Navvye/Olympiad-Problems-CSV huggingface dataset:
2011 Korea - Final Round
There is a chessboard with $m$ columns and $n$ rows. In each blanks, an integer is given. If a rectangle $R$ (in this chessboard) has an integer $h$ satisfying the following two conditions, we call $R$ as a 'shelf'.

(i) All integers contained in $R$ are bigger than $h$.

(ii) All integers in blanks, which are not contained in $R$ but meet with $R$ at a vertex or a side, are not bigger than $h$.

Assume that all integers are given to make shelves as much as possible. Find the number of shelves.
2010 Korea - Final Round
Given an arbitrary triangle $ ABC$, denote by $ P,Q,R$ the intersections of the incircle with sides $ BC, CA, AB$ respectively. Let the area of triangle $ ABC$ be $ T$, and its perimeter $ L$. Prove that the inequality

$$\left(\frac {AB}{PQ}\right)^3 +\left(\frac {BC}{QR}\right)^3 +\left(\frac {CA}{RP}\right)^3 \geq \frac {2}{\sqrt {3}} \cdot \frac {L^2}{T}$$

holds.
2010 Korea - Final Round
Let $ I$ be the incentre and $ O$ the circumcentre of a given acute triangle $ ABC$. The incircle is tangent to $ BC$ at $ D$. Assume that $ \angle B < \angle C$ and the segments $ AO$ and $ HD$ are parallel, where $H$ is the orthocentre of triangle $ABC$. Let the intersection of the line $ OD$ and $ AH$ be $ E$. If the midpoint of $ CI$ is $ F$, prove that $ E,F,I,O$ are concyclic.
2010 Korea - Final Round
There are $ n$ websites $ 1,2,\ldots,n$ ($ n \geq 2$). If there is a link from website $ i$ to $ j$, we can use this link so we can move website $ i$ to $ j$.

For all $ i \in \left\{1,2,\ldots,n - 1 \right\}$, there is a link from website $ i$ to $ i+1$.

Prove that we can add less or equal than $ 3(n - 1)\log_{2}(\log_{2} n)$ links so that for all integers $ 1 \leq i < j \leq n$, starting with website $ i$, and using at most three links to website $ j$. (If we use a link, website's number should increase. For example, No.7 to 4 is impossible).



Sorry for my bad English.
2010 Korea - Final Round
Given is a trapezoid $ ABCD$ where $ AB$ and $ CD$ are parallel, and $ A,B,C,D$ are clockwise in this order. Let $ \Gamma_1$ be the circle with center $ A$ passing through $ B$, $ \Gamma_2$ be the circle with center $ C$ passing through $ D$. The intersection of line $ BD$ and $ \Gamma_1$ is $ P$ $ ( \ne B,D)$. Denote by $ \Gamma$ the circle with diameter $ PD$, and let $ \Gamma$ and $ \Gamma_1$ meet at $ X$$ ( \ne P)$. $ \Gamma$ and $ \Gamma_2$ meet at $ Y$. If the circumcircle of triangle $ XBY$ and $ \Gamma_2$ meet at $ Q$, prove that $ B,D,Q$ are collinear.
2010 Korea - Final Round
On a circular table are sitting $ 2n$ people, equally spaced in between. $ m$ cookies are given to these people, and they give cookies to their neighbors according to the following rule.



(i) One may give cookies only to people adjacent to himself.

(ii) In order to give a cookie to one's neighbor, one must eat a cookie.



Select arbitrarily a person $ A$ sitting on the table. Find the minimum value $ m$ such that there is a strategy in which $ A$ can eventually receive a cookie, independent of the distribution of cookies at the beginning.
2010 Korea - Final Round
An arbitrary prime $ p$ is given. If an integer sequence $ (n_1 , n_2 , \cdots , n_k )$ satisfying the conditions

- For all $ i= 1, 2, \cdots , k$, $ n_i \geq \frac{p+1}{2}$

- For all $ i= 1, 2, \cdots , k$, $ p^{n_i} - 1$ is divisible by $ n_{i+1}$, and $ \frac{p^{n_i} - 1}{n_{i+1}}$ is coprime to $ n_{i+1}$. Let $ n_{k+1} = n_1$.

exists not for $ k=1$, but exists for some $ k \geq 2$, then call the prime a good prime.

