Navvye/Olympiad-Problems-CSV huggingface dataset:
2013 China National Olympiad
Find all positive real numbers $t$ with the following property: there exists an infinite set $X$ of real numbers such that the inequality $$ \max\{|x-(a-d)|,|y-a|,|z-(a+d)|\}>td$$ holds for all (not necessarily distinct) $x,y,z\in X$, all real numbers $a$ and all positive real numbers $d$.
2013 China National Olympiad
Let $n \geqslant 2$ be an integer. There are $n$ finite sets ${A_1},{A_2},\ldots,{A_n}$ which satisfy the condition

$$\left| {{A_i}\Delta {A_j}} \right| = \left| {i - j} \right| \quad \forall i,j \in \left\{ {1,2,...,n} \right\}.$$

Find the minimum of $\sum\limits_{i = 1}^n {\left| {{A_i}} \right|} $.
2013 China National Olympiad
For any positive integer $n$ and $0 \leqslant i \leqslant n$, denote $C_n^i \equiv c(n,i)\pmod{2}$, where $c(n,i) \in \left\{ {0,1} \right\}$. Define

$$f(n,q) = \sum\limits_{i = 0}^n {c(n,i){q^i}}$$

where $m,n,q$ are positive integers and $q + 1 \ne {2^\alpha }$ for any $\alpha  \in \mathbb N$. Prove that if $f(m,q)\left| {f(n,q)} \right.$, then $f(m,r)\left| {f(n,r)} \right.$ for any positive integer $r$.
2013 China National Olympiad
Let $m,n$ be positive integers. Find the minimum positive integer $N$ which satisfies the following condition. If there exists a set $S$ of integers that contains a complete residue system module $m$ such that $| S | = N$, then there exists a nonempty set $A \subseteq S$ so that $n\mid {\sum\limits_{x \in A} x }$.
2012 China National Olympiad
In the triangle $ABC$, $\angle A$ is biggest. On the circumcircle of $\triangle ABC$, let $D$ be the midpoint of $\widehat{ABC}$ and $E$ be the midpoint of $\widehat{ACB}$. The circle $c_1$ passes through $A,B$ and is tangent to $AC$ at $A$, the circle $c_2$ passes through $A,E$ and is tangent $AD$ at $A$. $c_1$ and $c_2$ intersect at $A$ and $P$. Prove that $AP$ bisects $\angle BAC$.



Diagram[asy]

/* File unicodetex not found. */

/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */

import graph; size(14.4cm);

real labelscalefactor = 0.5; /* changes label-to-point distance */

pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */

pen dotstyle = black; /* point style */

real xmin = -5.23, xmax = 9.18, ymin = -2.97, ymax = 4.82; /* image dimensions */

/* draw figures */

draw(circle((-1.32,1.36), 2.98));

draw(circle((3.56,1.53), 3.18));

draw((0.92,3.31)--(-2.72,-1.27));

draw(circle((0.08,0.25), 3.18));

draw((-2.72,-1.27)--(3.13,-0.65));

draw((3.13,-0.65)--(0.92,3.31));

draw((0.92,3.31)--(2.71,-1.54));

draw((-2.41,-1.74)--(0.92,3.31));

draw((0.92,3.31)--(1.05,-0.43));

/* dots and labels */

dot((-1.32,1.36),dotstyle);

dot((0.92,3.31),dotstyle);

label("$A$", (0.81,3.72), NE * labelscalefactor);

label("$c_1$", (-2.81,3.53), NE * labelscalefactor);

dot((3.56,1.53),dotstyle);

label("$c_2$", (3.43,3.98), NE * labelscalefactor);

dot((1.05,-0.43),dotstyle);

label("$P$", (0.5,-0.43), NE * labelscalefactor);

dot((-2.72,-1.27),dotstyle);

label("$B$", (-3.02,-1.57), NE * labelscalefactor);

dot((2.71,-1.54),dotstyle);

label("$E$", (2.71,-1.86), NE * labelscalefactor);

dot((3.13,-0.65),dotstyle);

label("$C$", (3.39,-0.9), NE * labelscalefactor);

dot((-2.41,-1.74),dotstyle);

label("$D$", (-2.78,-2.07), NE * labelscalefactor);

clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);

