Navvye/Olympiad-Problems-CSV huggingface dataset:
2008 China Team Selection Test
Let $ P$ be the the isogonal conjugate of $ Q$ with respect to triangle $ ABC$, and $ P,Q$ are in the interior of triangle $ ABC$. Denote by $ O_{1},O_{2},O_{3}$ the circumcenters of triangle $ PBC,PCA,PAB$, $ O'_{1},O'_{2},O'_{3}$ the circumcenters of triangle $ QBC,QCA,QAB$, $ O$ the circumcenter of triangle $ O_{1}O_{2}O_{3}$, $ O'$ the circumcenter of triangle $ O'_{1}O'_{2}O'_{3}$. Prove that $ OO'$ is parallel to $ PQ$.
2008 China Team Selection Test
Prove that for arbitary integer $ n > 16$, there exists the set $ S$ that contains $ n$ positive integers and has the following property:if the subset $ A$ of $ S$ satisfies for arbitary $ a,a'\in A, a\neq a', a + a'\notin S$ holds, then $ |A|\leq4\sqrt n.$
2008 China Team Selection Test
Let $ n>m>1$ be odd integers, let $ f(x)=x^n+x^m+x+1$. Prove that $ f(x)$ can't be expressed as the product of two polynomials having integer coefficients and positive degrees.
2008 China Team Selection Test
Given a rectangle $ ABCD,$ let $ AB = b, AD = a ( a\geq b),$ three points $ X,Y,Z$ are put inside or on the boundary of the rectangle, arbitrarily. Find the maximum of the minimum of the distances between any two points among the three points. (Denote it by $ a,b$)
2008 China Team Selection Test
Let $ x,y,z$ be positive real numbers, show that $ \frac {xy}{z} + \frac {yz}{x} + \frac {zx}{y} > 2\sqrt [3]{x^3 + y^3 + z^3}.$
2008 China Team Selection Test
Let $ S$ be a set that contains $ n$ elements. Let $ A_{1},A_{2},\cdots,A_{k}$ be $ k$ distinct subsets of $ S$, where $ k\geq 2, |A_{i}| = a_{i}\geq 1 ( 1\leq i\leq k)$. Prove that the number of subsets of $ S$ that don't contain any $ A_{i} (1\leq i\leq k)$ is greater than or equal to $ 2^n\prod_{i = 1}^k(1 - \frac {1}{2^{a_{i}}}).$
2008 China Team Selection Test
Let $ ABC$ be an acute triangle, let $ M,N$ be the midpoints of  minor arcs $ \widehat{CA},\widehat{AB}$ of the circumcircle of triangle $ ABC,$ point $ D$ is the midpoint of segment $ MN,$ point $ G$ lies on minor arc $ \widehat{BC}.$ Denote by $ I,I_{1},I_{2}$ the incenters of triangle $ ABC,ABG,ACG$ respectively.Let $ P$ be the second intersection of the circumcircle of triangle $ GI_{1}I_{2}$ with the circumcircle of triangle $ ABC.$ Prove that three points $ D,I,P$ are collinear.
2008 China Team Selection Test
For a given integer $ n\geq 2,$ determine the necessary and sufficient conditions that real numbers $ a_{1},a_{2},\cdots, a_{n},$ not all zero satisfy such that there exist integers $ 0<x_{1}<x_{2}<\cdots<x_{n},$ satisfying $ a_{1}x_{1}+a_{2}x_{2}+\cdots+a_{n}x_{n}\geq 0.$
2008 China Team Selection Test
Let $ 0 < x_{1}\leq\frac {x_{2}}{2}\leq\cdots\leq\frac {x_{n}}{n}, 0 < y_{n}\leq y_{n - 1}\leq\cdots\leq y_{1},$ Prove that $ (\sum_{k = 1}^{n}x_{k}y_{k})^2\leq(\sum_{k = 1}^{n}y_{k})(\sum_{k = 1}^{n}(x_{k}^2 - \frac {1}{4}x_{k}x_{k - 1})y_{k}).$ where $ x_{0} = 0.$
2008 China Team Selection Test
Prove that in a plane, arbitrary $ n$ points can be overlapped by discs that the sum of all the diameters is less than $ n$, and the distances between arbitrary two are greater than $ 1$. (where the distances between two discs that have no common points are defined as that the distances between its centers subtract the sum of its radii; the distances between two discs that have  common points are zero)
2008 China Team Selection Test
Prove that for all $ n\geq 2,$ there exists $ n$-degree polynomial $ f(x) = x^n + a_{1}x^{n - 1} + \cdots + a_{n}$ such that

