Navvye/Olympiad-Problems-CSV huggingface dataset:
2012 China Western Mathematical Olympiad
$P$ is a point in the $\Delta ABC$, $\omega $ is the circumcircle of $\Delta ABC $. $BP \cap \omega  = \left\{ {B,{B_1}} \right\}$,$CP \cap \omega  = \left\{ {C,{C_1}} \right\}$, $PE \bot AC$，$PF \bot AB$. The radius of the inscribed circle and circumcircle of $\Delta ABC $ is $r,R$. Prove $\frac{{EF}}{{{B_1}{C_1}}} \geqslant \frac{r}{R}$.
2012 China Western Mathematical Olympiad
$O$ is the circumcenter of acute $\Delta ABC$, $H$ is the Orthocenter. $AD \bot BC$, $EF$ is  the perpendicular bisector of $AO$,$D,E$ on the $BC$. Prove that the circumcircle of $\Delta ADE$ through the midpoint of $OH$.
2012 China Western Mathematical Olympiad
Define a sequence $\{a_n\}$  by$$a_0=\frac{1}{2},\ a_{n+1}=a_{n}+\frac{a_{n}^2}{2012}, (n=0,\ 1,\ 2,\ \cdots),$$  find  integer $k$ such that   $a_{k}<1<a_{k+1}.$



(September 29, 2012, Hohhot)
2012 China Western Mathematical Olympiad
Let $n$ be a positive integer $\geq 2$ . Consider a $n$ by $n$ grid with all entries $1$. Define an operation on a square to be changing the signs of all squares adjacent to it but not the sign of its own. Find all $n$ such that it is possible after a finite sequence of operations to reach a $n$ by $n$ grid with all entries $-1$
2012 China Western Mathematical Olympiad
Find all  prime number  $p$,  such that  there exist an infinite number of  positive integer $n$  satisfying the following condition:

$p|n^{ n+1}+(n+1)^n.$



(September 29, 2012, Hohhot)
2011 China Western Mathematical Olympiad
Given that $0 < x,y < 1$, find the maximum value of  $\frac{xy(1-x-y)}{(x+y)(1-x)(1-y)}$
2011 China Western Mathematical Olympiad
Let $M$ be a subset of $\{1,2,3... 2011\}$ satisfying the following condition:

For any three elements in $M$, there exist two of them $a$ and $b$ such that $a|b$ or $b|a$.

Determine the maximum value of $|M|$ where $|M|$ denotes the number of elements in $M$
2011 China Western Mathematical Olympiad
Let $n \geq 2$ be a given integer

$a)$ Prove that one can arrange all the subsets of the set $\{1,2... ,n\}$ as a sequence of subsets $A_{1}, A_{2},\cdots , A_{2^{n}}$, such that $|A_{i+1}| = |A_{i}| + 1$ or $|A_{i}| - 1$ where $i = 1,2,3,\cdots , 2^{n}$ and $A_{2^{n} + 1} = A_{1}$

$b)$ Determine all possible values of the sum $\sum \limits_{i = 1}^{2^n} (-1)^{i}S(A_{i})$ where $S(A_{i})$ denotes the sum of all elements in $A_{i}$ and $S(\emptyset) = 0$, for any subset sequence $A_{1},A_{2},\cdots ,A_{2^n}$ satisfying the condition in $a)$
2011 China Western Mathematical Olympiad
In a circle $\Gamma_{1}$, centered at $O$, $AB$ and $CD$ are two unequal in length chords intersecting at $E$ inside $\Gamma_{1}$. A circle $\Gamma_{2}$, centered at $I$ is tangent to $\Gamma_{1}$ internally at $F$, and also tangent to $AB$ at $G$ and $CD$ at $H$. A line $l$ through $O$ intersects $AB$ and $CD$ at $P$ and $Q$ respectively such that $EP = EQ$. The line $EF$ intersects $l$ at $M$. Prove that the line through $M$ parallel to $AB$ is tangent to $\Gamma_{1}$
2011 China Western Mathematical Olympiad
Does there exist any odd integer $n \geq 3$ and $n$ distinct prime numbers $p_1 , p_2, \cdots p_n$ such that all $p_i + p_{i+1} (i = 1,2,\cdots , n$ and $p_{n+1} = p_{1})$ are perfect squares?
2011 China Western Mathematical Olympiad
Let $a,b,c > 0$, prove that