/* end of picture */[/asy]
2012 China National Olympiad
Let $p$ be a prime. We arrange the numbers in ${\{1,2,\ldots ,p^2} \}$ as a $p \times p$ matrix $A = ( a_{ij} )$. Next we can select any row or column and add $1$ to every number in it, or subtract $1$ from every number in it. We call the arrangement good if we can change every number of the matrix to $0$ in a finite number of such moves. How many good arrangements are there?
2012 China National Olympiad
Prove for any $M>2$, there exists an increasing sequence of positive integers $a_1<a_2<\ldots $ satisfying:

1) $a_i>M^i$ for any $i$;

2) There exists a positive integer $m$ and $b_1,b_2,\ldots ,b_m\in\left\{ -1,1\right\}$, satisfying $n=a_1b_1+a_2b_2+\ldots +a_mb_m$ if and only if $n\in\mathbb{Z}/ \{0\}$.
2012 China National Olympiad
Let $f(x)=(x + a)(x + b)$ where $a,b>0$. For any reals $x_1,x_2,\ldots ,x_n\geqslant 0$ satisfying $x_1+x_2+\ldots +x_n =1$, find the maximum of $F=\sum\limits_{1 \leqslant i < j \leqslant n} {\min \left\{ {f({x_i}),f({x_j})} \right\}} $.
2012 China National Olympiad
Consider a square-free even integer $n$ and a prime $p$, such that

1) $(n,p)=1$;

2) $p\le 2\sqrt{n}$;

3) There exists an integer $k$ such that $p|n+k^2$.

Prove that there exists pairwise distinct positive integers $a,b,c$ such that $n=ab+bc+ca$.



Proposed by Hongbing Yu
2012 China National Olympiad
Find the smallest positive integer $k$ such that, for any subset $A$ of $S=\{1,2,\ldots,2012\}$ with $|A|=k$, there exist three elements $x,y,z$ in $A$ such that $x=a+b$, $y=b+c$, $z=c+a$, where $a,b,c$ are in $S$ and are distinct integers.



Proposed by Huawei Zhu
2011 China National Olympiad
Let $a_1,a_2,\ldots,a_n$ are real numbers, prove that;

$$\sum_{i=1}^na_i^2-\sum_{i=1}^n a_i a_{i+1} \le \left\lfloor \frac{n}{2}\right\rfloor(M-m)^2.$$

where $a_{n+1}=a_1,M=\max_{1\le i\le n} a_i,m=\min_{1\le i\le n} a_i$.
2011 China National Olympiad
On the circumcircle of the acute triangle $ABC$, $D$ is the midpoint of $ \stackrel{\frown}{BC}$. Let $X$ be a point on $ \stackrel{\frown}{BD}$, $E$ the midpoint of $ \stackrel{\frown}{AX}$, and let $S$ lie on $ \stackrel{\frown}{AC}$. The lines $SD$ and $BC$ have intersection $R$, and the lines $SE$ and $AX$ have intersection $T$. If $RT \parallel DE$, prove that the incenter of the triangle $ABC$ is on the line $RT.$
2011 China National Olympiad
Let $A$ be a set consist of finite real numbers,$A_1,A_2,\cdots,A_n$ be nonempty sets of $A$, such that



(a) The sum of the elements of $A$ is $0,$



(b) For all $x_i \in A_i(i=1,2,\cdots,n)$,we have $x_1+x_2+\cdots+x_n>0$.



