Navvye/Olympiad-Problems-CSV huggingface dataset:
2002 China Team Selection Test
$ A_1$, $ B_1$ and $ C_1$ are the projections of the vertices $ A$, $ B$ and $ C$ of triangle $ ABC$ on the respective sides. If $ AB = c$, $ AC = b$, $ BC = a$ and $ AC_1 = 2t AB$, $ BA_1 = 2rBC$, $ CB_1 = 2 \mu AC$. Prove that:



$$ \frac {a^2}{b^2} \cdot \left( \frac {t}{1 - 2t} \right)^2 + \frac {b^2}{c^2} \cdot \left( \frac {r}{1 - 2r} \right)^2 + \frac {c^2}{a^2} \cdot \left( \frac {\mu}{1 - 2\mu} \right)^2 + 16tr \mu \geq 1

$$
2002 China Team Selection Test
Let $ p_i \geq 2$, $ i = 1,2, \cdots n$ be $ n$ integers such that any two of them are relatively prime. Let:



$$ P = \{ x = \sum_{i = 1}^{n} x_i \prod_{j = 1, j \neq i}^{n} p_j \mid x_i \text{is a non - negative integer}, i = 1,2, \cdots n \}

$$

Prove that the biggest integer $ M$ such that $ M \not\in P$ is greater than $ \displaystyle \frac {n - 2}{2} \cdot \prod_{i = 1}^{n} p_i$, and also find $ M$.
2002 China Team Selection Test
Given triangle $ ABC$ and $ AB=c$, $ AC=b$ and $ BC=a$ satisfying $ a \geq b \geq c$, $ BE$ and $ CF$ are two interior angle bisectors. $ P$ is a point inside triangle $ AEF$. $ R$ and $ Q$ are the projections of $ P$ on sides $ AB$ and $ AC$.



Prove that $ PR + PQ + RQ < b$.
2002 China Team Selection Test
For any two rational numbers $ p$ and $ q$ in the interval $ (0,1)$ and function $ f$, there is always $ \displaystyle f \left( \frac{p+q}{2} \right) \leq \frac{f(p) + f(q)}{2}$. Then prove that for any rational numbers $ \lambda, x_1, x_2 \in (0,1)$, there is always:



$$ f( \lambda x_1 + (1-\lambda) x_2 ) \leq \lambda f(x_i) + (1-\lambda) f(x_2)$$
2002 China Team Selection Test
Sequence $ \{ f_n(a) \}$ satisfies $ \displaystyle f_{n+1}(a) = 2 - \frac{a}{f_n(a)}$, $ f_1(a) = 2$, $ n=1,2, \cdots$. If there exists a natural number $ n$, such that $ f_{n+k}(a) = f_{k}(a), k=1,2, \cdots$, then we call the non-zero real $ a$ a $ \textbf{periodic point}$ of $ f_n(a)$.



Prove that the sufficient and necessary condition for $ a$ being a $ \textbf{periodic point}$ of $ f_n(a)$ is $ p_n(a-1)=0$, where $ \displaystyle p_n(x)=\sum_{k=0}^{\left[ \frac{n-1}{2} \right]} (-1)^k C_n^{2k+1}x^k$, here we define $ \displaystyle \frac{a}{0}= \infty$ and $ \displaystyle \frac{a}{\infty} = 0$.
2002 China Team Selection Test
Given a positive integer $ n$, for all positive integers $ a_1, a_2, \cdots, a_n$ that satisfy $ a_1 = 1$, $ a_{i + 1} \leq a_i + 1$, find $ \displaystyle \sum_{i = 1}^{n} a_1a_2 \cdots a_i$.
2002 China Team Selection Test
Circles $ \omega_{1}$ and $ \omega_{2}$ intersect at points $ A$ and $ B.$ Points $ C$ and $ D$ are on circles $ \omega_{1}$ and $ \omega_{2},$ respectively,

such that lines $ AC$ and $ AD$ are tangent to circles $ \omega_{2}$ and $ \omega_{1},$ respectively.

Let $ I_{1}$ and $ I_{2}$ be the incenters of triangles $ ABC$ and $ ABD,$ respectively. Segments $ I_{1}I_{2}$ and $ AB$ intersect at $ E$.

