Navvye/Olympiad-Problems-CSV huggingface dataset:
1999 Korea - Final Round
We are given two triangles. Prove, that if $\angle{C}=\angle{C'}$ and $\frac{R}{r}=\frac{R'}{r'}$, then they are similar.
1999 Korea - Final Round
Suppose $f(x)$ is a function satisfying $\left | f(m+n)-f(m)  \right | \leq \frac{n}{m}$ for all positive integers $m$,$n$. Show that for all positive integers $k$:

$$\sum_{i=1}^{k}\left |f(2^k)-f(2^i)  \right |\leq \frac{k(k-1)}{2}$$.
1999 Korea - Final Round
Find all intengers n such that $2^n - 1$ is a multiple of 3 and $(2^n - 1)/3$ is a divisor of $4m^2 + 1$ for some intenger m.
1999 Korea - Final Round
If the equation:



$f(\frac{x-3}{x+1}) + f(\frac{3+x}{1-x}) = x$



holds true for all real x but $\pm 1$, find $f(x)$.
1999 Korea - Final Round
A permutation $a_1,a_2,\cdots ,a_6$ of numbers $1,2,\cdots ,6$ can be transformed to $1,2,\cdots,6$ by transposing two numbers exactly four times. Find the number of such permutations.
1999 Korea - Final Round
Let $a_1, a_2, ..., a_{1999}$ be nonnegative real numbers satisfying the following conditions:

a. $a_1+a_2+...+a_{1999}=2$

b. $a_1a_2+a_2a_3+...+a_{1999}a_1=1$.

Let $S=a_1^ 2+a_2 ^ 2+...+a_{1999}^2$. Find the maximum and minimum values of $S$.
1998 Korea - Final Round
Find all pairwise relatively prime positive integers $l, m, n$ such that $$(l+m+n)\left( \frac{1}{l}+\frac{1}{m}+\frac{1}{n}\right)$$ is an integer.
1998 Korea - Final Round
Let $D$,$E$,$F$ be points on the sides $BC$,$CA$,$AB$ respectively of a triangle $ABC$. Lines $AD$,$BE$,$CF$ intersect the circumcircle of $ABC$ again at $P$,$Q$,$R$, respectively.Show that:

$$\frac{AD}{PD}+\frac{BE}{QE}+\frac{CF}{RF}\geq 9$$

and find the cases of equality.
1998 Korea - Final Round
Denote by $\phi(n)$ for all $n\in\mathbb{N}$ the number of positive integer smaller than $n$ and relatively prime to $n$. Also, denote by $\omega(n)$ for all $n\in\mathbb{N}$ the number of prime divisors of $n$. Given that $\phi(n)|n-1$ and $\omega(n)\leq 3$. Prove that $n$ is a prime number.
1998 Korea - Final Round
Let $ x,y,z$ be positive real numbers satisfying $ x+y+z=xyz$. Prove that:

$$\frac1{\sqrt{1+x^2}}+\frac1{\sqrt{1+y^2}}+\frac1{\sqrt{1+z^2}}\leq\frac{3}{2}$$
1998 Korea - Final Round
Let $I$ be the incenter of triangle $ABC$, $O_1$ a circle through $B$ tangent to $CI$, and $O_2$ a circle through $C$ tangent to $BI$. Prove that $O_1$,$O_2$ and the circumcircle of $ABC$ have a common point.
1998 Korea - Final Round
Let $F_n$ be the set of all bijective functions $f:\left\{1,2,\ldots,n\right\}\rightarrow\left\{1,2,\ldots,n\right\}$ that satisfy the conditions:



1. $f(k)\leq k+1$ for all $k\in\left\{1,2,\ldots,n\right\}$

2. $f(k)\neq k$ for all $k\in\left\{2,3,\ldots,n\right\}$



Find the probability that $f(1)\neq1$ for $f$ randomly chosen from $F_n$.
1997 Korea - Final Round
A word is a sequence of 0 and 1 of length 8. Let $ x$ and $ y$ be two words differing in exactly three places.

