Navvye/Olympiad-Problems-CSV huggingface dataset:
2002 China National Olympiad
Given the polynomial sequence $(p_{n}(x))$ satisfying $p_{1}(x)=x^{2}-1$, $p_{2}(x)=2x(x^{2}-1)$, and $p_{n+1}(x)p_{n-1}(x)=(p_{n}(x)^{2}-(x^{2}-1)^{2}$, for $n\geq 2$, let $s_{n}$ be the sum of the absolute values of the coefficients of $p_{n}(x)$. For each $n$, find a non-negative integer $k_{n}$ such that $2^{-k_{n}}s_n$ is odd.
2002 China National Olympiad
In a competition there are $18$ teams and in each round $18$ teams are divided into $9$ pairs where the $9$ matches are played coincidentally. There are $17$ rounds, so that each pair of teams play each other exactly once. After $n$ rounds, there always exists $4$ teams such that there was exactly one match played between these teams in those $n$ rounds. Find the maximum value of $n$.
2002 China National Olympiad
For every four points $P_{1},P_{2},P_{3},P_{4}$ on the plane, find the minimum value of $\frac{\sum_{1\le\ i<j\le\ 4}P_{i}P_{j}}{\min_{1\le\ i<j\le\ 4}(P_{i}P_{j})}$.
2002 China National Olympiad
Suppose that a point in the plane is called good if it has rational coordinates. Prove that all good points can be divided into three sets satisfying:

1) If the centre of the circle is good, then there are three points in the circle from each of the three sets.

2) There are no three collinear points that are from each of the three sets.
2002 China National Olympiad
Suppose that $c\in\left(\frac{1}{2},1\right)$. Find the least $M$ such that for every integer $n\ge 2$ and real numbers $0<a_1\le a_2\le\ldots \le a_n$, if $\frac{1}{n}\sum_{k=1}^{n}ka_{k}=c\sum_{k=1}^{n}a_{k}$, then we always have that $\sum_{k=1}^{n}a_{k}\le M\sum_{k=1}^{m}a_{k}$ where $m=[cn]$
2001 China National Olympiad
Let $a$ be real number with $\sqrt{2}<a<2$, and let $ABCD$ be a convex cyclic quadrilateral whose circumcentre $O$ lies in its interior. The quadrilateral's circumcircle $\omega$ has radius $1$, and the longest and shortest sides of the quadrilateral have length $a$ and $\sqrt{4-a^2}$, respectively. Lines $L_A,L_B,L_C,L_D$ are tangent to $\omega$ at $A,B,C,D$, respectively.



Let lines $L_A$ and $L_B$, $L_B$ and $L_C$,$L_C$ and $L_D$,$L_D$ and $L_A$ intersect at $A',B',C',D'$ respectively. Determine the minimum value of $\frac{S_{A'B'C'D'}}{S_{ABCD}}$.
2001 China National Olympiad
Let $X=\{1,2,\ldots,2001\}$. Find the least positive integer $m$ such that for each subset $W\subset X$ with $m$ elements, there exist $u,v\in W$ (not necessarily distinct) such that $u+v$ is of the form $2^{k}$, where $k$ is a positive integer.
2001 China National Olympiad
Let $P$ be a regular $n$-gon $A_1A_2\ldots A_n$. Find all positive integers $n$ such that for each permutation $\sigma (1),\sigma (2),\ldots ,\sigma (n)$ there exists $1\le i,j,k\le n$ such that the triangles $A_{i}A_{j}A_{k}$ and $A_{\sigma (i)}A_{\sigma (j)}A_{\sigma (k)}$ are both acute, both right or both obtuse.
2001 China National Olympiad
Let $a,b,c$ be positive integers such that $a,b,c,a+b-c,a+c-b,b+c-a,a+b+c$ are $7$ distinct primes. The sum of two of $a,b,c$ is $800$. If $d$ be the difference of the largest prime and the least prime among those $7$ primes, find the maximum value of $d$.
2001 China National Olympiad
Let $P_1P_2\ldots P_{24}$ be a regular $24$-sided polygon inscribed in a circle $\omega$ with circumference $24$. Determine the number of ways to choose sets of eight distinct vertices from these $24$ such that none of the arcs has length $3$ or $8$.
2001 China National Olympiad
Let $a=2001$. Consider the set $A$ of all pairs of integers $(m,n)$ with $n\neq0$ such that

(i) $m<2a$;

(ii) $2n|(2am-m^2+n^2)$;

(iii) $n^2-m^2+2mn\leq2a(n-m)$.