Prove that: $ \frac {1}{AE} = \frac {1}{AC} + \frac {1}{AD}$
2002 China Team Selection Test
Find all groups of positive integers $ (a,x,y,n,m)$ that satisfy $ a(x^n - x^m) = (ax^m - 4) y^2$ and $ m \equiv n \pmod{2}$ and $ ax$ is odd.
2002 China Team Selection Test
In acute triangle $ ABC$, show that:



$ \sin^3{A}\cos^2{(B - C)} + \sin^3{B}\cos^2{(C - A)} + \sin^3{C}\cos^2{(A - B)} \leq 3\sin{A} \sin{B} \sin{C}$



and find out when the equality holds.
2002 China Team Selection Test
$ m$ and $ n$ are positive integers. In a $ 8 \times 8$ chessboard, $ (m,n)$ denotes the number of grids a Horse can jump in a chessboard ($ m$ horizontal $ n$ vertical or $ n$ horizontal $ m$ vertical ). If a $ (m,n) \textbf{Horse}$ starts from one grid, passes every grid once and only once, then we call this kind of Horse jump route a $ \textbf{H Route}$. For example, the $ (1,2) \textbf{Horse}$ has its $ \textbf{H Route}$. Find the smallest positive integer $ t$, such that from any grid of the chessboard, the $ (t,t+1) \textbf{Horse}$ does not has any $ \textbf{H Route}$.
2002 China Team Selection Test
$ n$ sets $ S_1$, $ S_2$ $ \cdots$, $ S_n$ consists of non-negative numbers. $ x_i$ is the sum of all elements of $ S_i$, prove that there is a natural number $ k$, $ 1<k<n$, and:



$$ \sum_{i=1}^n x_i < \frac{1}{k+1} \left[ k \cdot \frac{n(n+1)(2n+1)}{6} - (k+1)^2 \cdot \frac{n(n+1)}{2} \right]$$

and there exists subscripts $ i$, $ j$, $ t$, and $ l$ (at least $ 3$ of them are distinct) such that $ x_i + x_j = x_t + x_l$.
2002 China Team Selection Test
Given that $ a_1=1$, $ a_2=5$, $ \displaystyle a_{n+1} = \frac{a_n \cdot a_{n-1}}{\sqrt{a_n^2 + a_{n-1}^2 + 1}}$. Find a expression of the general term of $ \{ a_n \}$.
2002 China Team Selection Test
$ \odot O_1$ and $ \odot O_2$ meet at points $ P$ and $ Q$. The circle through $ P$, $ O_1$ and $ O_2$ meets $ \odot O_1$ and $ \odot O_2$ at points $ A$ and $ B$. Prove that the distance from $ Q$ to the lines $ PA$, $ PB$ and $ AB$ are equal.



(Prove the following three cases: $ O_1$ and $ O_2$ are in the common space of $ \odot O_1$ and $ \odot O_2$; $ O_1$ and $ O_2$ are out of the common space of $ \odot O_1$ and $ \odot O_2$; $ O_1$ is in the common space of $ \odot O_1$ and $ \odot O_2$, $ O_2$ is out of the common space of $ \odot O_1$ and $ \odot O_2$.
2002 China Team Selection Test
There is a game. The magician let the participant think up a positive integer (at least two digits). For example, an integer $ \displaystyle\overline{a_1a_2 \cdots a_n}$ is rearranged as $ \overline{a_{i_1}a_{i_2} \cdots a_{i_n}}$, that is, $ i_1, i_2, \cdots, i_n$ is a permutation of $ 1,2, \cdots, n$. Then we get $ n!-1$ integers. The participant is asked to calculate the sum of the $ n!-1$ numbers, then tell the magician the sum $ S$. The magician claims to be able to know the original number when he is told the sum $ S$. Try to decide whether the magician can be successful or not.
2002 China Team Selection Test
Circle $ O$ is inscribed in a trapzoid $ ABCD$, $ \angle{A}$ and $ \angle{B}$ are all acute angles. A line through $ O$ intersects $ AD$ at $ E$ and $ BC$ at $ F$, and satisfies the following conditions:



(1) $ \angle{DEF}$ and $ \angle{CFE}$ are acute angles.



(2) $ AE+BF=DE+CF$.