(1) $ a_{1},a_{2},\cdots, a_{n}$ all are unequal to $ 0$;

(2) $ f(x)$ can't be factorized into the product of two polynomials having integer coefficients and positive degrees;

(3) for any integers $ x, |f(x)|$ isn't prime numbers.
2008 China Team Selection Test
Find all positive integers $ n$ having the following properties:in two-dimensional Cartesian coordinates, there exists a convex $ n$ lattice polygon whose lengths of all sides are odd numbers, and unequal to each other. (where lattice polygon is defined as polygon whose coordinates of all vertices are integers in Cartesian coordinates.)
2007 China Team Selection Test
Points $ A$ and $ B$ lie on the circle with center $ O.$ Let point $ C$ lies outside the circle; let $ CS$ and $ CT$ be tangents to the circle. $ M$ be the midpoint of minor arc $ AB$ of $ (O).$ $ MS,\,MT$ intersect $ AB$ at points $ E,\,F$ respectively. The lines passing through $ E,\,F$ perpendicular to $ AB$ cut $ OS,\,OT$ at $ X$ and $ Y$ respectively.

A line passed through $ C$ intersect the circle $ (O)$ at $ P,\,Q$ ($ P$ lies on segment $ CQ$). Let $ R$ be the intersection of $ MP$ and $ AB,$ and let $ Z$ be the circumcentre of triangle $ PQR.$

Prove that: $ X,\,Y,\,Z$ are collinear.
2007 China Team Selection Test
A rational number $ x$  is called good if it satisfies: $ x=\frac{p}{q}>1$ with $ p$, $ q$ being positive integers, $ \gcd (p,q)=1$ and there exists constant numbers $ \alpha$, $ N$ such that for any integer $ n\geq N$, $$ |\{x^n\}-\alpha|\leq\dfrac{1}{2(p+q)}$$ Find all the good numbers.
2007 China Team Selection Test
There are $ 63$ points arbitrarily on the circle $ \mathcal{C}$ with its diameter being $ 20$. Let $ S$ denote the number of triangles whose vertices are three of the $ 63$ points and the length of its sides is no less than $ 9$. Fine the maximum of $ S$.
2007 China Team Selection Test
Find all functions $ f: \mathbb{Q}^{+} \mapsto \mathbb{Q}^{+}$ such that:

$$ f(x) + f(y) + 2xy f(xy) = \frac {f(xy)}{f(x+y)}.$$
2007 China Team Selection Test
Let $ x_1, \ldots, x_n$ be $ n>1$ real numbers satisfying $ A=\left |\sum^n_{i=1}x_i\right |\not =0$ and $ B=\max_{1\leq i<j\leq n}|x_j-x_i|\not =0$. Prove that for any $ n$ vectors $ \vec{\alpha_i}$ in the plane, there exists a permutation $ (k_1, \ldots, k_n)$ of the numbers $ (1, \ldots, n)$ such that $$ \left |\sum_{i=1}^nx_{k_i}\vec{\alpha_i}\right | \geq \dfrac{AB}{2A+B}\max_{1\leq i\leq n}|\alpha_i|.$$
2007 China Team Selection Test
Let $ n$ be a positive integer, let $ A$ be a subset of $ \{1, 2, \cdots, n\}$, satisfying for any two numbers $ x, y\in A$, the least common multiple of $ x$, $ y$ not more than $ n$. Show that $ |A|\leq 1.9\sqrt {n} + 5$.
2007 China Team Selection Test
When all vertex angles of a convex polygon are equal, call it equiangular. Prove that $ p > 2$ is a prime number, if and only if the lengths of all sides of  equiangular $ p$ polygon are rational numbers, it is a regular $ p$ polygon.
2007 China Team Selection Test
Let $ I$ be the incenter of triangle $ ABC.$ Let $ M,N$ be the midpoints of $ AB,AC,$ respectively. Points $ D,E$ lie on $ AB,AC$ respectively such that $ BD=CE=BC.$ The line perpendicular to $ IM$ through $ D$ intersects the line perpendicular to $ IN$ through $ E$ at $ P.$ Prove that $ AP\perp BC.$
2007 China Team Selection Test
Prove that for any positive integer $ n$, there exists only $ n$ degree polynomial $ f(x),$ satisfying $ f(0) = 1$ and $ (x + 1)[f(x)]^2 - 1$ is an odd function.
2007 China Team Selection Test
$ u,v,w > 0$,such that $ u + v + w + \sqrt {uvw} = 4$

prove that $ \sqrt {\frac {uv}{w}} + \sqrt {\frac {vw}{u}} + \sqrt {\frac {wu}{v}}\geq u + v + w$
2007 China Team Selection Test
Find all positive integers $ n$ such that there exists sequence consisting of $ 1$ and $ - 1: a_{1},a_{2},\cdots,a_{n}$ satisfying $ a_{1}\cdot1^2 + a_{2}\cdot2^2 + \cdots + a_{n}\cdot n^2 = 0.$
2007 China Team Selection Test
Assume there are $ n\ge3$ points in the plane, Prove that there exist three points $ A,B,C$ satisfying $ 1\le\frac{AB}{AC}\le\frac{n+1}{n-1}.$
2007 China Team Selection Test
Let $ ABC$ be a triangle. Circle $ \omega$­ passes through points $ B$ and $ C.$ Circle $ \omega_{1}$ is tangent internally to $ \omega$­ and also to sides $ AB$ and $ AC$ at $ T,\, P,$ and $ Q,$ respectively. Let $ M$ be midpoint of arc $ BC\, ($containing $ T)$ of ­$ \omega.$ Prove that lines $ PQ,\,BC,$ and $ MT$ are concurrent.
2007 China Team Selection Test
Given an integer $ k > 1.$ We call a $ k -$digits decimal integer $ a_{1}a_{2}\cdots a_{k}$ is $ p -$monotonic, if for each of integers $ i$ satisfying $ 1\le i\le k - 1,$ when $ a_{i}$ is an odd number, $ a_{i} > a_{i + 1};$ when $ a_{i}$ is an even number, $ a_{i}<a_{i + 1}.$ Find the number of $ p -$monotonic $ k -$digits integers.
2007 China Team Selection Test
Show that there exists a positive integer $ k$ such that $ k \cdot 2^{n} + 1$ is composite for all $ n \in \mathbb{N}_{0}$.
2007 China Team Selection Test
Let $ a_{1},a_{2},\cdots,a_{n}$ be positive real numbers satisfying $ a_{1} + a_{2} + \cdots + a_{n} = 1$. Prove that

