Navvye/Olympiad-Problems-CSV huggingface dataset:
2015 Korea National Olympiad
A positive integer $n$ is given. If there exists sets $F_1, F_2, \cdots F_m$ satisfying the following conditions, prove that $m \le n$. (For sets $A, B$, $|A|$ is the number of elements of $A$. $A-B$ is the set of elements that are in $A$ but not $B$. $\text{min}(x,y)$ is the number that is not larger than the other.)



(i): For all $1 \le i \le m$, $F_i \subseteq \{1,2,\cdots,n\}$



(ii): For all $1 \le i < j \le m$, $\text{min}(|F_i-F_j|,|F_j-F_i|) = 1$
2015 Korea National Olympiad
For a positive integer $n$, $a_1, a_2, \cdots a_k$ are all positive integers without repetition that are not greater than $n$ and relatively prime to $n$. If $k>8$, prove the following. $$\sum_{i=1}^k |a_i-\frac{n}{2}|<\frac{n(k-4)}{2}$$
2014 Korea National Olympiad
For $x, y$ positive integers, $x^2-4y+1$ is a multiple of $(x-2y)(1-2y)$. Prove that $|x-2y|$ is a square number.
2014 Korea National Olympiad
Determine all the functions $f : \mathbb{R}\rightarrow\mathbb{R}$ that satisfies the following.



$f(xf(x)+f(x)f(y)+y-1)=f(xf(x)+xy)+y-1$
2014 Korea National Olympiad
$AB$ is a chord of $O$ and $AB$ is not a diameter of $O$. The tangent lines to $O$ at $A$ and $B$ meet at $C$. Let $M$ and $N$ be the midpoint of the segments $AC$ and $BC$, respectively. A circle passing through $C$ and tangent to $O$ meets line $MN$ at $P$ and $Q$. Prove that $\angle PCQ = \angle CAB$.
2014 Korea National Olympiad
There is a city with $n$ metro stations, each located at a vertex of a regular n-polygon. Metro Line 1 is a line which only connects two non-neighboring stations $A$ and $B$. Metro Line 2 is a cyclic line which passes through all the stations in a shape of regular n-polygon. For each line metro can run in any direction, and $A$ and $B$ are the stations which one can transfer into other line. The line between two neighboring stations is called 'metro interval'. For each station there is one stationmaster, and there are at least one female stationmaster and one male stationmaster. If $n$ is odd, prove that for any integer $k$ $(0<k<n)$ there is a path that starts from a station with a male stationmaster and ends at a station with a female stationmaster, passing through $k$ metro intervals.
2014 Korea National Olympiad
There is a convex quadrilateral $ ABCD $ which satisfies $ \angle A=\angle D $.

Let the midpoints of $ AB, AD, CD $ be $ L,M,N $.

Let's say the intersection point of $ AC, BD $ be $ E $ .

Let's say point $ F $ which lies on $ \overrightarrow{ME} $ satisfies $ \overline{ME}\times \overline{MF}=\overline{MA}^{2} $.



Prove that $ \angle LFM=\angle MFN $. :)
2014 Korea National Olympiad
How many one-to-one functions $f : \{1, 2, \cdots, 9\} \rightarrow \{1, 2, \cdots, 9\}$ satisfy (i) and (ii)?

(i) $f(1)>f(2)$, $f(9)<9$.

(ii) For each $i=3, 4, \cdots, 8$, if $f(1), \cdots, f(i-1)$ are smaller than $f(i)$, then $f(i+1)$ is also smaller than $f(i)$.
2014 Korea National Olympiad
Let $x, y, z$ be the real numbers that satisfies the following.



$(x-y)^2+(y-z)^2+(z-x)^2=8, x^3+y^3+z^3=1$



Find the minimum value of $x^4+y^4+z^4$.
2014 Korea National Olympiad
Prove that there exists a function $f : \mathbb{N} \rightarrow \mathbb{N}$ that satisfies the following



