Navvye/Olympiad-Problems-CSV huggingface dataset:
2018 China Team Selection Test
Suppose $a_i, b_i, c_i, i=1,2,\cdots ,n$, are $3n$ real numbers in the interval $\left [ 0,1 \right ].$ Define $$S=\left \{ \left ( i,j,k \right ) |\, a_i+b_j+c_k<1 \right \}, \; \; T=\left \{ \left ( i,j,k \right ) |\, a_i+b_j+c_k>2 \right \}.$$Now we know that $\left | S \right |\ge 2018,\, \left | T \right |\ge 2018.$ Try to find the minimal possible value of $n$.
2017 China Team Selection Test
Find out the maximum value of the numbers of edges of a solid regular octahedron that we can see from a point out of the regular octahedron.(We define we can see an edge $AB$ of the regular octahedron from point $P$ outside if and only if the intersection of non degenerate triangle $PAB$ and the solid regular octahedron is exactly edge $AB$.
2017 China Team Selection Test
Let $x>1$ ,$n$ be positive integer. Prove that$$\sum_{k=1}^{n}\frac{\{kx \}}{[kx]}<\sum_{k=1}^{n}\frac{1}{2k-1}$$Where $[kx ]$ be the integer part of $kx$ ,$\{kx \}$ be the decimal part of $kx$.
2017 China Team Selection Test
Suppose $S=\{1,2,3,...,2017\}$,for every subset $A$ of $S$,define a real number $f(A)\geq 0$ such that:

$(1)$ For any $A,B\subset S$,$f(A\cup B)+f(A\cap B)\leq f(A)+f(B)$;

$(2)$ For any $A\subset B\subset S$, $f(A)\leq f(B)$;

$(3)$ For any $k,j\in S$,$$f(\{1,2,\ldots,k+1\})\geq f(\{1,2,\ldots,k\}\cup \{j\});$$$(4)$ For the empty set $\varnothing$, $f(\varnothing)=0$.

Confirm that for any three-element subset $T$ of $S$,the inequality $$f(T)\leq \frac{27}{19}f(\{1,2,3\})$$holds.
2017 China Team Selection Test
Find out all the integer pairs $(m,n)$ such that there exist two monic polynomials $P(x)$ and $Q(x)$ ,with $\deg{P}=m$ and $\deg{Q}=n$,satisfy that $$P(Q(t))\not=Q(P(t))$$holds for any real number $t$.
2017 China Team Selection Test
In the non-isosceles triangle $ABC$,$D$ is the midpoint of side $BC$,$E$ is the midpoint of side $CA$,$F$ is the midpoint of side $AB$.The line(different from line $BC$) that is tangent to the inscribed circle of triangle $ABC$ and passing through point $D$ intersect line $EF$ at $X$.Define $Y,Z$ similarly.Prove that $X,Y,Z$ are collinear.
2017 China Team Selection Test
For a given positive integer $n$ and prime number $p$, find the minimum value of positive integer $m$ that satisfies the following property: for any polynomial $$f(x)=(x+a_1)(x+a_2)\ldots(x+a_n)$$($a_1,a_2,\ldots,a_n$ are positive integers), and for any non-negative integer $k$, there exists a non-negative integer $k'$ such that $$v_p(f(k))<v_p(f(k'))\leq v_p(f(k))+m.$$Note: for non-zero integer $N$,$v_p(N)$ is the largest non-zero integer $t$ that satisfies $p^t\mid N$.
2017 China Team Selection Test
Let $n$ be a positive integer. Let $D_n$ be the set of all divisors of $n$ and let $f(n)$ denote the smallest natural $m$ such that the elements of $D_n$ are pairwise distinct in mod $m$. Show that there exists a natural $N$ such that for all $n \geq N$, one has $f(n) \leq n^{0.01}$.
2017 China Team Selection Test
$2017$ engineers attend a conference. Any two engineers if they converse, converse with each other in either Chinese or English. No two engineers converse with each other more than once. It is known that within any four engineers, there was an even number of conversations and furthermore within this even number of conversations:



i) At least one conversation is in Chinese.

ii) Either no conversations are in English or the number of English conversations is at least that of Chinese conversations.