$$\left(a_{1}a_{2} + a_{2}a_{3} + \cdots + a_{n}a_{1}\right)\left(\frac {a_{1}}{a_{2}^2 + a_{2}} + \frac {a_{2}}{a_{3}^2 + a_{3}} + \cdots + \frac {a_{n}}{a_{1}^2 + a_{1}}\right)\ge\frac {n}{n + 1}$$
2007 China Team Selection Test
After multiplying out and simplifying polynomial $ (x - 1)(x^2 - 1)(x^3 - 1)\cdots(x^{2007} - 1),$ getting rid of all terms whose powers are greater than $ 2007,$ we acquire a new polynomial $ f(x).$ Find its degree and the coefficient of the term having the highest power. Find the degree of $ f(x) = (1 - x)(1 - x^{2})...(1 - x^{2007})$ $ (mod$ $ x^{2008}).$
2007 China Team Selection Test
Let $ n$ be positive integer, $ A,B\subseteq[0,n]$ are sets of integers satisfying $ \mid A\mid + \mid B\mid\ge n + 2.$ Prove that there exist $ a\in A, b\in B$ such that $ a + b$ is  a power of $ 2.$
2007 China Team Selection Test
Let convex quadrilateral $ ABCD$ be inscribed in a circle centers at $ O.$ The opposite sides $ BA,CD$ meet at $ H$, the diagonals $ AC,BD$ meet at $ G.$ Let $ O_{1},O_{2}$ be the circumcenters of triangles $ AGD,BGC.$ $ O_{1}O_{2}$ intersects $ OG$ at $ N.$ The line $ HG$ cuts the circumcircles of triangles $ AGD,BGC$ at $ P,Q$, respectively. Denote by $ M$ the midpoint of $ PQ.$ Prove that $ NO = NM.$
2007 China Team Selection Test
Given $ n$ points arbitrarily in the plane $ P_{1},P_{2},\ldots,P_{n},$ among them no three points are collinear. Each of $ P_{i}$ ($1\le i\le n$) is colored red or blue arbitrarily. Let $ S$ be the set of triangles having $ \{P_{1},P_{2},\ldots,P_{n}\}$ as vertices, and having the following property: for any two segments $ P_{i}P_{j}$ and $ P_{u}P_{v},$ the number of triangles having $ P_{i}P_{j}$ as side and the number of triangles having $ P_{u}P_{v}$ as side are the same in $ S.$ Find the least $ n$ such that  in $ S$ there exist two triangles, the vertices of each triangle having the same color.
2007 China Team Selection Test
Find the smallest constant $ k$ such that



$ \frac {x}{\sqrt {x + y}} + \frac {y}{\sqrt {y + z}} + \frac {z}{\sqrt {z + x}}\leq k\sqrt {x + y + z}$



for all positive $ x$, $ y$, $ z$.
2007 China Team Selection Test
Find all the pairs of positive integers $ (a,b)$ such that $ a^2 + b - 1$ is a power of prime number $ ; a^2 + b + 1$ can divide $ b^2 - a^3 - 1,$ but it can't divide $ (a + b - 1)^2.$
2007 China Team Selection Test
Let $ ABCD$ be the inscribed quadrilateral with the circumcircle $ \omega$.Let $ \zeta$ be another circle that internally tangent to

$ \omega$ and to the lines $ BC$ and $ AD$ at points $ M,N$ respectively.Let $ I_1,I_2$ be the incenters of the $ \triangle ABC$ and $ \triangle ABD$.Prove that $ M,I_1,I_2,N$ are collinear.
2007 China Team Selection Test
Consider a $ 7\times 7$ numbers table $ a_{ij} = (i^2 + j)(i + j^2), 1\le i,j\le 7.$ When we add arbitrarily each term of an arithmetical progression consisting of $ 7$ integers to corresponding to term of certain row (or column) in turn, call it an operation. Determine whether such that each row of numbers table is an arithmetical progression, after a finite number of operations.
2006 China Team Selection Test
$ABCD$ is a trapezoid with $AB || CD$. There are two circles $\omega_1$ and $\omega_2$ is the trapezoid such that $\omega_1$ is tangent to $DA$, $AB$, $BC$ and $\omega_2$ is tangent to $BC$, $CD$, $DA$. Let $l_1$ be a line passing through $A$ and tangent to $\omega_2$(other than $AD$), Let $l_2$ be a line passing through $C$ and tangent to $\omega_1$ (other than $CB$).