Prove that there exist $1\le k\le n,$ and $1\le i_1<i_2<\cdots<i_k\le n$, such that

$$|A_{i_1}\bigcup A_{i_2} \bigcup \cdots \bigcup A_{i_k}|<\frac{k}{n}|A|.$$



Where $|X|$ denote the numbers of the elements in set $X$.
2011 China National Olympiad
Let $n$ be an given positive integer, the set $S=\{1,2,\cdots,n\}$.For any nonempty set $A$ and $B$, find the minimum of $|A\Delta S|+|B\Delta S|+|C\Delta S|,$ where $C=\{a+b|a\in A,b\in B\}, X\Delta Y=X\cup Y-X\cap Y.$
2011 China National Olympiad
Let $a_i,b_i,i=1,\cdots,n$ are nonnegitive numbers,and $n\ge 4$,such that $a_1+a_2+\cdots+a_n=b_1+b_2+\cdots+b_n>0$.



Find the maximum of $\frac{\sum_{i=1}^n a_i(a_i+b_i)}{\sum_{i=1}^n b_i(a_i+b_i)}$
2011 China National Olympiad
Let $m,n$ be positive integer numbers. Prove that there exist infinite many couples of positive integer nubmers $(a,b)$ such that

$$a+b| am^a+bn^b , \quad\gcd(a,b)=1.$$
2010 China National Olympiad
Two circles $\Gamma_1$ and $\Gamma_2$ meet at $A$ and $B$. A line through $B$ meets $\Gamma_1$ and $\Gamma_2$ again at $C$ and $D$ repsectively. Another line through $B$ meets $\Gamma_1$ and $\Gamma_2$ again at $E$ and $F$ repsectively. Line $CF$ meets $\Gamma_1$ and $\Gamma_2$ again at $P$ and $Q$ respectively. $M$ and $N$ are midpoints of arc $PB$ and arc $QB$ repsectively. Show that if $CD = EF$, then $C,F,M,N$ are concyclic.
2010 China National Olympiad
Let $k$ be an integer $\geq 3$. Sequence $\{a_n\}$ satisfies that $a_k = 2k$ and for all $n > k$, we have

$$a_n = 

\begin{cases}

  a_{n-1}+1 & \text{if } (a_{n-1},n) = 1 \\
  2n & \text{if } (a_{n-1},n) > 1

\end{cases}

$$

Prove that there are infinitely many primes in the sequence $\{a_n - a_{n-1}\}$.
2010 China National Olympiad
Given complex numbers $a,b,c$, we have that $|az^2 + bz +c| \leq 1$ holds true for any complex number $z, |z| \leq 1$. Find the maximum value of $|bc|$.
2010 China National Olympiad
Let $m,n\ge 1$ and $a_1 < a_2 < \ldots < a_n$ be integers. Prove that there exists a subset $T$ of $\mathbb{N}$ such that

$$|T| \leq 1+ \frac{a_n-a_1}{2n+1}$$

and for every $i \in \{1,2,\ldots , m\}$, there exists $t \in T$ and $s \in [-n,n]$, such that $a_i=t+s$.
2010 China National Olympiad
There is a deck of cards placed at every points $A_1, A_2, \ldots , A_n$ and $O$, where $n \geq 3$. We can do one of the following two operations at each step:

$1)$ If there are more than 2 cards at some points $A_i$, we can withdraw three cards from that deck and place one each at $A_{i-1}, A_{i+1}$ and $O$. (Here $A_0=A_n$ and $A_{n+1}=A_1$);

$2)$ If there are more than or equal to $n$ cards at point $O$, we can withdraw $n$ cards from that deck and place one each at $A_1, A_2, \ldots , A_n$.