Prove that a prime is good iff it is not $ 2$.
2009 Korea - Final Round
$a,b,c$ are the length of three sides of a triangle. Let $A= \frac{a^2 +bc}{b+c}+\frac{b^2 +ca}{c+a}+\frac{c^2 +ab}{a+b}$, $B=\frac{1}{\sqrt{(a+b-c)(b+c-a)}}+\frac{1}{\sqrt{(b+c-a)(c+a-b)}}$$+\frac{1}{\sqrt{(c+a-b)(a+b-c)}}$. Prove that $AB \ge 9$.
2009 Korea - Final Round
$ABC$ is an obtuse triangle. (angle $B$ is obtuse) Its circumcircle is $O$. A tangent line for $O$ passing $C$ meets with $AB$ at $B_1$. Let $O_1$ be a circumcenter of triangle $AB_1C$. $B_2$ is a point on the segment $BB_1$. Let $C_1$ be a contact point of the tangent line for $O$ passing $B_2$, which is more closer to $C$. Let $O_2$ be a circumcenter of triangle $AB_2C_1$. Prove that if $OO_2$ and $AO_1$ is perpendicular, then five points $O,O_2,O_1,C_1,C$ are concyclic.
2009 Korea - Final Round
2008 white stones and 1 black stone are in a row. An 'action' means the following: select one black stone and change the color of neighboring stone(s).

Find all possible initial position of the black stone, to make all stones black by finite actions.
2009 Korea - Final Round
$ABC$ is an acute triangle. (angle $C$ is bigger than angle $B$) Let $O$ be a center of the circle which passes $B$ and tangents to $AC$ at $C$. $O$ meets the segment $AB$ at $D$. $CO$ meets the circle $(O)$ again at $P$, a line, which passes $P$ and parallel to $AO$, meets $AC$ at $E$, and $EB$ meets the circle $(O)$ again at $L$. A perpendicular bisector of $BD$ meets $AC$ at $F$ and $LF$ meets $CD$ at $K$. Prove that two lines $EK$ and $CL$ are parallel.
2009 Korea - Final Round
There is a $m \times (m-1)$ board. (i.e. there are $m+1$ horizontal lines and $m$ vertical lines)  A stone is put on an intersection of the lowest horizontal line. Now two players move this stone with the following rules.

(i) Each players move the stone to a neighboring intersection along a segment, by turns.

(ii) A segment, which is already passed by the stone, cannot be used more.

(iii) One who cannot move the stone anymore loses.

Prove that there is a winning strategy for the former player.
2009 Korea - Final Round
Find all pairs of two positive integers $(m,n)$ satisfying $ 3^m - 7^n = 2 $.
2008 Korea - Final Round
Hexagon $ABCDEF$ is inscribed in a circle $O$.

Let $BD \cap CF = G, AC \cap BE = H, AD \cap CE = I$

Following conditions are satisfied.

$BD \perp CF , CI=AI$



Prove that $CH=AH+DE$ is equivalent to $GH \times BD = BC \times DE$
2008 Korea - Final Round
Find all integer polynomials $f$ such that there are infinitely many pairs of relatively prime natural numbers $(a,b)$ so that $a+b  \mid f(a)+f(b)$.
2008 Korea - Final Round
Determine all functions $f : \mathbb{R}^+\rightarrow\mathbb{R}$ that satisfy the following

$f(1)=2008$, $|{f(x)}| \le x^2+1004^2$, $f\left (x+y+\frac{1}{x}+\frac{1}{y}\right )=f\left (x+\frac{1}{y}\right )+f\left (y+\frac{1}{x}\right ).$
2008 Korea - Final Round
For any positive integer $m\ge2$ define  $A_m=\{m+1, 3m+2, 5m+3, 7m+4, \ldots, (2k-1)m + k, \ldots\}$.



(1) For every $m\ge2$, prove that there exists a positive integer $a$ that satisfies $1\le a<m$ and $2^a\in A_m$ or $2^a+1\in A_m$.



(2) For a certain $m\ge2$, let $a, b$ be positive integers that satisfy $2^a\in A_m$, $2^b+1\in A_m$. Let $a_0, b_0$ be the least such pair $a, b$.

Find the relation between $a_0$ and $b_0$.
2008 Korea - Final Round
Quadrilateral $ABCD$ is inscribed in a circle $O$.