Prove that $l_1 || l_2$.
2006 China Team Selection Test
Find all positive integer pairs $(a,n)$ such that $\frac{(a+1)^n-a^n}{n}$ is an integer.
2006 China Team Selection Test
Given $n$ real numbers $a_1$, $a_2$ $\ldots$ $a_n$. ($n\geq 1$).  Prove that there exists real numbers $b_1$, $b_2$ $\ldots$ $b_n$ satisfying:

(a) For any $1 \leq i \leq n$, $a_i - b_i$ is a positive integer.



(b)$\sum_{1 \leq i < j \leq n} (b_i - b_j)^2 \leq \frac{n^2-1}{12}$
2006 China Team Selection Test
Two positive valued sequences $\{ a_{n}\}$ and $\{ b_{n}\}$ satisfy:

(a): $a_{0}=1 \geq a_{1}$, $a_{n}(b_{n+1}+b_{n-1})=a_{n-1}b_{n-1}+a_{n+1}b_{n+1}$, $n \geq 1$.



(b): $\sum_{i=1}^{n}b_{i}\leq n^{\frac{3}{2}}$, $n \geq 1$.



Find the general term of $\{ a_{n}\}$.
2006 China Team Selection Test
Let $\omega$ be the circumcircle of $\triangle{ABC}$. $P$ is an interior point of $\triangle{ABC}$. $A_{1}, B_{1}, C_{1}$ are the intersections of $AP, BP, CP$ respectively and $A_{2}, B_{2}, C_{2}$ are the symmetrical points of $A_{1}, B_{1}, C_{1}$ with respect to the midpoints of side $BC, CA, AB$.

Show that the circumcircle of $\triangle{A_{2}B_{2}C_{2}}$ passes through the orthocentre of $\triangle{ABC}$.
2006 China Team Selection Test
Let $a_{i}$ and $b_{i}$ ($i=1,2, \cdots, n$) be rational numbers such that for any real number $x$ there is:

$$x^{2}+x+4=\sum_{i=1}^{n}(a_{i}x+b)^{2}$$

Find the least possible value of $n$.
2006 China Team Selection Test
The centre of the circumcircle of quadrilateral $ABCD$ is $O$ and $O$ is not on any of the sides of $ABCD$. $P=AC \cap BD$. The circumecentres of $\triangle{OAB}$, $\triangle{OBC}$, $\triangle{OCD}$ and $\triangle{ODA}$ are $O_1$, $O_2$, $O_3$ and $O_4$ respectively.



Prove that $O_1O_3$, $O_2O_4$ and $OP$ are concurrent.
2006 China Team Selection Test
$x_{1}, x_{2}, \cdots, x_{n}$ are positive numbers such that $\sum_{i=1}^{n}x_{i}= 1$. Prove that $$\left( \sum_{i=1}^{n}\sqrt{x_{i}}\right) \left( \sum_{i=1}^{n}\frac{1}{\sqrt{1+x_{i}}}\right) \leq \frac{n^{2}}{\sqrt{n+1}}$$
2006 China Team Selection Test
$d$ and $n$ are positive integers such that $d \mid n$. The n-number sets $(x_1, x_2, \cdots x_n)$ satisfy the following condition:

(1) $0 \leq x_1 \leq x_2 \leq \cdots \leq x_n \leq n$



(2) $d \mid (x_1+x_2+ \cdots x_n)$



Prove that in all the n-number sets that meet the conditions, there are exactly half satisfy $x_n=n$.
2006 China Team Selection Test
Let $K$ and $M$ be points on the side $AB$ of a triangle $\triangle{ABC}$, and let $L$ and $N$ be points on the side $AC$. The point $K$ is between $M$ and $B$, and the point $L$ is between $N$ and $C$. If $\frac{BK}{KM}=\frac{CL}{LN}$, then prove that the orthocentres of the triangles $\triangle{ABC}$, $\triangle{AKL}$ and $\triangle{AMN}$ lie on one line.
2006 China Team Selection Test
Given three positive real numbers $ x$, $ y$, $ z$ such that $ x + y + z = 1$, prove that