For $(m, n)\in A$, let $$f(m,n)=\frac{2am-m^2-mn}{n}.$$

Determine the maximum and minimum values of $f$.
2000 China National Olympiad
The sides $a,b,c$ of triangle $ABC$ satisfy $a\le b\le c$. The circumradius and inradius of triangle $ABC$ are $R$ and $r$ respectively. Let $f=a+b-2R-2r$. Determine the sign of $f$ by the measure of angle $C$.
2000 China National Olympiad
A sequence $(a_n)$ is defined recursively by $a_1=0, a_2=1$ and for $n\ge 3$,

$$a_n=\frac12na_{n-1}+\frac12n(n-1)a_{n-2}+(-1)^n\left(1-\frac{n}{2}\right).$$

Find a closed-form expression for $f_n=a_n+2\binom{n}{1}a_{n-1}+3\binom{n}{2}a_{n-2}+\ldots +(n-1)\binom{n}{n-2}a_2+n\binom{n}{n-1}a_1$.
2000 China National Olympiad
A table tennis club hosts a series of doubles matches following several rules:

(i)  each player belongs to two pairs at most;

(ii) every two distinct pairs play one game against each other at most;

(iii) players in the same pair do not play against each other when they pair with others respectively.

Every player plays a certain number of games in this series. All these distinct numbers make up a set called the “set of games”. Consider a set $A=\{a_1,a_2,\ldots ,a_k\}$ of positive integers such that every element in $A$ is divisible by $6$. Determine the minimum number of players needed to participate in this series so that a schedule for which the corresponding set of games  is equal to set $A$ exists.
2000 China National Olympiad
Given an ordered $n$-tuple $A=(a_1,a_2,\cdots ,a_n)$ of real numbers, where $n\ge 2$, we define $b_k=\max{a_1,\ldots a_k}$ for each k. We define $B=(b_1,b_2,\cdots ,b_n)$ to be the “innovated tuple” of $A$. The number of distinct elements in $B$ is called the “innovated degree” of $A$.

Consider all permutations of $1,2,\ldots ,n$ as an ordered $n$-tuple. Find the arithmetic mean of the first term of the permutations whose innovated degrees are all equal to $2$
2000 China National Olympiad
Find all positive integers $n$ such that there exists integers $n_1,\ldots,n_k\ge 3$, for some integer $k$, satisfying

$$n=n_1n_2\cdots n_k=2^{\frac{1}{2^k}(n_1-1)\cdots (n_k-1)}-1.$$
2000 China National Olympiad
A test contains $5$ multiple choice questions which have $4$ options in each. Suppose each examinee chose one option for each question. There exists a number $n$, such that for any $n$ sheets among $2000$ sheets of answer papers, there are $4$ sheets of answer papers such that any two of them have at most $3$ questions with the same answers. Find the minimum value of $n$.
1999 China National Olympiad
Let $ABC$ be an acute triangle with $\angle C>\angle B$. Let $D$ be a point on $BC$ such that $\angle ADB$ is obtuse, and let $H$ be the orthocentre of triangle $ABD$. Suppose that $F$ is a point inside triangle $ABC$ that is on the circumcircle of triangle $ABD$. Prove that $F$ is the orthocenter of triangle $ABC$ if and only if $HD||CF$ and $H$ is on the circumcircle of triangle $ABC$.
1999 China National Olympiad
Let $a$ be a real number. Let $(f_n(x))_{n\ge 0}$ be a sequence of polynomials such that $f_0(x)=1$ and $f_{n+1}(x)=xf_n(x)+f_n(ax)$ for all non-negative integers $n$.

a) Prove that $f_n(x)=x^nf_n\left(x^{-1}\right)$ for all non-negative integers $n$.

