Navvye/Olympiad-Problems-CSV huggingface dataset:
2003 China Girls Math Olympiad
Let $ n \geq 2$ be an integer. Find the largest real number $ \lambda$ such that the inequality $$ a^2_n \geq \lambda \sum^{n-1}_{i=1} a_i + 2 \cdot a_n.$$ holds for any positive integers $ a_1, a_2, \ldots a_n$ satisfying $ a_1 < a_2 < \ldots < a_n.$
2003 China Girls Math Olympiad
Let the sides of a scalene triangle $ \triangle ABC$ be $ AB = c,$ $ BC = a,$ $ CA =b,$ and $ D, E , F$ be points on $ BC, CA, AB$ such that $ AD, BE, CF$ are angle bisectors of the triangle, respectively. Assume that $ DE = DF.$ Prove that



(1) $ \frac{a}{b+c} = \frac{b}{c+a} + \frac{c}{a+b}$



(2) $ \angle BAC > 90^{\circ}.$
2003 China Girls Math Olympiad
Let $ n$ be a positive integer, and $ S_n,$ be the set of all positive integer divisors of $ n$ (including 1 and itself). Prove that at most half of the elements in $ S_n$ have their last digits equal to 3.
2002 China Girls Math Olympiad
Find all positive integers $ n$ such $ 20n+2$ can divide $ 2003n + 2002.$
2002 China Girls Math Olympiad
There are $ 3n, n \in \mathbb{Z}^+$ girl students who took part in a summer camp. There were three girl students to be on duty every day. When the summer camp ended, it was found that any two of the $ 3n$ students had just one time to be on duty on the same day.



(1) When $ n=3,$ is there any arrangement satisfying the requirement above. Prove yor conclusion.



(2) Prove that $ n$ is an odd number.
2002 China Girls Math Olympiad
Find all positive integers $ k$ such that for any positive numbers $ a, b$ and $ c$ satisfying the inequality



$$ k(ab + bc + ca) > 5(a^2 + b^2 + c^2),$$



there must exist a triangle with $ a, b$ and $ c$ as the length of its three sides respectively.
2002 China Girls Math Olympiad
Circles $O_1$ and $O_2$ interest at two points $ B$ and $ C,$ and $ BC$ is the diameter of circle $O_1.$ Construct a tangent line of circle $O_1$ at $ C$ and intersecting circle $O_2$ at another point $ A.$ We join $ AB$ to intersect circle $O_1$ at point $ E,$ then join $ CE$ and extend it to intersect circle $O_2$ at point $ F.$ Assume $ H$ is an arbitrary point on line segment $ AF.$ We join $ HE$ and extend it to intersect circle $O_1$ at point $ G,$ and then join $ BG$ and extend it to intersect the extend line of $ AC$ at point $ D.$ Prove that $$ \frac{AH}{HF} = \frac{AC}{CD}.$$
2002 China Girls Math Olympiad
There are $ n \geq 2$ permutations $ P_1, P_2, \ldots, P_n$ each being an arbitrary permutation of $ \{1,\ldots,n\}.$ Prove that



$$ \sum^{n-1}_{i=1} \frac{1}{P_i + P_{i+1}} > \frac{n-1}{n+2}.$$
2002 China Girls Math Olympiad
Find all pairs of positive integers $ (x,y)$ such that

$$ x^y = y^{x - y}.

$$

Albania
2002 China Girls Math Olympiad
An acute triangle $ ABC$ has three heights $ AD, BE$ and $ CF$ respectively. Prove that the perimeter of triangle $ DEF$ is not over half of the perimeter of triangle $ ABC.$
2002 China Girls Math Olympiad
Assume that $ A_1, A_2, \ldots, A_8$ are eight points taken arbitrarily on a plane. For a directed line $ l$ taken arbitrarily on the plane, assume that projections of $ A_1, A_2, \ldots, A_8$ on the line are $ P_1, P_2, \ldots, P_8$ respectively. If the eight projections are pairwise disjoint, they can be arranged as $ P_{i_1}, P_{i_2}, \ldots, P_{i_8}$ according to the direction of line $ l.$ Thus we get one permutation for $ 1, 2, \ldots, 8,$ namely, $ i_1, i_2, \ldots, i_8.$ In the figure, this permutation is $ 2, 1, 8, 3, 7, 4, 6, 5.$ Assume that after these eight points are projected to every directed line on the plane, we get the number of different permutations as $ N_8 = N(A_1, A_2, \ldots, A_8).$ Find the maximal value of $ N_8.$
2022 China National Olympiad
Let $a$ and $b$ be two positive real numbers, and $AB$ a segment of length $a$ on a plane. Let $C,D$ be two variable points on the plane such that $ABCD$ is a non-degenerate convex quadrilateral with $BC=CD=b$ and $DA=a$. It is easy to see that there is a circle tangent to all four sides of the quadrilateral $ABCD$.

