Navvye/Olympiad-Problems-CSV huggingface dataset:
2016 China Northern MO
$a_1,a_2,\cdots,a_n$ are positive real numbers, $a_1+a_2+\cdots,a_n=1$. Prove that

$$\sum_{m=1}^n\frac{a_m}{\prod\limits_{k=1}^m(1+a_k)}\leq1-\frac{1}{2^n}.$$
2016 China Northern MO
In isosceles triangle $ABC$, $\angle CAB=\angle CBA=\alpha$, points $P,Q$ are on different sides of line $AB$, and $\angle CAP=\angle ABQ=\beta,\angle CBP=\angle BAQ=\gamma$. Prove that $P,C,Q$ are colinear.
2016 China Northern MO
Prove:

(a) There are infinitely many positive intengers $n$, satisfying:

$$\gcd(n,[\sqrt2n])=1.$$(b) There are infinitely many positive intengers $n$, satisfying:

$$\gcd(n,[\sqrt2n])>1.$$
2016 China Northern MO
Can we put intengers $1,2,\cdots,12$ on a circle, number them $a_1,a_2,\cdots,a_{12}$ in order. For any $1\leq i<j\leq12$, $|a_i-a_j|\neq|i-j|$?
2016 China Northern MO
Let $\theta_{i}\in(0,\frac{\pi}{2})(i=1,2,\cdots,n)$. Prove:

$$(\sum_{i=1}^n\tan\theta_{i})(\sum_{i=1}^n\cot\theta_{i})\geq(\sum_{i=1}^n\sin\theta_{i})^2+(\sum_{i=1}^n\cos\theta_{i})^2.$$
2016 China Northern MO
Four points $B,E,A,F$ lie on line $AB$ in order, four points $C,G,D,H$ lie on line $CD$ in order, satisfying:

$$\frac{AE}{EB}=\frac{AF}{FB}=\frac{DG}{GC}=\frac{DH}{HC}=\frac{AD}{BC}.$$Prove that $FH\perp EG$.
2016 China Northern MO
Define sequence $(a_n):a_n=2^n+3^n+6^n+1(n\in\mathbb{Z}_+)$.

Are there intenger $k\geq2$, satisfying that $\gcd(k,a_i)=1$ for all $k\in\mathbb{Z}_+$?

If yes, find the smallest $k$. If not, prove this.
2016 China Northern MO
Set $A=\{1,2,\cdots,n\}$. If there exists nonempty sets $B,C$, such that $B\cap C=\varnothing,B\cup C=A$. Sum of Squares of all elements in $B$ is $M$, Sum of Squares of all elements in $C$ is $N$, $M-N=2016$. Find the minimum value of $n$.
2016 China Northern MO
Inscribed Triangle $ABC$ on circle $\odot O$. Bisector of $\angle ABC$ intersects $\odot O$ at $D$. Two lines $PB$ and $PC$ that are tangent to $\odot O$ intersect at $P$. $PD$ intersects $AC$ at $E$, $\odot O$ at $F$. $M$ is the midpoint of $BC$. Prove that $M,F,C,E$ are concyclic.
2016 China Northern MO
$m(m>1)$ is an intenger, define $(a_n)$:

$a_0=m,a_{n}=\varphi(a_{n-1})$ for all positive intenger $n$.

If for all nonnegative intenger $k$, $a_{k+1}\mid a_k$, find all $m$ that is not larger than $2016$.

Note: $\varphi(n)$ means Euler Function.
2016 China Northern MO
$a_1=2,a_{n+1}=\frac{2^{n+1}a_n}{(n+\frac{1}{2})a_n+2^n}(n\in\mathbb{Z}_+)$

(a) Find $a_n$.

(b) Let $b_n=\frac{n^3+2n^2+2n+2}{n(n+1)(n^2+1)a_n}$.

Find $S_n=\sum_{i=1}^nb_i$.
2016 China Northern MO
Given a set $I=\{(x_1,x_2,x_3,x_4)|x_i\in\{1,2,\cdots,11\}\}$.