$$\frac{(a-b)^2}{(c+a)(c+b)} + \frac{(b-c)^2}{(a+b)(a+c)} + \frac{(c-a)^2}{(b+c)(b+a)} \geq \frac{(a-b)^2}{a^2+b^2+c^2}$$
2011 China Western Mathematical Olympiad
In triangle $ABC$ with $AB>AC$ and incenter $I$, the incircle touches $BC,CA,AB$ at $D,E,F$ respectively. $M$ is the midpoint of $BC$, and the altitude at $A$ meets $BC$ at $H$. Ray $AI$ meets lines $DE$ and $DF$ at $K$ and $L$, respectively. Prove that the points $M,L,H,K$ are concyclic.
2011 China Western Mathematical Olympiad
Find all pairs of integers $(a,b)$ such that $n|( a^n + b^{n+1})$ for all positive integer $n$
2010 China Western Mathematical Olympiad
Suppose that $m$ and $k$ are non-negative integers, and $p = 2^{2^m}+1$ is a prime number. Prove that

(a) $2^{2^{m+1}p^k} \equiv 1$ $(\text{mod } p^{k+1})$;

(b) $2^{m+1}p^k$ is the smallest positive integer $n$ satisfying the congruence equation $2^n \equiv 1$ $(\text{mod } p^{k+1})$.
2010 China Western Mathematical Olympiad
$AB$ is a diameter of a circle with center $O$. Let $C$ and $D$ be two different points on the circle on the same side of $AB$, and the lines tangent to the circle at points $C$ and $D$ meet at $E$. Segments $AD$ and $BC$ meet at $F$. Lines $EF$ and $AB$ meet at $M$. Prove that $E,C,M$ and $D$ are concyclic.
2010 China Western Mathematical Olympiad
Determine all possible values of positive integer $n$, such that there are $n$ different 3-element subsets $A_1,A_2,...,A_n$ of the set $\{1,2,...,n\}$, with $|A_i \cap A_j| \not= 1$ for all $i \not= j$.
2010 China Western Mathematical Olympiad
Let $a_1,a_2,..,a_n,b_1,b_2,...,b_n$ be non-negative numbers satisfying the following conditions simultaneously:



(1) $\displaystyle\sum_{i=1}^{n} (a_i + b_i) = 1$;



(2) $\displaystyle\sum_{i=1}^{n} i(a_i - b_i) = 0$;



(3) $\displaystyle\sum_{i=1}^{n} i^2(a_i + b_i) = 10$.



Prove that $\text{max}\{a_k,b_k\} \le \dfrac{10}{10+k^2}$ for all $1 \le k \le n$.
2010 China Western Mathematical Olympiad
Let $k$ be an integer and $k > 1$. Define a sequence $\{a_n\}$ as follows:



$a_0 = 0$,



$a_1 = 1$, and



$a_{n+1} = ka_n + a_{n-1}$ for $n = 1,2,...$.



Determine, with proof, all possible $k$ for which there exist non-negative integers $l,m (l \not= m)$ and positive integers $p,q$ such that $a_l + ka_p = a_m + ka_q$.
2010 China Western Mathematical Olympiad
$\Delta ABC$ is a right-angled triangle, $\angle C = 90^{\circ}$. Draw a circle centered at $B$ with radius $BC$. Let $D$ be a point on the side $AC$, and $DE$ is tangent to the circle at $E$. The line through $C$ perpendicular to $AB$ meets line $BE$ at $F$. Line $AF$ meets $DE$ at point $G$. The line through $A$ parallel to $BG$ meets $DE$ at $H$. Prove that $GE = GH$.
2010 China Western Mathematical Olympiad
There are $n$ $(n \ge 3)$ players in a table tennis tournament, in which any two players have a match. Player $A$ is called not out-performed by player $B$, if at least one of player $A$'s losers is not a $B$'s loser.