Let $ AB=a$, $ BC=b$, $ CD=c$, then use $ a,b,c$ to express $ AE$.
2002 China Team Selection Test
Find all non-negative integers $m$ and $n$, such that $(2^n-1) \cdot (3^n-1)=m^2$.
2002 China Team Selection Test
Given positive integer $ m \geq 17$, $ 2m$ contestants participate in a circular competition. In each round, we devide the $ 2m$ contestants into $ m$ groups, and the two contestants in one group play against each other. The groups are re-divided in the next round. The contestants compete for $ 2m-1$ rounds so that each contestant has played a game with all the $ 2m-1$ players. Find the least possible positive integer $ n$, so that there exists a valid competition and after $ n$ rounds, for any $ 4$ contestants, non of them has played with the others or there have been at least $ 2$ games played within those $ 4$.
2002 China Team Selection Test
$ A$ is a set of points on the plane, $ L$ is a line on the same plane. If $ L$ passes through one of the points in $ A$, then we call that $ L$ passes through $ A$.



(1) Prove that we can divide all the rational points into $ 100$ pairwisely non-intersecting point sets with infinity elements. If for any line on the plane, there are two rational points on it, then it passes through all the $ 100$ sets.



(2) Find the biggest integer $ r$, so that if we divide all the rational points on the plane into $ 100$ pairwisely non-intersecting point sets with infinity elements with any method, then there is at least one line that passes through $ r$ sets of the $ 100$ point sets.
2002 China Team Selection Test
Does there exist $ 2002$ distinct positive integers $ k_1, k_2, \cdots k_{2002}$ such that for any positive integer $ n \geq 2001$, one of $ k_12^n + 1, k_22^n + 1, \cdots, k_{2002}2^n + 1$ is prime?
2002 China Team Selection Test
$ c$ is a positive integer, $ \alpha, \beta, \gamma$ are pairwise distinct positive integers, and $$  \{ \begin{array}{c} \displaystyle a= \alpha + \beta + \gamma \\ \\ b= \alpha \cdot \beta + \beta \cdot \gamma + \gamma \cdot \alpha \\ \\ c^2 = \alpha\beta\gamma \end{array} $$

A number $ \lambda$ satisfies the condition $ \lambda^4 -2a\lambda^2 + 8c\lambda + a^2 - 4b = 0$.



Prove that $ \lambda$ is an integer if and only if $ \alpha, \beta, \gamma$ are all perfect squares.
2001 China Team Selection Test
$E$ and $F$ are interior points of convex quadrilateral $ABCD$ such that $AE = BE$, $CE = DE$, $\angle AEB = \angle CED$, $AF = DF$, $BF = CF$, $\angle AFD = \angle BFC$.  Prove that $\angle AFD + \angle AEB = \pi$.
2001 China Team Selection Test
$a$ and $b$ are natural numbers such that $b > a > 1$, and $a$ does not divide $b$.  The sequence of natural numbers $\{b_n\}_{n=1}^\infty$ satisfies $b_{n + 1} \geq 2b_n \forall n \in \mathbb{N}$.  Does there exist a sequence $\{a_n\}_{n=1}^\infty$ of natural numbers such that for all $n \in \mathbb{N}$, $a_{n + 1} - a_n \in \{a, b\}$, and for all $m, l \in \mathbb{N}$ ($m$ may be equal to $l$), $a_m + a_l \not\in \{b_n\}_{n=1}^\infty$?
2001 China Team Selection Test
For a given natural number $k > 1$, find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for all $x, y \in \mathbb{R}$, $f[x^k + f(y)] = y +[f(x)]^k$.
2001 China Team Selection Test
For a given natural number $n > 3$, the real numbers $x_1, x_2, \ldots, x_n, x_{n + 1}, x_{n + 2}$ satisfy the conditions $0

< x_1 < x_2 < \cdots < x_n < x_{n + 1} < x_{n + 2}$.  Find the minimum possible value of

$$\frac{(\sum _{i=1}^n \frac{x_{i + 1}}{x_i})(\sum _{j=1}^n \frac{x_{j + 2}}{x_{j +