$A\subseteq I$, satisfying that for any $(x_1,x_2,x_3,x_4),(y_1,y_2,y_3,y_4)\in A$, there exists $i,j(1\leq i<j\leq4)$, $(x_i-x_j)(y_i-y_j)<0$. Find the maximum value of $|A|$.
2014 China Northern MO
Define a positive number sequence sequence $\{a_n\}$ by  $$a_{1}=1,(n^2+1)a^2_{n-1}=(n-1)^2a^2_{n}.$$Prove that$$\frac{1}{a^2_1}+\frac{1}{a^2_2}+\cdots +\frac{1}{a^2_n}\le 1+\sqrt{1-\frac{1}{a^2_n}}

.$$
2014 China Northern MO
Determine whether there exist an infinite number of  positive integers  $x,y $ satisfying the condition: $x^2+y \mid x+y^2.$  Please prove it.
2014 China Northern MO
Let $x,y,z,w $ be  real numbers such that $x+2y+3z+4w=1$. Find the minimum of $x^2+y^2+z^2+w^2+(x+y+z+w)^2$.
2014 China Northern MO
Prove that there exist infinitely many positive integers $n$ such that  $3^n+2$ and $5^n+2$ are all composite numbers.
2013 China Northern MO
If $a_1,a_2,\cdots,a_{2013}\in[-2,2]$ and $a_1+a_2+\cdots+a_{2013}=0$ , find the maximum of $a^3_1+a^3_2+\cdots+a^3_{2013}$.
2013 China Northern MO
Find all non-integers $x$ such that $x+\frac{13}{x}=[x]+\frac{13}{[x]} .  $where$[x]$ mean the greatest integer $n$ , where $n\leq x.$
2013 China Northern MO
Suppose that $\{a_n\}$ is a sequence such that $a_{n+1}=(1+\frac{k}{n})a_{n}+1$ with $a_{1}=1$.Find all positive integers $k$ such that  any $a_n$ be integer.
2007 China Northern MO
Let $ ABC$ be acute triangle. The circle with diameter $ AB$ intersects $ CA,\, CB$ at $ M,\, N,$ respectively. Draw $ CT\perp AB$ and intersects above circle at $ T$, where $ C$ and $ T$ lie on the same side of $ AB$. $ S$ is a point on $ AN$ such that $ BT = BS$. Prove that $ BS\perp SC$.
2007 China Northern MO
Let $ a,\, b,\, c$ be side lengths of a triangle and $ a+b+c = 3$. Find the minimum of

$$ a^{2}+b^{2}+c^{2}+\frac{4abc}{3}$$
2007 China Northern MO
Sequence $ \{a_{n}\}$ is defined by $ a_{1}= 2007,\, a_{n+1}=\frac{a_{n}^{2}}{a_{n}+1}$ for $ n \ge 1.$ Prove that $ [a_{n}] =2007-n$ for $ 0 \le n \le 1004,$ where $ [x]$ denotes the largest integer no larger than $ x.$
2007 China Northern MO
For every point on the plane, one of $ n$ colors are colored to it such that:



$ (1)$ Every color is used infinitely many times.



$ (2)$ There exists one line such that all points on this lines are colored exactly by one of two colors.



Find the least value of $ n$ such that there exist four concyclic points with pairwise distinct colors.
2007 China Northern MO
Let $ \alpha$, $ \beta$ be acute angles. Find the maximum value of

$$ \frac{\left(1-\sqrt{\tan\alpha\tan\beta}\right)^{2}}{\cot\alpha+\cot\beta}$$
2007 China Northern MO
Let $ f$ be a function given by $ f(x) = \lg(x+1)-\frac{1}{2}\cdot\log_{3}x$.



a) Solve the equation $ f(x) = 0$.