b) Find an explicit expression for $f_n(x)$.
1999 China National Olympiad
There are $99$ space stations. Each pair of space stations is connected by a tunnel. There are $99$ two-way main tunnels, and all the other tunnels are strictly one-way tunnels. A group of $4$ space stations is called connected if one can reach each station in the group from every other station in the group without using any tunnels other than the $6$ tunnels which connect them. Determine the maximum number of connected groups.
1999 China National Olympiad
Let $m$ be a positive integer. Prove that there are integers $a, b, k$, such that both $a$ and $b$ are odd, $k\geq0$ and

$$2m=a^{19}+b^{99}+k\cdot2^{1999}$$
1999 China National Olympiad
Determine the maximum value of $\lambda$ such that if $f(x) = x^3 +ax^2 +bx+c$ is a cubic polynomial with all its roots nonnegative, then $$f(x)\geq\lambda(x -a)^3$$ for all $x\geq0$. Find the equality condition.
1999 China National Olympiad
A $4\times4\times4$ cube is composed of $64$ unit cubes. The faces of $16$ unit cubes are to be coloured red. A colouring is called interesting if there is exactly $1$ red unit cube in every $1\times1\times 4$ rectangular box composed of $4$ unit cubes. Determine the number of interesting colourings.
1998 China National Olympiad
Let $ABC$ be a non-obtuse triangle satisfying $AB>AC$ and $\angle B=45^{\circ}$. The circumcentre $O$ and incentre $I$ of  triangle $ABC$ are such that $\sqrt{2}\ OI=AB-AC$. Find the value of $\sin A$.
1998 China National Olympiad
Given a positive integer $n>1$, determine with proof if there exist $2n$ pairwise different positive integers $a_1,\ldots ,a_n,b_1,\ldots b_n$ such that  $a_1+\ldots +a_n=b_1+\ldots +b_n$ and

$$n-1>\sum_{i=1}^{n}\frac{a_i-b_i}{a_i+b_i}>n-1-\frac{1}{1998}.$$
1998 China National Olympiad
Let $S=\{1,2,\ldots ,98\}$. Find the least natural number $n$ such that we can pick out $10$ numbers in any $n$-element subset of $S$ satisfying the following condition: no matter how we equally divide the $10$ numbers into two groups, there exists a number in one group such that it is coprime to the other numbers in that group, meanwhile there also exists a number in the other group such that it is not coprime to any of the other numbers in the same group.
1998 China National Olympiad
Find all natural numbers $n>3$, such that $2^{2000}$ is divisible by $1+C^1_n+C^2_n+C^3_n$.
1998 China National Olympiad
Let $D$ be a point inside acute triangle $ABC$ satisfying the condition

$$DA\cdot DB\cdot AB+DB\cdot DC\cdot BC+DC\cdot DA\cdot CA=AB\cdot BC\cdot CA.$$

Determine (with proof) the geometric position of point $D$.
1998 China National Olympiad
Let $x_1,x_2,\ldots ,x_n$ be real numbers, where $n\ge 2$, satisfying $\sum_{i=1}^{n}x^2_i+ \sum_{i=1}^{n-1}x_ix_{i+1}=1$ . For each $k$, find the maximal value of $|x_k|$.
1997 China National Olympiad
Let $x_1,x_2,\ldots ,x_{1997}$ be real numbers satisfying the following conditions:

i) $-\dfrac{1}{\sqrt{3}}\le x_i\le \sqrt{3}$ for $i=1,2,\ldots ,1997$;

ii) $x_1+x_2+\cdots +x_{1997}=-318 \sqrt{3}$ .

Determine (with proof) the maximum value of $x^{12}_1+x^{12}_2+\ldots +x^{12}_{1997}$ .
1997 China National Olympiad
Let $A_1B_1C_1D_1$ be an arbitrary convex quadrilateral. $P$ is a point inside the quadrilateral such that each angle enclosed by one edge and one ray which starts at one vertex on that edge and passes through point $P$ is acute. We recursively define points $A_k,B_k,C_k,D_k$ symmetric to $P$ with respect to lines $A_{k-1}B_{k-1}, B_{k-1}C_{k-1}, C_{k-1}D_{k-1},D_{k-1}A_{k-1}$ respectively for $k\ge 2$.