$ \frac {xy}{\sqrt {xy + yz}} + \frac {yz}{\sqrt {yz + zx}} + \frac {zx}{\sqrt {zx + xy}} \le \frac {\sqrt {2}}{2}$.
2006 China Team Selection Test
Find all second degree polynomial $d(x)=x^{2}+ax+b$ with integer coefficients, so that there exists an integer coefficient polynomial $p(x)$ and a non-zero integer coefficient polynomial $q(x)$ that satisfy: $$\left( p(x) \right)^{2}-d(x) \left( q(x) \right)^{2}=1, \quad \forall x \in \mathbb R.$$
2006 China Team Selection Test
Let $A$ be a non-empty subset of the set of all positive integers $N^*$. If any sufficient big positive integer can be expressed as the sum of $2$ elements in $A$(The two integers do not have to be different), then we call that $A$ is a divalent radical. For $x \geq 1$, let $A(x)$ be the set of all elements in $A$ that do not exceed $x$, prove that there exist a divalent radical $A$ and a constant number $C$ so that for every $x \geq 1$, there is always $\left| A(x) \right| \leq C \sqrt{x}$.
2006 China Team Selection Test
The function $f(n)$ satisfies $f(0)=0$, $f(n)=n-f \left( f(n-1) \right)$, $n=1,2,3 \cdots$. Find all polynomials $g(x)$ with real coefficient such that

$$ f(n)= [ g(n) ], \qquad n=0,1,2 \cdots $$

Where $[ g(n) ]$ denote the greatest integer that does not exceed $g(n)$.
2006 China Team Selection Test
Given positive integers $m$ and $n$ so there is a chessboard with $mn$ $1 \times 1$ grids. Colour the grids into red and blue (Grids that have a common side are not the same colour and the grid in the left corner at the bottom is red). Now the diagnol that goes from the left corner at the bottom to the top right corner is coloured into red and blue segments (Every segment has the same colour with the grid that contains it). Find the sum of the length of all the red segments.
2006 China Team Selection Test
Let the intersections of $\odot O_1$ and $\odot O_2$ be $A$ and $B$. Point $R$ is on arc $AB$ of $\odot O_1$ and $T$ is on arc $AB$ on $\odot O_2$. $AR$ and $BR$ meet $\odot O_2$ at $C$ and $D$; $AT$ and $BT$ meet $\odot O_1$ at $Q$ and $P$. If $PR$ and $TD$ meet at $E$ and $QR$ and $TC$ meet at $F$, then prove: $AE \cdot BT \cdot BR = BF \cdot AT \cdot AR$.
2006 China Team Selection Test
Prove that for any given positive integer $m$ and $n$, there is always a positive integer $k$ so that $2^k-m$ has at least $n$ different prime divisors.
2006 China Team Selection Test
$k$ and $n$ are positive integers that are greater than $1$. $N$ is the set of positive integers. $A_1, A_2, \cdots A_k$ are pairwise not-intersecting subsets of $N$ and $A_1 \cup A_2 \cup \cdots \cup A_k = N$.