b) Find the number of the subsets of the set $$ \{n | f(n^{2}-214n-1998) \geq 0,\ n \in\mathbb{Z}\}.$$
2007 China Northern MO
Let $ n$ be a positive integer and $ [ \ n ] = a.$ Find the largest integer $ n$ such that the following two conditions are satisfied:



$ (1)$ $ n$ is not a perfect square;



$ (2)$ $ a^{3}$ divides $ n^{2}$.
2007 China Northern MO
The inradius of triangle $ ABC$ is $ 1$ and the side lengths of $ ABC$ are all integers. Prove that triangle $ ABC$ is right-angled.
2019 China Western Mathematical Olympiad
Determine all the possible positive integer $n,$ such that $3^n+n^2+2019$ is a perfect square.
2019 China Western Mathematical Olympiad
Let $O,H$ be the circumcenter and orthocenter of acute triangle $ABC$ with $AB\neq AC$, respectively. Let $M$ be the midpoint of $BC$ and $K$ be the intersection of $AM$ and the circumcircle of $\triangle BHC$, such that $M$ lies between $A$ and $K$. Let $N$ be the intersection of $HK$ and $BC$. Show that if $\angle BAM=\angle CAN$, then $AN\perp OH$.
2019 China Western Mathematical Olympiad
Let $S=\{(i,j) \vert i,j=1,2,\ldots ,100\}$ be a set consisting of points on the coordinate plane. Each element of $S$ is colored one of four given colors. A subset $T$ of $S$ is called colorful if $T$ consists of exactly $4$ points with distinct colors, which are the vertices of a rectangle whose sides are parallel to the coordinate axes. Find the maximum possible number of colorful subsets $S$ can have, among all legitimate coloring patters.
2019 China Western Mathematical Olympiad
Let $n$ be a given integer such that $n\ge 2$. Find the smallest real number $\lambda$ with the following property: for any real numbers $x_1,x_2,\ldots ,x_n\in [0,1]$ , there exists integers $\varepsilon_1,\varepsilon_2,\ldots ,\varepsilon_n\in\{0,1\}$ such that the inequality $$\left\vert \sum^j_{k=i} (\varepsilon_k-x_k)\right\vert\le \lambda$$holds for all pairs of integers $(i,j)$ where $1\le i\le j\le n$.
2019 China Western Mathematical Olympiad
In acute-angled triangle $ABC,$ $AB>AC.$ Let $O,H$ be the circumcenter and orthocenter of $\triangle ABC,$ respectively. The line passing through $H$ and parallel to $AB$ intersects line $AC$ at $M,$ and the line passing through $H$ and parallel to $AC$ intersects line $AB$ at $N.$ $L$ is the reflection of the point $H$ in $MN.$ Line $OL$ and $AH$ intersect at $K.$ Prove that $K,M,L,N$ are concyclic.
2019 China Western Mathematical Olympiad
Let $a_1,a_2,\cdots,a_n  (n\ge 2)$ be positive numbers such that $a_1\leq a_2 \leq \cdots \leq a_n .$ Prove that

$$\sum_{1\leq i< j \leq n} (a_i+a_j)^2\left(\frac{1}{i^2}+\frac{1}{j^2}\right)\geq 4(n-1)\sum_{i=1}^{n}\frac{a^2_i}{i^2}.$$
2019 China Western Mathematical Olympiad
Prove that for any positive integer $k,$ there exist finitely many sets $T$ satisfying the following two properties:

$(1)T$ consists of finitely many prime numbers;

$(2)\textup{ }\prod_{p\in T} (p+k)$ is divisible by $ \prod_{p\in T} p.$
2019 China Western Mathematical Olympiad
We call a set $S$ a good set if $S=\{x,2x,3x\}(x\neq 0).$ For a given integer $n(n\geq 3),$ determine the largest possible number of the good subsets of a set containing $n$ positive integers.
2018 China Western Mathematical Olympiad
Real numbers $x_1, x_2, \dots, x_{2018}$ satisfy $x_i + x_j \geq (-1)^{i+j}$ for all $1 \leq i < j \leq 2018$.