Show that if the total number of cards is more than or equal to $n^2+3n+1$, we can make the number of cards at every points more than or equal to $n+1$ after finitely many steps.
2010 China National Olympiad
Suppose $a_1,a_2,a_3,b_1,b_2,b_3$ are distinct positive integers such that

$$(n + 1)a_1^n + na_2^n + (n - 1)a_3^n|(n + 1)b_1^n + nb_2^n + (n - 1)b_3^n$$

holds for all positive integers $n$. Prove that there exists $k\in N$ such that $ b_i = ka_i$ for $ i = 1,2,3$.
2009 China National Olympiad
Given an acute triangle $ PBC$ with $ PB\neq PC.$ Points $ A,D$ lie on $ PB,PC,$ respectively. $ AC$ intersects $ BD$ at point $ O.$ Let $ E,F$ be the feet of perpendiculars from $ O$ to $ AB,CD,$ respectively. Denote by $ M,N$  the midpoints of $ BC,AD.$

$ (1)$: If four points $ A,B,C,D$ lie on one circle, then $ EM\cdot FN = EN\cdot FM.$

$ (2)$: Determine whether the converse of $ (1)$ is true or not, justify your answer.
2009 China National Olympiad
Find all the pairs of prime numbers $ (p,q)$ such that $ pq|5^p+5^q.$
2009 China National Olympiad
Given two integers $ m,n$ satisfying $ 4 < m < n.$ Let $ A_{1}A_{2}\cdots A_{2n + 1}$ be a regular $ 2n+1$ polygon. Denote by $ P$ the set of its vertices. Find the number of convex $ m$ polygon whose vertices belongs to $ P$ and exactly has two acute angles.
2009 China National Olympiad
Given an integer $ n > 3.$ Let $ a_{1},a_{2},\cdots,a_{n}$  be real numbers satisfying $ min |a_{i} - a_{j}| = 1, 1\le i\le j\le n.$ Find the minimum value of $ \sum_{k = 1}^n|a_{k}|^3.$
2009 China National Olympiad
Let $ P$ be a convex $ n$ polygon each of which sides and diagnoals is colored with one of $ n$ distinct colors. For which $ n$ does: there exists a coloring method such that for any three of $ n$ colors, we can always find one triangle whose vertices is of $ P$' and whose sides is colored by the three colors respectively.
2009 China National Olympiad
Given an integer $ n > 3.$ Prove that there exists a set $ S$ consisting of $ n$ pairwisely distinct positive integers such that for any two different non-empty subset of $ S$:$ A,B, \frac {\sum_{x\in A}x}{|A|}$ and $ \frac {\sum_{x\in B}x}{|B|}$ are two composites which share no common divisors.
2008 China National Olympiad
Suppose $\triangle ABC$ is scalene. $O$ is the circumcenter and $A'$ is a point on the extension of segment $AO$ such that $\angle BA'A = \angle CA'A$. Let point $A_1$ and $A_2$ be foot of perpendicular from $A'$ onto $AB$ and $AC$. $H_{A}$ is the foot of perpendicular from $A$ onto $BC$. Denote $R_{A}$ to be the radius of circumcircle of $\triangle H_{A}A_1A_2$. Similiarly we can define $R_{B}$ and $R_{C}$. Show that:

$$\frac{1}{R_{A}} + \frac{1}{R_{B}} + \frac{1}{R_{C}} = \frac{2}{R}$$

where R is the radius of circumcircle of $\triangle ABC$.
2008 China National Olympiad
Given an integer $n\ge3$, prove that the set $X=\{1,2,3,\ldots,n^2-n\}$ can be divided into two non-intersecting subsets such that neither of them contains $n$ elements $a_1,a_2,\ldots,a_n$ with $a_1<a_2<\ldots<a_n$ and $a_k\le\frac{a_{k-1}+a_{k+1}}2$ for all $k=2,\ldots,n-1$.
2008 China National Olympiad
Given a positive integer $n$ and $x_1 \leq x_2 \leq \ldots \leq x_n, y_1 \geq y_2 \geq \ldots \geq y_n$, satisfying

$$\displaystyle\sum_{i = 1}^{n} ix_i = \displaystyle\sum_{i = 1}^{n} iy_i$$

Show that for any real number $\alpha$, we have

$$\displaystyle\sum_{i =1}^{n} x_i[i\alpha] \geq \displaystyle\sum_{i =1}^{n} y_i[i\alpha]$$



