Navvye/Olympiad-Problems-CSV huggingface dataset:
2005 China Team Selection Test
Let $\omega$ be the circumcircle of acute triangle $ABC$. Two tangents of $\omega$ from $B$ and $C$ intersect at $P$, $AP$ and $BC$ intersect at $D$. Point $E$, $F$ are on $AC$ and $AB$ such that $DE \parallel BA$ and $DF \parallel CA$.

(1) Prove that $F,B,C,E$ are concyclic.



(2) Denote $A_{1}$ the centre of the circle passing through $F,B,C,E$. $B_{1}$, $C_{1}$ are difined similarly. Prove that $AA_{1}$, $BB_{1}$, $CC_{1}$ are concurrent.
2005 China Team Selection Test
Let $n$ be a positive integer, set $S_n = \{ (a_1,a_2,\cdots,a_{2^n}) \mid a_i=0 \ \text{or} \ 1, 1 \leq i \leq 2^n\}$. For any two elements $a=(a_1,a_2,\cdots,a_{2^n})$ and $b=(b_1,b_2,\cdots,b_{2^n})$ of $S_n$, define

$$ d(a,b)= \sum_{i=1}^{2^n} |a_i - b_i| $$

We call $A \subseteq S_n$ a $\textsl{Good Subset}$ if $d(a,b) \geq 2^{n-1}$ holds for any two distinct elements $a$ and $b$ of $A$. How many elements can the $\textsl{Good Subset}$ of $S_n$ at most have?
2005 China Team Selection Test
Find all positive integers $m$ and $n$ such that the inequality:

$$ [ (m+n) \alpha ] + [ (m+n) \beta ] \geq [ m \alpha ] + [n \beta] + [ n(\alpha+\beta)] $$

is true for any real numbers $\alpha$ and $\beta$. Here $[x]$ denote the largest integer no larger than real number $x$.
2005 China Team Selection Test
Given prime number $p$. $a_1,a_2 \cdots a_k$ ($k \geq 3$) are integers not divible by $p$ and have different residuals when divided by $p$. Let

$$ S_n= \{ n \mid 1 \leq n \leq p-1, (na_1)_p < \cdots < (na_k)_p \} $$

Here $(b)_p$ denotes the residual when integer $b$ is divided by $p$. Prove that $|S|< \frac{2p}{k+1}$.
2005 China Team Selection Test
Find the least positive integer $n$ ($n\geq 3$), such that among any $n$ points (no three are collinear) in the plane, there exist three points which are the vertices of a non-isoscele triangle.
2005 China Team Selection Test
Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m - 1$ and $ b^n - 1$ have the same prime divisors, then $ b + 1$ is a power of 2.
2005 China Team Selection Test
In acute angled triangle $ABC$, $BC=a$,$CA=b$,$AB=c$, and $a>b>c$. $I,O,H$ are the incentre, circumcentre and orthocentre of $\triangle{ABC}$ respectively. Point $D \in BC$, $E \in CA$ and $AE=BD$, $CD+CE=AB$. Let the intersectionf of $BE$ and $AD$ be $K$. Prove that $KH \parallel IO$ and $KH = 2IO$.
2005 China Team Selection Test
$n$ is a positive integer, $F_n=2^{2^{n}}+1$. Prove that for $n \geq 3$, there exists a prime factor of $F_n$ which is larger than $2^{n+2}(n+1)$.
2004 China Team Selection Test
Let $\angle XOY = \frac{\pi}{2}$; $P$ is a point inside $\angle XOY$ and we have $OP = 1; \angle XOP = \frac{\pi}{6}.$ A line passes $P$ intersects the Rays $OX$ and $OY$ at $M$ and $N$. Find the maximum value of $OM + ON - MN.$
2004 China Team Selection Test
Let u be a fixed positive integer. Prove that the equation $n! = u^{\alpha} - u^{\beta}$ has a finite number of solutions $(n, \alpha, \beta).$
2004 China Team Selection Test
Let $k \geq 2, 1 < n_1 < n_2 < \ldots < n_k$ are positive integers, $a,b \in \mathbb{Z}^+$ satisfy $$ \prod^k_{i=1} \left( 1 - \frac{1}{n_i} \right) \leq \frac{a}{b} < \prod^{k-1}_{i=1} \left( 1 - \frac{1}{n_i} \right)  $$

