Given six consecutive positive integers, prove that there exists a prime such that one and only one of these six integers is divisible by this prime.

Find the limit, when $n$ tends to the infinity, of $$\frac{\sum_{k=0}^{n} {{2n} \choose {2k}} 3^k} {\sum_{k=0}^{n-1} {{2n} \choose {2k+1}} 3^k}$$

Quadrilateral $ABCD$ is a trapezoid, $AD = 15$, $AB = 50$, $BC = 20$, and the altitude is $12$. What is the area of the trapezoid? [asy] pair A,B,C,D; A=(3,20); B=(35,20); C=(47,0); D=(0,0); draw(A--B--C--D--cycle); dot((0,0)); dot((3,20)); dot((35,20)); dot((47,0)); label("A",A,N); label("B",B,N); label("C",C,S); label("D",D,S); draw((19,20)--(19,0)); dot((19,20)); dot((19,0)); draw((19,3)--(22,3)--(22,0)); label("12",(21,10),E); label("50",(19,22),N); label("15",(1,10),W); label("20",(41,12),E);[/asy] $ \textbf{(A)}600\qquad\textbf{(B)}650\qquad\textbf{(C)}700\qquad\textbf{(D)}750\qquad\textbf{(E)}800 $

Prove that if $n$ and $k$ are positive integers such that $1<k<n-1$,Then the binomial coefficient $\binom nk$ is divisible by at least two different primes.

The parallelogram $ABCD$ has $AB=a,AD=1,$ $\angle BAD=A$, and the triangle $ABD$ has all angles acute. Prove that circles radius $1$ and center $A,B,C,D$ cover the parallelogram if and only \[a\le\cos A+\sqrt3\sin A.\]

For every positive integer $x$, let $k(x)$ denote the number of composite numbers that do not exceed $x$. Find all positive integers $n$ for which $(k (n))! $ lcm $(1, 2,..., n)> (n - 1) !$ .

Find the natural number $A$ such that there are $A$ integer solutions to $x+y\geq A$ where $0\leq x \leq 6$ and $0\leq y \leq 7$. [i]Proposed by David Tang[/i]

(a) Let $\gcd(m, k) = 1$. Prove that there exist integers $a_1, a_2, . . . , a_m$ and $b_1, b_2, . . . , b_k$ such that each product $a_ib_j$ ($i = 1, 2, \cdots ,m; \ j = 1, 2, \cdots, k$) gives a different residue when divided by $mk.$ (b) Let $\gcd(m, k) > 1$. Prove that for any integers $a_1, a_2, . . . , a_m$ and $b_1, b_2, . . . , b_k$ there must be two products $a_ib_j$ and $a_sb_t$ ($(i, j) \neq (s, t)$) that give the same residue when divided by $mk.$ [i]Proposed by Hungary.[/i]

Four right triangles, each with the sides $1$ and $2$, are assembled to a figure as shown. How large a fraction does the area of the small circle make up of that of the big one? [img]https://1.bp.blogspot.com/-XODK1XKCS0Q/XzXDtcA-xAI/AAAAAAAAMWA/zSLPpf3IcX0rgaRtOxm_F2begnVdUargACLcBGAsYHQ/s0/2010%2BMohr%2Bp1.png[/img]

Let $\triangle ABC$ be an acute triangle, and let $D$ be the projection of $A$ on $BC$. Let $M,N$ be the midpoints of $AB$ and $AC$ respectively. Let $\Gamma_1$ and $\Gamma_2$ be the circumcircles of $\triangle BDM$ and $\triangle CDN$ respectively, and let $K$ be the other intersection point of $\Gamma_1$ and $\Gamma_2$. Let $P$ be an arbitrary point on $BC$ and $E,F$ are on $AC$ and $AB$ respectively such that $PEAF$ is a parallelogram. Prove that if $MN$ is a common tangent line of $\Gamma_1$ and $\Gamma_2$, then $K,E,A,F$ are concyclic.

Prove that there exists an infinite number of relatively prime pairs $(m, n)$ of positive integers such that the equation \[x^3-nx+mn=0\] has three distint integer roots.

Let $D$ be a regular dodecagon in the plane. How many squares are there in the plane at least two vertices in common with the vertices of $D$?

Let $x_1, x_2, ..., x_n$ be positive reals with sum $1$. Show that $$\frac{x_1^2}{x_1 + x_2}+ \frac{x_2^2}{x_2 + x_3} +... + \frac{x_{n-1}^2}{x_{n-1} + x_n} + \frac{x_n^2}{x_n + x_1} \ge \frac12$$

Five sets of brothers and sisters attend an activity of $k$ groups, stipulate that: (1)Anyone cannot be in the same group with his/her sister/brother. (2)Anyone has been in the same group with any other people who is not his/her sister/brother. (3)Only one person has attended moe than one group. Then, the minimun value of $k$ is________.

There are five guys named Alan, Bob, Casey, Dan, and Eric. Each one either always tells the truth or always lies. You overhear the following discussion between them: Alan: [i]"All of us are truth-tellers."[/i] Bob: [i]"No, only Alan and I are truth-tellers."[/i] Casey: [i]"You are both liars."[/i] Dan:[i] "If Casey is a truth-teller, then Eric is too."[/i] Eric: [i]"An odd number of us are liars."[/i] Who are the liars?