Determine, with proof, all possible values of $n$, such that the following case could happen: after finishing all the matches, every player is not out-performed by any other player.
2010 China Western Mathematical Olympiad
Determine all possible values of integer $k$ for which there exist positive integers $a$ and $b$ such that $\dfrac{b+1}{a} + \dfrac{a+1}{b} = k$.
2009 China Western Mathematical Olympiad
Let $M$ be the set of the real numbers except for finitely many elements. Prove that for every positive integer $n$ there exists a polynomial $f(x)$ with $\deg f = n$, such that all the coefficients and the $n$ real roots of $f$ are all in $M$.
2009 China Western Mathematical Olympiad
Given an integer $n\ge\ 3$, find the least positive integer $k$, such that there exists a set $A$ with $k$ elements, and $n$ distinct reals $x_{1},x_{2},\ldots,x_{n}$ such that $x_{1}+x_{2}, x_{2}+x_{3},\ldots, x_{n-1}+x_{n}, x_{n}+x_{1}$ all belong to $A$.
2009 China Western Mathematical Olympiad
Let $H$ be the orthocenter of acute triangle $ABC$ and $D$ the midpoint of $BC$. A line through $H$ intersects $AB,AC$ at $F,E$ respectively, such that $AE=AF$. The ray $DH$ intersects the circumcircle of $\triangle ABC$ at $P$. Prove that $P,A,E,F$ are concyclic.
2009 China Western Mathematical Olympiad
Prove that for every given positive integer $k$, there exist infinitely many $n$, such that $2^{n}+3^{n}-1, 2^{n}+3^{n}-2,\ldots, 2^{n}+3^{n}-k$ are all composite numbers.
2009 China Western Mathematical Olympiad
Define a sequence $(x_{n})_{n\geq 1}$ by taking $x_{1}\in\left\{5,7\right\}$; when $k\ge 1$, $x_{k+1}\in\left\{5^{x_{k}},7^{x_{k}}\right\}$. Determine all possible last two digits of $x_{2009}$.
2009 China Western Mathematical Olympiad
Given an acute triangle $ABC$, $D$ is a point on $BC$. A circle with diameter $BD$ intersects line $AB,AD$ at $X,P$ respectively (different from $B,D$).The circle with diameter $CD$ intersects $AC,AD$ at $Y,Q$ respectively (different from $C,D$). Draw two lines through $A$ perpendicular to $PX,QY$, the feet are $M,N$ respectively.Prove that $\triangle AMN$ is similar to $\triangle ABC$ if and only if $AD$ passes through the circumcenter of $\triangle ABC$.
2009 China Western Mathematical Olympiad
A total of $n$ people compete in a mathematical match which contains $15$ problems where $n>12$. For each problem, $1$ point is given for a right answer and $0$ is given for a wrong answer. Analysing each possible situation, we find that if the sum of points every group of $12$ people get is no less than $36$, then there are at least $3$ people that got the right answer of a certain problem, among the $n$ people. Find the least possible $n$.
2009 China Western Mathematical Olympiad
The real numbers $a_{1},a_{2},\ldots ,a_{n}$ where $n\ge 3$ are such that $\sum_{i=1}^{n}a_{i}=0$ and $2a_{k}\le\ a_{k-1}+a_{k+1}$ for all $k=2,3,\ldots ,n-1$. Find the least $f(n)$ such that, for all $k\in\left\{1,2,\ldots ,n\right\}$, we have $|a_{k}|\le f(n)\max\left\{|a_{1}|,|a_{n}|\right\}$.
2008 China Western Mathematical Olympiad
A sequence of real numbers $ \{a_{n}\}$ is defineds by $ a_{0}\neq 0,1$, $ a_1=1-a_0$,$ a_{n+1}=1-a_n(1-a_n)$, $ n=1,2,...$.



Prove that for any positive integer $ n$, we have

$ a_{0}a_{1}...a_{n}(\frac{1}{a_0}+\frac{1}{a_1}+...+\frac{1}{a_n})=1$
2008 China Western Mathematical Olympiad
In triangle $ ABC$, $ AB=AC$, the inscribed circle $ I$ touches $ BC, CA, AB$ at points $ D,E$ and $ F$ respectively. $ P$ is a point on arc $ EF$ opposite $ D$. Line $ BP$ intersects circle $ I$ at another point $ Q$, lines $ EP$, $ EQ$ meet line $ BC$ at $ M, N$ respectively. Prove that