Consider the sequence of quadrilaterals $A_iB_iC_iD_i$.

i) Among the first 12 quadrilaterals, which are similar to the 1997th quadrilateral and which are not?

ii) Suppose the 1997th quadrilateral is cyclic. Among the first 12 quadrilaterals, which are cyclic and which are not?
1997 China National Olympiad
Prove that there are infinitely many natural numbers $n$ such that we can divide $1,2,\ldots ,3n$ into three sequences $(a_n),(b_n)$ and $(c_n)$, with $n$ terms in each, satisfying the following conditions:

i) $a_1+b_1+c_1= a_2+b_2+c_2=\ldots =a_n+b_n+c_n$ and $a_1+b_1+c_1$ is divisible by $6$;

ii) $a_1+a_2+\ldots +a_n= b_1+b_2+\ldots +b_n=c_1+c_2+\ldots +c_n,$ and $a_1+a_2+\ldots +a_n$ is divisible by $6$.
1997 China National Olympiad
Consider a cyclic quadrilateral $ABCD$. The extensions of its sides $AB,DC$ meet at the point $P$ and the extensions of its sides $AD,BC$ meet at the point $Q$. Suppose $\quad QE,QF$ are tangents to the circumcircle of quadrilateral $ABCD$ at $E,F$ respectively. Show that $P,E,F$ are collinear.
1997 China National Olympiad
Let $A=\{1,2,3,\cdots ,17\}$. A mapping $f:A\rightarrow A$ is defined as follows: $f^{[1]}(x)=f(x), f^{[k+1]}(x)=f(f^{[k]}(x))$ for $k\in\mathbb{N}$. Suppose that $f$ is bijective and that there exists a natural number $M$ such that:

i) when $m<M$ and $1\le i\le 16$, we have $f^{[m]}(i+1)- f^{[m]}(i) \not=\pm 1\pmod{17}$ and  $f^{[m]}(1)- f^{[m]}(17) \not=\pm 1\pmod{17}$;

ii) when $1\le i\le 16$, we have $f^{[M]}(i+1)- f^{[M]}(i)=\pm 1 \pmod{17}$ and $f^{[M]}(1)- f^{[M]}(17)=\pm 1\pmod{17}$.

Find the maximal value of $M$.
1997 China National Olympiad
Let $(a_n)$ be a sequence of non-negative real numbers satisfying $a_{n+m}\le a_n+a_m$ for all non-negative integers $m,n$.

Prove that if $n\ge m$ then $a_n\le ma_1+\left(\dfrac{n}{m}-1\right)a_m$ holds.
1996 China National Olympiad
Let $\triangle{ABC}$ be a triangle with orthocentre $H$. The tangent lines from $A$ to the circle with diameter $BC$ touch this circle at $P$ and $Q$. Prove that $H,P$ and $Q$ are collinear.
1996 China National Olympiad
Find the smallest positive integer $ K$ such that every $ K$-element subset of $ \{1,2,...,50 \}$ contains two distinct elements $ a,b$ such that $ a+b$ divides $ ab$.
1996 China National Olympiad
Suppose that the function $f:\mathbb{R}\to\mathbb{R}$ satisfies

$$f(x^3 + y^3)=(x+y)(f(x)^2-f(x)f(y)+f(y)^2)$$

for all $x,y\in\mathbb{R}$.

Prove that $f(1996x)=1996f(x)$ for all $x\in\mathbb{R}$.
1996 China National Olympiad
$8$ singers take part in a festival. The organiser wants to plan $m$ concerts. For every concert there are $4$ singers who go on stage, with the restriction that the times of which every two singers go on stage in a concert are all equal. Find a schedule that minimises $m$.
1996 China National Olympiad
Let $n$ be a natural number. Suppose that $x_0=0$ and that $x_i>0$ for all $i\in\{1,2,\ldots ,n\}$. If $\sum_{i=1}^nx_i=1$ , prove that

$$1\leq\sum_{i=1}^{n} \frac{x_i}{\sqrt{1+x_0+x_1+\ldots +x_{i-1}}\sqrt{x_i+\ldots+x_n}} < \frac{\pi}{2}  $$
1996 China National Olympiad
In the triangle $ABC$, $\angle{C}=90^{\circ},\angle {A}=30^{\circ}$ and $BC=1$. Find the minimum value of the longest side of all inscribed triangles (i.e. triangles with vertices on each of three sides) of the triangle $ABC$.
1995 China National Olympiad
Let $a_1,a_2,\cdots ,a_n; b_1,b_2,\cdots ,b_n (n\ge 3)$ be real numbers satisfying the following conditions:

(1) $a_1+a_2+\cdots +a_n= b_1+b_2+\cdots +b_n $;

(2) $0<a_1=a_2, a_i+a_{i+1}=a_{i+2}$ ($i=1,2,\cdots ,n-2$);

(3) $0<b_1\le b_2, b_i+b_{i+1}\le b_{i+2}$ ($i=1,2,\cdots ,n-2$).