Find the precise locus of the point $I$.
2022 China National Olympiad
Find the largest real number $\lambda$ with the following property: for any positive real numbers $p,q,r,s$ there exists a complex number $z=a+bi$($a,b\in \mathbb{R})$  such that $$ |b|\ge \lambda |a| \quad \text{and} \quad (pz^3+2qz^2+2rz+s) \cdot (qz^3+2pz^2+2sz+r) =0.$$
2022 China National Olympiad
Find all positive integers $a$ such that there exists a set $X$ of $6$ integers satisfying the following conditions: for every $k=1,2,\ldots ,36$ there exist $x,y\in X$ such that $ax+y-k$ is divisible by $37$.
2022 China National Olympiad
A conference is attended by $n (n\ge 3)$ scientists.  Each scientist has some friends in this conference (friendship is mutual and no one is a friend of him/herself). Suppose that no matter how we partition the scientists into two nonempty groups, there always exist two scientists in the same group who are friends, and there always exist two scientists in different groups who are friends.

A proposal is introduced on the first day of the conference. Each of the scientists' opinion on the proposal can be expressed as a non-negative integer. Everyday from the second day onwards, each scientists' opinion is changed to the integer part of the average of his/her friends' opinions from the previous day.

Prove that after a period of time, all scientists have the same opinion on the proposal.
2022 China National Olympiad
On a blank piece of paper, two points with distance $1$ is given. Prove that one can use (only) straightedge and compass to construct on this paper a straight line, and two points on it whose distance is $\sqrt{2021}$ such that, in the process of constructing it, the total number of circles or straight lines drawn is at most $10.$



Remark: Explicit steps of the construction should be given. Label the circles and straight lines in the order that they appear. Partial credit may be awarded depending on the total number of circles/lines.
2022 China National Olympiad
For integers $0\le a\le n$, let $f(n,a)$ denote the number of coefficients in the expansion of $(x+1)^a(x+2)^{n-a}$ that is divisible by $3.$ For example, $(x+1)^3(x+2)^1=x^4+5x^3+9x^2+7x+2$, so $f(4,3)=1$. For each positive integer $n$, let $F(n)$ be the minimum of $f(n,0),f(n,1),\ldots ,f(n,n)$.



(1) Prove that there exist infinitely many positive integer $n$ such that $F(n)\ge \frac{n-1}{3}$.



(2) Prove that for any positive integer $n$, $F(n)\le \frac{n-1}{3}$.
2021 China National Olympiad
Let $\{ z_n \}_{n \ge 1}$ be a sequence of complex numbers, whose odd terms are real, even terms are purely imaginary, and for every positive integer $k$, $|z_k z_{k+1}|=2^k$. Denote $f_n=|z_1+z_2+\cdots+z_n|,$ for $n=1,2,\cdots$

(1) Find the minimum of $f_{2020}$.

(2) Find the minimum of $f_{2020} \cdot f_{2021}$.
2021 China National Olympiad
Let $m>1$ be an integer. Find the smallest positive integer $n$, such that for any integers $a_1,a_2,\ldots ,a_n; b_1,b_2,\ldots ,b_n$ there exists integers $x_1,x_2,\ldots ,x_n$ satisfying the following two conditions:



i) There exists $i\in \{1,2,\ldots ,n\}$ such that $x_i$ and $m$ are coprime



ii) $\sum^n_{i=1} a_ix_i \equiv \sum^n_{i=1} b_ix_i \equiv 0 \pmod m$
2021 China National Olympiad
Let  $n$ be  positive integer  such that  there are exactly 36 different prime numbers that divides $n.$ For $k=1,2,3,4,5,$ $c_n$ be the number of integers that are mutually prime numbers to $n$ in the interval $[\frac{(k-1)n}{5},\frac{kn}{5}] .$ $c_1,c_2,c_3,c_4,c_5$ is not exactly the same.Prove that$$\sum_{1\le i<j\le 5}(c_i-c_j)^2\geq 2^{36}.$$
2021 China National Olympiad
In acute triangle $ABC (AB>AC)$, $M$ is the midpoint of minor arc $BC$, $O$ is the circumcenter of $(ABC)$ and $AK$ is its diameter. The line parallel to $AM$ through $O$ meets segment $AB$ at $D$, and $CA$ extended at $E$. Lines $BM$ and $CK$ meet at $P$, lines $BK$ and $CM$ meet at $Q$. Prove that $\angle OPB+\angle OEB =\angle OQC+\angle ODC$.
2021 China National Olympiad
$P$ is a convex polyhedron such that:



(1) every vertex belongs to exactly $3$ faces.



(1) For every natural number $n$, there are even number of faces with $n$ vertices.



An ant walks along the edges of $P$ and forms a non-self-intersecting cycle, which divides the faces of this polyhedron into two sides, such that for every natural number $n$, the number of faces with $n$ vertices on each side are the same. (assume this is possible)



Show that the number of times the ant turns left is the same as the number of times the ant turn right.
2021 China National Olympiad
Find $f: \mathbb{Z}_+ \rightarrow \mathbb{Z}_+$, such that for any $x,y \in \mathbb{Z}_+$, $$f(f(x)+y)\mid x+f(y).$$
2020 China National Olympiad
Let $a_1,a_2,\cdots,a_{41}\in\mathbb{R},$ such that $a_{41}=a_1, \sum_{i=1}^{40}a_i=0,$ and for any $i=1,2,\cdots,40, |a_i-a_{i+1}|\leq 1.$ Determine the greatest possible value of

$(1)a_{10}+a_{20}+a_{30}+a_{40};$

$(2)a_{10}\cdot a_{20}+a_{30}\cdot a_{40}.$
2020 China National Olympiad
In triangle $ABC$, $AB>AC.$ The bisector of $\angle BAC$ meets $BC$ at $D.$ $P$ is on line $DA,$ such that $A$ lies between $P$ and $D$. $PQ$ is tangent to $\odot(ABD)$ at $Q.$ $PR$ is tangent to $\odot(ACD)$ at $R.$ $CQ$ meets $BR$ at $K.$ The line parallel to $BC$ and passing through $K$ meets $QD,AD,RD$ at $E,L,F,$ respectively. Prove that $EL=KF.$
2020 China National Olympiad
Let $S$ be a set, $|S|=35$. A set $F$ of mappings from $S$ to itself is called to be satisfying property $P(k)$, if for any $x,y\in S$, there exist $f_1, \cdots, f_k \in F$ (not necessarily different), such that $f_k(f_{k-1}(\cdots (f_1(x))))=f_k(f_{k-1}(\cdots (f_1(y))))$.