Prove that for some $i \in \{ 1,2,\cdots,k \}$, there exsits infinity many non-factorable n-th degree polynomials so that coefficients of one polynomial are pairwise distinct and all the coeficients are in $A_i$.
2006 China Team Selection Test
$H$ is the orthocentre of $\triangle{ABC}$. $D$, $E$, $F$ are on the circumcircle of $\triangle{ABC}$ such that $AD \parallel BE \parallel CF$. $S$, $T$, $U$ are the semetrical points of $D$, $E$, $F$ with respect to $BC$, $CA$, $AB$. Show that $S, T, U, H$ lie on the same circle.
2006 China Team Selection Test
Given positive integer $n$, find the biggest real number $C$ which satisfy the condition that if the sum of the reciprocals of a set of integers (They can be the same.) that are greater than $1$ is less than $C$, then we can divide the set of numbers into no more than $n$ groups so that the sum of reciprocals of every group is less than $1$.
2006 China Team Selection Test
For a positive integer $M$, if there exist integers $a$, $b$, $c$ and $d$ so that:

$$ M \leq a < b \leq c < d \leq M+49, \qquad ad=bc $$then we call $M$ a GOOD number, if not then $M$ is BAD. Please find the greatest GOOD number and the smallest BAD number.
2006 China Team Selection Test
$k$ is an odd number that is greater or equal than $3$. Prove that there exist a $k$-th degree integer valued polynomial with non-integer coefficients that has the following properties:

(1) $f(0)=0$, $f(1)=1$.

(2) There exist infinity many positive integer $n$ so that if the equation:

$$ n= f(x_1)+\cdots+f(x_s) $$

has integer solutions $x_1 \cdots x_s$, then $s \geq 2^k-1$.
2006 China Team Selection Test
Given positive integers $m$, $a$, $b$, $(a,b)=1$. $A$ is a non-empty subset of the set of all positive integers, so that for every positive integer $n$ there is $an \in A$ and $bn \in A$. For all $A$ that satisfy the above condition, find the minimum of the value of $\left| A \cap \{ 1,2, \cdots,m \} \right|$
2006 China Team Selection Test
$\triangle{ABC}$ can cover a convex polygon $M$.Prove that there exsit a triangle which is congruent to $\triangle{ABC}$ such that it can also cover $M$ and has one side line paralel to or superpose one side line of $M$.
2005 China Team Selection Test
Convex quadrilateral $ABCD$ is cyclic in circle $(O)$, $P$ is the intersection of the diagonals $AC$ and $BD$. Circle $(O_{1})$ passes through $P$ and $B$, circle $(O_{2})$ passes through $P$ and $A$, Circles $(O_{1})$ and $(O_{2})$ intersect at $P$ and $Q$. $(O_{1})$, $(O_{2})$ intersect $(O)$ at another points $E$, $F$ (besides $B$, $A$), respectively. Prove that $PQ$, $CE$, $DF$ are concurrent or parallel.
2005 China Team Selection Test
Given positive integer $n (n \geq 2)$, find the largest positive integer $\lambda$ satisfying :

For $n$ bags, if every bag contains some balls whose weights are all integer powers of $2$ (the weights of balls in a bag may not be distinct), and the total weights of balls in every bag are equal, then there exists a weight among these balls such that the total number of balls with this weight is at least $\lambda$.
2005 China Team Selection Test
Let $n$ be a positive integer, and  $a_j$, for $j=1,2,\ldots,n$ are complex numbers. Suppose $I$ is an arbitrary nonempty subset of $\{1,2,\ldots,n\}$, the inequality $\left|-1+ \prod_{j\in I} (1+a_j) \right| \leq \frac 12$ always holds.



Prove that $\sum_{j=1}^n |a_j| \leq 3$.
2005 China Team Selection Test
Let $a_{1}$, $a_{2}$, …, $a_{6}$; $b_{1}$, $b_{2}$, …, $b_{6}$ and $c_{1}$, $c_{2}$, …, $c_{6}$ are all permutations of $1$, $2$, …, $6$, respectively. Find the minimum value of $\sum_{i=1}^{6}a_{i}b_{i}c_{i}$.
2005 China Team Selection Test
Let n be a positive integer,and x be a positive real number. Prove that