(1) $\{f(n) : n\in\mathbb{N}\}$ is a finite set; and

(2) For nonzero integers $x_1, x_2, \ldots, x_{1000}$ that satisfy $f(\left|x_1\right|)=f(\left|x_2\right|)=\cdots=f(\left|x_{1000}\right|)$, then $x_1+2x_2+2^2x_3+2^3x_4+2^4x_5+\cdots+2^{999}x_{1000}\ne 0$.
2013 Korea National Olympiad
Let $P$ be a point on segment $BC$. $Q, R$ are points on $AC, AB$ such that $PQ \parallel AB $ and $ PR \parallel AC$. $O, O_{1}, O_{2} $ are the circumcenters of triangle $ABC, BPR, PCQ$. The circumcircles of $BPR, PCQ $ meet at point $K (\ne P)$. Prove that $OO_{1} = KO_{2} $.
2013 Korea National Olympiad
Let $ a, b, c>0 $ such that $ ab+bc+ca=3 $. Prove that

$$ \sum_{cyc} { \frac{ (a+b)^{3} }{ {(2(a+b)(a^2 + b^2))}^{\frac{1}{3}}} \ge 12 }$$
2013 Korea National Olympiad
Prove that there exist monic polynomial $f(x) $ with degree of 6 and having integer coefficients such that

(1) For all integer $m$, $f(m) \ne 0$.

(2) For all positive odd integer $n$, there exist positive integer $k$ such that $f(k)$ is divided by $n$.
2013 Korea National Olympiad
$\{a_n\}$ is a positive integer sequence such that $ a_{i+2} = a_{i+1} + a_{i} (i \ge 1) $. For positive integer $n$, define $\{b_n\}$ as

$$ b_n = \frac{1}{a_{2n+1}} \sum_{i=1}^{4n-2} { a_i } $$

Prove that $b_n$ is positive integer, and find the general form of $b_n$.
2013 Korea National Olympiad
Find all functions $f : \mathbb{N} \rightarrow \mathbb{N} $ satisfying

$$ f(mn) = \operatorname{lcm} (m,n) \cdot \gcd( f(m), f(n) ) $$

for all positive integer $m,n$.
2013 Korea National Olympiad
Let $ O $ be circumcenter of triangle $ABC$. For a point $P$ on segmet $BC$, the circle passing through $ P, B $ and tangent to line $AB $ and the circle passing through $ P, C $ and tangent to line $AC $ meet at point $ Q ( \ne P ) $. Let $ D, E $ be foot of perpendicular from $Q$ to $ AB, AC$. ($D \ne B, E \ne C $) Two lines $DE $ and $ BC $ meet at point $R$. Prove that $ O, P, Q $ are collinear if and only if $ A, R, Q $ are collinear.
2013 Korea National Olympiad
For positive integer $k$, define integer sequence $\{ b_n \}, \{ c_n \} $ as follows:

$$ b_1 = c_1 = 1 $$

$$ b_{2n} = kb_{2n-1} + (k-1)c_{2n-1},  c_{2n} = b_{2n-1} + c_{2n-1} $$

$$ b_{2n+1} = b_{2n} + (k-1)c_{2n},  c_{2n+1} = b_{2n} + kc_{2n} $$

Let $a_k = b_{2014} $. Find the value of

$$ \sum_{k=1}^{100} { (a_k - \sqrt{{a_k}^2-1} )^{ \frac{1}{2014}} }$$
2013 Korea National Olympiad
For positive integer $a,b,c,d$ there are $a+b+c+d$ points on plane which none of three are collinear. Prove there exist two lines $l_1, l_2 $ such that

(1) $l_1, l_2 $ are not parallel.

(2) $l_1, l_2 $ do not pass through any of  $a+b+c+d$ points.

(3) There are $ a, b, c, d $ points on each region separated by two lines $l_1, l_2 $.
2012 Korea National Olympiad
Let $ ABC $ be an obtuse triangle with $ \angle A > 90^{\circ} $. Let circle $ O $ be the circumcircle of $ ABC $. $ D $ is a point lying on segment $ AB $ such that $ AD = AC $. Let $ AK $ be the diameter of circle $ O $. Two lines $ AK $ and $ CD $ meet at $ L $. A circle passing through $ D, K, L $ meets with circle $ O $ at $ P ( \ne K ) $ . Given that $ AK = 2, \angle BCD = \angle BAP = 10^{\circ} $, prove that $ DP = \sin ( \frac{ \angle A}{2} )$.
2012 Korea National Olympiad
There are $n$ students $ A_1 , A_2 , \cdots , A_n $ and some of them shaked hands with each other. ($ A_i $ and $ A_j$ can shake hands more than one time.) Let the student $ A_i $ shaked hands $ d_i $ times. Suppose $ d_1 + d_2 + \cdots + d_n > 0 $. Prove that there exist $ 1 \le i < j \le n $ satisfying the following conditions:

(a) Two students  $ A_i $ and $ A_j $ shaked hands each other.