Find the least positive integer $m$, such that if $F$ satisfies property $P(2019)$, then it also satisfies property $P(m)$.
2020 China National Olympiad
Find the largest positive constant $C$ such that the following is satisfied: Given $n$ arcs (containing their endpoints) $A_1,A_2,\ldots ,A_n$ on the circumference of a circle, where among all sets of three arcs $(A_i,A_j,A_k)$ $(1\le i< j< k\le n)$, at least half of them has $A_i\cap A_j\cap A_k$ nonempty, then there exists $l>Cn$, such that we can choose $l$ arcs among $A_1,A_2,\ldots ,A_n$, whose intersection is nonempty.
2020 China National Olympiad
Given any positive integer $c$, denote $p(c)$ as the largest prime factor of $c$. A sequence $\{a_n\}$  of positive integers satisfies $a_1>1$ and $a_{n+1}=a_n+p(a_n)$ for all $n\ge 1$. Prove that there must exist at least one perfect square in sequence $\{a_n\}$.
2020 China National Olympiad
Does there exist positive reals $a_0, a_1,\ldots ,a_{19}$, such that the polynomial $P(x)=x^{20}+a_{19}x^{19}+\ldots +a_1x+a_0$ does not have any real roots, yet all polynomials formed from swapping any two coefficients $a_i,a_j$ has at least one real root?
2019 China National Olympiad
Let $a,b,c,d,e\geq  -1$ and $a+b+c+d+e=5.$ Find the maximum and minimum value of $S=(a+b)(b+c)(c+d)(d+e)(e+a).$
2019 China National Olympiad
Call a set of 3 positive integers $\{a,b,c\}$ a Pythagorean set if $a,b,c$  are the lengths of the 3 sides of a right-angled triangle. Prove that for any 2 Pythagorean sets $P,Q$, there exists a positive integer $m\ge 2$ and Pythagorean sets $P_1,P_2,\ldots ,P_m$ such that $P=P_1, Q=P_m$, and $\forall 1\le i\le m-1$, $P_i\cap P_{i+1}\neq \emptyset$.
2019 China National Olympiad
Let $O$ be the circumcenter of $\triangle ABC$($AB<AC$), and $D$ be a point on the internal angle bisector of $\angle BAC$. Point $E$ lies on $BC$, satisfying $OE\parallel AD$, $DE\perp BC$. Point $K$ lies on $EB$ extended such that $EK=EA$. The circumcircle of $\triangle ADK$ meets $BC$ at $P\neq K$, and meets the circumcircle of $\triangle ABC$ at $Q\neq A$. Prove that $PQ$ is tangent to the circumcircle of $\triangle ABC$.
2019 China National Olympiad
Given an ellipse that is not a circle.

(1) Prove that the rhombus tangent to the ellipse at all four of its sides with minimum area  is unique.

(2) Construct this rhombus using a compass and a straight edge.
2019 China National Olympiad
Given is an $n\times n$ board, with an integer written in each grid. For each move, I can choose any grid, and add $1$ to all $2n-1$ numbers in its row and column. Find the largest $N(n)$, such that for any initial choice of integers, I can make a finite number of moves so that there are at least $N(n)$ even numbers on the board.
2019 China National Olympiad
The point $P_1, P_2,\cdots ,P_{2018} $ is placed inside or on the boundary of a given regular pentagon. Find all placement methods are made so that $$S=\sum_{1\leq i<j\leq 2018}|P_iP_j| ^2$$takes the maximum value.
2018 China National Olympiad
Let $n$ be a positive integer. Let $A_n$ denote the set of primes $p$ such that there exists positive integers $a,b$ satisfying

$$\frac{a+b}{p} \text{ and } \frac{a^n + b^n}{p^2}$$are both integers that are relatively prime to $p$. If $A_n$ is finite, let $f(n)$ denote $|A_n|$.



a) Prove that $A_n$ is finite if and only if $n \not = 2$.



b) Let $m,k$ be odd positive integers and let $d$ be their gcd. Show that

$$f(d) \leq f(k) + f(m) - f(km) \leq 2 f(d).$$
2018 China National Olympiad
Let $n$ and $k$ be positive integers and let

$$T = \{ (x,y,z) \in \mathbb{N}^3 \mid 1 \leq x,y,z \leq n \}$$be the length $n$ lattice cube. Suppose that $3n^2 - 3n + 1 + k$ points of $T$ are colored red such that if $P$ and $Q$ are red points and $PQ$ is parallel to one of the coordinate axes, then the whole line segment $PQ$ consists of only red points.



Prove that there exists at least $k$ unit cubes of length $1$, all of whose vertices are colored red.
2018 China National Olympiad
Let $q$ be a positive integer which is not a perfect cube. Prove that there exists a positive constant $C$ such that for all natural numbers $n$, one has

$$\{ nq^{\frac{1}{3}} \} + \{ nq^{\frac{2}{3}} \} \geq Cn^{-\frac{1}{2}}$$where $\{ x \}$ denotes the fractional part of $x$.
2018 China National Olympiad
$ABCD$ is a cyclic quadrilateral whose diagonals intersect at $P$. The circumcircle of $\triangle APD$ meets segment $AB$ at points $A$ and $E$. The circumcircle of $\triangle BPC$ meets segment $AB$ at points $B$ and $F$. Let $I$ and $J$ be the incenters of $\triangle ADE$ and $\triangle BCF$, respectively. Segments $IJ$ and $AC$ meet at $K$. Prove that the points $A,I,K,E$ are cyclic.
2018 China National Olympiad
Let $n \geq 3$ be an odd number and suppose that each square in a $n \times n$ chessboard is colored either black or white. Two squares are considered adjacent if they are of the same color and share a common vertex and two squares $a,b$ are considered connected if there exists a sequence of squares $c_1,\ldots,c_k$ with $c_1 = a, c_k = b$ such that $c_i, c_{i+1}$ are adjacent for $i=1,2,\ldots,k-1$.