Let $AB\cap CD=E$ and $P\in BC, EP\perp BC$, $R\in AD, ER\perp AD$, $EP\cap AD=Q, ER\cap BC=S$

Let $K$ be the midpoint of $QS$



Prove that $E, K, O$ are collinear.
2008 Korea - Final Round
There is $n\times n$ chessboard. Each square has a number between $0$ and $k$. There is a button for each row and column, which increases the number of $n$ numbers of the row or column the button represents(if the number of the square is $k$, then it becomes $0$). If certain button is pressed, call it 'operation.'



And we have a chessboard which is filled with 0(for all squares). After some 'operation's, the numbers of squares are different now. Prove that we can make all of the number $0$ within $kn$ 'operation's.
2007 Korea - Final Round
Let $ O$ be the circumcenter of an acute triangle $ ABC$ and let $ k$ be the circle with center $ P$ that is tangent to $ O$ at $ A$ and tangent to side $ BC$ at $ D$. Circle $ k$ meets $ AB$ and $ AC$ again at $ E$ and $ F$ respectively. The lines $ OP$ and $ EP$ meet $ k$ again at $ I$ and $ G$. Lines $ BO$ and $ IG$ intersect at $ H$. Prove that $ \frac{{DF}^2}{AF}=GH$.
2007 Korea - Final Round
Given a $ 4\times 4$ squares table. How many ways that we can fill the table with $ \{0,1\}$ such that two neighbor squares (have one common side) have product which is equal to $ 0$?
2007 Korea - Final Round
Find all triples of $ (x, y, z)$ of positive intergers satisfying $ 1+{4}^{x}+{4}^{y}=z^2$.
2007 Korea - Final Round
Find all pairs $ (p, q)$ of primes such that $ {p}^{p}+{q}^{q}+1$ is divisible by $ pq$.
2007 Korea - Final Round
For the vertex $ A$ of a triangle $ ABC$, let $ l_a$ be the distance between the projections on $ AB$ and $ AC$ of the intersection of the angle bisector of ∠$ A$ with side $ BC$. Define $ l_b$ and $ l_c$ analogously. If $ l$ is the perimeter of triangle $ ABC$, prove that $ \frac{l_a l_b l_c}{l^3}\le\frac{1}{64}$.
2007 Korea - Final Round
Let f:N→N be a function satisfying $ kf(n)\le f(kn)\le kf(n) + k - 1$ for all $ k, n\in N$.



(a)Prove that $ f(a) + f(b)\le f(a + b)\le f(a) + f(b) + 1$ for all $ a, b\in N$.



(b)If $ f$ satisfies $ f(2007n)\le 2007f(n) + 200$ for every $ n\in N$, show that there exists $ c\in N$ such that $ f(2007c) = 2007f(c)$.
2006 Korea - Final Round
In a triangle $ABC$ with $AB\not = AC$, the incircle touches the sides $BC, CA, AB$ at $D, E, F$ , respectively. Line $AD$ meets the incircle again at $P$ . The line $EF$ and the line through $P$ perpendicular to $AD$ meet at $Q$. Line $AQ$ intersects $DE$ at $X$ and $DF$ at $Y$ . Prove that $A$ is the midpoint of $XY$.
2006 Korea - Final Round
For a positive integer $a$, let $S_{a}$ be the set of primes $p$ for which there exists an odd integer $b$ such that $p$ divides $(2^{2^{a}})^{b}-1.$ Prove that for every $a$ there exist inﬁnitely many primes that are not contained in $S_{a}$.
2006 Korea - Final Round
Three schools $A, B$ and $C$ , each with ﬁve players denoted $a_{i}, b_{i}, c_{i}$ respectively, take part in a chess tournament. The tournament is held following the rules:

(i) Players from each school have matches in order with respect to indices, and defeated players are eliminated; the ﬁrst match is between $a_{1}$ and $b_{1}$.

(ii) If $y_{j}\in Y$ defeats $x_{i}\in X$ , his next opponent should be from the remaining school if not all of its players are eliminated; otherwise his next oponent is $x_{i+1}$ . The tournament is over when two schools are completely eliminated.

(iii) When $x_{i}$ wins a match, its school wins $10^{i-1}$ points.