Find the minimum possible value of $\sum_{i=1}^{2018} ix_i$.
2018 China Western Mathematical Olympiad
Let $n \geq 2$ be an integer. Positive reals $x_1, x_2, \cdots, x_n$ satisfy $x_1x_2 \cdots x_n = 1$.

Show: $$\{x_1\} + \{x_2\} + \cdots + \{x_n\} < \frac{2n-1}{2}$$Where $\{x\}$ denotes the fractional part of $x$.
2018 China Western Mathematical Olympiad
Let $M = \{1,2,\cdots , 10\}$, and let $T$ be a set of 2-element subsets of $M$. For any two different elements $\{a,b\}, \{x,y\}$ in $T$, the integer $(ax+by)(ay+bx)$ is not divisible by 11. Find the maximum size of $T$.
2018 China Western Mathematical Olympiad
In acute angled $\triangle ABC$, $AB > AC$, points $E, F$ lie on $AC, AB$ respectively, satisfying $BF+CE = BC$. Let $I_B, I_C$ be the excenters of $\triangle ABC$ opposite $B, C$ respectively, $EI_C, FI_B$ intersect at $T$, and let $K$ be the midpoint of arc $BAC$. Let $KT$ intersect the circumcircle of $\triangle ABC$ at $K,P$. Show $T,F,P,E$ concyclic.
2018 China Western Mathematical Olympiad
In acute triangle $ABC,$ $AB<AC,$ $O$ is the circumcenter of the triangle. $M$ is the midpoint of segment $BC,$ $(AOM)$ intersects the line $AB$ again at $D$ and intersects the segment $AC$ at $E.$

Prove that $DM=EC.$
2018 China Western Mathematical Olympiad
Let $n \geq 2$ be an integer. Positive reals satisfy $a_1\geq a_2\geq \cdots\geq a_n.$

Prove that $$\left(\sum_{i=1}^n\frac{a_i}{a_{i+1}}\right)-n \leq \frac{1}{2a_1a_n}\sum_{i=1}^n(a_i-a_{i+1})^2,$$where $a_{n+1}=a_1.$
2018 China Western Mathematical Olympiad
Let $p$ and $c$ be an prime and a composite, respectively. Prove that there exist two integers $m,n,$ such that

