Navvye/Olympiad-Problems-CSV huggingface dataset:
2014 China Team Selection Test
Let the function  $f:N^*\to N^*$ such that

(1) $(f(m),f(n))\le (m,n)^{2014} , \forall m,n\in N^*$;

(2) $n\le f(n)\le n+2014 , \forall n\in N^*$

Show that:  there exists the positive integers $N$ such that $ f(n)=n $, for each integer $n \ge N$.

(High School Affiliated to Nanjing Normal University )
2014 China Team Selection Test
For any real numbers sequence $\{x_n\}$ ,suppose that $\{y_n\}$ is a sequence such that:

$y_1=x_1, y_{n+1}=x_{n+1}-(\sum\limits_{i = 1}^{n} {x^2_i})^{ \frac{1}{2}}$ ${(n \ge 1})$  .

Find the smallest positive number  $\lambda$ such that  for any real numbers sequence $\{x_n\}$ and all positive integers $m$ , have $\frac{1}{m}\sum\limits_{i = 1}^{m} {x^2_i}\le\sum\limits_{i = 1}^{m} {\lambda^{m-i}y^2_i} .$

(High School Affiliated to Nanjing Normal University )
2014 China Team Selection Test
Let $a_1<a_2<...<a_t$ be $t$ given positive integers where no three form an arithmetic progression. For $k=t,t+1,...$ define $a_{k+1}$ to be the smallest positive integer larger than $a_k$ satisfying the condition that no three of $a_1,a_2,...,a_{k+1}$ form an arithmetic progression. For any $x\in\mathbb{R}^+$ define $A(x)$ to be the number of terms in $\{a_i\}_{i\ge 1}$ that are at most $x$. Show that there exist $c>1$ and $K>0$ such that $A(x)\ge c\sqrt{x}$ for any $x>K$.
2014 China Team Selection Test
Let $n\ge 2$ be a positive integer. Fill up a $n\times n$ table with the numbers $1,2,...,n^2$ exactly once each. Two cells are termed adjacent if they have a common edge. It is known that for any two adjacent cells, the numbers they contain differ by at most $n$. Show that there exist a $2\times 2$ square of adjacent cells such that the diagonally opposite pairs sum to the same number.
2014 China Team Selection Test
Prove that for any positive integers $k$ and $N$, $$\left(\frac{1}{N}\sum\limits_{n=1}^{N}(\omega (n))^k\right)^{\frac{1}{k}}\leq k+\sum\limits_{q\leq N}\frac{1}{q},$$where $\sum\limits_{q\leq N}\frac{1}{q}$ is the summation over of prime powers $q\leq N$ (including $q=1$).

Note: For integer $n>1$, $\omega (n)$ denotes number of distinct prime factors of $n$, and $\omega (1)=0$.
2014 China Team Selection Test
Given a fixed positive integer $a\geq 9$. Prove: There exist finitely many positive integers $n$, satisfying:

(1)$\tau (n)=a$

(2)$n|\phi (n)+\sigma (n)$

Note: For positive integer $n$, $\tau (n)$ is the number of positive divisors of $n$, $\phi (n)$ is the number of positive integers $\leq n$ and relatively prime with $n$, $\sigma (n)$ is the sum of positive divisors of $n$.
2014 China Team Selection Test
$A$ is the set of points of a convex $n$-gon on a plane. The distinct pairwise distances between any $2$ points in $A$ arranged in descending order is $d_1>d_2>...>d_m>0$. Let the number of unordered pairs of points in $A$ such that their distance is $d_i$ be exactly $\mu _i$, for $i=1, 2,..., m$.

Prove: For any positive integer $k\leq m$, $\mu _1+\mu _2+...+\mu _k\leq (3k-1)n$.
2014 China Team Selection Test
Given circle $O$ with radius $R$, the inscribed triangle $ABC$ is an acute scalene triangle, where $AB$ is the largest side. $AH_A, BH_B,CH_C$ are heights on $BC,CA,AB$. Let $D$ be the symmetric point of $H_A$ with respect to $H_BH_C$, $E$ be the symmetric point of $H_B$ with respect to $H_AH_C$. $P$ is the intersection of $AD,BE$, $H$ is the orthocentre of $\triangle ABC$. Prove: $OP\cdot OH$ is fixed, and find this value in terms of $R$.