Show that there exists $673$ engineers such that any two of them conversed with each other in Chinese.
2017 China Team Selection Test
Let $ABCD$ be a quadrilateral and let $l$ be a line. Let $l$ intersect the lines $AB,CD,BC,DA,AC,BD$ at points $X,X',Y,Y',Z,Z'$ respectively. Given that these six points on $l$ are in the order $X,Y,Z,X',Y',Z'$, show that the circles with diameter $XX',YY',ZZ'$ are coaxal.
2017 China Team Selection Test
An integer  $n>1$ is given .  Find the smallest positive number $m$ satisfying the following conditions： for any set $\{a,b\}$ $\subset \{1,2,\cdots,2n-1\}$ ,there are non-negative integers   $ x, y$  ( not all zero)  such  that   $2n|ax+by$ and $x+y\leq m.$
2017 China Team Selection Test
Let $ \varphi(x)$ be a cubic polynomial with integer coefficients. Given that $ \varphi(x)$ has have 3 distinct real roots $u,v,w $ and $u,v,w $  are not  rational number. there are  integers  $ a, b,c$  such that $u=av^2+bv+c$.  Prove that $b^2 -2b -4ac - 7$ is a square number .
2017 China Team Selection Test
Let $M$ be a subset of $\mathbb{R}$ such that the following conditions are satisfied:



a) For any $x \in M, n \in \mathbb{Z}$, one has that $x+n \in \mathbb{M}$.

b) For any $x \in M$, one has that $-x \in M$.

c) Both $M$ and $\mathbb{R}$ \ $M$ contain an interval of length larger than $0$.



For any real $x$, let $M(x) = \{ n \in \mathbb{Z}^{+} | nx \in M \}$. Show that if $\alpha,\beta$ are reals such that $M(\alpha) = M(\beta)$, then we must have one of $\alpha + \beta$ and $\alpha - \beta$ to be rational.
2017 China Team Selection Test
Let $n \geq 4$ be a natural and let $x_1,\ldots,x_n$ be non-negative reals such that $x_1 + \cdots + x_n = 1$. Determine the maximum value of $x_1x_2x_3 + x_2x_3x_4 + \cdots + x_nx_1x_2$.
2017 China Team Selection Test
Let $ABCD$ be a non-cyclic convex quadrilateral. The feet of perpendiculars from $A$ to $BC,BD,CD$ are $P,Q,R$ respectively, where $P,Q$ lie on segments $BC,BD$ and $R$ lies on $CD$ extended. The feet of perpendiculars from $D$ to $AC,BC,AB$ are $X,Y,Z$ respectively, where $X,Y$ lie on segments $AC,BC$ and $Z$ lies on $BA$ extended. Let the orthocenter of $\triangle ABD$ be $H$. Prove that the common chord of circumcircles of $\triangle PQR$ and $\triangle XYZ$ bisects $BH$.
2017 China Team Selection Test
Let $X$ be a set of $100$ elements. Find the smallest possible $n$ satisfying the following condition: Given a sequence of $n$ subsets of $X$, $A_1,A_2,\ldots,A_n$, there exists $1 \leq i < j < k \leq n$ such that

$$A_i \subseteq A_j \subseteq A_k \text{ or } A_i \supseteq A_j \supseteq A_k.$$
2017 China Team Selection Test
Show that there exists a degree $58$ monic polynomial