Find the maximal number $M$ such that there exists a coloring admitting $M$ pairwise disconnected squares.
2018 China National Olympiad
Let $n > k$ be two natural numbers and let $a_1,\ldots,a_n$ be real numbers in the open interval $(k-1,k)$. Let $x_1,\ldots,x_n$ be positive reals such that for any subset $I \subset \{1,\ldots,n \}$ satisfying $|I| = k$, one has

$$\sum_{i \in I} x_i \leq \sum_{i \in I} a_i.$$Find the largest possible value of $x_1 x_2 \cdots x_n$.
2017 China National Olympiad
The sequences $\{u_{n}\}$ and $\{v_{n}\}$ are defined by $u_{0} =u_{1} =1$ ,$u_{n}=2u_{n-1}-3u_{n-2}$ $(n\geq2)$ ,  $v_{0} =a, v_{1} =b , v_{2}=c$ ,$v_{n}=v_{n-1}-3v_{n-2}+27v_{n-3}$ $(n\geq3)$. There exists a positive integer $N$ such that when $n> N$, we have $u_{n}\mid v_{n}$ . Prove that  $3a=2b+c$.
2017 China National Olympiad
In acute triangle $ABC$, let $\odot O$ be its circumcircle, $\odot I$ be its incircle. Tangents at $B,C$ to $\odot O$ meet at $L$, $\odot I$ touches $BC$ at $D$. $AY$ is perpendicular to $BC$ at $Y$, $AO$ meets $BC$ at $X$, and $OI$ meets $\odot O$ at $P,Q$. Prove that $P,Q,X,Y$ are concyclic if and only if $A,D,L$ are collinear.
2017 China National Olympiad
Consider a rectangle $R$ partitioned into $2016$ smaller rectangles such that the sides of each smaller rectangle is parallel to one of the sides of the original rectangle. Call the corners of each rectangle a vertex. For any segment joining two vertices, call it basic if no other vertex lie on it. (The segments must be part of the partitioning.) Find the maximum/minimum possible number of basic segments over all possible partitions of $R$.
2017 China National Olympiad
Let $n \geq 2$ be a natural number. For any two permutations of $(1,2,\cdots,n)$, say $\alpha = (a_1,a_2,\cdots,a_n)$ and $\beta = (b_1,b_2,\cdots,b_n),$ if there exists a natural number $k \leq n$ such that

$$b_i = \begin{cases} a_{k+1-i}, & \text{ }1 \leq i \leq k; \\ a_i, & \text{} k < i \leq n, \end{cases}$$we call $\alpha$ a friendly permutation of $\beta$.



Prove that it is possible to enumerate all possible permutations of $(1,2,\cdots,n)$ as $P_1,P_2,\cdots,P_m$ such that for all $i = 1,2,\cdots,m$, $P_{i+1}$ is a friendly permutation of $P_i$ where $m = n!$ and $P_{m+1} = P_1$.
2017 China National Olympiad
Let $D_n$ be the set of divisors of $n$. Find all natural $n$ such that it is possible to split $D_n$ into two disjoint sets $A$ and $G$, both containing at least three elements each, such that the elements in $A$ form an arithmetic progression while the elements in $G$ form a geometric progression.
2017 China National Olympiad
Given an integer $n \geq2$ and real numbers $a,b$ such that $0<a<b$. Let $x_1,x_2,\ldots, x_n\in [a,b]$ be real numbers. Find the maximum value of $$\frac{\frac{x^2_1}{x_2}+\frac{x^2_2}{x_3}+\cdots+\frac{x^2_{n-1}}{x_n}+\frac{x^2_n}{x_1}}{x_1+x_2+\cdots +x_{n-1}+x_n}.$$
2016 China National Olympiad
Let $a_1,a_2,\cdots, a_{31} ;b_1,b_2, \cdots, b_{31}$ be  positive integers such that