(Edited)
2014 China Team Selection Test
Find the smallest positive constant $c$ satisfying: For any simple graph $G=G(V,E)$, if $|E|\geq c|V|$, then $G$ contains $2$ cycles with no common vertex, and one of them contains a chord.



Note: The cycle of graph $G(V,E)$ is a set of distinct vertices ${v_1,v_2...,v_n}\subseteq V$, $v_iv_{i+1}\in E$ for all $1\leq i\leq n$ $(n\geq 3, v_{n+1}=v_1)$; a cycle containing a chord is the cycle ${v_1,v_2...,v_n}$, such that there exist $i,j, 1< i-j< n-1$, satisfying $v_iv_j\in E$.
2014 China Team Selection Test
Let $k$ be a fixed even positive integer, $N$ is the product of $k$ distinct primes $p_1,...,p_k$, $a,b$ are two positive integers, $a,b\leq N$. Denote

$S_1=\{d|$ $d|N, a\leq d\leq b, d$ has even number of prime factors$\}$,

$S_2=\{d|$ $d|N, a\leq d\leq b, d$ has odd number of prime factors$\}$,

Prove: $|S_1|-|S_2|\leq C^{\frac{k}{2}}_k$
2014 China Team Selection Test
Let the circumcenter of triangle $ABC$ be $O$. $H_A$ is the projection of $A$ onto $BC$. The extension of $AO$ intersects the circumcircle of $BOC$ at $A'$. The projections of $A'$ onto $AB, AC$ are $D,E$, and $O_A$ is the circumcentre of triangle $DH_AE$. Define $H_B, O_B, H_C, O_C$ similarly.

Prove: $H_AO_A, H_BO_B, H_CO_C$ are concurrent
2014 China Team Selection Test
Let $A_1A_2...A_{101}$ be a regular $101$-gon, and colour every vertex red or blue. Let $N$ be the number of obtuse triangles satisfying the following: The three vertices of the triangle must be vertices of the $101$-gon, both the vertices with acute angles have the same colour, and the vertex with obtuse angle have different colour.

$(1)$ Find the largest possible value of $N$.

$(2)$ Find the number of ways to colour the vertices such that maximum $N$ is acheived. (Two colourings a different if for some $A_i$ the colours are different on the two colouring schemes).
2014 China Team Selection Test
Show that  there are no 2-tuples $ (x,y)$ of positive integers satisfying the equation $ (x+1) (x+2)\cdots (x+2014)= (y+1) (y+2)\cdots (y+4028).$
2014 China Team Selection Test
Let $k$ be a fixed odd integer, $k>3$. Prove: There exist infinitely many positive integers $n$, such that there are two positive integers $d_1, d_2$ satisfying $d_1,d_2$ each dividing $\frac{n^2+1}{2}$, and $d_1+d_2=n+k$.
2014 China Team Selection Test
Let $n$ be a given integer which is greater than $1$ .  Find the greatest constant $\lambda(n)$ such that for any non-zero complex $z_1,z_2,\cdots,z_n$ ,have that $$\sum_{k=1}^n |z_k|^2\geq \lambda(n)\min\limits_{1\le k\le n}\{|z_{k+1}-z_k|^2\},$$ where $z_{n+1}=z_1$.
2014 China Team Selection Test
For positive integer $k>1$, let $f(k)$ be the number of ways of factoring $k$ into product of positive integers greater than $1$ (The order of factors are not countered, for example $f(12)=4$, as $12$ can be factored in these $4$ ways: $12,2\cdot 6,3\cdot 4, 2\cdot 2\cdot 3$.

Prove: If $n$ is a positive integer greater than $1$, $p$ is a prime factor of $n$, then $f(n)\leq \frac{n}{p}$
2013 China Team Selection Test
The quadrilateral $ABCD$ is inscribed in circle $\omega$. $F$ is the intersection point of $AC$ and $BD$. $BA$ and $CD$ meet at $E$. Let the projection of $F$ on $AB$ and $CD$ be $G$ and $H$, respectively. Let $M$ and $N$ be the midpoints of $BC$ and $EF$, respectively. If the circumcircle of $\triangle MNG$ only meets segment $BF$ at $P$, and the circumcircle of $\triangle MNH$ only meets segment $CF$ at $Q$, prove that $PQ$ is parallel to $BC$.
2013 China Team Selection Test
For the positive integer $n$, define $f(n)=\min\limits_{m\in\Bbb Z}\left|\sqrt2-\frac mn\right|$.  Let $\{n_i\}$ be a strictly increasing sequence of positive integers. $C$ is a constant such that $f(n_i)<\dfrac C{n_i^2}$ for all $i\in\{1,2,\ldots\}$. Show that there exists a real number $q>1$ such that $n_i\geqslant q^{i-1}$ for all $i\in\{1,2,\ldots \}$.
2013 China Team Selection Test
There are$n$ balls numbered $1,2,\cdots,n$, respectively. They are painted with $4$ colours, red, yellow, blue, and green, according to the following rules:

First, randomly line them on a circle.