At the end of the tournament, schools $A, B, C$ scored $P_{A}, P_{B}, P_{C}$ respectively. Find the remainder of the number of possible triples $(P_{A}, P_{B}, P_{C})$ upon division by $8.$
2006 Korea - Final Round
Given three distinct real numbers $a_{1}, a_{2}, a_{3}$ , deﬁne $b_{j}= (1+\frac{a_{j}a_{i}}{a_{j}-a_{i}})(1+\frac{a_{j}a_{k}}{a_{j}-a_{k}})$, where $\{i, j, k\}= \{1, 2, 3\}$.

Prove that $1+|a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}| \leq (1+|a_{1}|)(1+|a_{2}|)(1+|a_{3}|)$ and ﬁnd

the cases of equality.
2006 Korea - Final Round
In a convex hexagon $ABCDEF$ triangles $ABC , CDE , EFA$ are similar. Find conditions on these triangles under which triangle $ACE$ is equilateral if and only if so is $BDF.$
2006 Korea - Final Round
A positive integer $N$ is said to be $n-$ good if

(i) $N$ has at least $n$ distinct prime divisors, and

(ii) there exist distinct positive divisors $1, x_{2}, . . . , x_{n}$ whose sum is $N$ .

Show that there exists an $n-$ good number for each $n\geq 6$.
2005 Korea - Final Round
Find all natural numbers that can be expressed in a unique way as a sum of ﬁve or less perfect squares.
2005 Korea - Final Round
Let $(a_{n})_{n=1}^{\infty}$ be a sequence of positive real numbers and let $\alpha_{n}$ be the

arithmetic mean of $a_{1},..., a_{n}$ . Prove that for all positive integers $N$ ,

$$\sum_{n=1}^{N}\alpha_{n}^{2}\leq 4\sum_{n=1}^{N}a_{n}^{2}. $$
2005 Korea - Final Round
In a trapezoid $ABCD$ with $AD \parallel BC , O_{1}, O_{2}, O_{3}, O_{4}$ denote the circles with diameters $AB, BC, CD, DA$, respectively. Show that there exists a circle with center inside the trapezoid which is tangent to all the four circles $O_{1},..., O_{4}$ if and only if $ABCD$ is a parallelogram.
2005 Korea - Final Round
In the following, the point of intersection of two lines $ g$ and $ h$ will be abbreviated as $ g\cap h$.



Suppose $ ABC$ is a triangle in which $ \angle A = 90^{\circ}$ and $ \angle B > \angle C$. Let $ O$ be the circumcircle of the triangle $ ABC$. Let $ l_{A}$ and $ l_{B}$ be the tangents to the circle $ O$ at $ A$ and $ B$, respectively.



Let $ BC \cap l_{A} = S$ and $ AC \cap l_{B} = D$. Furthermore, let $ AB \cap DS = E$, and let $ CE \cap l_{A} = T$. Denote by $ P$ the foot of the perpendicular from $ E$ on $ l_{A}$. Denote by $ Q$ the point of intersection of the line $ CP$ with the circle $ O$ (different from $ C$). Denote by $ R$ be the point of intersection of the line $ QT$ with the circle $ O$ (different from $ Q$). Finally, define $ U = BR \cap l_{A}$. Prove that

$$ \frac {SU \cdot SP}{TU \cdot TP} = \frac {SA^{2}}{TA^{2}}.

$$
2005 Korea - Final Round
Find all positive integers $m$ and $n$ such that both $3^{m}+1$ and $3^{n}+1$ are divisible by $mn$.
2005 Korea - Final Round
A set $P$ consists of $2005$ distinct prime numbers. Let $A$ be the set of all possible products of $1002$ elements of $P$ , and $B$ be the set of all products of $1003$ elements of $P$ . Find a one-to-one correspondance $f$ from $A$ to $B$ with the property that $a$ divides $f (a)$ for all $a \in A.$
2004 Korea - Final Round
An isosceles triangle with $AB=AC$ has an inscribed circle $O$, which touches its sides $BC,CA,AB$ at $K,L,M$ respectively. The lines $OL$ and $KM$ intersect at $N$; the lines $BN$ and $CA$ intersect at $Q$. Let $P$ be the foot of the perpendicular from $A$ on $BQ$. Suppose that $BP=AP+2\cdot PQ$. Then, what values can the ratio $\frac{AB}{BC}$ assume?
2004 Korea - Final Round
Prove that the equation $3y^2 = x^4 + x$ has no positive integer solutions.
2004 Korea - Final Round
2004 computers make up a network using several cables. If for a subset $S$ in the set of all computers, there isn't a cable that connects two computers in $S$, $S$ is called independant. One lets the arbitrary independant set consists at most 50 computers, and uses the least number of cables.