(b) $ \frac{(d_1 + d_2 + \cdots + d_n ) ^2 }{n^2 } \le d_i d_j $
2012 Korea National Olympiad
Find all triples $(m,p,q)$ where $ m $ is a positive integer and $ p , q $ are primes.

$$ 2^m p^2 + 1 = q^5 $$
2012 Korea National Olympiad
$a,b,c$ are positive numbers such that $ a^2 + b^2 + c^2 = 2abc + 1 $. Find the maximum value of

$$ (a-2bc)(b-2ca)(c-2ab) $$
2012 Korea National Olympiad
$ p >3 $ is a prime number such that $ p | 2^{p-1} -1  $ and $ p \not |  2^x - 1 $ for $ x = 1, 2, \cdots , p-2 $. Let $ p = 2k+3 $. Now we define sequence $ \{ a_n \} $ as

$$ a_i = a_{i+k}= 2^i ( 1 \le i \le k ) , \  a_{j+2k} = a_j a_{j+k} \ ( j \ge 1 ) $$

Prove that there exist $2k$ consecutive terms of sequence $ a_{x+1} , a_{x+2} , \cdots , a_{x+2k} $ such that for all $ 1 \le i < j \le 2k $, $ a_{x+i} \not \equiv a_{x+j} \ (mod \ p) $.
2012 Korea National Olympiad
Let $ w $ be the incircle of triangle $ ABC $. Segments $ BC, CA $ meet with $ w $ at points $ D, E$. A line passing through $ B $ and parallel to $ DE $ meets $ w $ at $ F $ and $ G $. ($ F $ is nearer to $ B $ than $ G $.) Line $ CG $ meets $ w $ at $ H ( \ne G ) $. A line passing through $ G $ and parallel to $ EH $ meets with line $ AC $ at $ I $. Line $ IF $ meets with circle $ w $ at $ J (\ne F ) $. Lines $ CJ $ and $ EG $ meets at $ K $. Let $ l $ be the line passing through $ K $ and parallel to $ JD $. Prove that $ l, IF, ED $ meet at one point.
2012 Korea National Olympiad
Let $ \{ a_1 , a_2 , \cdots, a_{10} \} = \{ 1, 2, \cdots , 10 \} $ . Find the maximum value of

$$ \sum_{n=1}^{10}(na_n ^2 - n^2 a_n ) $$
2012 Korea National Olympiad
Let $ p \equiv 3 \pmod{4}$ be a prime. Define $T = \{ (i,j) \mid i, j \in \{ 0, 1, \cdots , p-1 \} \} \smallsetminus \{ (0,0) \} $.  For arbitrary subset $ S ( \ne \emptyset ) \subset T $, prove that there exist subset $ A \subset S $ satisfying following conditions:

(a) $ (x_i , y_i ) \in A ( 1 \le i \le 3) $ then $ p \not | x_1 + x_2 - y_3  $ or $ p \not | y_1 + y_2 + x_3 $.

(b) $ 8 n(A) > n(S) $
2011 Korea National Olympiad
Two circles $ O, O'$ having same radius meet at two points, $ A,B (A \not = B) $. Point $ P,Q $ are each on circle $ O $ and $ O' $ $(P \not = A,B ~ Q\not = A,B )$. Select the point $ R $ such that $ PAQR $ is a parallelogram. Assume that $ B, R, P, Q $ is cyclic. Now prove that $ PQ = OO' $.
2011 Korea National Olympiad
Let $x, y$ be positive integers such that $\gcd(x,y)=1$ and $x+3y^2$ is a perfect square. Prove that $x^2+9y^4$ can't be a perfect square.
2011 Korea National Olympiad
Let $a,b,c,d$ real numbers such that $a+b+c+d=19$ and $a^2+b^2+c^2+d^2=91$. Find the maximum value of

$$ \frac{1}{a} +\frac{1}{b} +\frac{1}{c} +\frac{1}{d} $$
2011 Korea National Olympiad
Let $k,n$ be positive integers. There are $kn$ points $P_1, P_2, \cdots, P_{kn}$ on a circle. We can color each points with one of color $ c_1, c_2, \cdots , c_k $. In how many ways we can color the points satisfying the following conditions?



(a) Each color is used $ n $ times.