Prove that: $$ \prod^k_{i=1} n_i \geq (4 \cdot a)^{2^k - 1}.  $$
2004 China Team Selection Test
Points $D,E,F$ are on the sides $BC, CA$ and $AB$, respectively which satisfy $EF || BC$, $D_1$ is a point on $BC,$ Make $D_1E_1 || D_E, D_1F_1 || DF$ which intersect $AC$ and $AB$ at $E_1$ and $F_1$, respectively. Make $\bigtriangleup PBC \sim \bigtriangleup DEF$ such that $P$ and $A$ are on the same side of $BC.$ Prove that $E, E_1F_1, PD_1$ are concurrent.



[Edit by Darij: See my post #4 below for a possible correction of this problem. However, I am not sure that it is in fact the problem given at the TST... Does anyone have a reliable translation?]
2004 China Team Selection Test
Let $p_1, p_2, \ldots, p_{25}$ are primes which don’t exceed 2004. Find the largest integer $T$ such that every positive integer $\leq T$ can be expressed as sums of distinct divisors of $(p_1\cdot p_2 \cdot \ldots \cdot p_{25})^{2004}.$
2004 China Team Selection Test
Let $a, b, c$ be sides of a triangle whose perimeter does not exceed $2 \cdot \pi.$, Prove that $\sin a, \sin b, \sin c$ are sides of a triangle.
2004 China Team Selection Test
Using $ AB$ and $ AC$ as diameters, two semi-circles are constructed respectively outside the acute triangle $ ABC$. $ AH \perp BC$ at $ H$, $ D$ is any point on side $ BC$ ($ D$ is not coinside with $ B$ or $ C$ ), through $ D$, construct $ DE \parallel AC$ and $ DF \parallel AB$ with $ E$ and $ F$ on the two semi-circles respectively. Show that $ D$, $ E$, $ F$ and $ H$ are concyclic.
2004 China Team Selection Test
Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that each contestant solved at most six problems, and for each pair of a girl and a boy, there was at least one problem that was solved by both the girl and the boy. Show that there is a problem that was solved by at least three girls and at least three boys.
2004 China Team Selection Test
Find all positive integer $ n$ satisfying the following condition: There exist positive integers $ m$, $ a_1$, $ a_2$, $ \cdots$, $ a_{m-1}$, such that $ \displaystyle n = \sum_{i=1}^{m-1} a_i(m-a_i)$, where $ a_1$, $ a_2$, $ \cdots$, $ a_{m-1}$ may not distinct and $ 1 \leq a_i \leq m-1$.
2004 China Team Selection Test
Given integer $ n$ larger than $ 5$, solve the system of equations:

$$  \{ \begin{array}{c} \displaystyle x_1+ x_2+ x_3 + \cdots + x_n = n+2 \\ \\ x_1 + 2x_2 + 3x_3 + \cdots + nx_n = 2n+2 \\ \\ x_1 + 2^2x_2 + 3^2 x_3 + \cdots + n^2x_n = n^2 + n +4 \\ \\ x_1+ 2^3x_2 + 3^3x_3+ \cdots + n^3x_n = n^3 + n + 8 \end{array} $$

where $ x_i \geq 0$, $ i=1,2, \cdots n$.
2004 China Team Selection Test
Convex quadrilateral $ ABCD$ is inscribed in a circle, $ \angle{A}=60^o$, $ BC=CD=1$, rays $ AB$ and $ DC$ intersect at point $ E$, rays $ BC$ and $ AD$ intersect each other at point $ F$. It is given that the perimeters of triangle $ BCE$ and triangle $ CDF$ are both integers. Find the perimeter of quadrilateral $ ABCD$.
2004 China Team Selection Test
$ S$ is a non-empty subset of the set $ \{ 1, 2, \cdots, 108 \}$, satisfying:



(1) For any two numbers $ a,b \in S$ ( may not distinct), there exists $ c \in S$, such that $ \gcd(a,c)=\gcd(b,c)=1$.