(1) Let $c(L)$ be the number of cables which connects the computer $L$. Prove that for two computers $A,B$, $c(A)=c(B)$ if there is a cable which connects $A$ and $B$, $|c(A)-c(B)|\leq 1$ otherwise.



(2) Determine the number of used cables.
2004 Korea - Final Round
On a circle there are $n$ points such that every point has a distinct number. Determine the number of ways of  choosing $k$ points such that for any point there are at least 3 points between this point and the nearest point. (clockwise) ($n,k\geq 2$)
2004 Korea - Final Round
An acute triangle $ABC$ has circumradius $R$, inradius $r$. $A$ is the biggest angle among $A,B,C$. Let $M$ be the midpoint of $BC$, and $X$ be the intersection of two lines  that touches circumcircle of $ABC$ and goes through $B,C$ respectively. Prove the following inequality : $ \frac{r}{R} \geq \frac{AM}{AX}$.
2004 Korea - Final Round
For prime number $p$, let $f_p(x)=x^{p-1} +x^{p-2} + \cdots + x + 1$.



(1) When $p$ divides $m$, prove that there exists a prime number that is coprime with $m(m-1)$ and divides $f_p(m)$.



(2) Prove that there are infinitely many positive integers $n$ such that $pn+1$ is prime number.
2003 Korea - Final Round
Some computers of a computer room have a following network. Each computers are connected by three cable to three computers. Two arbitrary computers can exchange data directly or indirectly (through other computers). Now let's remove $K$ computers so that there are two computers, which can not exchange data, or there is one computer left. Let $k$ be the minimum value of $K$. Let's remove $L$ cable from original network so that there are two computers, which can not exchange data. Let $l$ be the minimum value of $L$. Show that $k=l$.
2003 Korea - Final Round
Let $M$ be the intersection of two diagonal, $AC$ and $BD$, of a rhombus $ABCD$, where angle $A<90^\circ$. Construct $O$ on segment $MC$ so that $OB<OC$ and let $t=\frac{MA}{MO}$, provided that $O \neq M$. Construct a circle that has $O$ as centre and goes through $B$ and $D$. Let the intersections between the circle and $AB$ be $B$ and $X$. Let the intersections between the circle and $BC$ be $B$ and $Y$. Let the intersections of $AC$ with $DX$ and $DY$ be $P$ and $Q$, respectively. Express $\frac{OQ}{OP}$ in terms of $t$.
2003 Korea - Final Round
Show that the equation, $2x^4+2x^2y^2+y^4=z^2$, does not have integer solution when $x \neq 0$.
2003 Korea - Final Round
Let $P$, $Q$, and $R$ be the points where the incircle of a triangle $ABC$ touches the sides $AB$, $BC$, and $CA$, respectively.

Prove the inequality $\frac{BC} {PQ} + \frac{CA} {QR} + \frac{AB} {RP} \geq 6$.
2003 Korea - Final Round
For a positive integer, $m$, answer the following questions.



1) Show that $2^{m+1}+1$ is a prime number, when $2^{m+1}+1$ is a factor of $3^{2^m}+1$.



2) Is converse of 1) true?
2003 Korea - Final Round
There are $n$ distinct points on a circumference. Choose one of the points. Connect this point and the $m$th point from the chosen point counterclockwise with a segment. Connect this $m$th point and the $m$th point from this $m$th point counterclockwise with a segment. Repeat such steps until no new segment is constructed. From the intersections of the segments, let the number of the intersections - which are in the circle - be $I$. Answer the following questions ($m$ and $n$ are positive integers that are relatively prime and they satisfy $6 \leq 2m < n$).



1) When the $n$ points take different positions, express the maximum value of $I$ in terms of $m$ and $n$.