(b) $ \forall  i \not = j $, if $ P_a $ and $ P_b $ is colored with color $ c_i $ , and $ P_c $ and $ P_d $ is colored with color $ c_j $, then the segment $ P_a P_b $ and segment $ P_c P_d $ doesn't meet together.
2011 Korea National Olympiad
Find the number of positive integer $ n < 3^8 $ satisfying the following condition.



"The number of positive integer $k (1 \leq k \leq \frac {n}{3})$ such that  $ \frac{n!}{(n-3k)! \cdot k! \cdot 3^{k+1}} $ is not a integer"  is $ 216 $.
2011 Korea National Olympiad
Let $ABC$ be a triangle and its incircle meets $BC, AC, AB$ at $D, E$ and $F$ respectively. Let point $ P $ on the incircle and inside $ \triangle AEF $. Let $ X=PB \cap DF , Y=PC \cap DE, Q=EX \cap FY $. Prove that the points $ A$ and $Q$ lies on $DP$ simultaneously or located opposite sides from $DP$.
2011 Korea National Olympiad
There are $n$ students each having $r$ positive integers. Their $nr$ positive integers are all different. Prove that we can divide the students into $k$ classes satisfying the following conditions.



(a) $ k \le 4r $



(b) If a student $A$ has the number $m$, then the student $B$ in the same class can't have a number $l$ such that

$$ (m-1)! < l < (m+1)!+1  $$
2011 Korea National Olympiad
Let $ x_1, x_2, \cdots, x_{25} $ real numbers such that $ 0 \le x_i \le i (i=1, 2, \cdots, 25) $. Find the maximum value of

$$x_{1}^{3}+x_{2}^{3}+\cdots +x_{25}^{3} - ( x_1x_2x_3 + x_2x_3x_4 + \cdots x_{25}x_1x_2 ) $$
2010 Korea National Olympiad
Prove that $ 7^{2^{20}} + 7^{2^{19}} + 1 $ has at least $ 21 $ distinct prime divisors.
2010 Korea National Olympiad
Let $ a, b, c $ be positive real numbers such that $ ab+bc+ca=1 $. Prove that

$$ \sqrt{ a^2 + b^2 + \frac{1}{c^2}} + \sqrt{ b^2 + c^2 + \frac{1}{a^2}} + \sqrt{ c^2 + a^2 + \frac{1}{b^2}} \ge \sqrt{33} $$
2010 Korea National Olympiad
Let $ I $ be the incenter of triangle $ ABC $. The incircle touches $ BC, CA, AB$ at points $ P, Q, R $. A circle passing through $ B , C $ is tangent to the circle $I$ at point $ X $, a circle passing through $ C , A $ is tangent to the circle $I$ at point $ Y $, and a circle passing through $ A , B $ is tangent to the circle $I$ at point $ Z $, respectively. Prove that three lines $ PX, QY, RZ $ are concurrent.
2010 Korea National Olympiad
There are $ n ( \ge 4 ) $ people and some people shaked hands each other. Two people can shake hands at most 1 time. For arbitrary four people $ A, B, C, D$ such that $ (A,B), (B,C), (C,D) $ shaked hands, then one of $ (A,C), (A,D), (B,D) $ shaked hand each other. Prove the following statements.

(a) Prove that $ n $ people can be divided into two groups, $ X, Y ( \ne \emptyset )$ , such that for all $ (x,y) $ where $ x \in X $ and $ y \in Y $, $ x $ and $ y $ shaked hands or $ x $ and $ y $ didn't shake hands.

(b) There exist two people $ A , B $ such that the set of people who are not $ A $ and $ B $ that shaked hands with $ A $ is same wiith the set of people who are not $ A $ and $ B $ that shaked hands with $ B $.
2010 Korea National Olympiad
$ x, y, z $ are positive real numbers such that $ x+y+z=1 $. Prove that