(2) For any two numbers $ a,b \in S$ ( may not distinct), there exists $ c' \in S$, $ c' \neq a$, $ c' \neq b$, such that $ \gcd(a, c') > 1$, $ \gcd(b,c') >1$.



Find the largest possible value of $ |S|$.
2004 China Team Selection Test
Let $ m_1$, $ m_2$, $ \cdots$, $ m_r$ (may not distinct) and $ n_1$, $ n_2$ $ \cdots$, $ n_s$ (may not distinct) be two groups of positive integers such that for any positive integer $ d$ larger than $ 1$, the numbers of which can be divided by $ d$ in group $ m_1$, $ m_2$, $ \cdots$, $ m_r$ (including repeated numbers) are no less than that in group $ n_1$, $ n_2$ $ \cdots$, $ n_s$  (including repeated numbers).



Prove that $ \displaystyle \frac{m_1 \cdot m_2 \cdots m_r}{n_1 \cdot n_2 \cdots n_s}$ is integer.
2004 China Team Selection Test
Two equal-radii circles with centres $ O_1$ and $ O_2$ intersect each other at $ P$ and $ Q$, $ O$ is the midpoint of the common chord $ PQ$. Two lines $ AB$ and $ CD$ are drawn through $ P$ ( $ AB$ and $ CD$ are not coincide with $ PQ$ ) such that $ A$ and $ C$ lie on circle $ O_1$ and $ B$ and $ D$ lie on circle $ O_2$. $ M$ and $ N$ are the mipoints of segments $ AD$ and $ BC$ respectively. Knowing that $ O_1$ and $ O_2$ are not in the common part of the two circles, and $ M$, $ N$ are not coincide with $ O$.



Prove that $ M$, $ N$, $ O$ are collinear.
2004 China Team Selection Test
Given arbitrary positive integer $ a$ larger than $ 1$, show that for any positive integer $ n$, there always exists a n-degree integral coefficient polynomial $ p(x)$, such that $ p(0)$, $ p(1)$, $ \cdots$, $ p(n)$ are pairwise distinct positive integers, and all have the form of $ 2a^k+3$, where $ k$ is also an integer.
2004 China Team Selection Test
Find the largest value of the real number $ \lambda$, such that as long as point $ P$ lies in the acute triangle $ ABC$ satisfying $ \angle{PAB}=\angle{PBC}=\angle{PCA}$, and rays $ AP$, $ BP$, $ CP$ intersect the circumcircle of triangles $ PBC$, $ PCA$, $ PAB$ at points $ A_1$, $ B_1$, $ C_1$ respectively, then $ S_{A_1BC}+ S_{B_1CA}+ S_{C_1AB} \geq \lambda S_{ABC}$.
2004 China Team Selection Test
Find the largest positive real $ k$, such that for any positive reals $ a,b,c,d$, there is always:

$$ (a+b+c) \left[ 3^4(a+b+c+d)^5 + 2^4(a+b+c+2d)^5 \right] \geq kabcd^3$$
2004 China Team Selection Test
Find all positive integer $ m$ if there exists prime number $ p$ such that $ n^m-m$ can not be divided by $ p$ for any integer $ n$.
2004 China Team Selection Test
Given non-zero reals $ a$, $ b$, find all functions $ f: \mathbb{R} \longmapsto \mathbb{R}$, such that for every $ x, y \in \mathbb{R}$, $ y \neq 0$, $ f(2x) = af(x) + bx$ and $ \displaystyle f(x)f(y) = f(xy) + f \left( \frac {x}{y} \right)$.
2004 China Team Selection Test
Let $ k$ be a positive integer. Set $ A \subseteq \mathbb{Z}$ is called a $ \textbf{k - set}$ if there exists $ x_1, x_2, \cdots, x_k \in \mathbb{Z}$ such that for any $ i \neq j$, $ (x_i + A) \cap (x_j + A) = \emptyset$, where $ x + A = \{ x + a \mid a \in A \}$. Prove that if $ A_i$ is $ \textbf{k}_i\textbf{ - set}$($ i = 1,2, \cdots, t$), and $ A_1 \cup A_2 \cup \cdots \cup A_t = \mathbb{Z}$, then $ \displaystyle \frac {1}{k_1} + \frac {1}{k_2} + \cdots + \frac {1}{k_t} \geq 1$.
2004 China Team Selection Test
In convex quadrilateral $ ABCD$, $ AB=a$, $ BC=b$, $ CD=c$, $ DA=d$, $ AC=e$, $ BD=f$. If $ \max \{a,b,c,d,e,f \}=1$, then find the maximum value of $ abcd$.
2004 China Team Selection Test
Given sequence $ \{ c_n \}$ satisfying the conditions that $ c_0=1$, $ c_1=0$, $ c_2=2005$, and  $ c_{n+2}=-3c_n - 4c_{n-1} +2008$, ($ n=1,2,3, \cdots$). Let $ \{ a_n \}$ be another sequence such that $ a_n=5(c_{n+1} - c_n) \cdot (502 - c_{n-1} - c_{n-2}) + 4^n \times 2004 \times 501$, ($ n=2,3, \cdots$).



Is $ a_n$ a perfect square for every $ n > 2$?
2004 China Team Selection Test
There are $ n \geq 5$ pairwise different points in the plane. For every point, there are just four points whose distance from which is $ 1$. Find the maximum value of $ n$.
2004 China Team Selection Test
The largest one of numbers $ p_1^{\alpha_1}, p_2^{\alpha_2}, \cdots, p_t^{\alpha_t}$ is called a  $ \textbf{Good Number}$ of positive integer $ n$, if $ \displaystyle n= p_1^{\alpha_1} \cdot p_2^{\alpha_2}  \cdots p_t^{\alpha_t}$, where $ p_1$, $ p_2$, $ \cdots$, $ p_t$ are pairwisely different primes and $ \alpha_1, \alpha_2, \cdots, \alpha_t$ are positive integers. Let $ n_1, n_2, \cdots, n_{10000}$ be $ 10000$ distinct positive integers such that the $ \textbf{Good Numbers}$ of $ n_1, n_2, \cdots, n_{10000}$ are all equal.



Prove that there exist integers $ a_1, a_2, \cdots, a_{10000}$ such that any two of the following $ 10000$ arithmetical progressions $ \{ a_i, a_i + n_i, a_i + 2n_i, a_i + 3n_i, \cdots \}$($ i=1,2, \cdots 10000$) have no common terms.
2003 China Team Selection Test
$ABC$ is an acute-angled triangle. Let $D$ be the point on $BC$ such that $AD$ is the bisector of $\angle A$. Let $E, F$ be the feet of perpendiculars from $D$ to $AC,AB$ respectively. Suppose the lines $BE$ and $CF$ meet at $H$. The circumcircle of triangle $AFH$ meets $BE$ at $G$ (apart from $H$). Prove that the triangle constructed from $BG$, $GE$ and $BF$ is right-angled.
2003 China Team Selection Test
Suppose $A\subseteq \{0,1,\dots,29\}$. It satisfies that for any integer $k$ and any two members $a,b\in A$($a,b$ is allowed to be same), $a+b+30k$ is always not the product of two consecutive integers. Please find $A$ with largest possible cardinality.
2003 China Team Selection Test
Suppose $A\subset \{(a_1,a_2,\dots,a_n)\mid a_i\in \mathbb{R},i=1,2\dots,n\}$. For any $\alpha=(a_1,a_2,\dots,a_n)\in A$ and $\beta=(b_1,b_2,\dots,b_n)\in A$, we define