2) Prove that $I \geq n$. Prove that there is a case, which is $I=n$, when $m=3$ and $n$ is arbitrary even number that satisfies the condition.
2002 Korea - Final Round
For a prime $p$ of the form $12k+1$ and $\mathbb{Z}_p=\{0,1,2,\cdots,p-1\}$, let

$$\mathbb{E}_p=\{(a,b) \mid a,b \in \mathbb{Z}_p,\quad  p\nmid 4a^3+27b^2\}$$

For $(a,b), (a',b') \in \mathbb{E}_p$ we say that $(a,b)$ and $(a',b')$ are equivalent if there is a non zero element $c\in \mathbb{Z}_p$ such that $p\mid (a' -ac^4)$ and $p\mid (b'-bc^6)$. Find the maximal number of inequivalent elements in $\mathbb{E}_p$.
2002 Korea - Final Round
Find all functions $f:\mathbb{R}\to \mathbb{R}$ satisfying $f(x-y)=f(x)+xy+f(y)$ for every $x \in \mathbb{R}$ and every $y \in \{f(x) \mid x\in \mathbb{R}\}$, where $\mathbb{R}$ is the set of real numbers.
2002 Korea - Final Round
The following facts are known in a mathematical contest:



(a) The number of problems tested was

$n\ge 4$

(b) Each problem was solved by exactly four contestants.

(c) For each pair of problems, there is exactly one contestant who solved both problems





Assuming the number of contestants is greater than or equal to $4n$, find the minimum value of $n$ for which there always exists a contestant who solved all the problems.
2002 Korea - Final Round
For $n \ge 3$, let $S=a_1+a_2+\cdots+a_n$ and $T=b_1b_2\cdots b_n$ for positive real numbers $a_1,a_2,\ldots,a_n, b_1,b_2 ,\ldots,b_n$, where the numbers $b_i$ are pairwise distinct.



(a) Find the number of distinct real zeroes of the polynomial

$$f(x)=(x-b_1)(x-b_2)\cdots(x-b_n)\sum_{j=1}^n \frac{a_j}{x-b_j}$$

(b) Prove the inequality

$$\frac1{n-1}\sum_{j=1}^n \left(1-\frac{a_j}{S}\right)b_j > \left(\frac{T}{S}\sum_{j=1}^{n} \frac{a_j}{b_j}\right)^{\frac1{n-1}}$$
2002 Korea - Final Round
Let $ABC$ be an acute triangle and let $\omega$ be its circumcircle. Let the perpendicular line from $A$ to $BC$ meet $\omega$ at $D$. Let $P$ be a point on $\omega$, and let $Q$ be the foot of the perpendicular line from $P$ to the line $AB$. Prove that if $Q$ is on the outside of $\omega$ and $2\angle QPB = \angle PBC$, then $D,P,Q$ are collinear.
2002 Korea - Final Round
Let $p_n$ be the $n^{\mbox{th}}$ prime counting from the smallest prime $2$ in increasing order. For example, $p_1=2, p_2=3, p_3 =5, \cdots$



(a) For a given $n \ge 10$, let $r$ be the smallest integer satisfying

$$2\le r \le n-2, \quad n-r+1 < p_r$$

and define $N_s=(sp_1p_2\cdots p_{r-1})-1$ for $s=1,2,\ldots, p_r$. Prove that there exists $j, 1\le j \le p_r$, such that none of $p_1,p_2,\cdots, p_n$ divides $N_j$.



(b) Using the result of (a), find all positive integers $m$ for which

$$p_{m+1}^2 < p_1p_2\cdots p_m$$
2001 Korea - Final Round
Given an odd prime $p$, find all functions $f:Z \rightarrow Z$ satisfying the following two conditions:



(i) $f(m)=f(n)$ for all $m,n \in Z$ such that $m\equiv n\pmod p$;

(ii) $f(mn)=f(m)f(n)$ for all $m,n \in Z$.
2001 Korea - Final Round
Let $P$ be a given point inside a convex quadrilateral $O_1O_2O_3O_4$. For each $i = 1,2,3,4$, consider the lines $l$ that pass through $P$ and meet the rays $O_iO_{i-1}$ and $O_iO_{i+1}$ (where $O_0 = O_4$ and $O_5 = O_1$) at distinct points $A_i(l)$ and $B_i(l)$, respectively. Denote $f_i(l) = PA_i(l) \cdot PB_i(l)$. Among all such lines $l$, let $l_i$ be the one that minimizes $f_i$. Show that if $l_1 = l_3$ and $l_2 = l_4$, then the quadrilateral $O_1O_2O_3O_4$ is a parallelogram.
2001 Korea - Final Round
Let $x_1,x_2, \cdots,x_n$ and $y_1,y_2, \cdots ,y_n$ be arbitrary real numbers satisfying $x_1^2+x_2^2+\cdots+x_n^2=y_1^2+y_2^2+\cdots+y_n^2=1$. Prove that