Here $[\beta]$ denotes the greastest integer not larger than $\beta$.
2008 China National Olympiad
Let $A$ be an infinite subset of $\mathbb{N}$, and $n$ a fixed integer. For any prime $p$ not dividing $n$, There are infinitely many elements of $A$ not divisible by $p$. Show that for any integer $m >1, (m,n) =1$, There exist finitely many elements of $A$, such that their sum is congruent to 1 modulo $m$ and congruent to 0 modulo $n$.
2008 China National Olympiad
Find the smallest integer $n$ satisfying the following condition: regardless of how one colour the vertices of a regular $n$-gon with either red, yellow or blue, one can always find an isosceles trapezoid whose vertices are of the same colour.
2008 China National Olympiad
Find all triples $(p,q,n)$ that satisfy

$$q^{n+2} \equiv 3^{n+2} (\mod p^n) ,\quad p^{n+2} \equiv 3^{n+2} (\mod q^n)$$

where $p,q$ are odd primes and $n$ is an positive integer.
2007 China National Olympiad
Given complex numbers $a, b, c$, let $|a+b|=m, |a-b|=n$. If $mn \neq 0$, Show that

$$\max \{|ac+b|,|a+bc|\} \geq \frac{mn}{\sqrt{m^2+n^2}}$$
2007 China National Olympiad
Show that:

1) If $2n-1$ is a prime number, then for any $n$ pairwise distinct positive integers $a_1, a_2, \ldots , a_n$, there exists $i, j \in \{1, 2, \ldots , n\}$ such that

$$\frac{a_i+a_j}{(a_i,a_j)} \geq 2n-1$$

2) If $2n-1$ is a composite number, then there exists $n$ pairwise distinct positive integers $a_1, a_2, \ldots , a_n$, such that for any $i, j \in \{1, 2, \ldots , n\}$ we have

$$\frac{a_i+a_j}{(a_i,a_j)} < 2n-1$$



Here $(x,y)$ denotes the greatest common divisor of $x,y$.
2007 China National Olympiad
Let $a_1, a_2, \ldots , a_{11}$ be 11 pairwise distinct positive integer with sum less than 2007. Let S be the sequence of  $1,2, \ldots ,2007$. Define an operation to be 22 consecutive applications of the following steps on the sequence $S$: on $i$-th step, choose a number from the sequense $S$ at random, say $x$. If $1 \leq i \leq 11$, replace $x$ with $x+a_i$ ; if $12 \leq i \leq 22$, replace $x$ with $x-a_{i-11}$ . If the result of operation on the sequence $S$ is an odd permutation of $\{1, 2, \ldots , 2007\}$, it is an odd operation; if the result of operation on the sequence $S$ is an even permutation of $\{1, 2, \ldots , 2007\}$, it is an even operation. Which is larger, the number of odd operation or the number of even permutation? And by how many?



Here $\{x_1, x_2, \ldots , x_{2007}\}$ is an even permutation of $\{1, 2, \ldots ,2007\}$ if the product $\prod_{i > j} (x_i - x_j)$ is positive, and an odd one otherwise.
2007 China National Olympiad
Let $O, I$ be the circumcenter and incenter of triangle $ABC$. The incircle of $\triangle ABC$ touches $BC, CA, AB$ at points $D, E, F$ repsectively. $FD$ meets $CA$ at $P$, $ED$ meets $AB$ at $Q$. $M$ and $N$ are midpoints of $PE$ and $QF$ respectively. Show that $OI \perp MN$.
2007 China National Olympiad
Let $\{a_n\}_{n \geq 1}$ be a bounded sequence satisfying

$$a_n < \displaystyle\sum_{k=a}^{2n+2006} \frac{a_k}{k+1} + \frac{1}{2n+2007} \quad \forall \quad n = 1, 2, 3, \ldots $$