$$ \gamma(\alpha,\beta)=(|a_1-b_1|,|a_2-b_2|,\dots,|a_n-b_n|), $$ $$ D(A)=\{\gamma(\alpha,\beta)\mid\alpha,\beta\in A\}. $$ Please show that $|D(A)|\geq |A|$.
2003 China Team Selection Test
Find all functions $f: \mathbb{Z}^+\to \mathbb{R}$, which satisfies $f(n+1)\geq f(n)$ for all $n\geq 1$ and $f(mn)=f(m)f(n)$ for all $(m,n)=1$.
2003 China Team Selection Test
Suppose $A=\{1,2,\dots,2002\}$ and $M=\{1001,2003,3005\}$. $B$ is an non-empty subset of $A$. $B$ is called a $M$-free set if the sum of any two numbers in $B$ does not belong to $M$. If $A=A_1\cup A_2$, $A_1\cap A_2=\emptyset$ and $A_1,A_2$ are $M$-free sets, we call the ordered pair $(A_1,A_2)$ a $M$-partition of $A$. Find the number of $M$-partitions of $A$.
2003 China Team Selection Test
Let $ \left(x_{n}\right)$ be a real sequence satisfying $ x_{0}=0$, $ x_{2}=\sqrt[3]{2}x_{1}$, and $ x_{n+1}=\frac{1}{\sqrt[3]{4}}x_{n}+\sqrt[3]{4}x_{n-1}+\frac{1}{2}x_{n-2}$ for every integer $ n\geq 2$, and such that $ x_{3}$ is a positive integer. Find the minimal number of integers belonging to this sequence.
2003 China Team Selection Test
$x$, $y$ and $z$ are positive reals such that $x+y+z=xyz$. Find the minimum value of:

$$ x^7(yz-1)+y^7(zx-1)+z^7(xy-1) $$
2003 China Team Selection Test
In triangle $ABC$, the medians and bisectors corresponding to sides $BC$, $CA$, $AB$ are $m_a$, $m_b$, $m_c$ and $w_a$, $w_b$, $w_c$ respectively. $P=w_a \cap m_b$, $Q=w_b \cap m_c$, $R=w_c \cap m_a$. Denote the areas of triangle $ABC$ and $PQR$ by $F_1$ and $F_2$ respectively. Find the least positive constant $m$ such that $\frac{F_1}{F_2}<m$ holds for any $\triangle{ABC}$.
2003 China Team Selection Test
There is a frog in every vertex of a regular 2n-gon with circumcircle($n \geq 2$). At certain time, all frogs jump to the neighborhood vertices simultaneously (There can be more than one frog in one vertex). We call it as $\textsl{a way of jump}$. It turns out that there is $\textsl{a way of jump}$ with respect to 2n-gon, such that the line connecting any two distinct vertice having frogs on it after the jump, does not pass through the circumcentre of the 2n-gon. Find all possible values of $n$.
2003 China Team Selection Test
Let $ ABCD$ be a quadrilateral which has an incircle centered at $ O$. Prove that

$$ OA\cdot OC+OB\cdot OD=\sqrt{AB\cdot BC\cdot CD\cdot DA}$$
2003 China Team Selection Test
Let $x<y$ be positive integers and $P=\frac{x^3-y}{1+xy}$. Find all integer values that $P$ can take.
2003 China Team Selection Test
The $ n$ roots of a complex coefficient polynomial $ f(z) = z^n + a_1z^{n - 1} + \cdots + a_{n - 1}z + a_n$ are $ z_1, z_2, \cdots, z_n$. If $ \sum_{k = 1}^n |a_k|^2 \leq 1$, then prove that $ \sum_{k = 1}^n |z_k|^2 \leq n$.
2003 China Team Selection Test
$m$ and $n$ are positive integers. Set $A=\{ 1, 2, \cdots, n \}$. Let set $B_{n}^{m}=\{ (a_1, a_2 \cdots, a_m) \mid a_i \in A, i= 1, 2, \cdots, m \}$ satisfying:

(1) $|a_i - a_{i+1}| \neq n-1$, $i=1,2, \cdots, m-1$; and



(2) at least three of $a_1, a_2, \cdots, a_m$ ($m \geq 3$) are pairwise distince.