$a_1< a_2<\cdots< a_{31}\leq2015$ , $ b_1< b_2<\cdots<b_{31}\leq2015$ and $a_1+a_2+\cdots+a_{31}=b_1+b_2+\cdots+b_{31}.$

Find the maximum value of $S=|a_1-b_1|+|a_2-b_2|+\cdots+|a_{31}-b_{31}|.$
2016 China National Olympiad
In $\triangle AEF$, let $B$ and $D$ be on segments $AE$ and $AF$ respectively, and let $ED$ and $FB$ intersect at $C$. Define $K,L,M,N$ on segments $AB,BC,CD,DA$ such that $\frac{AK}{KB}=\frac{AD}{BC}$ and its cyclic equivalents. Let the incircle of $\triangle AEF$ touch $AE,AF$ at $S,T$ respectively; let the incircle of $\triangle CEF$ touch $CE,CF$ at $U,V$ respectively.

Prove that $K,L,M,N$ concyclic implies $S,T,U,V$ concyclic.
2016 China National Olympiad
Let $p$ be an odd prime and $a_1, a_2,...,a_p$ be integers. Prove that the following two conditions are equivalent:



1) There exists a polynomial $P(x)$ with degree $\leq \frac{p-1}{2}$ such that $P(i) \equiv a_i \pmod p$ for all $1 \leq i \leq p$



2) For any natural $d \leq \frac{p-1}{2}$,

$$ \sum_{i=1}^p (a_{i+d} - a_i )^2 \equiv 0 \pmod p$$where indices are taken $\pmod p$
2016 China National Olympiad
Let $n \geq 2$ be a positive integer and define $k$ to be the number of primes $\leq n$. Let $A$ be a subset of $S = \{2,...,n\}$ such that $|A| \leq k$ and no two elements in $A$ divide each other. Show that one can find a set $B$ such that $|B| = k$, $A \subseteq B \subseteq S$ and no two elements in $B$ divide each other.
2016 China National Olympiad
Let $ABCD$ be a convex quadrilateral. Show that there exists a square $A'B'C'D'$ (Vertices maybe ordered clockwise or counter-clockwise) such that $A \not = A', B \not = B', C \not = C', D \not = D'$ and $AA',BB',CC',DD'$ are all concurrent.
2016 China National Olympiad
Let $G$ be a complete directed graph with $100$ vertices such that for any two vertices $x,y$ one can find a directed path from $x$ to $y$.



a) Show that for any such $G$, one can find a $m$ such that for any two vertices $x,y$ one can find a directed path of length $m$ from $x$ to $y$ (Vertices can be repeated in the path)



b) For any graph $G$ with the properties above, define $m(G)$ to be smallest possible $m$ as defined in part a). Find the minimim value of $m(G)$ over all such possible $G$'s.
2015 China National Olympiad
Let $z_1,z_2,...,z_n$ be complex numbers satisfying $|z_i - 1| \leq r$ for some $r$ in $(0,1)$. Show that



$$ \left | \sum_{i=1}^n z_i \right | \cdot \left | \sum_{i=1}^n \frac{1}{z_i} \right | \geq n^2(1-r^2).$$
2015 China National Olympiad
Let $ A, B, D, E, F, C $ be six points lie on a circle (in order) satisfy $ AB=AC $ .

Let $ P=AD \cap BE, R=AF \cap CE, Q=BF \cap CD, S=AD \cap BF, T=AF \cap CD $ .

Let $ K $ be a point lie on $ ST $ satisfy $ \angle QKS=\angle ECA $ .