Prove that the number of words differing from each of $ x$ and $ y$ in at least five places is 188.
1997 Korea - Final Round
The incircle of a triangle $ A_1A_2A_3$ is centered at $ O$ and meets the segment $ OA_j$ at $ B_j$ , $ j = 1, 2, 3$. A circle with center $ B_j$ is tangent to the two sides of the triangle having $ A_j$ as an endpoint and intersects the segment $ OB_j$ at $ C_j$. Prove that

$$ \frac{OC_1+OC_2+OC_3}{A_1A_2+A_2A_3+A_3A_1} \leq \frac{1}{4\sqrt{3}}$$

and find the conditions for equality.
1997 Korea - Final Round
Find all pairs of functions $ f, g: \mathbb R \to \mathbb R$ such that



(i) if

$ x < y$, then $ f(x) < f(y)$;



(ii) $ f(xy) = g(y)f(x) + f(y)$ for all $ x, y \in \mathbb R$.
1997 Korea - Final Round
Given a positive integer $ n$, find the number of $ n$-digit natural numbers consisting of digits 1, 2, 3 in which any two adjacent digits are either distinct or both equal to 3.
1997 Korea - Final Round
For positive numbers $ a_1,a_2,\dots,a_n$, we define

$$ A=\frac{a_1+a_2+\cdots+a_n}{n}, \quad G=\sqrt[n]{a_1\cdots a_n}, \quad H=\frac{n}{a_1^{-1}+\cdots+a_n^{-1}}$$

Prove that



(i) $ \frac{A}{H}\leq -1+2\left(\frac{A}{G}\right)^n$, for n even



(ii) $ \frac{A}{H}\leq -\frac{n-2}{n}+\frac{2(n-1)}{n}\left(\frac{A}{G}\right)^n$, for $ n$ odd
1997 Korea - Final Round
Let $ p_1,p_2,\dots,p_r$ be distinct primes, and let $ n_1,n_2,\dots,n_r$ be arbitrary natural numbers. Prove that the number of pairs of integers $ (x, y)$ such that

$$ x^3 + y^3 = p_1^{n_1}p_2^{n_2}\cdots p_r^{n_r}$$

does not exceed $ 2^{r+1}$.
1993 Korea - Final Round
Consider a $9 \times 9$ array of white squares. Find the largest $n \in\mathbb N$ with the property: No matter how one chooses $n$ out of 81 white squares and color in black, there always remains a $1 \times 4$ array of white squares (either vertical or horizontal).
1993 Korea - Final Round
Let be given a triangle $ABC$ with $BC = a, CA = b, AB = c$. Find point $P$ in the plane for which $aAP^{2}+bBP^{2}+cCP^{2}$ is minimum, and compute this minimum.
1993 Korea - Final Round
Find the smallest $x \in\mathbb{N}$ for which $\frac{7x^{25}-10}{83}$ is an integer.
1993 Korea - Final Round
An integer which is the area of a right-angled triangle with integer sides is called Pythagorean. Prove that for every positive integer $n > 12$ there exists a Pythagorean number between $n$ and $2n.$
1993 Korea - Final Round
Given $n \in\mathbb{N}$, ﬁnd all continuous functions $f : \mathbb{R}\to \mathbb{R}$ such that for all $x\in\mathbb{R},$

$$\sum_{k=0}^{n}\binom{n}{k}f(x^{2^{k}})=0. $$
1993 Korea - Final Round
Consider a triangle $ABC$ with $BC = a, CA = b, AB = c.$ Let $D$ be the midpoint of $BC$ and $E$ be the intersection of the bisector of $A$ with $BC$ . The circle through $A, D, E$ meets $AC, AB$ again at $F, G$ respectively. Let $H\not = B$ be a point on $AB$ with $BG = GH$ . Prove that triangles $EBH$ and $ABC$ are similar and ﬁnd the ratio of their areas.
2021 Korea National Olympiad
Let $ABC$ be an acute triangle and $D$ be an intersection of the angle bisector of $A$ and side $BC$. Let $\Omega$ be a circle tangent to the circumcircle of triangle $ABC$ and side $BC$ at $A$ and $D$, respectively. $\Omega$ meets the sides $AB, AC$ again at $E, F$, respectively. The perpendicular line to $AD$, passing through $E, F$ meets $\Omega$ again at $G, H$, respectively. Suppose that $AE$ and $GD$ meet at $P$, $EH$ and $GF$ meet at $Q$, and $HD$ and $AF$ meet at $R$. Prove that $\dfrac{\overline{QF}}{\overline{QG}}=\dfrac{\overline{HR}}{\overline{PG}}$.
2021 Korea National Olympiad
For positive integers $n, k, r$, denote by $A(n, k, r)$ the number of integer tuples $(x_1, x_2, \ldots, x_k)$ satisfying the following conditions.