Find $|B_n^m|$ and $|B_6^3|$.
2003 China Team Selection Test
Can we find positive reals $a_1, a_2, \cdots, a_{2002}$ such that for any positive integer $k$ ($1 \leq k \leq 2002$), every complex root $z$ of polynomial $a_{k+2001}x^{2001}+a_{k+2000}x^{2000}+ \cdots + a_{k+1}x+a_k$ satisfy the condition $|\text{Im} z| \leq |\text{Re} z|$, where $a_{2002+i}=a_i, i=1,2, \cdots, 2001$.
2003 China Team Selection Test
Let $x_0+\sqrt{2003}y_0$ be the minimum positive integer root of Pell function $x^2-2003y^2=1$. Find all the positive integer solutions $(x,y)$ of the equation, such that $x_0$ is divisible by any prime factor of $x$.
2003 China Team Selection Test
In triangle $ABC$, $AB > BC > CA$, $AB=6$, $\angle{B}-\angle{C}=90^o$. The incircle touches $BC$ at $E$ and $EF$ is a diameter of the incircle. Radical $AF$ intersect $BC$ at $D$. $DE$ equals to the circumradius of $\triangle{ABC}$. Find $BC$ and $AC$.
2003 China Team Selection Test
Find all functions $f,g$:$R \to R$ such that $f(x+yg(x))=g(x)+xf(y)$ for $x,y \in R$.
2003 China Team Selection Test
Let $A= \{a_1,a_2, \cdots, a_n \}$ and $B=\{b_1,b_2 \cdots, b_n \}$ be two positive integer sets and $|A \cap B|=1$. $C= \{ \text{all the 2-element subsets of A} \} \cup \{ \text{all the 2-element subsets of B} \}$. Function $f: A \cup B \to \{ 0, 1, 2, \cdots 2 C_n^2 \}$ is injective. For any $\{x,y\} \in C$, denote $|f(x)-f(y)|$ as the $\textsl{mark}$ of $\{x,y\}$. If $n \geq 6$, prove that at least two elements in $C$ have the same $\textsl{mark}$.
2003 China Team Selection Test
Let $S$ be the set of points inside and on the boarder of a regular haxagon with side length 1. Find the least constant $r$, such that there exists one way to colour all the points in $S$ with three colous so that the distance between any two points with same colour is less than $r$.
2003 China Team Selection Test
Denote by $\left(ABC\right)$ the circumcircle of a triangle $ABC$.

Let $ABC$ be an isosceles right-angled triangle with $AB=AC=1$ and $\measuredangle CAB=90^{\circ}$. Let $D$ be the midpoint of the side $BC$, and let $E$ and $F$ be two points on the side $BC$.

Let $M$ be the point of intersection of the circles $\left(ADE\right)$ and $\left(ABF\right)$ (apart from $A$).

Let $N$ be the point of intersection of the line $AF$ and the circle $\left(ACE\right)$ (apart from $A$).

Let $P$ be the point of intersection of the line $AD$ and the circle $\left(AMN\right)$.

Find the length of $AP$.
2003 China Team Selection Test
Sequence $\{ a_n \}$ satisfies: $a_1=3$, $a_2=7$, $a_n^2+5=a_{n-1}a_{n+1}$, $n \geq 2$. If $a_n+(-1)^n$ is prime, prove that there exists a nonnegative integer $m$ such that $n=3^m$.
2003 China Team Selection Test
Let $g(x)= \sum_{k=1}^{n} a_k \cos{kx}$, $a_1,a_2, \cdots, a_n, x \in R$. If $g(x) \geq -1$ holds for every $x \in R$, prove that $\sum_{k=1}^{n}a_k \leq n$.
2003 China Team Selection Test
Positive integer $n$ cannot be divided by $2$ and $3$, there are no nonnegative integers $a$ and $b$ such that $|2^a-3^b|=n$. Find the minimum value of $n$.
2003 China Team Selection Test
(1) $D$ is an arbitary point in $\triangle{ABC}$. Prove that:

$$ \frac{BC}{\min{AD,BD,CD}} \geq \{ \begin{array}{c} \displaystyle 2\sin{A}, \ \angle{A}< 90^o \\ \\ 2, \ \angle{A} \geq 90^o \end{array}  $$