Show that

$$a_n < \frac{1}{n} \quad \forall \quad n = 1, 2, 3, \ldots$$
2007 China National Olympiad
Find a number $n \geq 9$ such that for any $n$ numbers, not necessarily distinct, $a_1,a_2, \ldots , a_n$, there exists 9 numbers $a_{i_1}, a_{i_2}, \ldots , a_{i_9}, (1 \leq i_1 < i_2 < \ldots < i_9 \leq n)$ and $b_i \in {4,7}, i =1, 2, \ldots , 9$ such that $b_1a_{i_1} + b_2a_{i_2} + \ldots + b_9a_{i_9}$ is a multiple of 9.
2006 China National Olympiad
Let $a_1,a_2,\ldots,a_k$ be  real  numbers  and $a_1+a_2+\ldots+a_k=0$.  Prove that

$$ \max_{1\leq i \leq k} a_i^2 \leq \frac{k}{3} \left( (a_1-a_2)^2+(a_2-a_3)^2+\cdots +(a_{k-1}-a_k)^2\right).  $$
2006 China National Olympiad
For positive integers $a_1,a_2 ,\ldots,a_{2006}$ such that $\frac{a_1}{a_2},\frac{a_2}{a_3},\ldots,\frac{a_{2005}}{a_{2006}}$ are pairwise distinct, find the minimum possible amount of distinct positive integers in the set$\{a_1,a_2,...,a_{2006}\}$.
2006 China National Olympiad
Positive integers $k, m, n$ satisfy $mn=k^2+k+3$, prove that at least one of the equations $x^2+11y^2=4m$ and $x^2+11y^2=4n$ has an odd solution.
2006 China National Olympiad
In a right angled-triangle $ABC$, $\angle{ACB} = 90^o$. Its incircle $O$ meets $BC$, $AC$, $AB$ at $D$,$E$,$F$ respectively. $AD$ cuts $O$ at $P$. If $\angle{BPC} = 90^o$, prove $AE + AP = PD$.
2006 China National Olympiad
Let $\{a_n\}$ be a sequence such that: $a_1 = \frac{1}{2}$, $a_{k+1}=-a_k+\frac{1}{2-a_k}$ for all $k = 1, 2,\ldots$. Prove that

$$ \left(\frac{n}{2(a_1+a_2+\cdots+a_n)}-1\right)^n \leq \left(\frac{a_1+a_2+\cdots+a_n}{n}\right)^n\left(\frac{1}{a_1}-1\right)\left(\frac{1}{a_2}-1\right)\cdots \left(\frac{1}{a_n}-1\right). $$
2006 China National Olympiad
Suppose $X$ is a set with $|X| = 56$. Find the minimum value of $n$, so that for any 15 subsets of $X$, if the cardinality of the union of any 7 of them is greater or equal to $n$, then there exists 3 of them whose intersection is nonempty.
2005 China National Olympiad
Suppose  $\theta_{i}\in(-\frac{\pi}{2},\frac{\pi}{2}), i = 1,2,3,4$. Prove that, there exist $x\in \mathbb{R}$, satisfying two inequalities \begin{eqnarray*} \cos^2\theta_1\cos^2\theta_2-(\sin\theta\sin\theta_2-x)^2 &\geq& 0, \\ \cos^2\theta_3\cos^2\theta_4-(\sin\theta_3\sin\theta_4-x)^2 & \geq & 0 \end{eqnarray*}

if and only if $$ \sum^4_{i=1}\sin^2\theta_i\leq2(1+\prod^4_{i=1}\sin\theta_i + \prod^4_{i=1}\cos\theta_i). $$
2005 China National Olympiad
A circle meets the three sides $BC,CA,AB$ of a triangle $ABC$ at points $D_1,D_2;E_1,E_2; F_1,F_2$ respectively. Furthermore, line segments $D_1E_1$ and $D_2F_2$ intersect at point $L$, line segments $E_1F_1$ and $E_2D_2$ intersect at point $M$, line segments $F_1D_1$ and $F_2E_2$ intersect at point $N$. Prove that the lines $AL,BM,CN$ are concurrent.
2005 China National Olympiad
As the graph, a pond is divided into 2n (n $\geq$ 5) parts. Two parts are called neighborhood if they have a common side or arc. Thus every part has three neighborhoods. Now there are 4n+1 frogs at the pond. If there are three or more frogs at one part, then three of the frogs of the part will jump to the three neighborhoods repsectively. Prove that for some time later, the frogs at the pond will uniformily distribute. That is, for any part either there are frogs at the part or there are frogs at the each of its neighborhoods.