$x_1 \ge x_2 \ge \cdots \ge x_k \ge 0$

$x_1+x_2+ \cdots +x_k = n$

$x_1-x_k \le r$



For all positive integers $m, s, t$, prove that $$A(m, s, t)=A(m, t, s).$$
2021 Korea National Olympiad
Show that for any positive integers $k$ and $1 \leq a \leq 9$, there exists $n$ such that satisfies the below statement.



When $2^n=a_0+10a_1+10^2a_2+ \cdots +10^ia_i+ \cdots $ $(0 \leq a_i \leq 9$ and $a_i$ is integer), $a_k$ is equal to $a$.
2021 Korea National Olympiad
For a positive integer $n$, there are two countries $A$ and $B$ with $n$ airports each and $n^2-2n+ 2$ airlines operating between the two countries. Each airline operates at least one flight. Exactly one flight by one of the airlines operates between each airport in $A$ and each airport in $B$, and that flight operates in both directions. Also, there are no flights between two airports in the same country. For two different airports $P$ and $Q$, denote by "$(P, Q)$-travel route" the list of airports $T_0, T_1, \ldots, T_s$ satisfying the following conditions.



$T_0=P,\ T_s=Q$

$T_0, T_1, \ldots, T_s$ are all distinct.

There exists an airline that operates between the airports $T_i$ and $T_{i+1}$ for all $i = 0, 1, \ldots, s-1$.



Prove that there exist two airports $P, Q$ such that there is no or exactly one $(P, Q)$-travel route.



Graph WordingConsider a complete bipartite graph $G(A, B)$ with $\vert A \vert = \vert B \vert = n$. Suppose there are $n^2-2n+2$ colors and each edge is colored by one of these colors. Define $(P, Q)-path$ a path from $P$ to $Q$ such that all of the edges in the path are colored the same. Prove that there exist two vertices $P$ and $Q$ such that there is no or only one $(P, Q)-path$.
2021 Korea National Olympiad
A real number sequence $a_1, \cdots ,a_{2021}$ satisfies the below conditions.

$$a_1=1, a_2=2, a_{n+2}=\frac{2a_{n+1}^2}{a_n+a_{n+1}} (1\leq n \leq 2019)$$Let the minimum of $a_1, \cdots ,a_{2021}$ be $m$, and the maximum of $a_1, \cdots ,a_{2021}$ be $M$.

Let a 2021 degree polynomial  $$P(x):=(x-a_1)(x-a_2) \cdots (x-a_{2021})$$$|P(x)|$ is maximum in $[m, M]$ when $x=\alpha$. Show that $1<\alpha <2$.
2021 Korea National Olympiad
Let $ABC$ be an obtuse triangle with $\angle A > \angle B > \angle C$, and let $M$ be a midpoint of the side $BC$. Let $D$ be a point on the arc $AB$ of the circumcircle of triangle $ABC$ not containing $C$. Suppose that the circle tangent to $BD$ at $D$ and passing through $A$ meets the circumcircle of triangle $ABM$ again at $E$ and $\overline{BD}=\overline{BE}$. $\omega$, the circumcircle of triangle $ADE$, meets $EM$ again at $F$.



Prove that lines $BD$ and $AE$ meet on the line tangent to $\omega$ at $F$.
2020 Korea National Olympiad
Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that

$$x^2f(x)+yf(y^2)=f(x+y)f(x^2-xy+y^2)$$for all $x,y\in\mathbb{R}$.
2020 Korea National Olympiad
$H$ is the orthocenter of an acute triangle $ABC$, and let $M$ be the midpoint of $BC$. Suppose $(AH)$ meets $AB$ and $AC$ at $D,E$ respectively. $AH$ meets $DE$ at $P$, and the line through $H$ perpendicular to $AH$ meets $DM$ at $Q$. Prove that $P,Q,B$ are collinear.
2020 Korea National Olympiad
There are n boys and m girls at Daehan Mathematical High School.

Let $d(B)$ a number of girls who know Boy $B$ each other, and let $d(G)$ a number of boys who know Girl $G$ each other.

Each girl knows at least one boy each other.