$$ \sqrt{ \frac{x}{1-x} } +  \sqrt{ \frac{y}{1-y} } +  \sqrt{ \frac{z}{1-z} } > 2 $$
2010 Korea National Olympiad
Let $ ABCD$ be a cyclic convex quadrilateral. Let $ E $ be the intersection of lines $ AB, CD $. $ P $ is the intersection of line passing $ B $ and perpendicular to $ AC $, and line passing $ C $ and perpendicular to $ BD$. $ Q $ is the intersection of line passing $ D $ and perpendicular to $ AC $, and line passing $ A $ and perpendicular to $ BD $. Prove that three points $ E, P, Q $ are collinear.
2010 Korea National Olympiad
There are $ 2000 $ people, and some of them have called each other. Two people can call each other at most $1$ time. For any two groups of three people $ A$ and $  B $  which $ A \cap B = \emptyset $, there exist one person from each of $A$ and $B$ that haven't called each other. Prove that the number of two people called each other is less than $ 201000 $.
2010 Korea National Olympiad
There are $ 2010 $ people sitting around a round table. First, we give one person $ x $ a candy. Next, we give candies to $1$ st person, $1+2$ th person, $ 1+2+3$ th person, $\cdots$ , and $1+2+\cdots + 2009 $ th person clockwise from $ x $. Find the number of people who get at least one candy.
2009 Korea National Olympiad
Let $I, O$ be the incenter and the circumcenter of triangle $ABC$, and $D,E,F$ be the circumcenters of triangle $ BIC, CIA, AIB$. Let $ P, Q, R$ be the midpoints of segments $ DI, EI, FI $. Prove that the circumcenter of triangle $PQR $, $M$, is the midpoint of segment $IO$.
2009 Korea National Olympiad
Let $ a,b,c$ be positive real numbers. Prove that

$$ \frac{a^3}{c(a^2 + bc)} + \frac{b^3}{a(b^2 + ca)} + \frac{c^3}{b(c^2 +ab )} \ge \frac{3}{2} . $$
2009 Korea National Olympiad
Let $n$ be a positive integer. Suppose that the diophantine equation

$$z^n = 8 x^{2009} + 23 y^{2009} $$

uniquely has an integer solution $(x,y,z)=(0,0,0)$. Find the possible minimum value of $n$.
2009 Korea National Olympiad
There are $n ( \ge 3) $ students in a class. Some students are friends each other, and friendship is always mutual. There are $ s ( \ge 1 ) $ couples of two students who are friends, and $ t ( \ge 1 ) $ triples of three students who are each friends. For two students $ x, y $ define $ d(x,y)$ be the number of students who are both friends with $ x $ and $ y $. Prove that there exist three students $ u, v, w $ who are each friends and satisfying

$$ d(u,v) + d(v,w) + d(w,u) \ge \frac{9t}{s} . $$
2009 Korea National Olympiad
Let $ A = \{ 1, 2, 3, \cdots , 12 \} $. Find the number of one-to-one function $ f :A \to A $ satisfying following condition: for all $ i \in A $, $ f(i)-i $ is not a multiple of  $ 3 $.
2009 Korea National Olympiad
Let $ABC$ be a triangle and $ P, Q ( \ne A, B, C ) $ are the points lying on segments $ BC , CA $. Let $ I, J, K $ be the incenters of triangle $ ABP, APQ, CPQ $. Prove that $ PIJK $ is a convex quadrilateral.
2009 Korea National Olympiad
For all positive integer $ n \ge 2 $, prove that $ 2^n -1 $ can't be a divisor of $ 3^n -1 $.
2009 Korea National Olympiad
For a positive integer $n$, define a function $ f_n (x) $ at an interval $ [ 0, n+1 ] $ as

$$ f_n (x) = ( \sum_{i=1} ^ {n}  | x-i | )^2 - \sum_{i=1} ^{n} (x-i)^2  . $$

Let $ a_n $ be the minimum value of $f_n (x) $. Find the value of

$$ \sum_{n=1}^{11} (-1)^{n+1} a_n . $$
2008 Korean National Olympiad
Let $V=[(x,y,z)|0\le x,y,z\le 2008]$ be a set of points in a 3-D space.

If the distance between two points is either $1, \sqrt{2}, 2$, we color the two points differently.

How many colors are needed to color all points in $V$?
2008 Korean National Olympiad
We have $x_i >i$ for all $1 \le i \le n$.

Find the minimum value of $\frac{(\sum_{i=1}^n x_i)^2}{\sum_{i=1}^n \sqrt{x^2_i - i^2}}$
2008 Korean National Olympiad
Points $A,B,C,D,E$ lie in a counterclockwise order on a circle $O$, and $AC = CE$

$P=BD \cap AC$, $Q=BD \cap CE$

Let $O_1$ be the circle which is tangent to $\overline {AP}, \overline {BP}$ and arc $AB$ (which doesn't contain $C$)

Let $O_2$ be the circle which is tangent $\overline {DQ}, \overline {EQ}$ and arc $DE$ (which doesn't contain $C$)

Let $O_1 \cap O = R, O_2 \cap O = S, RP \cap QS = X$

Prove that $XC$ bisects $\angle ACE$
2008 Korean National Olympiad
We define $A, B, C$ as a partition of $\mathbb{N}$ if $A,B,C$ satisfies the following.