Invalid image file
2005 China National Olympiad
The sequence $\{a_n\}$ is defined by: $a_1=\frac{21}{16}$, and for $n\ge2$,$$ 2a_n-3a_{n-1}=\frac{3}{2^{n+1}}. $$Let $m$ be an integer with $m\ge2$. Prove that: for $n\le m$, we have$$ \left(a_n+\frac{3}{2^{n+3}}\right)^{\frac{1}{m}}\left(m-\left(\frac{2}{3}\right)^{{\frac{n(m-1)}{m}}}\right)<\frac{m^2-1}{m-n+1}. $$
2005 China National Olympiad
There are 5 points in a rectangle (including its boundary) with area 1, no three of them are in the same line. Find the minimum number of triangles with the area not more than $\frac 1{4}$, vertex of which are three of the five points.
2005 China National Olympiad
Find all nonnegative integer solutions $(x,y,z,w)$ of the equation$$2^x\cdot3^y-5^z\cdot7^w=1.$$
2004 China National Olympiad
Let $EFGH,ABCD$ and $E_1F_1G_1H_1$ be three convex quadrilaterals satisfying:



i) The points $E,F,G$ and $H$ lie on the sides $AB,BC,CD$ and $DA$ respectively, and $\frac{AE}{EB}\cdot\frac{BF}{FC}\cdot \frac{CG}{GD}\cdot \frac{DH}{HA}=1$;

ii) The points $A,B,C$ and $D$ lie on sides $H_1E_1,E_1F_1,F_1,G_1$ and $G_1H_1$ respectively, and $E_1F_1||EF,F_1G_1||FG,G_1H_1||GH,H_1E_1||HE$.



Suppose that $\frac{E_1A}{AH_1}=\lambda$. Find an expression for $\frac{F_1C}{CG_1}$ in terms of $\lambda$.



Xiong Bin
2004 China National Olympiad
Let $c$ be a positive integer. Consider the sequence $x_1,x_2,\ldots$ which satisfies $x_1=c$ and, for $n\ge 2$,

$$x_n=x_{n-1}+\left\lfloor\frac{2x_{n-1}-(n+2)}{n}\right\rfloor+1$$

where $\lfloor x\rfloor$ denotes the largest integer not greater than $x$. Determine an expression for $x_n$ in terms of $n$ and $c$.



Huang Yumin
2004 China National Olympiad
Let $M$ be a set consisting of $n$ points in the plane, satisfying:

i) there exist $7$ points in $M$ which constitute the vertices of a convex heptagon;

ii) if for any $5$ points in $M$ which constitute the vertices of a convex pentagon, then there is a point in $M$ which lies in the interior of the pentagon.

Find the minimum value of $n$.



Leng Gangsong
2004 China National Olympiad
For a given real number $a$ and a positive integer $n$, prove that:

i) there exists exactly one sequence of real numbers $x_0,x_1,\ldots,x_n,x_{n+1}$ such that

$$\begin{cases} x_0=x_{n+1}=0,\\ \frac{1}{2}(x_i+x_{i+1})=x_i+x_i^3-a^3,\ i=1,2,\ldots,n.\end{cases}$$

ii) the sequence $x_0,x_1,\ldots,x_n,x_{n+1}$ in i) satisfies $|x_i|\le |a|$ where $i=0,1,\ldots,n+1$.



