of the median $CM$ and the line $AD$. Prove that $\vartriangle ABC$ is isosceles if and only if $CN = 2AM$.
2017 Thailand Mathematical Olympiad
Show that no pairs of integers $(m, n)$ satisfy $2560m^2 + 5m + 6 = n^5$.

.
2017 Thailand Mathematical Olympiad
Let $a, b, c$ be side lengths of a right triangle. Determine the minimum possible value of $\frac{a^3 + b^3 + c^3}{abc}$.
2017 Thailand Mathematical Olympiad
Determine all functions $f$ on the set of positive rational numbers such that $f(xf(x) + f(y)) = f(x)^2 + y$ for all positive rational numbers $x, y$.
2017 Thailand Mathematical Olympiad
A lattice point is defined as a point on the plane with integer coordinates. Show that for all positive integers $n$, there is a circle on the plane with exactly n lattice points in its interior (not including its boundary).
2016 Thailand Mathematical Olympiad
Let $ABC$ be a triangle with $AB \ne AC$. Let the angle bisector of $\angle BAC$ intersects $BC$ at $P$ and intersects the perpendicular bisector of segment $BC$ at $Q$. Prove that $\frac{PQ}{AQ} =\left( \frac{BC}{AB + AC}\right)^2$
2016 Thailand Mathematical Olympiad
Let $M$ be a positive integer, and $A = \{1, 2,... , M + 1\}$. Show that if $f$ is a bijection from $A$ to $A$ then

$\sum_{n=1}^{M} \frac{1}{f(n) + f(n + 1)} > \frac{M}{M + 3}$
2016 Thailand Mathematical Olympiad
Determine all functions $f : R \to R$ satisfying $f (f(x)f(y) + f(y)f(z) + f(z)f(x))= f(x) + f(y) + f(z)$ for all real numbers $x, y, z$.
2016 Thailand Mathematical Olympiad
Each point on the plane is colored either red, green, or blue. Prove that there exists an isosceles triangle whose vertices all have the same color.
2016 Thailand Mathematical Olympiad
given $p_1,p_2,...$ be a sequence of integer and $p_1=2$,

for positive integer $n$, $p_{n+1}$ is the least prime factor of     $np_1^{1!}p_2^{2!}...p_n^{n!}+1 $

prove that all primes appear in the sequence



(Proposed by Beatmania)
2016 Thailand Mathematical Olympiad
Let $m$ and $n$ be positive integers. Prove that if $m^{4^n+1} - 1$ is a prime number, then there exists an integer $t \ge 0$ such that $n = 2^t$.
2016 Thailand Mathematical Olympiad
Given $P(x)=a_{2016}x^{2016}+a_{2015}x^{2015}+...+a_1x+a_0$

be a polynomial with real coefficients and $a_{2016} \neq 0$

satisfies



$|a_1+a_3+...+a_{2015}| > |a_0+a_2+...+a_{2016}|$



Prove that $P(x)$ has an odd number of complex roots with absolute value less than $1$ (count multiple roots also)



edited: complex roots
2016 Thailand Mathematical Olympiad
Let $\vartriangle ABC$ be an acute triangle with incenter $I$. The line passing through $I$ parallel to $AC$ intersects $AB$ at $M$, and the line passing through $I$ parallel to $AB$ intersects $AC$ at $N$. Let the line $MN$ intersect the circumcircle of $\vartriangle ABC$ at $X$ and $Y$ . Let $Z$ be the midpoint of arc $BC$ (not containing $A$). Prove that $I$ is the orthocenter of $\vartriangle XY Z$
2016 Thailand Mathematical Olympiad
A real number $a \ne 0$ is given. Determine all functions $f : R \to R$ satisfying $f(x)f(y) + f(x + y) = axy$ for all real numbers $x, y$.
2016 Thailand Mathematical Olympiad
A Pattano coin is a coin which has a blue side and a yellow side. A positive integer not exceeding $100$ is written on each side of every coin (the sides may have different integers).

Two Pattano coins are identical  if the number on the blue side of both coins are equal and the number on the yellow side of both coins are equal.

Two Pattano coins are pairable  if the number on the blue side of both coins are equal or the number on the yellow side of both coins are equal.

Given $2559$ Pattano coins such that no two coins are identical. Show that at least one Pattano coin is pairable with at least $50$ other coins
2015 Thailand Mathematical Olympiad
Let $p$ be a prime, and let $a_1, a_2, a_3, . . .$ be a sequence of positive integers so that $a_na_{n+2} = a^2_{n+1} + p$ for all positive integers $n$. Show that $a_{n+1}$ divides $a_n + a_{n+2}$ for all positive integers $n$.
2015 Thailand Mathematical Olympiad
Let $a, b, c$ be positive reals with $abc = 1$. Prove the inequality

$$\frac{a^5}{a^3 + 1}+\frac{b^5}{b^3 + 1}+\frac{c^5}{c^3 + 1} \ge \frac32$$and determine all values of a, b, c for which equality is attained
2015 Thailand Mathematical Olympiad
Let $P = \{(x, y) | x, y \in \{0, 1, 2,... , 2015\}\}$ be a set of points on the plane. Straight wires of unit length are placed to connect points in $P$ so that each piece of wire connects exactly two points in $P$, and each point in $P$ is an endpoint of exactly one wire.

Prove that no matter how the wires are placed, it is always possible to draw a straight line parallel to either the horizontal or vertical axis passing through midpoints of at least $506$ pieces of wire.