$$(x_1y_2-x_2y_1)^2 \le 2\left|1-\sum_{k=1}^n x_ky_k\right|$$

and find all cases of equality.
2001 Korea - Final Round
For given positive integers $n$ and $N$, let $P_n$ be the set of all polynomials $f(x)=a_0+a_1x+\cdots+a_nx^n$ with integer coefficients such that:



(a)

$|a_j| \le N$ for $j = 0,1, \cdots ,n$;

(b) The set $\{ j \mid a_j = N\}$ has at most two elements.





Find the number of elements of the set $\{f(2N) \mid f(x) \in P_n\}$.
2001 Korea - Final Round
In a triangle $ABC$ with $\angle B < 45^{\circ}$, $D$ is a point on $BC$ such that the incenter of $\triangle ABD$ coincides with the circumcenter $O$ of $\triangle ABC$. Let $P$ be the intersection point of the tangent lines to the circumcircle $\omega$ of $\triangle AOC$ at points $A$ and $C$. The lines $AD$ and $CO$ meet at $Q$. The tangent to $\omega$ at $O$ meets $PQ$ at $X$. Prove that $XO=XD$.
2001 Korea - Final Round
For a positive integer $n \ge 5$, let $a_i,b_i (i = 1,2, \cdots ,n)$ be integers satisfying the

following two conditions:



(a) The pairs

$(a_i,b_i)$ are distinct for $i = 1,2,\cdots,n$;

(b) $|a_1b_2-a_2b_1| = |a_2b_3-a_3b_2| = \cdots = |a_nb_1-a_1b_n| = 1$.





Prove that there exist indices $i,j$ such that $1<|i-j|<n-1$ and $|a_ib_j-a_jb_i|=1$.
2000 Korea - Final Round
Prove that for any prime $p$, there exist integers $x,y,z,$ and $w$ such that $x^2+y^2+z^2-wp=0$ and $0<w<p$
2000 Korea - Final Round
Determine all function $f$ from the set of real numbers to itself such that for every $x$ and $y$,

$$f(x^2-y^2)=(x-y)(f(x)+f(y))$$
2000 Korea - Final Round
A rectangle $ABCD$ is inscribed in a circle with centre $O$. The exterior bisectors of $\angle ABD$ and $\angle ADB$ intersect at $P$; those of $\angle DAB$ and $\angle DBA$ intersect at $Q$; those of $\angle ACD$ and $\angle ADC$ intersect at $R$; and those of $\angle DAC$ and $\angle DCA$ intersect at $S$. Prove that $P,Q,R$, and $S$ are concyclic.
2000 Korea - Final Round
Let $p$ be a prime such that $p \equiv 1 (\text {mod}4)$. Evaluate

$$\sum_{k=1}^{p-1} \left( \left \lfloor \frac{2k^2}{p}\right \rfloor   - 2 \left \lfloor {\frac{k^2}{p}}\right \rfloor \right)$$
2000 Korea - Final Round
Prove that an $m \times n$ rectangle can be constructed using copies of the following shape if and only if $mn$ is a multiple of $8$ where $m>1$ and $n>1$

[asy]

draw ((0,0)--(0,1));

draw ((0,0)--(1.5,0));

draw ((0,1)--(.5,1));

draw ((.5,1)--(.5,0));

draw ((0,.5)--(1.5,.5));

draw ((1.5,.5)--(1.5,0));

draw ((1,.5)--(1,0));

[/asy]
2000 Korea - Final Round
The real numbers $a,b,c,x,y,$ and $z$ are such that $a>b>c>0$ and $x>y>z>0$. Prove that

$$\frac {a^2x^2}{(by+cz)(bz+cy)}+\frac{b^2y^2}{(cz+ax)(cx+az)}+\frac{c^2z^2}{(ax+by)(ay+bx)}\ge \frac{3}{4}$$