1}})}{(\sum _{k=1}^n \frac{x_{k + 1} x_{k + 2}}{x_{k + 1}^2 + x_k

x_{k + 2}})(\sum _{l=1}^n \frac{x_{l + 1}^2 + x_l x_{l + 2}}{x_l

x_{l + 1}})}$$ and find all $(n + 2)$-tuplets of real numbers $(x_1, x_2, \ldots, x_n, x_{n + 1}, x_{n + 2})$ which gives this value.
2001 China Team Selection Test
In the equilateral $\bigtriangleup ABC$, $D$ is a point on side $BC$. $O_1$ and $I_1$ are the circumcenter and incenter of $\bigtriangleup ABD$ respectively, and $O_2$ and $I_2$ are the circumcenter and incenter of $\bigtriangleup ADC$ respectively.  $O_1I_1$ intersects $O_2I_2$ at $P$.  Find the locus of point $P$ as $D$ moves along $BC$.
2001 China Team Selection Test
Let $F = \max_{1 \leq x \leq 3} |x^3 - ax^2 - bx - c|$.  When $a$, $b$, $c$ run over all the real numbers, find the smallest possible value of $F$.
2000 China Team Selection Test
Let $ABC$ be a triangle such that $AB = AC$. Let $D,E$ be points on $AB,AC$ respectively such that $DE = AC$. Let $DE$ meet the circumcircle of triangle $ABC$ at point $T$. Let $P$ be a point on $AT$. Prove that $PD + PE = AT$ if and only if $P$ lies on the circumcircle of triangle $ADE$.
2000 China Team Selection Test
Given positive integers $k, m, n$ such that $1 \leq  k \leq  m \leq  n$. Evaluate



$$\sum^{n}_{i=0} \frac{(-1)^i}{n+k+i} \cdot \frac{(m+n+i)!}{i!(n-i)!(m+i)!}.$$
2000 China Team Selection Test
For positive integer $a  \geq 2$, denote $N_a$ as the number of positive integer $k$ with the following property: the sum of squares of digits of $k$ in base a representation equals $k$. Prove that:



a.) $N_a$ is odd;



b.) For every positive integer $M$, there exist a positive integer $a \geq  2$ such that $N_a \geq  M$.
2000 China Team Selection Test
Let $F$ be the set of all polynomials $\Gamma$ such that all the coefficients of $\Gamma (x)$ are integers and $\Gamma (x) = 1$ has integer roots. Given a positive intger $k$, find the smallest integer $m(k) > 1$ such that there exist $\Gamma \in F$ for which $\Gamma (x) = m(k)$ has exactly $k$ distinct integer roots.
2000 China Team Selection Test
a.) Let $a,b$ be real numbers. Define sequence $x_k$ and $y_k$ such that



$$x_0 = 1, y_0 = 0, x_{k+1} = a \cdot x_k - b \cdot y_l,  \quad y_{k+1} = x_k - a \cdot y_k \text{ for } k = 0,1,2, \ldots $$



Prove that



$$x_k = \sum^{[k/2]}_{l=0} (-1)^l \cdot a^{k - 2 \cdot l} \cdot \left(a^2 + b \right)^l \cdot \lambda_{k,l}$$



where $\lambda_{k,l} = \sum^{[k/2]}_{m=l} \binom{k}{2 \cdot m} \cdot \binom{m}{l}$



b.) Let $u_k = \sum^{[k/2]}_{l=0} \lambda_{k,l} $. For positive integer $m,$ denote the remainder of $u_k$ divided by $2^m$ as $z_{m,k}$. Prove that $z_{m,k},$ $k = 0,1,2, \ldots$ is a periodic function, and find the smallest period.
2000 China Team Selection Test
Let $n$ be a positive integer. Denote $M = \{(x, y)|x, y \text{ are integers }, 1 \leq  x, y \leq  n\}$. Define function $f$ on $M$ with the following properties:



a.) $f(x, y)$ takes non-negative integer value; 

b.) $\sum^n_{y=1} f(x, y) = n - 1$ for $1 \eq  x \leq n$;

c.) If $f(x_1, y_1)f(x2, y2) > 0$, then $(x_1 -  x_2)(y_1 -  y_2) \geq 0.$



Find $N(n)$, the number of functions $f$ that satisfy all the conditions. Give the explicit value of $N(4)$.
1999 China Team Selection Test
For non-negative real numbers $x_1, x_2, \ldots, x_n$ which satisfy $x_1 + x_2 + \cdots + x_n = 1$, find the largest possible value of $\sum_{j = 1}^{n} (x_j^{4} - x_j^{5})$.
1999 China Team Selection Test
Find all prime numbers $p$ which satisfy the following condition: For any prime $q < p$, if $p = kq + r, 0 \leq r < q$, there does not exist an integer $q > 1$ such that $a^{2} \mid r$.
1999 China Team Selection Test
Let $S = \lbrace 1, 2, \ldots, 15 \rbrace$.  Let $A_1, A_2, \ldots, A_n$ be $n$ subsets of $S$ which satisfy the following conditions:





I. $|A_i| = 7, i = 1, 2, \ldots, n$;



II. $|A_i \cap A_j| \leq 3, 1 \leq i < j \leq n$



III. For any 3-element subset $M$ of $S$, there exists $A_k$ such

that $M \subset A_k$.