(1) $ P, F, B, M$ concyclic

(2)$ \frac{EM}{EN} = \frac{BD}{BP}$



(P.S. Can anyone help me with using GeoGebra, the incircle function of the plugin doesn't work with my computer.)
2008 China Western Mathematical Olympiad
Given an integer $ m\geq$ 2, m positive integers $ a_1,a_2,...a_m$. Prove that there exist infinitely many positive integers n, such that $ a_{1}1^{n} + a_{2}2^{n} + ... + a_{m}m^{n}$ is composite.
2008 China Western Mathematical Olympiad
Given an integer $ m\geq 2$, and two real numbers $ a,b$ with $ a > 0$ and $ b\neq 0$. The sequence $ \{x_n\}$ is such that $ x_1 = b$ and $ x_{n + 1} = ax^{m}_{n} + b$, $ n = 1,2,...$. Prove that

(1)when $ b < 0$ and m is even, the sequence is bounded if and only if $ ab^{m - 1}\geq - 2$;

(2)when $ b < 0$ and m is odd, or when $ b > 0$ the sequence is bounded if and only if $ ab^{m - 1}\geq\frac {(m - 1)^{m - 1}}{m^m}$.
2008 China Western Mathematical Olympiad
Four frogs are positioned at four points on a straight line such that the distance between any two neighbouring points is 1 unit length. Suppose the every frog can jump to its corresponding point of reflection, by taking any one of the other 3 frogs as the reference point. Prove that, there is no such case that the distance between any two neighbouring points, where the frogs stay, are all equal to 2008 unit length.
2008 China Western Mathematical Olympiad
Given $ x,y,z\in (0,1)$ satisfying that

$ \sqrt{\frac{1 - x}{yz}} + \sqrt{\frac{1 - y}{xz}} + \sqrt{\frac{1 - z}{xy}} = 2$.

Find the maximum value of $ xyz$.
2008 China Western Mathematical Olympiad
For a given positive integer $n$, find the greatest positive integer $k$, such that there exist three sets of $k$ non-negative distinct integers, $A=\{x_1,x_2,\cdots,x_k\}, B=\{y_1,y_2,\cdots,y_k\}$ and $C=\{z_1,z_2,\cdots,z_k\}$ with $ x_j+y_j+z_j=n$ for any $ 1\leq j\leq k$.



[Moderator edit: LaTeXified]
2008 China Western Mathematical Olympiad
Let P be an interior point of a regular n-gon $ A_1 A_2 ...A_n$, the lines $ A_i P$ meet the regular n-gon at another point $ B_i$, where $ i=1,2,...,n$. Prove that sums of all $ PA_i\geq$ sum of all $ PB_i$.
2007 China Western Mathematical Olympiad
Let set $ T = \{1,2,3,4,5,6,7,8\}$. Find the number of all nonempty subsets $ A$ of  $ T$ such that $ 3|S(A)$ and $ 5\nmid S(A)$, where $ S(A)$ is the sum of all the elements in $ A$.
2007 China Western Mathematical Olympiad
Let $ C$ and $ D$ be two intersection points of circle $ O_1$ and circle $ O_2$. A line, passing through $ D$, intersects the circle $ O_1$ and the circle $ O_2$ at the points $ A$ and $ B$ respectively. The points $ P$ and $ Q$ are on circles $ O_1$ and $ O_2$ respectively. The lines $ PD$ and $ AC$ intersect at $ H$, and the lines $ QD$ and $ BC$ intersect at $ M$. Suppose that $ O$ is the circumcenter of the triangle $ ABC$. Prove that $ OD\perp MH$ if and only if $ P,Q,M$ and $ H$ are concyclic.
2007 China Western Mathematical Olympiad
Let $ a,b,c$ be real numbers such that $ a+b+c=3$. Prove that

$$\frac{1}{5a^2-4a+11}+\frac{1}{5b^2-4b+11}+\frac{1}{5c^2-4c+11}\leq\frac{1}{4}$$
2007 China Western Mathematical Olympiad
Let $ O$ be an interior point of the triangle $ ABC$. Prove that there exist positive integers $ p,q$ and $ r$ such that

$$ |p\cdot\overrightarrow{OA} + q\cdot\overrightarrow{OB} + r\cdot\overrightarrow{OC}|<\frac{1}{2007}$$
2007 China Western Mathematical Olympiad
Is there a triangle with sides of integer lengths such that the length of the shortest side is $ 2007$ and that the largest angle is twice the smallest?
2007 China Western Mathematical Olympiad
Find all natural numbers $n$ such that there exist $ x_1,x_2,\ldots,x_n,y\in\mathbb{Z}$ where $x_1,x_2,\ldots,x_n,y\neq 0$ satisfying:

$$x_1 + x_2 + \ldots + x_n = 0$$$$ny^2 = x_1^2 + x_2^2 + \ldots + x_n^2$$
2007 China Western Mathematical Olympiad
Let $ P$ be an interior point of an acute angled triangle $ ABC$. The lines $ AP,BP,CP$ meet $ BC,CA,AB$ at points $ D,E,F$ respectively. Given that triangle $ \triangle DEF$ and $ \triangle ABC$ are similar, prove that $ P$ is the centroid of $ \triangle ABC$.
2007 China Western Mathematical Olympiad
A circular disk is partitioned into $ 2n$ equal sectors by $ n$ straight lines through its center. Then, these $ 2n$ sectors are colored in such a way that exactly $ n$ of the sectors are colored in blue, and the other $ n$ sectors are colored in red. We number the red sectors with numbers from $ 1$ to $ n$ in counter-clockwise direction (starting at some of these red sectors), and then we number the blue sectors with numbers from $ 1$ to $ n$ in clockwise direction (starting at some of these blue sectors).



Prove that one can find a half-disk which contains sectors numbered with all the numbers from $ 1$ to $ n$ (in some order). (In other words, prove that one can find $ n$ consecutive sectors which are numbered by all numbers $ 1$, $ 2$, ..., $ n$ in some order.)



Problem 8 from CWMO 2007$ n$ white and $ n$ black balls are placed at random on the circumference of a circle.Starting from a certain white ball,number all white balls in a clockwise direction by $ 1,2,\dots,n$. Likewise number all black balls by $ 1,2,\dots,n$ in anti-clockwise direction starting from a certain black ball.Prove that there exists a chain of $ n$ balls whose collection of numbering forms the set $ \{1,2,3\dots,n\}$.
2006 China Western Mathematical Olympiad
Let $n$ be a positive integer with $n \geq 2$, and $0<a_{1}, a_{2},...,a_{n}< 1$. Find the maximum value of the sum

$\sum_{i=1}^{n}(a_{i}(1-a_{i+1}))^{\frac{1}{6}}$

where $a_{n+1}=a_{1}$
2006 China Western Mathematical Olympiad
Find the smallest positive real $k$ satisfying the following condition: for any given four DIFFERENT real numbers $a,b,c,d$, which are not less than $k$, there exists a permutation $(p,q,r,s)$ of $(a,b,c,d)$, such that the equation $(x^{2}+px+q)(x^{2}+rx+s)=0$ has four different real roots.
2006 China Western Mathematical Olympiad
In $\triangle PBC$, $\angle PBC=60^{o}$, the tangent at point $P$ to the circumcircle$g$ of $\triangle PBC$ intersects with line $CB$ at $A$. Points $D$ and $E$ lie on the line segment $PA$ and $g$ respectively, satisfying $\angle DBE=90^{o}$, $PD=PE$. $BE$ and $PC$ meet at $F$. It is known that lines $AF,BP,CD$ are concurrent.

a) Prove that $BF$ bisect $\angle PBC$

b) Find $\tan \angle PCB$
2006 China Western Mathematical Olympiad
Assuming that the positive integer $a$ is not a perfect square, prove that for any positive integer n, the sum ${S_{n}=\sum_{i=1}^{n}\{a^{\frac{1}{2}}\}^{i}}$ is irrational.
2006 China Western Mathematical Olympiad
Let $S=\{n|n-1,n,n+1$ can be expressed as the sum of the square of two positive integers.$\}$. Prove that if $n$ in $S$, $n^{2}$ is also in $S$.
2006 China Western Mathematical Olympiad
$AB$ is a diameter of the circle $O$, the point $C$ lies on the line $AB$ produced. A line passing though $C$ intersects with the circle $O$ at the point $D$ and $E$. $OF$ is a diameter of circumcircle $O_{1}$ of $\triangle BOD$. Join $CF$ and produce, cutting the circle $O_{1}$ at $G$. Prove that points $O,A,E,G$ are concyclic.
2006 China Western Mathematical Olympiad
Let $k$ be a positive integer not less than 3 and $x$ a real number. Prove that if $\cos (k-1)x$ and $\cos kx$ are rational, then there exists a positive integer $n>k$, such that both $\cos (n-1)x$ and $\cos nx$ are rational.
2006 China Western Mathematical Olympiad
Given a positive integer $ n\geq 2$, let $ B_{1}$, $ B_{2}$, ..., $ B_{n}$ denote $ n$ subsets of a set $ X$ such that each $ B_{i}$ contains exactly two elements. Find the minimum value of $ \left|X\right|$ such that for any such choice of subsets $ B_{1}$, $ B_{2}$, ..., $ B_{n}$, there exists a subset $ Y$ of $ X$ such that:

(1) $ \left|Y\right| = n$;

(2) $ \left|Y \cap B_{i}\right|\leq 1$ for every $ i\in\left\{1,2,...,n\right\}$.
2005 China Western Mathematical Olympiad
It is known that $a^{2005} + b^{2005}$ can be expressed as the polynomial of $a + b$ and $ab$. Find the coefficients' sum of this polynomial.
2005 China Western Mathematical Olympiad
Given three points $P$, $A$, $B$ and a circle such that the lines $PA$ and $PB$ are tangent to the circle at the points $A$ and $B$, respectively. A line through the point $P$ intersects that circle at two points $C$ and $D$. Through the point $B$, draw a line parallel to $PA$; let this line intersect the lines $AC$ and $AD$ at the points $E$ and $F$, respectively. Prove that $BE = BF$.
2005 China Western Mathematical Olympiad
Set $S = \{1, 2, 3, ..., 2005\}$. If among any $n$ pairwise coprime numbers in $S$ there exists at least a prime number, find the minimum of $n$.
2005 China Western Mathematical Olympiad
Given is the positive integer $n > 2$. Real numbers $\mid x_i \mid \leq 1$ ($i = 1, 2, ..., n$) satisfying $\mid \sum_{i=1}^{n}x_i \mid > 1$. Prove that there exists positive integer $k$ such that $\mid \sum_{i=1}^{k}x_i - \sum_{i=k+1}^{n}x_i \mid \leq 1$.
2005 China Western Mathematical Olympiad
Circles $C(O_1)$ and $C(O_2)$ intersect at points $A$, $B$. $CD$ passing through point $O_1$ intersects $C(O_1)$ at point $D$ and tangents $C(O_2)$ at point $C$. $AC$ tangents $C(O_1)$ at $A$. Draw $AE \bot CD$, and $AE$ intersects $C(O_1)$ at $E$. Draw $AF \bot DE$, and $AF$ intersects $DE$ at $F$. Prove that $BD$ bisects $AF$.
2005 China Western Mathematical Olympiad
In isosceles right-angled triangle $ABC$, $CA = CB = 1$. $P$ is an arbitrary point on the sides of $ABC$. Find the maximum of $PA \cdot PB \cdot PC$.
2005 China Western Mathematical Olympiad
If $a,b,c$ are positive reals such that $a+b+c=1$, prove that  $$ 10(a^3+b^3+c^3)-9(a^5+b^5+c^5)\geq 1 . $$
2005 China Western Mathematical Olympiad
For $n$ people, if it is known that



(a) there exist two people knowing each other among any three people, and



(b) there exist two people not knowing each other among any four people.



Find the maximum of $n$.



Here, we assume that if $A$ knows $B$, then $B$ knows $A$.
2004 China Western Mathematical Olympiad
Find all integers $n$, such that the following number is a

perfect square $$N= n^4 + 6n^3 + 11n^2 +3n+31. $$
2004 China Western Mathematical Olympiad
Let $ABCD$ be a convex quadrilateral, $I_1$ and $I_2$ be the incenters of triangles $ABC$ and $DBC$ respectively. The line $I_1I_2$ intersects the lines $AB$ and $DC$ at points $E$ and $F$ respectively. Given that $AB$ and $CD$ intersect in $P$, and $PE=PF$, prove that the points $A$, $B$, $C$, $D$ lie on a circle.
2004 China Western Mathematical Olympiad
Find all reals $ k$ such that

$$ a^3 + b^3 + c^3 + d^3 + 1\geq k(a + b + c + d)

$$

holds for all $ a,b,c,d\geq - 1$.



Edited by orl.
2004 China Western Mathematical Olympiad
Let $\mathbb{N}$ be the set of positive integers. Let $n\in \mathbb{N}$ and let $d(n)$ be the number of divisors of $n$. Let $\varphi(n)$ be the Euler-totient function (the number of co-prime positive integers with $n$, smaller than $n$).