$$0<m-n<\frac{\textup{lcm}(n+1,n+2,\cdots,m)}{\textup{lcm}(n,n+1,\cdots,m-1)}=p^c.$$
2018 China Western Mathematical Olympiad
Let $n,k$ be positive integers, satisfying $n$ is even, $k\geq 2$ and $n>4k.$ There are $n$ points on the circumference of a circle. If the endpoints of $\frac{n}{2}$ chords in a circle that do not intersect with each other are exactly the $n$ points, we call these chords a matching.Determine the maximum of integer $m,$ such that for any matching, there exists $k$ consecutive points, satisfying all the endpoints of at least $m$ chords are in the $k$ points.
2017 China Western Mathematical Olympiad
Let $p$ be a prime and $n$ be a positive integer such that $p^2$ divides $\prod_{k=1}^n (k^2+1)$. Show that $p<2n$.
2017 China Western Mathematical Olympiad
Let $n$ be a positive integer such that there exist positive integers $x_1,x_2,\cdots ,x_n$ satisfying $$x_1x_2\cdots x_n(x_1 + x_2 + \cdots + x_n)=100n.$$Find the greatest possible value of $n$.
2017 China Western Mathematical Olympiad
In triangle $ABC$, let $D$ be a point on $BC$. Let $I_1$ and $I_2$ be the incenters of triangles $ABD$ and $ACD$ respectively. Let $O_1$ and $O_2$ be the circumcenters of triangles $AI_1D$ and $AI_2D$ respectively. Let lines $I_1O_2$ and $I_2O_1$ meet at $P$. Show that $PD\perp BC$.
2017 China Western Mathematical Olympiad
Let $n$ and $k$ be given integers such that $n\ge k\ge 2$. Alice and Bob play a game on an $n$ by $n$ table with white cells. They take turns to pick a white cell and color it black. Alice moves first. The game ends as soon as there is at least one black cell in every $k$ by $k$ square after a player moves, who is declared the winner of the game. Who has the winning strategy?
2017 China Western Mathematical Olympiad
Let $a_1,a_2,\cdots ,a_9$ be $9$ positive integers (not necessarily distinct) satisfying: for all $1\le i<j<k\le 9$, there exists $l (1\le l\le 9)$ distinct from $i,j$ and $j$ such that $a_i+a_j+a_k+a_l=100$. Find the number of $9$-tuples $(a_1,a_2,\cdots ,a_9)$ satisfying the above conditions.
2017 China Western Mathematical Olympiad
In acute triangle $ABC$, let $D$ and $E$ be points on sides $AB$ and $AC$ respectively. Let segments $BE$ and $DC$ meet at point $H$. Let $M$ and $N$ be the midpoints of segments $BD$ and $CE$ respectively. Show that $H$ is the orthocenter of triangle $AMN$ if and only if $B,C,E,D$ are concyclic and $BE\perp CD$.
2017 China Western Mathematical Olympiad
Let $n=2^{\alpha} \cdot q$ be a positive integer, where $\alpha$ is a nonnegative integer and $q$ is an odd number. Show that for any positive integer $m$, the number of integer solutions to the equation $x_1^2+x_2^2+\cdots +x_n^2=m$ is divisible by $2^{\alpha +1}$.
2017 China Western Mathematical Olympiad
Let $a_1,a_2,\cdots,a_n>0$ $(n\geq 2)$. Prove that$$\sum_{i=1}^n max\{a_1,a_2,\cdots,a_i \} \cdot  min

\{a_i,a_{i+1},\cdots,a_n\}\leq \frac{n}{2\sqrt{n-1}}\sum_{i=1}^n a^2_i$$
2016 China Western Mathematical Olympiad
Let $a,b,c,d$ be real numbers such that $abcd>0$. Prove that:There exists a permutation $x,y,z,w$ of $a,b,c,d$ such that $$2(xy+zw)^2>(x^2+y^2)(z^2+w^2)$$.
2016 China Western Mathematical Olympiad
Let $\astrosun O_1$ and $\astrosun O_2$ intersect at $P$ and $Q$, their common external tangent touches $\astrosun O_1$ and $\astrosun O_2$ at $A$ and $B$ respectively. A circle $\Gamma$ passing through $A$ and $B$ intersects $\astrosun O_1$, $\astrosun O_2$ at $D$, $C$. Prove that $\displaystyle \frac{CP}{CQ}=\frac{DP}{DQ}$
2016 China Western Mathematical Olympiad
Let $n$ and $k$ be integers with $k\leq n-2$. The absolute value of the sum of elements of any $k$-element subset of $\{a_1,a_2,\cdots,a_n\}$ is less than or equal to 1. Show that: If $|a_1|\geq1$, then for any $2\leq i \leq n$, we have:



$$|a_1|+|a_i|\leq2$$
2016 China Western Mathematical Olympiad
For an $n$-tuple of integers, define a transformation to be:



$$(a_1,a_2,\cdots,a_{n-1},a_n)\rightarrow (a_1+a_2, a_2+a_3, \cdots, a_{n-1}+a_n, a_n+a_1)$$