$$P(x) = x^{58} + a_1x^{57} + \cdots + a_{58}$$such that $P(x)$ has exactly $29$ positive real roots and $29$ negative real roots and that $\log_{2017} |a_i|$ is a positive integer for all $1 \leq i \leq 58$.
2017 China Team Selection Test
Show that there exists a positive real $C$ such that for any naturals $H,N$ satisfying $H \geq 3, N \geq e^{CH}$, for any subset of $\{1,2,\ldots,N\}$ with size $\lceil \frac{CHN}{\ln N} \rceil$, one can find $H$ naturals in it such that the greatest common divisor of any two elements is the greatest common divisor of all $H$ elements.
2017 China Team Selection Test
Every cell of a $2017\times 2017$ grid is colored either black or white, such that every cell has at least one side in common with another cell of the same color. Let $V_1$ be the set of all black cells, $V_2$ be the set of all white cells. For set $V_i (i=1,2)$, if two cells share a common side, draw an edge with the centers of the two cells as endpoints, obtaining graphs $G_i$. If both $G_1$ and $G_2$ are connected paths (no cycles, no splits), prove that the center of the grid is one of the endpoints of $G_1$ or $G_2$.
2017 China Team Selection Test
Prove that :$$\sum_{k=0}^{58}C_{2017+k}^{58-k}C_{2075-k}^{k}=\sum_{p=0}^{29}C_{4091-2p}^{58-2p}$$
2017 China Team Selection Test
In $\varDelta{ABC}$,the excircle of $A$ is tangent to segment $BC$,line $AB$ and $AC$ at $E,D,F$ respectively.$EZ$ is the diameter of the circle.$B_1$ and $C_1$ are on $DF$, and $BB_1\perp{BC}$,$CC_1\perp{BC}$.Line $ZB_1,ZC_1$ intersect $BC$ at $X,Y$ respectively.Line $EZ$ and line $DF$ intersect at $H$,$ZK$ is perpendicular to $FD$ at $K$.If $H$ is the orthocenter of $\varDelta{XYZ}$,prove that:$H,K,X,Y$ are concyclic.
2017 China Team Selection Test
Find the numbers of ordered array $(x_1,...,x_{100})$ that satisfies the following conditions:

($i$)$x_1,...,x_{100}\in\{1,2,..,2017\}$;

($ii$)$2017|x_1+...+x_{100}$;

($iii$)$2017|x_1^2+...+x_{100}^2$.
2017 China Team Selection Test
Given integer $d>1,m$,prove that there exists integer $k>l>0$, such that $$(2^{2^k}+d,2^{2^l}+d)>m.$$
2017 China Team Selection Test
Given integer $m\geq2$,$x_1,...,x_m$ are non-negative real numbers,prove that:$$(m-1)^{m-1}(x_1^m+...+x_m^m)\geq(x_1+...+x_m)^m-m^mx_1...x_m$$and please find out when the equality holds.
2017 China Team Selection Test
A plane has no vertex of a regular dodecahedron on it,try to find out how many edges at most may the plane intersect the regular dodecahedron?
2017 China Team Selection Test
Given $n\ge 3$. consider a sequence $a_1,a_2,...,a_n$, if $(a_i,a_j,a_k)$ with i+k=2j (i<j<k) and $a_i+a_k\ne 2a_j$, we call such a triple a $NOT-AP$ triple. If a sequence has at least one $NOT-AP$ triple, find the least possible number of the $NOT-AP$ triple it contains.
2017 China Team Selection Test
Find the least positive number m such that for any polynimial f(x) with real coefficients, there is a polynimial g(x) with real coefficients (degree not greater than m) such that there exist 2017 distinct number $a_1,a_2,...,a_{2017}$ such that $g(a_i)=f(a_{i+1})$ for i=1,2,...,2017 where indices taken modulo 2017.
2017 China Team Selection Test
For a rational point (x,y), if xy is an integer that divided by 2 but not 3, color (x,y) red, if xy is an integer that divided by 3 but not 2, color (x,y) blue. Determine whether there is a line segment in the plane such that it contains exactly 2017 blue points and 58 red points.
2017 China Team Selection Test
Given a circle with radius 1 and 2 points C, D given on it. Given a constant l with $0<l\le 2$. Moving chord of the circle AB=l and ABCD is a non-degenerated convex quadrilateral. AC and BD intersects at P. Find the loci of the circumcenters of triangles ABP and BCP.
2017 China Team Selection Test
A(x,y), B(x,y), and C(x,y) are three homogeneous real-coefficient polynomials of x and y with degree 2, 3, and 4 respectively. we know that there is a real-coefficient polinimial R(x,y) such that $B(x,y)^2-4A(x,y)C(x,y)=-R(x,y)^2$. Proof that there exist 2 polynomials F(x,y,z) and G(x,y,z) such that $F(x,y,z)^2+G(x,y,z)^2=A(x,y)z^2+B(x,y)z+C(x,y)$ if for any x, y, z real numbers $A(x,y)z^2+B(x,y)z+C(x,y)\ge 0$
2017 China Team Selection Test
We call a graph with n vertices $k-flowing-chromatic$ if:

1. we can place a chess on each vertex and any two neighboring (connected by an edge) chesses have different colors.