Prove that there exist Boy $B$ and Girl $G$ who knows each other in condition that $\frac{d(B)}{d(G)}\ge\frac{m}{n}$.
2020 Korea National Olympiad
Find a pair of coprime positive integers $(m,n)$ other than $(41,12)$ such that $m^2-5n^2$ and $m^2+5n^2$ are both perfect squares.
2020 Korea National Olympiad
For some positive integer $n$, there exists $n$ different positive integers $a_1, a_2, ..., a_n$ such that

$(1)$ $a_1=1, a_n=2000$

$(2)$ $\forall i\in \mathbb{Z}$ $s.t.$ $2\le i\le n, a_i -a_{i-1}\in \{-3,5\}$

Determine the maximum value of n.
2020 Korea National Olympiad
Let $ABCDE$ be a convex pentagon such that quadrilateral $ABDE$ is a parallelogram and quadrilateral $BCDE$ is inscribed in a circle. The circle with center $C$ and radius $CD$ intersects the line $BD, DE$ at points $F, G(\neq D)$, and points $A, F, G$ is on line l. Let $H$ be the intersection point of line $l$ and segment $BC$.

Consider the set of circle $\Omega$ satisfying the following condition.



Circle $\Omega$ passes through $A, H$ and intersects the sides $AB, AE$ at point other than $A$.



Let $P, Q(\neq A)$ be the intersection point of circle $\Omega$ and sides $AB, AE$.

Prove that $AP+AQ$ is constant.
2019 Korea National Olympiad
The sequence ${a_1, a_2, ..., a_{2019}}$ satisfies the following condition.

$a_1=1, a_{n+1}=2019a_{n}+1$

Now let $x_1, x_2, ..., x_{2019}$ real numbers such that $x_1=a_{2019}, x_{2019}=a_1$ (The others are arbitary.)

Prove that $\sum_{k=1}^{2018} (x_{k+1}-2019x_k-1)^2 \ge \sum_{k=1}^{2018} (a_{2019-k}-2019a_{2020-k}-1)^2$
2019 Korea National Olympiad
Triangle $ABC$ is an acute triangle with distinct sides. Let $I$ the incenter, $\Omega$ the circumcircle, $E$ the $A$-excenter of triangle $ABC$. Let $\Gamma$ the circle centered at $E$ and passes $A$. $\Gamma$ and $\Omega$ intersect at point $D(\neq A)$, and the perpendicular line of $BC$ which passes $A$ meets $\Gamma$ at point $K(\neq A)$. $L$ is the perpendicular foot from $I$ to $AC$. Now if $AE$ and $DK$ intersects at $F$, prove that $BE\cdot CI=2\cdot CF\cdot CL$.
2019 Korea National Olympiad
Suppose that positive integers $m,n,k$ satisfy the equations $$m^2+1=2n^2, 2m^2+1=11k^2.$$Find the residue when $n$ is divided by $17$.
2019 Korea National Olympiad
Let $(x_1, y_1, z_1), (x_2, y_2, z_2), \cdots, (x_{19}, y_{19}, z_{19})$ be integers. Prove that there exist pairwise distinct subscripts $i, j, k$ such that $x_i+x_j+x_k$, $y_i+y_j+y_k$, $z_i+z_j+z_k$ are all multiples of $3$.
2019 Korea National Olympiad
Find all functions $f$ such that $f:\mathbb{R}\rightarrow \mathbb{R}$ and $f(f(x)-x+y^2)=yf(y)$
2019 Korea National Olympiad
In acute triangle $ABC$, $AB>AC$. Let $I$ the incenter, $\Omega$ the circumcircle of triangle $ABC$, and $D$ the foot of perpendicular from $A$ to $BC$. $AI$ intersects $\Omega$ at point $M(\neq A)$, and the line which passes $M$ and perpendicular to $AM$ intersects $AD$ at point $E$. Now let $F$ the foot of perpendicular from $I$ to $AD$.

Prove that $ID\cdot AM=IE\cdot AF$.
2019 Korea National Olympiad
For prime $p\equiv 1\pmod{7} $, prove that there exists some positive integer $m$ such that $m^3+m^2-2m-1$ is a multiple of $p$.
2019 Korea National Olympiad
There are two countries $A$ and $B$, where each countries have $n(\ge 2)$ airports. There are some two-way flights among airports of $A$ and $B$, so that each airport has exactly $3$ flights. There might be multiple flights among two airports; and there are no flights among airports of the same country. A travel agency wants to plan an exotic traveling course which travels through all $2n$ airports exactly once, and returns to the initial airport. If $N$ denotes the number of all exotic traveling courses, then prove that $\frac{N}{4n}$ is an even integer.