Prove that $ \frac{SK}{KT}=\frac{PQ}{QR} $
2015 China National Olympiad
Let $n \geq 5$ be a positive integer and let $A$ and $B$ be sets of integers satisfying the following conditions:



i) $|A| = n$, $|B| = m$ and $A$ is a subset of $B$

ii) For any distinct $x,y \in B$, $x+y \in B$ iff $x,y \in A$



Determine the minimum value of $m$.
2015 China National Olympiad
Determine all integers $k$ such that there exists infinitely many positive integers $n$ not satisfying

$$n+k |\binom{2n}{n}$$
2015 China National Olympiad
Given $30$ students such that each student has at most $5$ friends and for every $5$ students there is a pair of students that are not friends, determine the maximum $k$ such that for all such possible configurations, there exists $k$ students who are all not friends.
2015 China National Olympiad
Let $a_1,a_2,...$ be a sequence of non-negative integers such that for any $m,n$ $$ \sum_{i=1}^{2m} a_{in}  \leq m.$$ Show that there exist $k,d$ such that $$ \sum_{i=1}^{2k} a_{id} = k-2014.$$
2014 China National Olympiad
Let $ABC$ be a triangle with $AB>AC$. Let $D$ be the foot of the internal angle bisector of $A$. Points $F$ and $E$ are on $AC,AB$ respectively such that $B,C,F,E$ are concyclic. Prove that the circumcentre of $DEF$ is the incentre of $ABC$ if and only if $BE+CF=BC$.
2014 China National Olympiad
For the integer $n>1$, define $D(n)=\{ a-b\mid ab=n, a>b>0, a,b\in\mathbb{N} \}$. Prove that for any integer $k>1$, there exists pairwise distinct positive integers $n_1,n_2,\ldots,n_k$ such that $n_1,\ldots,n_k>1$ and $|D(n_1)\cap D(n_2)\cap\cdots\cap D(n_k)|\geq 2$.
2014 China National Olympiad
Prove that: there exists only one function $f:\mathbb{N^*}\to\mathbb{N^*}$ satisfying:

i) $f(1)=f(2)=1$;

ii)$f(n)=f(f(n-1))+f(n-f(n-1))$ for $n\ge 3$.

For each integer $m\ge 2$, find the value of $f(2^m)$.
2014 China National Olympiad
Let $n=p_1^{a_1}p_2^{a_2}\cdots p_t^{a_t}$ be the prime factorisation of $n$. Define $\omega(n)=t$ and $\Omega(n)=a_1+a_2+\ldots+a_t$. Prove or disprove:

For any fixed positive integer $k$ and positive reals $\alpha,\beta$, there exists a positive integer $n>1$ such that

i) $\frac{\omega(n+k)}{\omega(n)}>\alpha$

ii) $\frac{\Omega(n+k)}{\Omega(n)}<\beta$.
2014 China National Olympiad
Let $f:X\rightarrow X$, where $X=\{1,2,\ldots ,100\}$, be a function satisfying:

1) $f(x)\neq x$ for all $x=1,2,\ldots,100$;

2) for any subset $A$ of $X$ such that $|A|=40$, we have $A\cap f(A)\neq\emptyset$.

Find the minimum $k$ such that for any such function $f$, there exist a subset $B$ of $X$, where $|B|=k$, such that $B\cup f(B)=X$.
2014 China National Olympiad
For non-empty number sets $S, T$, define the sets $S+T=\{s+t\mid s\in S, t\in T\}$ and $2S=\{2s\mid s\in S\}$.

Let $n$ be a positive integer, and $A, B$ be two non-empty subsets of $\{1,2\ldots,n\}$. Show that there exists a subset $D$ of $A+B$ such that

1) $D+D\subseteq  2(A+B)$,

2) $|D|\geq\frac{|A|\cdot|B|}{2n}$,

where $|X|$ is the number of elements of the finite set $X$.
2013 China National Olympiad
Two circles $K_1$ and $K_2$ of different radii intersect at two points $A$ and $B$, let $C$ and $D$ be two points on $K_1$ and $K_2$, respectively, such that $A$ is the midpoint of the segment $CD$. The extension of $DB$ meets $K_1$ at another point $E$, the extension of $CB$ meets $K_2$ at another point $F$. Let $l_1$ and $l_2$ be the perpendicular bisectors of $CD$ and $EF$, respectively.

i) Show that $l_1$ and $l_2$ have a unique common point (denoted by $P$).

ii) Prove that the lengths of $CA$, $AP$ and $PE$ are the side lengths of a right triangle.
2013 China National Olympiad
Find all nonempty sets $S$ of integers such that $3m-2n \in  S$ for all (not necessarily distinct) $m,n \in S$.