Then let any three clockwise consecutive balls numbered $i, j, k$, in order.

1) If $i>j>k$, then the ball $j$ is painted in red;

2) If $i<j<k$, then the ball $j$ is painted in yellow;

3) If $i<j, k<j$, then the ball $j$ is painted in blue;

4) If $i>j, k>j$, then the ball $j$ is painted in green.

And now each permutation of the balls determine a painting method.

We call two painting methods distinct, if there exists a ball, which is painted with two different colours in that two methods.



Find out the number of all distinct painting methods.
2013 China Team Selection Test
Let $n$ and $k$ be two integers which are greater than $1$. Let $a_1,a_2,\ldots,a_n,c_1,c_2,\ldots,c_m$ be non-negative real numbers such that

i) $a_1\ge a_2\ge\ldots\ge a_n$ and $a_1+a_2+\ldots+a_n=1$;

ii) For any integer $m\in\{1,2,\ldots,n\}$, we have that $c_1+c_2+\ldots+c_m\le m^k$.

Find the maximum of $c_1a_1^k+c_2a_2^k+\ldots+c_na_n^k$.
2013 China Team Selection Test
Let $P$ be a given point inside the triangle $ABC$. Suppose $L,M,N$ are the midpoints of $BC, CA, AB$ respectively and $$PL: PM: PN= BC: CA: AB.$$ The extensions of $AP, BP, CP$ meet the circumcircle of $ABC$ at $D,E,F$ respectively. Prove that the circumcentres of $APF, APE, BPF, BPD, CPD, CPE$ are concyclic.
2013 China Team Selection Test
Find all positive real numbers $r<1$ such that there exists a set $\mathcal{S}$ with the given properties:

i) For any real number $t$, exactly one of $t, t+r$ and $t+1$ belongs to $\mathcal{S}$;

ii) For any real number $t$, exactly one of $t, t-r$ and $t-1$ belongs to $\mathcal{S}$.
2013 China Team Selection Test
For a positive integer $k\ge 2$ define $\mathcal{T}_k=\{(x,y)\mid x,y=0,1,\ldots, k-1\}$ to be a collection of $k^2$ lattice points on the cartesian  coordinate plane. Let $d_1(k)>d_2(k)>\cdots$ be the decreasing sequence of the distinct distances between any two points in $T_k$. Suppose $S_i(k)$ be the number of distances equal to $d_i(k)$.

Prove that for any three positive integers $m>n>i$ we have $S_i(m)=S_i(n)$.
2013 China Team Selection Test
Prove that: there exists a positive constant $K$, and an integer series $\{a_n\}$, satisfying:

$(1)$ $0<a_1<a_2<\cdots <a_n<\cdots $;

$(2)$ For any positive integer $n$, $a_n<1.01^n K$;

$(3)$ For any finite number of distinct terms in $\{a_n\}$, their sum is not a perfect square.
2013 China Team Selection Test
Let $A$ be a set consisting of 6 points in the plane. denoted $n(A)$ as the number of the unit circles  which meet  at  least three points of $A$. Find the maximum of $n(A)$
2013 China Team Selection Test
For a positive integer $N>1$ with unique factorization $N=p_1^{\alpha_1}p_2^{\alpha_2}\dotsb p_k^{\alpha_k}$, we define

$$\Omega(N)=\alpha_1+\alpha_2+\dotsb+\alpha_k.$$

Let $a_1,a_2,\dotsc, a_n$ be positive integers and $p(x)=(x+a_1)(x+a_2)\dotsb (x+a_n)$ such that for all positive integers $k$, $\Omega(P(k))$ is even. Show that $n$ is an even number.
2013 China Team Selection Test
Find the greatest positive integer $m$ with the following property:

For every permutation $a_1, a_2, \cdots, a_n,\cdots$ of the set of positive integers, there exists positive integers $i_1<i_2<\cdots <i_m$ such that $a_{i_1}, a_{i_2}, \cdots, a_{i_m}$ is an arithmetic progression with an odd common difference.
2013 China Team Selection Test
Let $n>1$ be an integer and let $a_0,a_1,\ldots,a_n$ be non-negative real numbers. Definite $S_k=\sum_{i=0}^k \binom{k}{i}a_i$ for $k=0,1,\ldots,n$. Prove that$$\frac{1}{n} \sum_{k=0}^{n-1} S_k^2-\frac{1}{n^2}\left(\sum_{k=0}^{n} S_k\right)^2\le \frac{4}{45} (S_n-S_0)^2.$$
2013 China Team Selection Test
Let $n\ge 2$ be an integer. $a_1,a_2,\dotsc,a_n$ are arbitrarily chosen positive integers with $(a_1,a_2,\dotsc,a_n)=1$. Let $A=a_1+a_2+\dotsb+a_n$ and $(A,a_i)=d_i$. Let $(a_2,a_3,\dotsc,a_n)=D_1, (a_1,a_3,\dotsc,a_n)=D_2,\dotsc, (a_1,a_2,\dotsc,a_{n-1})=D_n$.

Find the minimum of $\prod\limits_{i=1}^n\dfrac{A-a_i}{d_iD_i}$
2013 China Team Selection Test
The circumcircle of triangle $ABC$ has centre $O$. $P$ is the midpoint of $\widehat{BAC}$ and $QP$ is the diameter. Let $I$ be the incentre of $\triangle ABC$ and let $D$ be the intersection of $PI$ and $BC$. The circumcircle of $\triangle AID$ and the extension of $PA$ meet at $F$. The point $E$ lies on the line segment $PD$ such that $DE=DQ$. Let $R,r$ be the radius of the inscribed circle and circumcircle of $\triangle ABC$, respectively.

Show that if $\angle AEF=\angle APE$, then $\sin^2\angle BAC=\dfrac{2r}R$
2013 China Team Selection Test
$101$ people,  sitting at a round table in any order,  had  $1,2,... , 101$ cards,  respectively.

A transfer is someone give  one  card  to one of  the two people  adjacent to him.

Find the  smallest positive integer $k$ such that   there always  can  through  no more than  $ k $  times transfer,  each person  hold cards of the same number, regardless of  the sitting order.
2013 China Team Selection Test
Let $p$ be a prime number and $a, k$ be positive integers such that $p^a<k<2p^a$. Prove that there exists a positive integer $n$ such that $$n<p^{2a},  C_n^k\equiv n\equiv k\pmod {p^a}.$$
2013 China Team Selection Test
Let $k\ge 2$ be an integer and let $a_1 ,a_2 ,\cdots ,a_n,b_1 ,b_2 ,\cdots ,b_n$ be non-negative real numbers. Prove that$$\left(\frac{n}{n-1}\right)^{n-1}\left(\frac{1}{n} \sum_{i=1}^{n} a_i^2\right)+\left(\frac{1}{n} \sum_{i=1}^{n} b_i\right)^2\ge\prod_{i=1}^{n}(a_i^{2}+b_i^{2})^{\frac{1}{n}}.$$
2013 China Team Selection Test
A point $(x,y)$ is a lattice point if $x,y\in\Bbb Z$. Let $E=\{(x,y):x,y\in\Bbb Z\}$. In the coordinate plane, $P$ and $Q$ are both sets of points in and on the boundary of a convex polygon with vertices on lattice points. Let $T=P\cap Q$. Prove that if $T\ne\emptyset$ and $T\cap E=\emptyset$, then $T$ is a  non-degenerate convex quadrilateral region.
2012 China Team Selection Test
Complex numbers ${x_i},{y_i}$ satisfy $\left| {{x_i}} \right| = \left| {{y_i}} \right| = 1$ for $i=1,2,\ldots ,n$. Let $x=\frac{1}{n}\sum\limits_{i=1}^n{{x_i}}$, $y=\frac{1}{n}\sum\limits_{i=1}^n{{y_i}}$ and $z_i=x{y_i}+y{x_i}-{x_i}{y_i}$. Prove that $\sum\limits_{i=1}^n{\left| {{z_i}}\right|}\leqslant n$.
2012 China Team Selection Test
Given a scalene triangle $ABC$. Its incircle touches $BC,AC,AB$ at $D,E,F$ respectvely. Let $L,M,N$ be the symmetric points of $D$ with $EF$,of $E$ with $FD$,of $F$ with $DE$,respectively. Line $AL$ intersects $BC$ at $P$,line $BM$ intersects $CA$ at $Q$,line $CN$ intersects $AB$ at $R$. Prove that $P,Q,R$ are collinear.
2012 China Team Selection Test
Let $x_n=\binom{2n}{n}$ for all $n\in\mathbb{Z}^+$. Prove there exist infinitely many finite sets $A,B$ of positive integers, satisfying $A \cap B = \emptyset $, and $$\frac{{\prod\limits_{i \in A} {{x_i}} }}{{\prod\limits_{j\in B}{{x_j}} }}=2012.$$
2012 China Team Selection Test
Given two circles ${\omega _1},{\omega _2}$, $S$ denotes all $\Delta ABC$ satisfies that ${\omega _1}$ is the circumcircle of $\Delta ABC$, ${\omega _2}$ is the $A$- excircle of $\Delta ABC$ , ${\omega _2}$ touches $BC,CA,AB$ at $D,E,F$.