(i) $A, B, C \not= \phi$ (ii) $A \cap B = B \cap C = C \cap A = \phi$ (iii) $A \cup B \cup C = \mathbb{N}$.



Prove that the partition of $\mathbb{N}$ satisfying the following does not exist.

(i) $\forall$ $a \in A, b \in B$, we have $a+b+2008 \in C$

(ii) $\forall$ $b \in B, c \in C$, we have $b+c+2008 \in A$

(iii) $\forall$ $c \in C, a \in A$, we have $c+a+2008 \in B$
2008 Korean National Olympiad
Let $p$ be a prime where $p \ge 5$.

Prove that $\exists n$ such that $1+ (\sum_{i=2}^n \frac{1}{i^2})(\prod_{i=2}^n i^2) \equiv 0 \pmod p$
2008 Korean National Olympiad
Let $ABCD$ be inscribed in a circle $\omega$.

Let the line parallel to the tangent to $\omega$ at $A$ and passing $D$ meet $\omega$ at $E$.

$F$ is a point on $\omega$ such that lies on the different side of $E$ wrt $CD$.

If $AE \cdot AD \cdot CF = BE \cdot BC \cdot DF$ and $\angle CFD = 2\angle AFB$,

Show that the tangent to $\omega$ at $A, B$ and line $EF$ concur at one point.

($A$ and $E$ lies on the same side of $CD$)
2008 Korean National Olympiad
Prove that the only function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying the following is $f(x)=x$.

(i) $\forall x \not= 0$, $f(x) = x^2f(\frac{1}{x})$.

(ii) $\forall x, y$, $f(x+y) = f(x)+f(y)$.

(iii) $f(1)=1$.
2008 Korean National Olympiad
For fixed positive integers $s, t$, define $a_n$ as the following.

$a_1 = s, a_2 = t$, and $\forall n \ge 1$, $a_{n+2} = \lfloor \sqrt{a_n+(n+2)a_{n+1}+2008} \rfloor$.

Prove that the solution set of $a_n \not= n$, $n \in \mathbb{N}$ is finite.
2007 Korea National Olympiad
Consider the string of length $ 6$ composed of three characters $ a$, $ b$, $ c$. For each string, if two $ a$s are next to each other, or two $ b$s are next to each other, then replace $ aa$ by $ b$, and replace $ bb$ by $ a$. Also, if $ a$ and $ b$ are next to each other, or two $ c$s are next to each other, remove all two of them (i.e. delete $ ab$, $ ba$, $ cc$). Determine the number of strings that can be reduced to $ c$, the string of length 1, by the reducing processes mentioned above.
2007 Korea National Olympiad
$ A_{1}B_{1}B_{2}A_{2}$ is a convex quadrilateral, and $ A_{1}B_{1}\neq A_{2}B_{2}$. Show that there exists a point $ M$ such that

$$\frac{A_{1}B_{1}}{A_{2}B_{2}}=\frac{MA_{1}}{MA_{2}}=\frac{MB_{1}}{MB_{2}}$$
2007 Korea National Olympiad
Let $ S$ be the set of all positive integers whose all digits are $ 1$ or $ 2$. Denote $ T_{n}$ as the set of all integers which is divisible by $ n$, then find all positive integers $ n$ such that $ S\cap T_{n}$ is an infinite set.
2007 Korea National Olympiad
Two real sequence $ \{x_{n}\}$ and $ \{y_{n}\}$ satisfies following recurrence formula;



$ x_{0}= 1$, $ y_{0}= 2007$

$ x_{n+1}= x_{n}-(x_{n}y_{n}+x_{n+1}y_{n+1}-2)(y_{n}+y_{n+1})$,

$ y_{n+1}= y_{n}-(x_{n}y_{n}+x_{n+1}y_{n+1}-2)(x_{n}+x_{n+1})$



Then show that for all nonnegative integer $ n$, $ {x_{n}}^{2}\leq 2007$.
2007 Korea National Olympiad
For all positive reals $ a$, $ b$, and $ c$, what is the value of positive constant $ k$ satisfies the following inequality?