Prove that $a_{n-1}+a_n\le b_{n-1}+b_n$.
1995 China National Olympiad
Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function satisfying the following conditions:

(1) $f(1)=1$;

(2) $\forall n\in \mathbb{N}$, $3f(n) f(2n+1) =f(2n) ( 1+3f(n) )$;

(3) $\forall n\in \mathbb{N}$, $f(2n) < 6 f(n)$.

Find all solutions of equation $f(k) +f(l)=293$, where $k<l$.

($\mathbb{N}$ denotes the set of all natural numbers).
1995 China National Olympiad
Find the minimun value of $\sum_{i=1}^{10} \sum_{j=1}^{10} \sum_{k=1}^{10}|k(x+y-10i)(3x-6y-36j)(19x+95y-95k)|$ , where $x,y$ are integers.
1995 China National Olympiad
Given four spheres with their radii equal to $2,2,3,3$ respectively, each sphere externally touches the other spheres. Suppose that there is another sphere that is externally tangent to all those four spheres, determine the radius of this sphere.
1995 China National Olympiad
Let $a_1,a_2,\cdots ,a_{10}$ be pairwise distinct natural numbers with their sum equal to 1995. Find the minimal value of $a_1a_2+a_2a_3+\cdots +a_9a_{10}+a_{10}a_1$.
1995 China National Olympiad
Let $n(n>1)$ be an odd. We define $x_k=(x^{(k)}_1,x^{(k)}_2,\cdots ,x^{(k)}_n)$ as follow:

$x_0=(x^{(0)}_1,x^{(0)}_2,\cdots ,x^{(0)}_n)=(1,0,\cdots ,0,1)$;

$ x^{(k)}_i =\begin{cases}0, \quad x^{(k-1)}_i=x^{(k-1)}_{i+1},\\ 1, \quad x^{(k-1)}_i\not= x^{(k-1)}_{i+1},\end{cases} $

$i=1,2,\cdots ,n$, where $x^{(k-1)}_{n+1}= x^{(k-1)}_1$.

Let $m$ be a positive integer satisfying $x_0=x_m$. Prove that $m$ is divisible by $n$.
1994 China National Olympiad
Let $ABCD$ be a trapezoid with $AB\parallel CD$. Points $E,F$ lie on segments $AB,CD$ respectively. Segments $CE,BF$ meet at $H$, and segments $ED,AF$ meet at $G$. Show that $S_{EHFG}\le \dfrac{1}{4}S_{ABCD}$. Determine, with proof, if the conclusion still holds when $ABCD$ is just any convex quadrilateral.
1994 China National Olympiad
There are $m$ pieces of candy held in $n$ trays($n,m\ge 4$).  An operation is defined as follow: take out one piece of candy from any two trays respectively, then put them in a third tray. Determine, with proof, if we can move all candies to a single tray by finite operations.
1994 China National Olympiad
Find all functions $f:[1,\infty )\rightarrow [1,\infty)$ satisfying the following conditions:

(1) $f(x)\le 2(x+1)$;

(2) $f(x+1)=\dfrac{1}{x}[(f(x))^2-1]$ .
1994 China National Olympiad
Let $f(z)=c_0z^n+c_1z^{n-1}+ c_2z^{n-2}+\cdots +c_{n-1}z+c_n$ be a polynomial with complex coefficients. Prove that there exists a complex number $z_0$ such that $|f(z_0)|\ge |c_0|+|c_n|$, where $|z_0|\le 1$.
1994 China National Olympiad
For arbitrary natural number $n$, prove that $\sum^n_{k=0}C^k_n2^kC^{[(n-k)/2]}_{n-k}=C^n_{2n+1}$, where $C^0_0=1$ and $[\dfrac{n-k}{2}]$ denotes the integer part of $\dfrac{n-k}{2}$.
1994 China National Olympiad
Let $M$ be a point which has coordinates $(p\times 1994,7p\times 1994)$ in the Cartesian plane ($p$ is a prime). Find the number of right-triangles satisfying the following conditions:

(1) all vertexes of the triangle are lattice points, moreover $M$ is on the right-angled corner of the triangle;

(2) the origin ($0,0$) is the incenter of the triangle.
1993 China National Olympiad
Given an odd $n$, prove that there exist $2n$ integers $a_1,a_2,\cdots ,a_n$; $b_1,b_2,\cdots ,b_n$, such that for any integer $k$ ($0<k<n$), the following $3n$ integers:

$a_i+a_{i+1}, a_i+b_i, b_i+b_{i+k}$ ($i=1,2,\cdots ,n; a_{n+1}=a_1, b_{n+j}=b_j, 0<j<n$) are of different remainders on division by $3n$.
1993 China National Olympiad
Given a natural number $k$ and a real number $a (a>0)$,  find the maximal value of $a^{k_1}+a^{k_2}+\cdots +a^{k_r}$, where $k_1+k_2+\cdots +k_r=k$ ($k_i\in \mathbb{N} ,1\le r \le k$).
1993 China National Olympiad
Let $K, K_1$ be two circles with the same center and their radii equal to $R$ and $R_1 (R_1>R)$ respectively. Quadrilateral $ABCD$ is inscribed in circle $K$. Quadrilateral $A_1B_1C_1D_1$ is inscribed in circle $K_1$ where $A_1,B_1,C_1,D_1$ lie on rays $CD,DA,AB,BC$ respectively. Show that $\dfrac{S_{A_1B_1C_1D_1}}{S_{ABCD}}\ge \dfrac{R^2_1}{R^2}$.
1993 China National Olympiad
We are given a set $S=\{z_1,z_2,\cdots ,z_{1993}\}$, where $z_1,z_2,\cdots ,z_{1993}$ are nonzero complex numbers (also viewed as nonzero vectors in the plane). Prove that we can divide $S$ into some groups such that the following conditions are satisfied:

(1) Each element in $S$ belongs and only belongs to one group;

(2) For any group $p$, if we use $T(p)$ to denote the sum of all memebers in $p$, then for any memeber $z_i (1\le i \le 1993)$ of $p$, the angle between $z_i$ and $T(p)$ does not exceed $90^{\circ}$;

(3) For any two groups $p$ and $q$, the angle between $T(p)$ and $T(q)$ exceeds $90^{\circ}$ (use the notation introduced in (2)).
1993 China National Olympiad
$10$ students bought some books in a bookstore. It is known that every student bought exactly three kinds of books, and any two of them shared at least one kind of book. Determine, with proof, how many students bought the most popular book at least? (Note: the most popular book means most students bought this kind of book)
1993 China National Olympiad
Let  $f: (0,+\infty)\rightarrow (0,+\infty)$ be a function satisfying the following condition: for arbitrary positive real numbers $x$ and $y$, we have $f(xy)\le f(x)f(y)$. Show that for arbitrary positive real number $x$ and natural number $n$, inequality $f(x^n)\le f(x)f(x^2)^{\dfrac{1}{2}}\dots f(x^n)^{\dfrac{1}{n}}$ holds.
1992 China National Olympiad
Let equation $x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\dots +a_1x+a_0=0$ with real coefficients satisfy $0<a_0\le a_1\le a_2\le \dots \le a_{n-1}\le 1$. Suppose that $\lambda$ ($|\lambda|>1$) is a complex root of the equation, prove that $\lambda^{n+1}=1$.
1992 China National Olympiad
Given nonnegative real numbers $x_1,x_2,\dots ,x_n$, let $a=min\{x_1, x_2,\dots ,x_n\}$. Prove that the following inequality holds:

$$ \sum^{n}_{i=1}\dfrac{1+x_i}{1+x_{i+1}}\le n+\dfrac{1}{(1+a)^2}\sum^{n}_{i=1}(x_i-a)^2 \quad\quad (x_{n+1}=x_1),$$