$S$ is not empty, prove that the centroid of $\Delta DEF$ is a fixed point.
2012 China Team Selection Test
For a positive integer $n$, denote by $\tau (n)$ the number of its positive divisors. For a positive integer $n$, if  $\tau (m) < \tau (n)$ for all $m < n$, we call $n$ a good number. Prove that for any positive integer $k$, there are only finitely many good numbers not divisible by $k$.
2012 China Team Selection Test
$n$ being a given integer, find all functions $f\colon \mathbb{Z} \to \mathbb{Z}$, such that for all integers $x,y$ we have $f\left( {x + y + f(y)} \right) = f(x) + ny$.
2012 China Team Selection Test
In a simple graph $G$, we call $t$ pairwise adjacent vertices a $t$-clique. If a vertex is connected with all other vertices in the graph, we call it a central vertex. Given are two integers $n,k$ such that $\dfrac {3}{2} \leq \dfrac{1}{2} n < k < n$. Let $G$ be a graph on $n$ vertices such that

(1) $G$ does not contain a $(k+1)$-clique;

(2) if we add an arbitrary edge to $G$, that creates a $(k+1)$-clique.

Find the least possible number of central vertices in $G$.
2012 China Team Selection Test
Prove that there exists a positive real number $C$ with the following property: for any integer $n\ge 2$ and any subset $X$ of the set $\{1,2,\ldots,n\}$ such that $|X|\ge 2$, there exist $x,y,z,w \in X$(not necessarily distinct) such that

$$0<|xy-zw|<C\alpha ^{-4}$$

where $\alpha =\frac{|X|}{n}$.
2012 China Team Selection Test
Let $a_1<a_2$ be two given integers. For any integer $n\ge 3$, let $a_n$ be the smallest integer which is larger than $a_{n-1}$ and can be uniquely represented as $a_i+a_j$, where $1\le i<j\le n-1$. Given that there are only a finite number of even numbers in $\{a_n\}$, prove that the sequence $\{a_{n+1}-a_{n}\}$ is eventually periodic, i.e. that there exist positive integers $T,N$ such that for all integers $n>N$, we have

$$a_{T+n+1}-a_{T+n}=a_{n+1}-a_{n}.$$
2012 China Team Selection Test
Given an integer $n\ge 2$. Prove that there only exist a finite number of n-tuples of positive integers $(a_1,a_2,\ldots,a_n)$ which simultaneously satisfy the following three conditions:



$a_1>a_2>\ldots>a_n$;

$\gcd (a_1,a_2,\ldots,a_n)=1$;