$ \frac{a}{c+kb}+\frac{b}{a+kc}+\frac{c}{b+ka}\geq\frac{1}{2007}$ .
2007 Korea National Olympiad
$ ABC$ is a triangle which is not isosceles. Let the circumcenter and orthocenter of $ ABC$ be $ O$, $ H$, respectively,

and the altitudes of $ ABC$ be $ AD$, $ BC$, $ CF$. Let $ K\neq A$ be the intersection of $ AD$ and circumcircle of triangle $ ABC$, $ L$ be the intersection of $ OK$ and $ BC$, $ M$ be the midpoint of $ BC$, $ P$ be the intersection of $ AM$ and the line that passes $ L$ and perpendicular to $ BC$, $ Q$ be the intersection of $ AD$ and the line that passes $ P$ and parallel to $ MH$, $ R$ be the intersection of line $ EQ$ and $ AB$, $ S$ be the intersection of $ FD$ and $ BE$.

If $ OL = KL$, then prove that two lines $ OH$ and $ RS$ are orthogonal.
2007 Korea National Olympiad
In each $ 2007^{2}$ unit squares on chess board whose size is $ 2007\times 2007$,

there lies one coin each square such that their "heads" face upward.

Consider the process that flips four consecutive coins on the same row, or flips four consecutive coins on the same column.

Doing this process finite times, we want to make the "tails" of all of coins face upward, except one that lies in the $ i$th row and $ j$th column.

Show that this is possible if and only if both of $ i$ and $ j$ are divisible by $ 4$.
2007 Korea National Olympiad
For all positive integer $ n\geq 2$, prove that product of all prime numbers less or equal than $ n$ is smaller than $ 4^{n}$.
2006 Korea National Olympiad
Given that for reals $a_1,\cdots, a_{2004},$ equation $x^{2006}-2006x^{2005}+a_{2004}x^{2004}+\cdots +a_2x^2+a_1x+1=0$ has $2006$ positive real solution, find the maximum possible value of $a_1.$
2006 Korea National Olympiad
Alice and Bob are playing "factoring game." On the paper, $270000(=2^43^35^4)$ is written and each person picks one number from the paper(call it $N$) and erase $N$ and writes integer $X,Y$ such that $N=XY$ and $\text{gcd}(X,Y)\ne1.$ Alice goes first and the person who can no longer make this factoring loses. If two people use optimal strategy, prove that Alice always win.
2006 Korea National Olympiad
For three positive integers $a,b$ and $c,$ if $\text{gcd}(a,b,c)=1$ and $a^2+b^2+c^2=2(ab+bc+ca),$ prove that all of $a,b,c$ is perfect square.
2006 Korea National Olympiad
On the circle $O,$ six points $A,B,C,D,E,F$ are  on the circle counterclockwise. $BD$ is the diameter of the circle and it is perpendicular to $CF.$ Also, lines $CF, BE, AD$ is concurrent. Let $M$ be the foot of altitude from $B$ to $AC$ and let $N$ be the foot of altitude from $D$ to $CE.$ Prove that the area of $\triangle MNC$ is less than half the area of $\square ACEF.$
2006 Korea National Olympiad
Find all positive integers $n$ such that $\phi(n)$ is the fourth power of some prime.
2006 Korea National Olympiad
Prove that for any positive real numbers $x,y$ and $z,$

$xyz(x+2)(y+2)(z+2)\le(1+\frac{2(xy+yz+zx)}{3})^3$
2006 Korea National Olympiad
Points $A,B,C,D,E,F$ is on the circle $O.$ A line $\ell$ is tangent to $O$ at $E$ is parallel to $AC$ and $DE>EF.$ Let $P,Q$ be the intersection of $\ell$ and $BC,CD$ ,respectively and let $R,S$ be the intersection of $\ell$ and $CF,DF$ ,respectively. Show that $PQ=RS$ if and only if $QE=ER.$
2006 Korea National Olympiad
$27$ students are given a number from $1$ to $27.$ How many ways are there to divide $27$ students into $9$ groups of $3$ with the following condition?



(i) The sum of students number in each group is $1\pmod{3}$

(ii) There are no such two students where their numbering differs by $3.$
2005 Korea National Olympiad
For two positive integers a and b, which are relatively prime, find all integer that can be the great common divisor of $a+b$ and $\frac{a^{2005}+b^{2005}}{a+b}$.