and equality occurs if and only if $x_1=x_2=\dots =x_n$.
1992 China National Olympiad
Given a $9\times 9$ grid, we assign either $+1$ or $-1$ to each square on the grid. We define an adjustment as follow: for each square on the grid, we make a product of all numbers of those squares which share a common side with the square (excluding itself).Then we have $81$ products. Next we replace all the squares’ values with their corresponding products. Determine if we can make all values in the grid equal to $1$ through finite adjustments.
1992 China National Olympiad
A convex quadrilateral $ABCD$ is inscribed in a circle with center $O$. The diagonals $AC$, $BD$ of $ABCD$ meet at $P$. Circumcircles of $\triangle ABP$ and $\triangle CDP$ meet at $P$ and $Q$ ($O,P,Q$ are pairwise distinct). Show that $\angle OQP=90^{\circ}$.
1992 China National Olympiad
Find the maximum possible number of edges of a simple graph with $8$ vertices and without any quadrilateral. (a simple graph is an undirected graph that has no loops (edges connected at both ends to the same vertex) and no more than one edge between any two different vertices.)
1992 China National Olympiad
Let sequence $\{a_1,a_2,\dots \}$ with integer terms satisfy the following conditions:

1) $a_{n+1}=3a_n-3a_{n-1}+a_{n-2}, n=2,3,\dots$ ;

2) $2a_1=a_0+a_2-2$ ;

3) for arbitrary natural number $m$, there exist $m$ consecutive terms $a_k, a_{k-1}, \dots ,a_{k+m-1}$ among the sequence such that all such $m$ terms are perfect squares.

Prove that all terms of the sequence $\{a_1,a_2,\dots \}$ are perfect squares.
1991 China National Olympiad
We are given a convex quadrilateral $ABCD$ in the plane.

(i) If there exists a point $P$ in the plane such that the areas of $\triangle ABP, \triangle BCP, \triangle CDP, \triangle DAP$ are equal, what condition must be satisfied by the quadrilateral $ABCD$?

(ii) Find (with proof) the maximum possible number of such point $P$ which satisfies the condition in (i).
1991 China National Olympiad
Given $I=[0,1]$ and $G=\{(x,y)|x,y \in I\}$, find all functions $f:G\rightarrow I$, such that $\forall x,y,z \in I$ we have:

i. $f(f(x,y),z)=f(x,f(y,z))$;

ii. $f(x,1)=x, f(1,y)=y$;

iii. $f(zx,zy)=z^kf(x,y)$.

($k$ is a positive real number irrelevant to $x,y,z$.)
1991 China National Olympiad
There are $10$ birds on the ground. For any $5$ of them, there are at least $4$ birds on a circle. Determine the least possible number of birds on the circle with the most birds.
1991 China National Olympiad
Find all positive integer solutions $(x,y,z,n)$ of equation $x^{2n+1}-y^{2n+1}=xyz+2^{2n+1}$, where $n\ge 2$ and $z \le 5\times 2^{2n}$.
1991 China National Olympiad
Find all natural numbers $n$, such that $\min_{k\in \mathbb{N}}(k^2+[n/k^2])=1991$. ($[n/k^2]$ denotes the integer part of $n/k^2$.)
1991 China National Olympiad
A football is covered by some polygonal pieces of leather which are sewed up by three different colors threads. It features as follows:

i) any edge of a polygonal piece of leather is sewed up with an equal-length edge of another polygonal piece of leather by a certain color thread;

ii) each node on the ball is vertex to exactly three polygons, and the three threads joint at the node are of different colors.

Show that we can assign to each node on the ball a complex number (not equal to $1$), such that the product of the numbers assigned to the vertices of any polygonal face is equal to $1$.
1990 China National Olympiad
Given a convex quadrilateral $ABCD$, side $AB$ is not parallel to side $CD$. The circle $O_1$ passing through $A$ and $B$ is tangent to side $CD$ at $P$. The circle $O_2$ passing through $C$ and $D$ is tangent to side $AB$ at $Q$. Circle $O_1$ and circle $O_2$ meet at $E$ and $F$. Prove that $EF$ bisects segment $PQ$ if and only if $BC\parallel AD$.