(Here, note that two exotic traveling courses are different if their starting place are different.)
2018 Korea National Olympiad
Let there be an acute triangle $\triangle ABC$ with incenter $I$. $E$ is the foot of the perpendicular from $I$ to $AC$. The line which passes through $A$ and is perpendicular to $BI$ hits line $CI$ at $K$. The line which passes through $A$ and is perpendicular to $CI$ hits the line which passes through $C$ and is perpendicular to $BI$ at $L$. Prove that $E, K, L$ are colinear.
2018 Korea National Olympiad
For a positive integer $n$, denote $p(n)$ to be the number of nonnegative integer tuples $(x,y,z,w)$ such that $x+y+2z+3w=n-1$. Also, denote $q(n)$ to be the number of nonnegative integer tuples $(a,b,c,d)$ such that



(i). $a+b+c+d=n$.

(ii). $a \ge b$, $c \ge d$, $a \ge d$.

(iii). $b < c$.



Prove that for all $n$, $p(n) = q(n)$.
2018 Korea National Olympiad
Denote $f(x) = x^4 + 2x^3 - 2x^2 - 4x+4$. Prove that there are infinitely many primes $p$ that satisfies the following.



For all positive integers $m$, $f(m)$ is not a multiple of $p$.
2018 Korea National Olympiad
Find all real values of $K$ which satisfies the following.



Let there be a sequence of real numbers $\{a_n\}$ which satisfies the following for all positive integers $n$.



(i). $0 < a_n < n^K$.

(ii). $a_1 + a_2 + \cdots + a_n < \sqrt{n}$.



Then, there exists a positive integer $N$ such that for all integers $n>N$, $$a^{2018}_1 + a^{2018}_2 + \cdots +a^{2018}_n < \frac{n}{2018}$$
2018 Korea National Olympiad
Let there be a convex quadrilateral $ABCD$. The angle bisector of $\angle A$ meets the angle bisector of $\angle B$, the angle bisector of $\angle D$ at $P, Q$ respectively. The angle bisector of $\angle C$ meets the angle bisector of $\angle D$, the angle bisector of $\angle B$ at $R, S$ respectively. $P, Q, R, S$ are all distinct points. $PR$ and $QS$ meets perpendicularly at point $Z$. Denote $l_A, l_B, l_C, l_D$ as the exterior angle bisectors of $\angle A, \angle B, \angle C, \angle D$. Denote $E = l_A \cap l_B$, $F= l_B \cap l_C$, $G = l_C \cap l_D$, and $H= l_D \cap l_A$. Let $K, L, M, N$ be the midpoints of $FG, GH, HE, EF$ respectively.



Prove that the area of quadrilateral $KLMN$ is equal to $ZM \cdot ZK + ZL \cdot ZN$.
2018 Korea National Olympiad
Let $n \ge 3$ be a positive integer. For every set $S$ with $n$ distinct positive integers, prove that there exists a bijection $f: \{1,2, \cdots n\} \rightarrow S$ which satisfies the following condition.



For all $1 \le i < j < k \le n$, $f(j)^2 \neq f(i) \cdot f(k)$.
2018 Korea National Olympiad
Let there be a figure with $9$ disks and $11$ edges, as shown below.



We will write a real number in each and every disk. Then, for each edge, we will write the square of the difference between the two real numbers written in the two disks that the edge connects. We must write $0$ in disk $A$, and $1$ in disk $I$. Find the minimum sum of all real numbers written in $11$ edges.
2018 Korea National Olympiad
Let there be positive integers $a, c$. Positive integer $b$ is a divisor of $ac-1$. For a positive rational number $r$ which is less than $1$, define the set $A(r)$ as follows.

$$A(r) = \{m(r-ac)+nab| m, n \in \mathbb{Z} \}$$Find all rational numbers $r$ which makes the minimum positive rational number in $A(r)$ greater than or equal to $\frac{ab}{a+b}$.
2017 Korea National Olympiad
Denote $U$ as the set of $20$ diagonals of the regular polygon $P_1P_2P_3P_4P_5P_6P_7P_8$.