2. we can choose a hamilton cycle $v_1,v_2,\cdots , v_n$, and move the chess on $v_i$ to $v_{i+1}$ with $i=1,2,\cdots ,n$ and $v_{n+1}=v_1$, such that any two neighboring chess also have different colors.

3. after some action of step 2 we can make all the chess reach each of the n vertices.

Let T(G) denote the least number k such that G is k-flowing-chromatic.

If such k does not exist, denote T(G)=0.

denote $\chi (G)$ the chromatic number of G.

Find all the positive number m such that there is a graph G with $\chi (G)\le m$ and $T(G)\ge 2^m$ without a cycle of length small than 2017.
2016 China Team Selection Test
$ABCDEF$ is a cyclic hexagon with $AB=BC=CD=DE$. $K$ is a point on segment $AE$ satisfying $\angle BKC=\angle KFE, \angle CKD = \angle KFA$. Prove that $KC=KF$.
2016 China Team Selection Test
Find the smallest positive number $\lambda $ , such that for any complex numbers ${z_1},{z_2},{z_3}\in\{z\in C\big| |z|<1\}$ ,if  $z_1+z_2+z_3=0$, then $$\left|z_1z_2 +z_2z_3+z_3z_1\right|^2+\left|z_1z_2z_3\right|^2 <\lambda .$$
2016 China Team Selection Test
Let $n \geq 2$ be a natural. Define

$$X = \{ (a_1,a_2,\cdots,a_n) | a_k \in \{0,1,2,\cdots,k\}, k = 1,2,\cdots,n \}$$.

For any two elements $s = (s_1,s_2,\cdots,s_n) \in X, t = (t_1,t_2,\cdots,t_n) \in X$, define

$$s \vee t = (\max \{s_1,t_1\},\max \{s_2,t_2\}, \cdots , \max \{s_n,t_n\} )$$$$s \wedge t = (\min \{s_1,t_1 \}, \min \{s_2,t_2,\}, \cdots, \min \{s_n,t_n\})$$Find the largest possible size of a proper subset $A$ of $X$ such that for any $s,t \in A$, one has $s \vee t \in A, s \wedge t \in A$.
2016 China Team Selection Test
Let $c,d \geq 2$ be naturals. Let $\{a_n\}$ be the sequence satisfying $a_1 = c, a_{n+1} = a_n^d + c$ for $n = 1,2,\cdots$.

Prove that for any $n \geq 2$, there exists a prime number $p$ such that $p|a_n$ and $p \not | a_i$ for $i = 1,2,\cdots n-1$.
2016 China Team Selection Test
Refer to the diagram below. Let $ABCD$ be a cyclic quadrilateral with center $O$. Let the internal angle bisectors of $\angle A$ and $\angle C$ intersect at $I$ and let those of $\angle B$ and $\angle D$ intersect at $J$. Now extend $AB$ and $CD$ to intersect $IJ$ and $P$ and $R$ respectively and let $IJ$ intersect $BC$ and $DA$ at $Q$ and $S$ respectively. Let the midpoints of $PR$ and $QS$ be $M$ and $N$ respectively. Given that $O$ does not lie on the line $IJ$, show that $OM$ and $ON$ are perpendicular.
2016 China Team Selection Test
Let $m,n$ be naturals satisfying $n \geq m \geq 2$ and let $S$ be a set consisting of $n$ naturals. Prove that $S$ has at least $2^{n-m+1}$ distinct subsets, each whose sum is divisible by $m$. (The zero set counts as a subset).
2016 China Team Selection Test
$P$ is a point in the interior of acute triangle $ABC$. $D,E,F$ are the reflections of $P$ across $BC,CA,AB$ respectively. Rays $AP,BP,CP$ meet the circumcircle of $\triangle ABC$ at $L,M,N$ respectively. Prove that the circumcircles of $\triangle PDL,\triangle PEM,\triangle PFN$ meet at a point $T$ different from $P$.
2016 China Team Selection Test
Find the smallest positive number $\lambda$, such that for any $12$ points on the plane $P_1,P_2,\ldots,P_{12}$(can overlap), if the distance between any two of them does not exceed $1$, then $\sum_{1\le i<j\le 12} |P_iP_j|^2\le \lambda$.
2016 China Team Selection Test
Let $P$ be a finite set of primes, $A$ an infinite set of positive integers, where every element of $A$ has a prime factor not in $P$. Prove that there exist an infinite subset $B$ of $A$, such that the sum of elements in any finite subset of $B$ has a prime factor not in $P$.
2016 China Team Selection Test
Set positive integer $m=2^k\cdot t$, where $k$ is a non-negative integer, $t$ is an odd number, and let $f(m)=t^{1-k}$. Prove that for any positive integer $n$ and for any positive odd number $a\le n$, $\prod_{m=1}^n f(m)$ is a multiple of $a$.
2016 China Team Selection Test
Does there exist two infinite positive integer sets $S,T$, such that any positive integer $n$ can be uniquely expressed in the form