[x] here means Gauss function.
2005 China Team Selection Test
Let $\alpha$ be given positive real number, find all the functions $f: N^{+} \rightarrow R$ such that $f(k + m) = f(k) + f(m)$ holds for any positive integers $k$, $m$ satisfying $\alpha m \leq k \leq (\alpha + 1)m$.
2005 China Team Selection Test
Triangle $ABC$ is inscribed in circle $\omega$. Circle $\gamma$ is tangent to $AB$ and $AC$ at points $P$ and $Q$ respectively. Also circle $\gamma$ is tangent to circle $\omega$ at point $S$. Let the intesection of $AS$ and $PQ$ be $T$. Prove that $\angle{BTP}=\angle{CTQ}$.
2005 China Team Selection Test
Let $a$, $b$, $c$ be nonnegative reals such that $ab+bc+ca = \frac{1}{3}$. Prove that

$$\frac{1}{a^{2}-bc+1}+\frac{1}{b^{2}-ca+1}+\frac{1}{c^{2}-ab+1}\leq 3 $$
2005 China Team Selection Test
Let $a_1,a_2 \cdots a_n$ and $x_1, x_2 \cdots x_n$ be integers and $r \geq 2$ be an integer. It is known that $\sum_{j=0}^{n} a_j x_j^k =0$ for $k=1,2, \cdots r$. Prove that $\sum_{j=0}^{n} a_j x_j^m \equiv 0 (\mod m)$ for all $m \in \{ r+1, r+2, \cdots, 2r+1 \}$
2005 China Team Selection Test
Let $k$ be a positive integer. Prove that one can partition the set $\{ 0,1,2,3, \cdots ,2^{k+1}-1 \}$ into two disdinct subsets $\{ x_1,x_2, \cdots, x_{2k} \}$ and $\{ y_1, y_2, \cdots, y_{2k} \}$ such that $\sum_{i=1}^{2^k} x_i^m =\sum_{i=1}^{2^k} y_i^m$ for all $m \in \{ 1,2, \cdots, k \}$.
2005 China Team Selection Test
Cyclic quadrilateral $ABCD$ has positive integer side lengths $AB$, $BC$, $CA$, $AD$. It is known that $AD=2005$, $\angle{ABC}=\angle{ADC} = 90^o$, and $\max \{ AB,BC,CD \} < 2005$. Determine the maximum and minimum possible values for the perimeter of $ABCD$.
2005 China Team Selection Test
Let $a,b,c,d >0$ and $abcd=1$. Prove that:

$$ \frac{1}{(1+a)^2}+\frac{1}{(1+b)^2}+\frac{1}{(1+c)^2}+\frac{1}{(1+d)^2} \geq 1 $$
2005 China Team Selection Test
Determine whether $\sqrt{1001^2+1}+\sqrt{1002^2+1}+ \cdots + \sqrt{2000^2+1}$ be a rational number or not?
2005 China Team Selection Test
We call a matrix $\textsl{binary matrix}$ if all its entries equal to $0$ or $1$. A binary matrix is $\textsl{Good}$ if it simultaneously satisfies the following two conditions:

(1) All the entries above the main diagonal (from left to right), not including the main diagonal, are equal.



(2) All the entries below the main diagonal (from left to right), not including the main diagonal, are equal.



Given positive integer $m$, prove that there exists a positive integer $M$, such that for any positive integer $n>M$ and a given $n \times n$ binary matrix $A_n$, we can select integers $1 \leq i_1 <i_2< \cdots < i_{n-m} \leq n$ and delete the $i_i$-th, $i_2$-th,$\cdots$, $i_{n-m}$-th rows and $i_i$-th, $i_2$-th,$\cdots$, $i_{n-m}$-th columns of $A_n$, then the resulting binary matrix $B_m$ is $\textsl{Good}$.
2005 China Team Selection Test
Prove that for any $n$ ($n \geq 2$) pairwise disdinct fractions in the inteval $(0,1)$, the sum of their denominators is no less than $\frac{1}{3} n^{\frac{3}{2}}$.