Find the smallest possible value of $n$.
1999 China Team Selection Test
A circle is tangential to sides $AB$ and $AD$ of convex quadrilateral $ABCD$ at $G$ and $H$ respectively, and cuts diagonal $AC$ at $E$ and $F$.  What are the necessary and sufficient conditions such that there exists another circle which passes through $E$ and $F$, and is tangential to $DA$ and $DC$ extended?
1999 China Team Selection Test
For a fixed natural number $m \geq 2$, prove that



a.) There exists integers $x_1, x_2, \ldots, x_{2m}$ such that $$x_i x_{m + i} = x_{i + 1} x_{m + i - 1} + 1, i = 1, 2, \ldots, m \hspace{2cm}(*)$$



b.) For any set of integers $\lbrace x_1, x_2, \ldots, x_{2m}$ which fulfils (*), an integral sequence $\ldots, y_{-k}, \ldots, y_{-1}, y_0, y_1, \ldots, y_k, \ldots$ can be constructed such that $y_k y_{m + k} = y_{k + 1} y_{m + k - 1} + 1, k = 0, \pm 1, \pm 2, \ldots$ such that $y_i = x_i, i = 1, 2, \ldots, 2m$.
1999 China Team Selection Test
For every permutation $ \tau$ of $ 1, 2, \ldots, 10$, $ \tau = (x_1, x_2, \ldots, x_{10})$, define $ S(\tau) = \sum_{k = 1}^{10} |2x_k - 3x_{k - 1}|$.  Let $ x_{11} = x_1$.  Find



I. The maximum and minimum values of $ S(\tau)$.



II. The number of $ \tau$ which lets $ S(\tau)$ attain its maximum.



III. The number of $ \tau$ which lets $ S(\tau)$ attain its minimum.
1998 China Team Selection Test
Find $k \in \mathbb{N}$ such that



a.) For any $n \in \mathbb{N}$, there does not exist $j \in \mathbb{Z}$ which satisfies the conditions $0 \leq j \leq n - k + 1$ and $\left(

\begin{array}{c}

n\\
j\end{array} \right), \left( \begin{array}{c}

n\\
j + 1\end{array} \right), \ldots, \left( \begin{array}{c}

n\\
j + k - 1\end{array} \right)$ forms an arithmetic progression.



b.) There exists $n \in \mathbb{N}$ such that there exists $j$ which satisfies $0 \leq j \leq n - k + 2$, and $\left(

\begin{array}{c}

n\\
j\end{array} \right), \left( \begin{array}{c}

n\\
j + 1\end{array} \right), \ldots , \left( \begin{array}{c}

n\\
j + k - 2\end{array} \right)$ forms an arithmetic progression.



Find all $n$ which satisfies part b.)
1998 China Team Selection Test
$n \geq 5$ football teams participate in a round-robin tournament.  For every game played, the winner receives 3 points, the loser receives 0 points, and in the event of a draw, both teams receive 1 point.  The third-from-bottom team has fewer points than all the teams ranked before it, and more points than the last 2 teams; it won more games than all the teams before it, but fewer games than the 2 teams behind it.  Find the smallest possible $n$.
1998 China Team Selection Test
For a fixed $\theta \in \lbrack 0, \frac{\pi}{2} \rbrack$, find the smallest $a \in \mathbb{R}^{+}$ which satisfies the following conditions:



I.  $\frac{\sqrt a}{\cos \theta} + \frac{\sqrt a}{\sin \theta} >

1$.