$$n=s_1t_1+s_2t_2+\ldots+s_kt_k$$,where $k$ is a positive integer dependent on $n$, $s_1<\ldots<s_k$ are elements of $S$, $t_1,\ldots, t_k$ are elements of $T$?
2016 China Team Selection Test
The diagonals of a cyclic quadrilateral $ABCD$ intersect at $P$, and there exist a circle $\Gamma$ tangent to the extensions of $AB,BC,AD,DC$ at $X,Y,Z,T$ respectively. Circle $\Omega$ passes through points $A,B$, and is externally tangent to circle $\Gamma$ at $S$. Prove that $SP\perp ST$.
2016 China Team Selection Test
Let $n$ be an integer greater than $1$, $\alpha$ is a real, $0<\alpha < 2$, $a_1,\ldots ,a_n,c_1,\ldots ,c_n$ are all positive numbers. For $y>0$, let

$$f(y)=\left(\sum_{a_i\le y} c_ia_i^2\right)^{\frac{1}{2}}+\left(\sum_{a_i>y} c_ia_i^{\alpha} \right)^{\frac{1}{\alpha}}.$$If positive number $x$ satisfies $x\ge f(y)$ (for some $y$), prove that $f(x)\le 8^{\frac{1}{\alpha}}\cdot x$.
2016 China Team Selection Test
In the coordinate plane the points with both coordinates being rational numbers are called rational points. For any positive integer $n$, is there a way to use $n$ colours to colour all rational points, every point is coloured one colour, such that any line segment with both endpoints being rational points contains the rational points of every colour?
2016 China Team Selection Test
In cyclic quadrilateral $ABCD$, $AB>BC$, $AD>DC$, $I,J$ are the incenters of $\triangle ABC$,$\triangle ADC$ respectively. The circle with diameter $AC$ meets segment $IB$ at $X$, and the extension of $JD$ at $Y$. Prove that if the four points $B,I,J,D$ are concyclic, then $X,Y$ are the reflections of each other across $AC$.
2016 China Team Selection Test
Let $a,b,b',c,m,q$ be positive integers, where $m>1,q>1,|b-b'|\ge a$. It is given that there exist a positive integer $M$ such that

$$S_q(an+b)\equiv S_q(an+b')+c\pmod{m}$$

holds for all integers $n\ge M$. Prove that the above equation is true for all positive integers $n$. (Here $S_q(x)$ is the sum of digits of $x$ taken in base $q$).
2016 China Team Selection Test
Let $S$ be a finite set of points on a plane, where no three points are collinear, and the convex hull of $S$, $\Omega$, is a $2016-$gon $A_1A_2\ldots A_{2016}$. Every point on $S$ is labelled one of the four numbers $\pm 1,\pm 2$, such that for $i=1,2,\ldots , 1008,$ the numbers labelled on points $A_i$ and $A_{i+1008}$ are the negative of each other.