Liang Yengde
2004 China National Olympiad
For a given positive integer $n\ge 2$, suppose positive integers $a_i$ where $1\le i\le n$ satisfy $a_1<a_2<\ldots <a_n$ and $\sum_{i=1}^n \frac{1}{a_i}\le 1$. Prove that, for any real number $x$, the following inequality holds

$$\left(\sum_{i=1}^n\frac{1}{a_i^2+x^2}\right)^2\le\frac{1}{2}\cdot\frac{1}{a_1(a_1-1)+x^2} $$



Li Shenghong
2004 China National Olympiad
Prove that every positive integer $n$, except a finite number of them, can be represented as a sum of $2004$ positive integers: $n=a_1+a_2+\cdots +a_{2004}$, where $1\le a_1<a_2<\cdots <a_{2004}$, and $a_i \mid a_{i+1}$ for all $1\le i\le 2003$.



Chen Yonggao
2003 China National Olympiad
Let $I$ and $H$ be the incentre and orthocentre of triangle $ABC$ respectively. Let $P,Q$ be the midpoints of $AB,AC$. The rays $PI,QI$ intersect $AC,AB$ at $R,S$ respectively. Suppose that $T$ is the circumcentre of triangle $BHC$. Let $RS$ intersect $BC$ at $K$. Prove that $A,I$ and $T$ are collinear if and only if $[BKS]=[CKR]$.



Shen Wunxuan
2003 China National Olympiad
Determine the maximal size of the set $S$ such that:

i) all elements of $S$ are natural numbers not exceeding $100$;

ii) for any two elements $a,b$ in $S$, there exists $c$ in $S$ such that $(a,c)=(b,c)=1$;

iii) for any two elements $a,b$ in $S$, there exists $d$ in $S$ such that $(a,d)>1,(b,d)>1$.



Yao Jiangang
2003 China National Olympiad
Given a positive integer $n$, find the least $\lambda>0$ such that for any $x_1,\ldots x_n\in \left(0,\frac{\pi}{2}\right)$, the condition $\prod_{i=1}^{n}\tan x_i=2^{\frac{n}{2}}$ implies $\sum_{i=1}^{n}\cos x_i\le\lambda$.



Huang Yumin
2003 China National Olympiad
Find all integer triples $(a,m,n)$ such that $a^m+1|a^n+203$ where $a,m>1$.



Chen Yonggao
2003 China National Olympiad
Ten people apply for a job. The manager decides to interview the candidates one by one according to the following conditions:

i) the first three candidates will not be employed;

ii) from the fourth candidates onwards, if a candidate's comptence surpasses the competence of all those who preceded him, then that candidate is employed;

iii) if the first nine candidates are not employed, then the tenth candidate will be employed.

We assume that none of the $10$ applicants have the same competence, and these competences can be ranked from the first to tenth. Let $P_k$ represent the probability that the $k$th-ranked applicant in competence is employed. Prove that:

i) $P_1>P_2>\ldots>P_8=P_9=P_{10}$;

ii) $P_1+P_2+P_3>0.7$

iii) $P_8+P_9+P_{10}\le 0.1$.



Su Chun
2003 China National Olympiad
Suppose $a,b,c,d$ are positive reals such that $ab+cd=1$ and $x_i,y_i$ are real numbers such that $x_i^2+y_i^2=1$ for $i=1,2,3,4$. Prove that

$$(ax_1+bx_2+cx_3+dx_4)^2+(ay_4+by_3+cy_2+dy_1)^2\le 2\left(\frac{a^2+b^2}{ab}+\frac{c^2+d^2}{cd}\right).$$



Li Shenghong
2002 China National Olympiad
the edges of triangle $ABC$ are $a,b,c$ respectively,$b<c$,$AD$ is the bisector of $\angle A$,point $D$ is on segment $BC$.

(1)find the property $\angle A$,$\angle B$,$\angle C$ have,so that there exists point $E,F$ on $AB,AC$ satisfy $BE=CF$,and $\angle NDE=\angle CDF$

(2)when such $E,F$ exist,express $BE$ with $a,b,c$