II. There exists $x \in \lbrack 1 - \frac{\sqrt a}{\sin \theta},

\frac{\sqrt a}{\cos \theta} \rbrack$ such that $\lbrack (1 -

x)\sin \theta - \sqrt{a - x^2 \cos^{2} \theta} \rbrack^{2} +

\lbrack x\cos \theta - \sqrt{a - (1 - x)^2 \sin^{2} \theta}

\rbrack^{2} \leq a$.
1998 China Team Selection Test
In acute-angled $\bigtriangleup ABC$, $H$ is the orthocenter, $O$ is the circumcenter and $I$ is the incenter. Given that $\angle C > \angle B > \angle A$, prove that $I$ lies within $\bigtriangleup BOH$.
1998 China Team Selection Test
Let $n$ be a natural number greater than 2. $l$ is a line on a plane. There are $n$ distinct points $P_1$, $P_2$, …, $P_n$ on $l$. Let the product of distances between $P_i$ and the other $n-1$ points be $d_i$ ($i = 1, 2,$ …, $n$). There exists a point $Q$, which does not lie on $l$, on the plane. Let the distance from $Q$ to $P_i$ be $C_i$ ($i = 1, 2,$ …, $n$). Find $S_n = \sum_{i = 1}^{n} (-1)^{n-i} \frac{c_i^2}{d_i}$.
1998 China Team Selection Test
For any $h = 2^{r}$ ($r$ is a non-negative integer), find all $k \in \mathbb{N}$ which satisfy the following condition: There exists an odd natural number $m > 1$ and $n \in \mathbb{N}$, such that $k \mid m^{h} - 1, m \mid n^{\frac{m^{h}-1}{k}} + 1$.
1997 China Team Selection Test
Given a real number $\lambda > 1$, let $P$ be a point on the arc $BAC$ of the circumcircle of $\bigtriangleup ABC$.  Extend $BP$ and $CP$ to $U$ and $V$ respectively such that $BU = \lambda BA$, $CV = \lambda CA$. Then extend $UV$ to $Q$ such that $UQ = \lambda UV$. Find the locus of point $Q$.
1997 China Team Selection Test
There are $ n$ football teams in a round-robin competition where every 2 teams meet once. The winner of each match receives 3 points while the loser receives 0 points. In the case of a draw, both teams receive 1 point each. Let $ k$ be as follows: $ 2 \leq k \leq n - 1$.  At least how many points must a certain team get in the competition so as to ensure that there are at most $ k - 1$ teams whose scores are not less than that particular team's score?
1997 China Team Selection Test
Prove that there exists $m \in \mathbb{N}$ such that there exists an integral sequence $\lbrace a_n \rbrace$ which satisfies:





I. $a_0 = 1, a_1 = 337$;



II. $(a_{n + 1} a_{n - 1} - a_n^2) + \frac{3}{4}(a_{n + 1} + a_{n - 1} - 2a_n) = m, \forall$ $n \geq 1$;



III. $\frac{1}{6}(a_n + 1)(2a_n + 1)$ is a perfect square $\forall$ $n \geq 1$.
1997 China Team Selection Test
Find all real-coefficient polynomials $f(x)$ which satisfy the following conditions:



i. $f(x) = a_0 x^{2n} + a_2 x^{2n - 2} + \cdots + a_{2n - 2}

x^2 + a_{2n}, a_0 > 0$;

ii. $\sum_{j=0}^n a_{2j} a_{2n - 2j} \leq \left(

\begin{array}{c}

2n\\
n\end{array} \right) a_0 a_{2n}$;

iii. All the roots of $f(x)$ are imaginary numbers with no real part.
1997 China Team Selection Test
Let $n$ be a natural number greater than 6.  $X$ is a set such that $|X| = n$. $A_1, A_2, \ldots, A_m$ are distinct 5-element subsets of $X$.  If $m > \frac{n(n - 1)(n - 2)(n - 3)(4n - 15)}{600}$, prove that there exists $A_{i_1}, A_{i_2}, \ldots, A_{i_6}$ $(1 \leq i_1 < i_2 < \cdots, i_6 \leq m)$, such that $\bigcup_{k = 1}^6 A_{i_k} = 6$.
1997 China Team Selection Test
There are 1997 pieces of medicine.  Three bottles $A, B, C$ can contain at most 1997, 97, 19 pieces of medicine respectively. At first, all 1997 pieces are placed in bottle $A$, and the three bottles are closed. Each piece of medicine can be split into 100 part. When a bottle is opened, all pieces of medicine in that bottle lose a part each. A man wishes to consume all the medicine. However, he can only open each of the bottles at most once each day, consume one piece of medicine, move some pieces between the bottles, and close them.  At least how many parts will be lost by the time he finishes consuming all the medicine?
1996 China Team Selection Test
Let side $BC$ of $\bigtriangleup ABC$ be the diameter of a semicircle which cuts $AB$ and $AC$ at $D$ and $E$ respectively. $F$ and $G$ are the feet of the perpendiculars from $D$ and $E$ to $BC$ respectively.  $DG$ and $EF$ intersect at $M$.  Prove that $AM \perp BC$.
1996 China Team Selection Test
$S$ is the set of functions $f:\mathbb{N} \to \mathbb{R}$ that satisfy the following conditions:



I. $f(1) = 2$

II. $f(n+1) \geq f(n) \geq \frac{n}{n + 1} f(2n)$ for $n = 1, 2, \ldots$



Find the smallest $M \in \mathbb{N}$ such that for any $f \in S$ and any $n \in \mathbb{N}, f(n) < M$.
1996 China Team Selection Test
Let $ M = \lbrace 2, 3, 4, \ldots\, 1000 \rbrace$.  Find the smallest $ n \in \mathbb{N}$ such that any $ n$-element subset of $ M$ contains 3 pairwise disjoint 4-element subsets $ S, T, U$ such that



I. For any 2 elements in $ S$, the larger number is a multiple of the smaller number.  The same applies for $ T$ and $ U$.



II. For any $ s \in S$ and $ t \in T$, $ (s,t) = 1$.



III.  For any $ s \in S$ and $ u \in U$, $ (s,u) > 1$.
1996 China Team Selection Test
3 countries $A, B, C$ participate in a competition where each country has 9 representatives.  The rules are as follows: every round of competition is between 1 competitor each from 2 countries.  The winner plays in the next round, while the loser is knocked out.  The remaining country will then send a representative to take on the winner of the previous round. The competition begins with $A$ and $B$ sending a competitor each.  If all competitors from one country have been knocked out, the competition continues between the remaining 2 countries until another country is knocked out.  The remaining team is the champion.





I. At least how many games does the champion team win?



II. If the champion team won 11 matches, at least how many matches were played?
1996 China Team Selection Test
Let $\alpha_1, \alpha_2, \ldots, \alpha_n, \beta_1, \beta_2, \ldots, \beta_n$ $(n \geq 4)$ be 2 sets of real numbers such that $\sum_{i=1}^{n} \alpha_i^2 < 1, \sum_{i=1}^{n} \beta_i^2 < 1$. Let $A^2 = 1 - \sum_{i=1}^{n} \alpha_i^2, B^2 = 1 - \sum_{i=1}^{n}

\beta_i^2, W = \frac{1}{2} (1 - \sum_{i=1}^{n} \alpha_i \beta_i)^2$. Find all real numbers $\lambda$ such that $x^n + \lambda (x^{n-1} + \cdots + x^3 + Wx^2 + ABx + 1) = 0$ only has real roots.
1996 China Team Selection Test
Does there exist non-zero complex numbers $a, b, c$ and natural number $h$ such that if integers $k, l, m$ satisfy $|k| + |l| + |m| \geq 1996$, then $|ka + lb + mc| > \frac {1}{h}$ is true?
1995 China Team Selection Test
Find the smallest prime number $p$ that cannot be represented in the form $|3^{a} - 2^{b}|$, where $a$ and $b$ are non-negative integers.
1995 China Team Selection Test
Given a fixed acute angle $\theta$ and a pair of internally tangent circles, let the line $l$ which passes through the point of tangency, $A$, cut the larger circle again at $B$ ($l$ does not pass through the centers of the circles).  Let $M$ be a point on the major arc $AB$ of the larger circle, $N$ the point where $AM$ intersects the smaller circle, and $P$ the point on ray $MB$ such that $\angle MPN = \theta$.  Find the locus of $P$ as $M$ moves on major arc $AB$ of the larger circle.
1995 China Team Selection Test
21 people take a test with 15 true or false questions.  It is known that every 2 people have at least 1 correct answer in common.  What is the minimum number of people that could have correctly answered the question which the most people were correct on?
1995 China Team Selection Test
Let $S = \lbrace A = (a_1, \ldots, a_s) \mid a_i = 0$ or $1, i = 1, \ldots, 8 \rbrace$.  For any 2 elements of $S$, $A = \lbrace a_1, \ldots, a_8\rbrace$ and $B = \lbrace b_1, \ldots, b_8\rbrace$.  Let $d(A,B) = \sum_{i=1}{8} |a_i - b_i|$. Call $d(A,B)$ the distance between $A$ and $B$.  At most how many elements can $S$ have such that the distance between any 2 sets is at least 5?