Find the number of sets $S$ which satisfies the following conditions.



1. $S$ is a subset of $U$.



2. If $P_iP_j \in S$ and $P_j P_k \in S$, and $i \neq k$, $P_iP_k \in S$.
2017 Korea National Olympiad
Find all primes $p$ such that there exist an integer $n$ and positive integers $k, m$ which satisfies the following.



$$ \frac{(mk^2+2)p-(m^2+2k^2)}{mp+2} = n^2$$
2017 Korea National Olympiad
Let there be a scalene triangle $ABC$, and its incircle hits $BC, CA, AB$ at $D, E, F$. The perpendicular bisector of $BC$ meets the circumcircle of $ABC$ at $P, Q$, where $P$ is on the same side with $A$ with respect to $BC$. Let the line parallel to $AQ$ and passing through $D$ meet $EF$ at $R$. Prove that the intersection between $EF$ and $PQ$ lies on the circumcircle of $BCR$.
2017 Korea National Olympiad
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be the function as

$$ f(x) = \begin{cases} \frac{1}{x-1}& (x > 1)\\ 1& (x=1)\\ \frac{x}{1-x} & (x<1) \end{cases} $$Let $x_1$ be a positive irrational number which is a zero of a quadratic polynomial with integer coefficients. For every positive integer $n$, let $x_{n+1} = f(x_n)$. Prove that there exists different positive integers $k$ and $\ell$ such that $x_k = x_\ell$.
2017 Korea National Olympiad
Given a prime $p$, show that there exist two integers $a, b$ which satisfies the following.



For all integers $m$, $m^3+ 2017am+b$ is not a multiple of $p$.
2017 Korea National Olympiad
In a quadrilateral $ABCD$, we have $\angle ACB = \angle ADB = 90$ and $CD < BC$. Denote $E$ as the intersection of $AC$ and $BD$, and let the perpendicular bisector of $BD$ hit $BC$ at $F$. The circle with center $F$ which passes through $B$ hits $AB$ at $P (\neq B)$ and $AC$ at $Q$. Let $M$ be the midpoint of $EP$. Prove that the circumcircle of $EPQ$ is tangent to $AB$ if and only if $B, M, Q$ are colinear.
2017 Korea National Olympiad
Find all real numbers $c$ such that there exists a function $f: \mathbb{R}_{ \ge 0} \rightarrow \mathbb{R}$ which satisfies the following.



For all nonnegative reals $x, y$, $f(x+y^2) \ge cf(x)+y$.



Here $\mathbb{R}_{\ge 0}$ is the set of all nonnegative reals.
2017 Korea National Olympiad
For a positive integer $n$, there is a school with $2n$ people. For a set $X$ of students in this school, if any two students in $X$ know each other, we call $X$ well-formed. If the maximum number of students in a well-formed set is no more than $n$, find the maximum number of well-formed set.



Here, an empty set and a set with one student is regarded as well-formed as well.
2016 Korea National Olympiad
$n$ is a positive integer. The number of solutions of $x^2+2016y^2=2017^n$ is $k$. Write $k$ with $n$.
2016 Korea National Olympiad
A non-isosceles triangle $\triangle ABC$ has its incircle tangent to $BC, CA, AB$ at points $D, E, F$.

Let the incenter be $I$. Say $AD$ hits the incircle again at $G$, at let the tangent to the incircle at $G$ hit $AC$ at $H$. Let $IH \cap AD = K$, and let the foot of the perpendicular from $I$ to $AD$ be $L$.



Prove that $IE \cdot IK= IC \cdot IL$.
2016 Korea National Olympiad
Acute triangle $\triangle ABC$ has area $S$ and perimeter $L$. A point $P$ inside $\triangle ABC$ has $dist(P,BC)=1, dist(P,CA)=1.5, dist(P,AB)=2$. Let $BC \cap AP = D$, $CA \cap BP = E$, $AB \cap CP= F$.

Let $T$ be the area of $\triangle DEF$. Prove the following inequality.



$$ \left( \frac{AD \cdot BE \cdot CF}{T} \right)^2 > 4L^2 + \left( \frac{AB \cdot BC \cdot CA}{24S} \right)^2 $$
2016 Korea National Olympiad
For a positive integer $n$, $S_n$ is the set of positive integer $n$-tuples $(a_1,a_2, \cdots ,a_n)$ which satisfies the following.