Draw triangles whose vertices are in $S$, such that any two triangles do not have any common interior points, and the union of these triangles is $\Omega$. Prove that there must exist a triangle, where the numbers labelled on some two of its vertices are the negative of each other.
2016 China Team Selection Test
Find all functions $f: \mathbb R^+ \rightarrow \mathbb R^+$ satisfying the following condition: for any three distinct real numbers $a,b,c$, a triangle can be formed with side lengths $a,b,c$, if and only if a triangle can be formed with side lengths $f(a),f(b),f(c)$.
2015 China Team Selection Test
The circle $\Gamma$ through $A$ of triangle $ABC$ meets sides $AB,AC$ at $E$,$F$ respectively, and circumcircle of $ABC$ at $P$. Prove: Reflection of $P$ across $EF$ is on $BC$ if and only if $\Gamma$ passes through $O$ (the circumcentre of $ABC$).
2015 China Team Selection Test
Let $a_1,a_2,a_3, \cdots $ be distinct positive integers, and $0<c<\frac{3}{2}$ . Prove that : There exist infinitely many positive integers $k$, such that $[a_k,a_{k+1}]>ck $.
2015 China Team Selection Test
Fix positive integers $k,n$. A candy vending machine has many different colours of candy, where there are $2n$ candies of each colour. A couple of kids each buys from the vending machine $2$ candies of different colours. Given that for any $k+1$ kids there are two kids who have at least one colour of candy in common, find the maximum number of kids.
2015 China Team Selection Test
Prove that :  For each integer $n \ge 3$,  there exists the positive integers   $a_1<a_2< \cdots <a_n$ , such that for $ i=1,2,\cdots,n-2 $ ,   With $a_{i},a_{i+1},a_{i+2}$  may be formed as a triangle side length , and the area of the triangle  is a positive integer.
2015 China Team Selection Test
FIx positive integer $n$. Prove: For any positive integers $a,b,c$ not exceeding $3n^2+4n$, there exist integers $x,y,z$ with absolute value not exceeding $2n$ and not all $0$, such that $ax+by+cz=0$
2015 China Team Selection Test
There are some players in a Ping Pong tournament, where every $2$ players play with each other at most once. Given:



(1) Each player wins at least $a$ players, and loses to at least $b$ players. ($a,b\geq 1$)



(2) For any two players $A,B$, there exist some players $P_1,...,P_k$ ($k\geq 2$) (where $P_1=A$,$P_k=B$), such that $P_i$ wins $P_{i+1}$ ($i=1,2...,k-1$).



Prove that there exist $a+b+1$ distinct players $Q_1,...Q_{a+b+1}$, such that $Q_i$ wins $Q_{i+1}$ ($i=1,...,a+b$)
2015 China Team Selection Test
For a positive integer $n$, and a non empty subset $A$ of $\{1,2,...,2n\}$, call $A$ good if the set $\{u\pm v|u,v\in A\}$ does not contain the set $\{1,2,...,n\}$. Find the smallest real number $c$, such that for any positive integer $n$, and any good subset $A$ of  $\{1,2,...,2n\}$, $|A|\leq cn$.
2015 China Team Selection Test
Let $a_1,a_2,a_3, \cdots ,a_n$ be positive real numbers. For the integers $n\ge 2$, prove that$$ \left (\frac{\sum_{j=1}^{n} \left (\prod_{k=1}^{j}a_k  \right )^{\frac{1}{j}}}{\sum_{j=1}^{n}a_j}  \right )^{\frac{1}{n}}+\frac{\left (\prod_{i=1}^{n}a_i  \right )^{\frac{1}{n}}}{\sum_{j=1}^{n} \left (\prod_{k=1}^{j}a_k  \right )^{\frac{1}{j}}}\le \frac{n+1}{n}$$
2015 China Team Selection Test
Let $ \triangle ABC $ be an acute triangle with circumcenter $ O $ and centroid $ G .$

Let $ D $ be the midpoint of $ BC $ and $ E\in \odot (BC) $ be a point inside $ \triangle ABC $ such that $ AE \perp BC  . $

Let $ F=EG \cap OD $ and $ K, L $ be the point lie on $ BC $ such that $ FK \parallel OB, FL \parallel OC . $

Let $ M \in AB $ be a point such that $ MK \perp BC $ and $ N \in AC $ be a point such that $ NL \perp BC . $

Let $ \omega $ be a circle tangent to $ OB, OC $ at $ B, C, $ respectively $ . $



Prove that $ \odot (AMN) $ is tangent to $ \omega $
2015 China Team Selection Test
Let $n$ be a positive integer, let $f_1(x),\ldots,f_n(x)$ be $n$ bounded real functions, and let $a_1,\ldots,a_n$ be $n$ distinct reals.