Find all ordered pairs of integers $(n,k)$ with $n,k\geq 2$, such that for any $n$-tuple of integers $(a_1,a_2,\cdots,a_{n-1},a_n)$, after a finite number of transformations, every element in the of the $n$-tuple is a multiple of $k$.
2016 China Western Mathematical Olympiad
Prove that there exist infinitely many positive integer triples $(a,b,c)$ such that $a ,b,c$ are pairwise relatively prime ,and $ab+c ,bc+a ,ca+b$ are pairwise relatively prime .
2016 China Western Mathematical Olympiad
Let $a_1,a_2,\ldots,a_n$ be non-negative real numbers ,$S_k= \sum\limits_{i=1}^{k}a_i $ $(1\le k\le n)$.Prove that$$\sum\limits_{i=1}^{n}\left(a_iS_i\sum\limits_{j=i}^{n}a^2_j\right)\le \sum\limits_{i=1}^{n}\left(a_iS_i\right)^2$$
2016 China Western Mathematical Olympiad
$ABCD$ is a cyclic quadrilateral, and $\angle BAC = \angle DAC$. $\astrosun I_1$ and $\astrosun I_2$ are the incircles of $\triangle ABD$ and $\triangle ADC$ respectively. Prove that one of the common external tangents of $\astrosun I_1$ and $\astrosun I_2$ is parallel to $BD$
2016 China Western Mathematical Olympiad
For any given integers $m,n$ such that $2\leq m<n$ and $(m,n)=1$. Determine the smallest positive integer $k$ satisfying the following condition: for any $m$-element subset $I$ of $\{1,2,\cdots,n\}$ if $\sum_{i\in I}i> k$, then there exists a sequence of $n$ real numbers $a_1\leq a_2 \leq \cdots \leq a_n$ such that



$$\frac1m\sum_{i\in I} a_i>\frac1n\sum_{i=1}^na_i$$
2015 China Western Mathematical Olympiad
Let the integer $n \ge 2$ , and $x_1,x_2,\cdots,x_n $ be real numbers such that $\sum_{k=1}^nx_k$ be integer  . $d_k=\underset{m\in {Z}}{\min}\left|x_k-m\right| $, $1\leq k\leq n$ .Find the maximum  value of $\sum_{k=1}^nd_k$.
2015 China Western Mathematical Olympiad
Two circles $ \left(\Omega_1\right),\left(\Omega_2\right) $ touch internally on the point $ T $. Let $ M,N $ be two points on the circle $ \left(\Omega_1\right) $ which are different from $ T $ and $ A,B,C,D $ be four points on $ \left(\Omega_2\right) $ such that the chords $ AB, CD $ pass through $ M,N $, respectively. Prove that if $ AC,BD,MN $ have a common point $ K $, then $ TK $ is the angle bisector of $ \angle MTN $.



* $ \left(\Omega_2\right) $ is bigger than $ \left(\Omega_1\right) $
2015 China Western Mathematical Olympiad
Let the integer $n \ge 2$ , and $x_1,x_2,\cdots,x_n $ be positive real numbers such that $\sum_{i=1}^nx_i=1$ .Prove that$$\left(\sum_{i=1}^n\frac{1}{1-x_i}\right)\left(\sum_{1\le i<j\le n} x_ix_j\right)\le \frac{n}{2}.$$
2015 China Western Mathematical Olympiad
For $100$ straight lines on a plane, let $T$ be the set of all right-angled triangles bounded by some $3$ lines. Determine, with proof, the maximum value of $|T|$.
2015 China Western Mathematical Olympiad
Let $a,b,c,d$ are lengths of the sides of a convex quadrangle with the area equal to $S$, set $S =\{x_1, x_2,x_3,x_4\}$ consists of permutations $x_i$ of $(a, b, c, d)$. Prove that $$S \leq \frac{1}{2}(x_1x_2+x_3x_4).$$
2015 China Western Mathematical Olympiad
For a sequence $a_1,a_2,...,a_m$ of real numbers, define the following sets

$$A=\{a_i | 1\leq i\leq m\}\  \text{and} \ B=\{a_i+2a_j | 1\leq i,j\leq m, i\neq j\}$$

Let $n$ be a given integer, and $n>2$. For any strictly increasing arithmetic sequence of positive integers, determine, with proof, the minimum number of elements of set $A\triangle B$, where $A\triangle B$ $= \left(A\cup B\right) \setminus \left(A\cap B\right).$
2015 China Western Mathematical Olympiad
Let $a\in (0,1)$, $f(z)=z^2-z+a, z\in \mathbb{C}$. Prove the following statement holds:

For any complex number z with $|z| \geq 1$, there exists a complex number $z_0$ with $|z_0|=1$, such that $|f(z_0)| \leq |f(z)|$.
2015 China Western Mathematical Olympiad
Let $k$ be a positive integer, and $n=\left(2^k\right)!$ .Prove that $\sigma(n)$ has at least a prime divisor larger than $2^k$, where  $\sigma(n)$ is the sum of all positive divisors of $n$.
2014 China Western Mathematical Olympiad
Let $x,y$ be positive real numbers .Find the minimum of $x+y+\frac{|x-1|}{y}+\frac{|y-1|}{x}$.
2014 China Western Mathematical Olympiad
Let $ AB$ be the diameter of semicircle $O$ ,

$C, D $ be points on the arc $AB$,

$P, Q$ be respectively the circumcenter  of $\triangle OAC $ and $\triangle OBD $  .

Prove that:$CP\cdot CQ=DP \cdot DQ$.[asy]

import cse5; import olympiad; unitsize(3.5cm); dotfactor=4; pathpen=black;

real h=sqrt(55/64);

pair A=(-1,0), O=origin, B=(1,0),C=shift(-3/8,h)*O,D=shift(4/5,3/5)*O,P=circumcenter(O,A,C), Q=circumcenter(O,D,B);

D(arc(O,1,0,180),darkgreen);

D(MP("A",A,W)--MP("C",C,N)--MP("P",P,SE)--MP("D",D,E)--MP("Q",Q,E)--C--MP("O",O,S)--D--MP("B",B,E)--cycle,deepblue);

D(O);

[/asy]
2014 China Western Mathematical Olympiad
Let $A_1,A_2,...$ be a sequence of sets such that for any positive integer $i$, there are only finitely many values of $j$ such that $A_j\subseteq A_i$. Prove that there is a sequence of positive integers $a_1,a_2,...$ such that for any pair $(i,j)$ to have $a_i\mid a_j\iff A_i\subseteq A_j$.
2014 China Western Mathematical Olympiad
Given a positive integer $n$, let $a_1,a_2,..,a_n$ be a sequence of nonnegative integers. A sequence of one or more consecutive terms of $a_1,a_2,..,a_n$ is called $dragon$ if their aritmetic mean is larger than 1. If a sequence is a $dragon$, then its first term is the $head$ and the last term is the $tail$. Suppose $a_1,a_2,..,a_n$ is the $head$ or/and $tail$ of some $dragon$ sequence; determine the minimum value of $a_1+a_2+\cdots +a_n$ in terms of $n$.
2014 China Western Mathematical Olympiad
Given a positive integer $m$, Prove that there exists a positive integers $n_0$ such that all first digits after the decimal points of $\sqrt{n^2+817n+m}$ in decimal representation are equal, for all integers $n>n_0$.
2014 China Western Mathematical Olympiad
Let  $n\ge 2$ is a given integer , $x_1,x_2,\ldots,x_n $ be real numbers such that

$(1) x_1+x_2+\ldots+x_n=0 $,

$(2) |x_i|\le 1$ $(i=1,2,\cdots,n)$.

Find the maximum of Min$\{|x_1-x_2|,|x_2-x_3|,\cdots,|x_{n-1}-x_n|\}$.
2014 China Western Mathematical Olympiad
In the plane, Point $ O$ is the center of the equilateral triangle $ABC$ , Points $P,Q$ such that $\overrightarrow{OQ}=2\overrightarrow{PO}$.