$a_1=\sum_{i=1}^{n}\gcd (a_i,a_{i+1})$,where $a_{n+1}=a_1$.
2012 China Team Selection Test
Given two integers $m,n$ which are greater than $1$. $r,s$ are two given positive real numbers such that $r<s$. For all $a_{ij}\ge 0$ which are not all zeroes,find the maximal value of the expression

$$f=\frac{(\sum_{j=1}^{n}(\sum_{i=1}^{m}a_{ij}^s)^{\frac{r}{s}})^{\frac{1}{r}}}{(\sum_{i=1}^{m})\sum_{j=1}^{n}a_{ij}^r)^{\frac{s}{r}})^{\frac{1}{s}}}.$$
2012 China Team Selection Test
Given an integer $n\ge 2$, a function $f:\mathbb{Z}\rightarrow \{1,2,\ldots,n\}$ is called good, if for any integer $k,1\le k\le n-1$ there exists an integer $j(k)$ such that for every integer $m$ we have

$$f(m+j(k))\equiv f(m+k)-f(m) \pmod{n+1}. $$

Find the number of good functions.
2012 China Team Selection Test
In an acute-angled $ABC$, $\angle A>60^{\circ}$, $H$ is its orthocenter. $M,N$ are two points on $AB,AC$ respectively, such that $\angle HMB=\angle HNC=60^{\circ}$. Let $O$ be the circumcenter of triangle $HMN$. $D$ is a point on the same side with $A$ of $BC$ such that $\triangle DBC$ is an equilateral triangle. Prove that $H,O,D$ are collinear.
2012 China Team Selection Test
Given an integer $k\ge 2$. Prove that there exist $k$ pairwise distinct positive integers $a_1,a_2,\ldots,a_k$ such that for any non-negative integers $b_1,b_2,\ldots,b_k,c_1,c_2,\ldots,c_k$ satisfying $a_1\le b_i\le 2a_i, i=1,2,\ldots,k$ and $\prod_{i=1}^{k}b_i^{c_i}<\prod_{i=1}^{k}b_i$, we have

$$k\prod_{i=1}^{k}b_i^{c_i}<\prod_{i=1}^{k}b_i.$$
2012 China Team Selection Test
Find the smallest possible value of a real number $c$ such that for any $2012$-degree monic polynomial

$$P(x)=x^{2012}+a_{2011}x^{2011}+\ldots+a_1x+a_0$$with real coefficients,  we can obtain a new polynomial $Q(x)$ by multiplying some of its coefficients by $-1$ such that every root $z$ of $Q(x)$ satisfies the inequality

$$ \left\lvert \operatorname{Im} z \right\rvert \le c \left\lvert \operatorname{Re} z \right\rvert. $$
2012 China Team Selection Test
Given an integer $n\ge 4$. $S=\{1,2,\ldots,n\}$. $A,B$ are two subsets of $S$ such that for every pair of $(a,b),a\in A,b\in B, ab+1$ is a perfect square. Prove that

$$\min \{|A|,|B|\}\le\log _2n.$$
2012 China Team Selection Test
Find all integers $k\ge 3$ with the following property: There exist integers $m,n$ such that $1<m<k$, $1<n<k$, $\gcd (m,k)=\gcd (n,k) =1$, $m+n>k$ and $k\mid (m-1)(n-1)$.
2012 China Team Selection Test
In some squares of a $2012\times 2012$ grid there are some beetles, such that no square contain more than one beetle. At one moment, all the beetles fly off the grid and then land on the grid again, also satisfying the condition that there is at most one beetle standing in each square. The vector from the centre of the square from which a beetle $B$ flies to the centre of the square on which it lands is called the translation vector of beetle $B$.

For all possible starting and ending configurations, find the maximum length of the sum of the  translation vectors of all beetles.
2011 China Team Selection Test
In $\triangle ABC$ we have $BC>CA>AB$. The nine point circle is tangent to the incircle, $A$-excircle, $B$-excircle and $C$-excircle at the points $T,T_A,T_B,T_C$ respectively. Prove that the segments $TT_B$ and lines $T_AT_C$ intersect each other.
2011 China Team Selection Test
Let $S$ be a set of $n$ points in the plane such that no four points are collinear. Let $\{d_1,d_2,\cdots ,d_k\}$ be the set of distances between pairs of distinct points in $S$, and let $m_i$ be the multiplicity of $d_i$, i.e. the number of unordered pairs $\{P,Q\}\subseteq S$ with $|PQ|=d_i$. Prove that $\sum_{i=1}^k m_i^2\leq n^3-n^2$.
2011 China Team Selection Test
A positive integer $n$ is known as an interesting number if $n$ satisfies

$${\  \{\frac{n}{10^k}} \} > \frac{n}{10^{10}} $$

for all $k=1,2,\ldots 9$.