(i). $a_1=1$.



(ii). $a_{i+1} \le a_i+1$.



For $k \le n$, define $N_k$ as the number of $n$-tuples $(a_1, a_2, \cdots a_n) \in S_n$ such that $a_k=1, a_{k+1}=2$.



Find the sum $N_1 + N_2+ \cdots N_{k-1}$.
2016 Korea National Olympiad
A non-isosceles triangle $\triangle ABC$ has incenter $I$ and the incircle hits $BC, CA, AB$ at $D, E, F$.

Let $EF$ hit the circumcircle of $CEI$ at $P \not= E$. Prove that $\triangle ABC = 2 \triangle ABP$.
2016 Korea National Olympiad
For a positive integer $n$, there are $n$ positive reals $a_1 \ge a_2 \ge a_3 \cdots \ge a_n$.

For all positive reals $b_1, b_2, \cdots b_n$, prove the following inequality.



$$\frac{a_1b_1+a_2b_2 + \cdots +a_nb_n}{a_1+a_2+ \cdots a_n} \le \text{max}\{ \frac{b_1}{1}, \frac{b_1+b_2}{2}, \cdots, \frac{b_1+b_2+ \cdots +b_n}{n} \}$$
2016 Korea National Olympiad
Let $N=2^a p_1^{b_1} p_2^{b_2} \ldots p_k^{b_k}$. Prove that there are $(b_1+1)(b_2+1)\ldots(b_k+1)$ number of $n$s which satisfies these two conditions.

$\frac{n(n+1)}{2}\le N$, $N-\frac{n(n+1)}{2}$ is divided by $n$.
2016 Korea National Olympiad
A subset $S \in \{0, 1, 2, \cdots , 2000\}$ satisfies $|S|=401$.



Prove that there exists a positive integer $n$ such that there are at least $70$ positive integers $x$ such that $x, x+n \in S$
2015 Korea National Olympiad
For a positive integer $m$, prove that the number of pairs of positive integers $(x,y)$ which satisfies the following two conditions is even or $0$.



(i): $x^2-3y^2+2=16m$



(ii): $2y \le x-1$
2015 Korea National Olympiad
Let the circumcircle of $\triangle ABC$ be $\omega$. A point $D$ lies on segment $BC$, and $E$ lies on segment $AD$. Let ray $AD \cap \omega = F$. A point $M$, which lies on $\omega$, bisects $AF$ and it is on the other side of $C$ with respect to $AF$. Ray $ME \cap \omega = G$, ray $GD \cap \omega = H$, and $MH \cap AD = K$. Prove that $B, E, C, K$ are cyclic.
2015 Korea National Olympiad
Reals $a,b,c,x,y$ satisfies $a^2+b^2+c^2+x^2+y^2=1$. Find the maximum value of $$(ax+by)^2+(bx+cy)^2$$
2015 Korea National Olympiad
For positive integers $n, k, l$, we define the number of $l$-tuples of positive integers $(a_1,a_2,\cdots a_l)$ satisfying the following as $Q(n,k,l)$.



(i): $n=a_1+a_2+\cdots +a_l$



(ii): $a_1>a_2>\cdots > a_l > 0$.



(iii): $a_l$ is an odd number.



(iv): There are $k$ odd numbers out of $a_i$.



For example, from $9=8+1=6+3=6+2+1$, we have $Q(9,1,1)=1$, $Q(9,1,2)=2$, $Q(9,1,3)=1$.



Prove that if $n>k^2$, $\sum_{l=1}^n Q(n,k,l)$ is $0$ or an even number.
2015 Korea National Olympiad
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all reals $x,y,z$, we have $$(f(x)+1)(f(y)+f(z))=f(xy+z)+f(xz-y)$$
2015 Korea National Olympiad
An isosceles trapezoid $ABCD$, inscribed in $\omega$, satisfies $AB=CD, AD<BC, AD<CD$.

A circle with center $D$ and passing $A$ hits $BD, CD, \omega$ at $E, F, P(\not= A)$, respectively.

Let $AP \cap EF = Q$, and $\omega$ meet $CQ$ and the circumcircle of $\triangle BEQ$ at $R(\not= C), S(\not= B)$, respectively.

Prove that $\angle BER= \angle FSC$.