Prove that$$|PA|+|PB|+|PC|\le |QA|+|QB|+|QC|.$$
2014 China Western Mathematical Olympiad
Given a real number $q$, $1 < q < 2$ define a sequence $ \{x_n\}$ as follows:

for any positive integer $n$, let

$$x_n=a_0+a_1 \cdot 2+ a_2 \cdot 2^2 + \cdots + a_k \cdot 2^k \qquad (a_i \in \{0,1\}, i = 0,1, \cdots m k)$$

be its binary representation, define

$$x_k= a_0 +a_1 \cdot q + a_2 \cdot q^2 + \cdots +a_k \cdot q^k.$$

Prove that for any positive integer $n$, there exists a positive integer $m$ such that $x_n < x_m \leq x_n+1$.
2013 China Western Mathematical Olympiad
Does there exist any integer $a,b,c$ such that $a^2bc+2,ab^2c+2,abc^2+2$  are perfect squares?
2013 China Western Mathematical Olympiad
Let the integer $n \ge 2$, and the real numbers $x_1,x_2,\cdots,x_n\in \left[0,1\right] $.Prove that$$\sum_{1\le k<j\le n} kx_kx_j\le \frac{n-1}{3}\sum_{k=1}^n kx_k.$$
2013 China Western Mathematical Olympiad
Let $ABC$ be a triangle, and $B_1,C_1$ be its excenters opposite $B,C$. $B_2,C_2$ are reflections of $B_1,C_1$ across midpoints of $AC,AB$. Let $D$ be the extouch at $BC$. Show that $AD$ is perpendicular to $B_2C_2$
2013 China Western Mathematical Olympiad
There are $n$ coins in a row, $n\geq 2$. If one of the coins is head, select an odd number of consecutive coins (or even 1 coin) with the one in head on the leftmost, and then flip all the selected coins upside down simultaneously. This is a $move$. No move is allowed if all $n$ coins are tails.

Suppose $m-1$ coins are heads at the initial stage, determine if there is a way to carry out $ \lfloor\frac {2^m}{3}\rfloor $ moves
2013 China Western Mathematical Olympiad
A nonempty set $A$ is called an $n$-level-good set if  $ A \subseteq \{1,2,3,\ldots,n\}$ and $|A| \le \min_{x\in A} x$ (where $|A|$ denotes the number of elements in $A$ and $\min_{x\in A} x$  denotes the minimum of the elements in $A$). Let $a_n$ be the number of $n$-level-good sets. Prove that for all positive integers $n$ we have $a_{n+2}=a_{n+1}+a_{n}+1$.
2013 China Western Mathematical Olympiad
Let $PA, PB$ be tangents to a circle centered at $O$, and $C$ a point on the minor arc $AB$. The perpendicular from $C$ to $PC$ intersects internal angle bisectors of $AOC,BOC$ at $D,E$. Show that $CD=CE$
2013 China Western Mathematical Olympiad
Label sides of a regular $n$-gon in clockwise direction in order 1,2,..,n. Determine all integers n ($n\geq 4$) satisfying the following conditions:

(1) $n-3$ non-intersecting diagonals in the $n$-gon are selected, which subdivide the $n$-gon into $n-2$ non-overlapping triangles;

(2) each of the chosen $n-3$ diagonals are labeled with an integer, such that the sum of labeled numbers on three sides of each triangles in (1) is equal to the others;
2013 China Western Mathematical Olympiad
Find all positive integers $a$ such that for any positive integer $n\ge 5$ we have  $2^n-n^2\mid a^n-n^a$.
2012 China Western Mathematical Olympiad
Find the smallest positive integer $m$ satisfying the following condition: for all  prime numbers $p$ such that $p>3$,have  $105|9^{ p^2}-29^p+m.$

(September 28, 2012, Hohhot)
2012 China Western Mathematical Olympiad
Show that among any $n\geq 3$ vertices of a regular $(2n-1)$-gon we can find $3$ of them forming an isosceles triangle.
2012 China Western Mathematical Olympiad
Let $A$ be a set of $n$ elements and $A_1, A_2, ... A_k$ subsets of $A$ such that for any $2$ distinct subsets $A_i, A_j$ either they are disjoint or one contains the other. Find the maximum value of $k$