Given $\vartriangle ABC$ with circumcircle $\Omega$. Assume $\omega_a, \omega_b, \omega_c$ are circles which tangent internally to $\Omega$ at $T_a,T_b, T_c $ and tangent to $BC,CA,AB$ at $P_a, P_b, P_c$, respectively. If $AT_a,BT_b,CT_c$ are collinear, prove that $AP_a,BP_b,CP_c$ are collinear.

For a positive integer $ n$, let $ \theta(n)$ denote the number of integers $ 0 \leq x < 2010$ such that $ x^2 \minus{} n$ is divisible by $ 2010$. Determine the remainder when $ \displaystyle \sum_{n \equal{} 0}^{2009} n \cdot \theta(n)$ is divided by $ 2010$.

On a line $\ell$ there exists $3$ points $A, B$, and $C$ where $B$ is located between $A$ and $C$. Let $\Gamma_1, \Gamma_2, \Gamma_3$ be circles with $AC, AB$, and $BC$ as diameter respectively; $BD$ is a segment, perpendicular to $\ell$ with $D$ on $\Gamma_1$. Circles $\Gamma_4, \Gamma_5, \Gamma_6$ and $\Gamma_7$ satisfies the following conditions: $\bullet$ $\Gamma_4$ touches $\Gamma_1, \Gamma_2$, and$ BD$. $\bullet$ $\Gamma_5$ touches $\Gamma_1, \Gamma_3$, and $BD$. $\bullet$ $\Gamma_6$ touches $\Gamma_1$ internally, and touches $\Gamma_2$ and $\Gamma_3$ externally. $\bullet$ $\Gamma_7$ passes through $B$ and the tangent points of $\Gamma_2$ with $\Gamma_6$, and $\Gamma_3$ with $\Gamma_6$. Show that the circles $\Gamma_4, \Gamma_5$, and $\Gamma_7$ are congruent.

Fix positive integers $n$ and $k\ge 2$. A list of $n$ integers is written in a row on a blackboard. You can choose a contiguous block of integers, and I will either add $1$ to all of them or subtract $1$ from all of them. You can repeat this step as often as you like, possibly adapting your selections based on what I do. Prove that after a finite number of steps, you can reach a state where at least $n-k+2$ of the numbers on the blackboard are all simultaneously divisible by $k$.

Let $\omega_1$ and $\omega_2$ with center $O_1$ and $O_2$ respectively, meet at points $A$ and $B$. Let $X$ and $Y$ be points on $\omega_1$. Lines $XA$ and $Y A$ meet $\omega_2$ at $Z$ and $W$, respectively, such that $A$ lies between $X$ and $Z$ and between $Y$ and $W$. Let $M$ be the midpoint of $O_1O_2$, $S$ be the midpoint of $XA$ and $T$ be the midpoint of $W A$. Prove that $MS = MT$ if and only if $X,~ Y ,~ Z$ and $W$ are concyclic.

For integers $m,n\geq 1$, let $A(n,m)$ be the number of sequences $(a_1,\cdots,a_{nm})$ of integers satisfying the following two properties: [list=a] [*]Each integer $k$ with $1\leq k\leq n$ occurs exactly $m$ times in the sequence $(a_1,\cdots,a_{nm})$. [*]If $i,j,$ and $k$ are integers such that $1\leq i\leq nm$ and $1\leq j\leq k\leq n$, then $j$ occurs in the sequence $(a_1,\cdots,a_i)$ at least as many times as $k$ does.[/list] For example, if $n=2$ and $m=5$, a possible sequence is $(a_1,\cdots,a_{10})=(1,1,2,1,2,2,1,2,1,2)$. On the other hand, the sequence $(a_1,\cdots,a_{10})=(1,2,1,2,2,1,1,1,2,2)$ does not satisfy property (2) for $i=5$, $j=1$, and $k=2$. Prove that $A(n,m)=A(m,n)$.

Let $ABC$ be an acute triangle. Find the locus of the centers of the rectangles which have their vertices on the sides of $ABC$.

Suppose that some real number $x$ satisfies \[\log_2 x + \log_8 x + \log_{64} x = \log_x 2 + \log_x 16 + \log_x 128.\] Given that the value of $\log_2 x + \log_x 2$ can be expressed as $\tfrac{a\sqrt{b}}{c}$, where $a$ and $c$ are coprime positive integers and $b$ is squarefree, compute $abc$.

For integer $ n,\ a_n$ is difined by $ a_n\equal{}\int_0^{\frac{\pi}{4}} (\cos x)^ndx$. (1) Find $ a_{\minus{}2},\ a_{\minus{}1}$. (2) Find the relation of $ a_n$ and $ a_{n\minus{}2}$. (3) Prove that $ a_{2n}\equal{}b_n\plus{}\pi c_n$ for some rational number $ b_n,\ c_n$, then find $ c_n$ for $ n<0$.

Circle $C$(the center is $C$.) is inside the $\angle XOY$ and it is tangent to the two sides of the angle. Let $C_1$ be the circle that passes through the center of $C$ and tangent to two sides of angle and let $A$ be one of the endpoint of diameter of $C_1$ that passes through $C$ and $B$ be the intersection of this diameter and circle $C.$ Prove that the cirlce that $A$ is the center and $AB$ is the radius is also tangent to the two sides of $\angle XOY.$