Find the number of interesting numbers.
2011 China Team Selection Test
Let one of the intersection points of two circles with centres $O_1,O_2$ be $P$. A common tangent touches the circles at $A,B$ respectively. Let the perpendicular from $A$ to the line $BP$ meet $O_1O_2$ at $C$. Prove that $AP\perp PC$.
2011 China Team Selection Test
Let $n$ be a positive integer and let $\alpha_n $ be the number of $1$'s within binary representation of $n$.



Show that for all positive integers $r$,

$$\ 2^{2n-\alpha_n} \huge\mid \sum_{k=-n}^{n} {C_{2n}}^{n+k}k^{2r}.$$
2011 China Team Selection Test
For a given integer $n\ge 2$, let $a_0,a_1,\ldots ,a_n$ be integers satisfying $0=a_0<a_1<\ldots <a_n=2n-1$. Find the smallest possible number of elements in the set $\{ a_i+a_j \mid 0\le i \le j \le n \}$.
2011 China Team Selection Test
Let $n\geq 2$ be a given integer. Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that

$$f(x-f(y))=f(x+y^n)+f(f(y)+y^n), \qquad \forall x,y \in \mathbb R.$$
2011 China Team Selection Test
Let $\ell$ be a positive integer, and let $m,n$ be positive integers with $m\geq n$, such that $A_1,A_2,\cdots,A_m,B_1,\cdots,B_m$ are $m+n$ pairwise distinct subsets of the set $\{1,2,\cdots,\ell\}$. It is known that $A_i\Delta B_j$ are pairwise distinct, $1\leq i\leq m, 1\leq j\leq n$, and runs over all nonempty subsets of $\{1,2,\cdots,\ell\}$. Find all possible values of $m,n$.
2011 China Team Selection Test
For any positive integer $d$, prove there are infinitely many positive integers $n$ such that  $d(n!)-1$ is a composite number.
2011 China Team Selection Test
Let $AA',BB',CC'$ be three diameters of the circumcircle of an acute triangle $ABC$. Let $P$ be an arbitrary point in the interior of $\triangle ABC$, and let $D,E,F$ be the orthogonal projection of $P$ on $BC,CA,AB$, respectively. Let $X$ be the point such that $D$ is the midpoint of $A'X$, let $Y$ be the point such that $E$ is the midpoint of $B'Y$, and similarly let $Z$ be the point such that $F$ is the midpoint of $C'Z$. Prove that triangle $XYZ$ is similar to  triangle $ABC$.
2011 China Team Selection Test
Let $\{b_n\}_{n\geq 1}^{\infty}$ be a sequence of positive integers. The sequence $\{a_n\}_{n\geq 1}^{\infty}$ is defined as follows: $a_1$ is a fixed positive integer and

$$a_{n+1}=a_n^{b_n}+1 ,\qquad \forall n\geq 1.$$

Find all positive integers $m\geq 3$ with the following property: If the sequence $\{a_n\mod m\}_{n\geq 1 }^{\infty}$ is eventually periodic, then there exist positive integers $q,u,v$ with $2\leq q\leq m-1$, such that the sequence $\{b_{v+ut}\mod q\}_{t\geq 1}^{\infty}$ is purely periodic.
2011 China Team Selection Test
Let $n$ be a positive integer. Find the largest real number $\lambda$ such that for all positive real numbers $x_1,x_2,\cdots,x_{2n}$ satisfying the inequality

$$\frac{1}{2n}\sum_{i=1}^{2n}(x_i+2)^n\geq \prod_{i=1}^{2n} x_i,$$

the following inequality also holds

$$\frac{1}{2n}\sum_{i=1}^{2n}(x_i+1)^n\geq \lambda\prod_{i=1}^{2n} x_i.$$
2011 China Team Selection Test
Let $n\geq 3$ be an integer. Find the largest real number $M$ such that for any positive real numbers $x_1,x_2,\cdots,x_n$, there exists an arrangement $y_1,y_2,\cdots,y_n$ of real numbers satisfying

$$\sum_{i=1}^n \frac{y_i^2}{y_{i+1}^2-y_{i+1}y_{i+2}+y_{i+2}^2}\geq M,$$

where $y_{n+1}=y_1,y_{n+2}=y_2$.
2011 China Team Selection Test
Let $n>1$ be an integer, and let $k$ be the number of distinct prime divisors of $n$. Prove that there exists an integer $a$, $1<a<\frac{n}{k}+1$, such that $n \mid a^2-a$.