Find all non-negative integers $c$ such that there exists $n\in\mathbb{N}$ such that $$ d(n) + \varphi(n) = n+c , $$ and for such $c$ find all values of $n$ satisfying the above relationship.
2004 China Western Mathematical Olympiad
The sequence $\{a_n\}_{n}$ satisfies the relations $a_1=a_2=1$ and for all positive integers $n$,

$$ a_{n+2} = \frac 1{a_{n+1}} + a_n . $$

Find $a_{2004}$.
2004 China Western Mathematical Olympiad
All the grids of a $m\times n$ chess board ($m,n\geq 3$), are colored either with red or with blue. Two adjacent grids (having a common side) are called a "good couple" if they have different colors. Suppose there are $S$ "good  couples". Explain how to determine whether $S$ is odd or even. Is it prescribed by

some specific color grids? Justify your answers.
2004 China Western Mathematical Olympiad
Let $\ell$ be the perimeter of an acute-angled triangle $ABC$ which is not an equilateral triangle. Let $P$ be a variable points inside the triangle $ABC$, and let $D,E,F$ be the projections of $P$ on the sides $BC,CA,AB$ respectively. Prove

that $$ 2(AF+BD+CE ) = \ell $$ if and only if $P$ is collinear with the incenter and the circumcenter of the triangle $ABC$.
2004 China Western Mathematical Olympiad
Suppose that $ a$, $ b$, $ c$ are positive real numbers, prove that

$$ 1 < \frac {a}{\sqrt {a^{2} + b^{2}}} + \frac {b}{\sqrt {b^{2} + c^{2}}} + \frac {c}{\sqrt {c^{2} + a^{2}}}\leq\frac {3\sqrt {2}}{2}

$$
2003 China Western Mathematical Olympiad
Place the numbers $ 1, 2, 3, 4, 5, 6, 7, 8$ at the vertices of a cuboid such that the sum of any $ 3$ numbers on a side is not less than $ 10$.  Find the smallest possible sum of the 4 numbers on a side.
2003 China Western Mathematical Olympiad
Let $ a_1, a_2, \ldots, a_{2n}$ be $ 2n$ real numbers satisfying the condition $ \sum_{i = 1}^{2n - 1} (a_{i + 1} - a_i)^2 = 1$.  Find the greatest possible value of $ (a_{n + 1} + a_{n + 2} + \ldots + a_{2n}) - (a_1 + a_2 + \ldots + a_n)$.
2003 China Western Mathematical Olympiad
Let $ n$ be a given positive integer.  Find the smallest positive integer $ u_n$ such that for any positive integer $ d$, in any $ u_n$ consecutive odd positive integers, the number of them that can be divided by $ d$ is not smaller than the number of odd integers among $ 1, 3, 5, \ldots, 2n - 1$ that can be divided by $ d$.
2003 China Western Mathematical Olympiad
Given that the sum of the distances from point $ P$ in the interior of a convex quadrilateral $ ABCD$ to the sides $ AB, BC, CD, DA$ is a constant, prove that $ ABCD$ is a parallelogram.
2003 China Western Mathematical Olympiad
The sequence $ \{a_n\}$ satisfies $ a_0 = 0, a_{n + 1} = ka_n + \sqrt {(k^2 - 1)a_n^2 + 1}, n = 0, 1, 2, \ldots$, where $ k$ is a fixed positive integer.  Prove that all the terms of the sequence are integral and that $ 2k$ divides $ a_{2n}, n = 0, 1, 2, \ldots$.
2003 China Western Mathematical Olympiad
A circle can be inscribed in the convex quadrilateral $ ABCD$.  The incircle touches the sides $ AB, BC, CD, DA$ at $ A_1, B_1, C_1, D_1$ respectively.  The points $ E, F, G, H$ are the midpoints of $ A_1B_1, B_1C_1, C_1D_1, D_1A_1$ respectively. Prove that the quadrilateral $ EFGH$ is a rectangle if and only if $ A, B, C, D$ are concyclic.
2003 China Western Mathematical Olympiad
The non-negative numbers $ x_1, x_2, \ldots, x_5$ satisfy $ \sum_{i = 1}^5 \frac {1}{1 + x_i} = 1$.  Prove that $ \sum_{i = 1}^5 \frac {x_i}{4 + x_i^2} \leq 1$.