(2)$E$ is an arbitary point in convex quadrilateral $ABCD$. Denote $k$ the ratio of the largest and least distances of any two points among $A$, $B$, $C$, $D$, $E$. Prove that $k \geq 2\sin{70^o}$. Can equality be achieved?
2003 China Team Selection Test
There are $n$($n\geq 3$) circles in the plane, all with radius $1$. In among any three circles, at least two have common point(s), then the total area covered by these $n$ circles is less than $35$.
2003 China Team Selection Test
Given an integer $a_1$($a_1 \neq -1$), find a real number sequence $\{ a_n \}$($a_i \neq 0, i=1,2,\cdots,5$) such that $x_1,x_2,\cdots,x_5$ and $y_1,y_2,\cdots,y_5$ satisfy $b_{i1}x_1+b_{i2}x_2+\cdots +b_{i5}x_5=2y_i$, $i=1,2,3,4,5$, then $x_1y_1+x_2y_2+\cdots+x_5y_5=0$, where $b_{ij}=\prod_{1 \leq k \leq i} (1+ja_k)$.
2003 China Team Selection Test
Given $S$ be the finite lattice (with integer coordinate) set in the $xy$-plane. $A$ is the subset of $S$ with most elements such that the line connecting any two points in $A$ is not parallel to $x$-axis or $y$-axis. $B$ is the subset of integer with least elements such that for any $(x,y)\in S$, $x \in B$ or $y \in B$ holds. Prove that $|A| \geq |B|$.
2003 China Team Selection Test
Triangle $ABC$ is inscribed in circle $O$. Tangent $PD$ is drawn from $A$, $D$ is on ray $BC$, $P$ is on ray $DA$. Line $PU$ ($U \in BD$) intersects circle $O$ at $Q$, $T$, and intersect $AB$ and $AC$ at $R$ and $S$ respectively. Prove that if $QR=ST$, then $PQ=UT$.
2003 China Team Selection Test
Let $S$ be a finite set. $f$ is a function defined on the subset-group $2^S$ of set $S$. $f$ is called $\textsl{monotonic decreasing}$ if when $X \subseteq Y\subseteq S$, then $f(X) \geq f(Y)$ holds. Prove that: $f(X \cup Y)+f(X \cap Y ) \leq f(X)+ f(Y)$ for $X, Y \subseteq S$ if and only if $g(X)=f(X \cup \{ a \}) - f(X)$ is a $\textsl{monotonic decreasing}$ funnction on the subset-group $2^{S \setminus \{a\}}$ of set $S \setminus \{a\}$ for any $a \in S$.
2003 China Team Selection Test
Let $a_{1},a_{2},...,a_{n}$ be positive real number $(n \geq 2)$,not all equal,such that $\sum_{k=1}^n a_{k}^{-2n}=1$,prove that:

$\sum_{k=1}^n a_{k}^{2n}-n^2.\sum_{1 \leq i<j \leq n}(\frac{a_{i}}{a_{j}}-\frac{a_{j}}{a_{i}})^2 >n^2$
2002 China Team Selection Test
Let $E$ and $F$ be the intersections of opposite sides of a convex quadrilateral $ABCD$. The two diagonals meet at $P$. Let $O$ be the foot of the perpendicular from $P$ to $EF$. Show that $\angle BOC=\angle AOD$.
2002 China Team Selection Test
Let $ \left(a_{n}\right)$ be the sequence of reals defined by $ a_{1}=\frac{1}{4}$ and the recurrence $ a_{n}= \frac{1}{4}(1+a_{n-1})^{2}, n\geq 2$. Find the minimum real $ \lambda$ such that for any non-negative reals $ x_{1},x_{2},\dots,x_{2002}$, it holds

$$ \sum_{k=1}^{2002}A_{k}\leq \lambda a_{2002}, $$

where $  A_{k}= \frac{x_{k}-k}{(x_{k}+\cdots+x_{2002}+\frac{k(k-1)}{2}+1)^{2}}, k\geq 1$.
2002 China Team Selection Test
Seventeen football fans were planning to go to Korea to watch the World Cup football match. They selected 17 matches. The conditions of the admission tickets they booked were such that