Show that there exists a real number $x$ such that $\sum^n_{i=1}f_i(x)-\sum^n_{i=1}f_i(x-a_i)<1$.
2015 China Team Selection Test
Set $S$ to be a subset of size $68$ of $\{1,2,...,2015\}$. Prove that there exist $3$ pairwise disjoint, non-empty subsets $A,B,C$ such that $|A|=|B|=|C|$ and $\sum_{a\in A}a=\sum_{b\in B}b=\sum_{c\in C}c$
2015 China Team Selection Test
Prove that there exist infinitely many integers $n$ such that $n^2+1$ is squarefree.
2015 China Team Selection Test
$\triangle{ABC}$ is isosceles with $AB = AC >BC$. Let $D$ be a point in its interior such that $DA = DB+DC$. Suppose that the perpendicular bisector of $AB$ meets the external angle bisector of $\angle{ADB}$ at $P$, and let $Q$ be the intersection of the perpendicular bisector of $AC$ and the external angle bisector of $\angle{ADC}$. Prove that $B,C,P,Q$ are concyclic.
2015 China Team Selection Test
Let $X$ be a non-empty and finite set, $A_1,...,A_k$ $k$ subsets of $X$, satisying:



(1) $|A_i|\leq 3,i=1,2,...,k$

(2) Any element of $X$ is an element of at least $4$ sets among $A_1,....,A_k$.



Show that one can select $[\frac{3k}{7}] $ sets from $A_1,...,A_k$ such that their union is $X$.
2015 China Team Selection Test
Let $a,b$ be two integers such that their gcd has at least two prime factors. Let $S =  \{ x \mid x \in \mathbb{N}, x \equiv a \pmod b \} $ and call $ y \in S$ irreducible if it cannot be expressed as product of two or more elements of $S$ (not necessarily distinct). Show there exists $t$ such that any element of $S$ can be expressed as product of at most $t$ irreducible elements.
2015 China Team Selection Test
Let $x_1,x_2,\cdots,x_n$ $(n\geq2)$ be a non-decreasing monotonous sequence of positive numbers such that $x_1,\frac{x_2}{2},\cdots,\frac{x_n}{n}$ is a non-increasing  monotonous sequence .Prove that

$$ \frac{\sum_{i=1}^{n} x_i }{n\left (\prod_{i=1}^{n}x_i  \right )^{\frac{1}{n}}}\le \frac{n+1}{2\sqrt[n]{n!}}$$
2015 China Team Selection Test
Let $G$ be the complete graph on $2015$ vertices. Each edge of $G$ is dyed red, blue or white. For a subset $V$ of vertices of $G$, and a pair of vertices $(u,v)$, define $$ L(u,v) = \{ u,v \} \cup \{ w | w \in V \ni \triangle{uvw} \text{ has exactly 2 red sides} \}$$Prove that, for any choice of $V$, there exist at least $120$ distinct values of $L(u,v)$.
2015 China Team Selection Test
For all natural numbers $n$, define $f(n) = \tau (n!) - \tau ((n-1)!)$, where $\tau(a)$ denotes the number of positive divisors of $a$. Prove that there exist infinitely many composite $n$, such that for all naturals $m < n$, we have $f(m) < f(n)$.
2014 China Team Selection Test
$ABCD$ is a cyclic quadrilateral, with diagonals $AC,BD$ perpendicular to each other. Let point $F$ be on side $BC$, the parallel line $EF$ to $AC$ intersect $AB$ at point $E$, line $FG$ parallel to $BD$ intersect $CD$ at $G$. Let the projection of $E$ onto $CD$ be $P$, projection of $F$ onto $DA$ be $Q$, projection of $G$ onto $AB$ be $R$. Prove that $QF$ bisects $\angle PQR$.
2014 China Team Selection Test
Let  $A$ be a finite set of positive numbers , $B=\{\frac{a+b}{c+d}  |a,b,c,d \in A  \}$.

Show that:  $\left | B \right | \ge  2\left | A \right |^2-1 $,

where $|X| $ be the number of elements of the finite set $X$.

(High School Affiliated to Nanjing Normal University )