- One person should book at most one admission ticket for one match;



- At most one match was same in the tickets booked by every two persons;



- There was one person who booked six tickets.



How many tickets did those football fans book at most?
2002 China Team Selection Test
Find all natural numbers $n (n \geq 2)$ such that there exists reals $a_1, a_2, \dots, a_n$ which satisfy $$ \{ |a_i - a_j| \mid 1\leq i<j \leq n\} = \left\{1,2,\dots,\frac{n(n-1)}{2}\right\}. $$



Let $A=\{1,2,3,4,5,6\}, B=\{7,8,9,\dots,n\}$. $A_i(i=1,2,\dots,20)$ contains eight numbers, three of which are chosen from $A$ and the other five numbers from $B$. $|A_i \cap A_j|\leq 2, 1\leq i<j\leq 20$. Find the minimum possible value of $n$.
2002 China Team Selection Test
Given an integer $k$. $f(n)$ is defined on negative integer set and its values are integers. $f(n)$ satisfies $$ f(n)f(n+1)=(f(n)+n-k)^2, $$ for $n=-2,-3,\cdots$. Find an expression of $f(n)$.
2002 China Team Selection Test
Let $$ f(x_1,x_2,x_3) = -2 \cdot (x_1^3+x_2^3+x_3^3) + 3 \cdot (x_1^2(x_2+x_3) + x_2^2 \cdot (x_1+x_3) + x_3^2 \cdot ( x_1+x_2 ) - 12x_1x_2x_3.  $$ For any reals $r,s,t$, we denote $$ g(r,s,t)=\max_{t\leq x_3\leq t+2} |f(r,r+2,x_3)+s|.  $$ Find the minimum value of $g(r,s,t)$.
2002 China Team Selection Test
Given $ n \geq 3$, $ n$ is a integer. Prove that:

$$ (2^n - 2) \cdot \sqrt{2i-1} \geq \left( \sum_{j=0}^{i-1}C_n^j + C_{n-1}^{i-1} \right) \cdot \sqrt{n}$$

where if $ n$ is even, then $ \displaystyle 1 \leq i \leq \frac{n}{2}$; if $ n$ is odd, then $ \displaystyle 1 \leq i \leq \frac{n-1}{2}$.
2002 China Team Selection Test
There are $ n$ points ($ n \geq 4$) on a sphere with radius $ R$, and not all of them lie on the same semi-sphere. Prove that among all the angles formed by any two of the $ n$ points and the sphere centre $ O$ ($ O$ is the vertex of the angle), there is at least one that is not less than $ \displaystyle 2 \arcsin{\frac{\sqrt{6}}{3}}$.
2002 China Team Selection Test
The positive integers $ \alpha, \beta, \gamma$ are the roots of a polynomial $ f(x)$ with degree $ 4$ and the coefficient of the first term is $ 1$. If there exists an integer such that $ f(-1)=f^2(s)$.



Prove that $ \alpha\beta$ is not a perfect square.
2002 China Team Selection Test
If $ P_n(x)=a_0 + a_1x + \cdots + a_nx^n$, $ n \geq 2$ is a real-coefficient polynomial. Prove that if there exists $ a > 0$, making $ \displaystyle P_n(x) = (x + a)^2 \left( \sum_{i=0}^{n-2} b_i x^i \right)$, $ b_i \in R^+$, $ i=0,1,2, \cdots, n-2$, and there is some $ i$, $ 1 \leq i \leq n-1$, making $ a_i^2 - 4a_{i-1}a_{i+1} \leq 0$.