Find all primes $p$ such that $4p^2+1$ and $6p^2+1$ are both primes.

A triangle has lattice points as vertices and contains no other lattice points. Prove that its area is $\frac{1}{2}$.

When a function $f(x)$ is differentiated $n$ times ,the function we get id denoted $f^n(x)$.If $f(x)=\dfrac {e^x}{x}$.Find the value of \[\lim_{n \to \infty} \dfrac {f^ {2n}(1)}{(2n)!}\]

Find all functions $f:\mathbb Z\to \mathbb Z$ such that for all surjective functions $g:\mathbb Z\to \mathbb Z$, $f+g$ is also surjective. (A function $g$ is surjective over $\mathbb Z$ if for all integers $y$, there exists an integer $x$ such that $g(x)=y$.) [i]Proposed by Sean Li[/i]

A chicken breeder went to check what price per chick he had charged the previous year. He found an invoice, half erased, which read: $72$ chickens sold for $*679*$ escudos” (the digits of the units and tens of thousands were illegible). What price did each chick sell for last year?

Let $a, b, c, d$ be real numbers, such that $ab(c+d)=cd(a+b)$. Prove that $\frac{a+1}{a^2+3}+\frac{b+1}{b^2+3} \geq \frac{c-1}{c^2+3}+\frac{d-1}{d^2+3}$.

Let $r$ be a positive integer, and let $a_0 , a_1 , \cdots $ be an infinite sequence of real numbers. Assume that for all nonnegative integers $m$ and $s$ there exists a positive integer $n \in [m+1, m+r]$ such that \[ a_m + a_{m+1} +\cdots +a_{m+s} = a_n + a_{n+1} +\cdots +a_{n+s} \] Prove that the sequence is periodic, i.e. there exists some $p \ge 1 $ such that $a_{n+p} =a_n $ for all $n \ge 0$.

Let $\overline{CH}$ be an altitude of $\triangle ABC$. Let $R$ and $S$ be the points where the circles inscribed in the triangles $ACH$ and $BCH$ are tangent to $\overline{CH}$. If $AB = 1995$, $AC = 1994$, and $BC = 1993$, then $RS$ can be expressed as $m/n$, where $m$ and $n$ are relatively prime integers. Find $m + n$

Find all $3$-digit numbers $\overline{abc}$ ($a,b \ne 0$) such that $\overline{bcd} \times  a = \overline{1a4d}$ for some integer $d$ from $1$ to $9$

A trapezium is given with parallel bases having lengths $1$ and $4$. Split it into two trapeziums by a cut, parallel to the bases, of length $3$. We now want to divide the two new trapeziums, always by means of cuts parallel to the bases, in $m$ and $n$ trapeziums, respectively, so that all the $m + n$ trapezoids obtained have the same area. Determine the minimum possible value for $m + n$ and the lengths of the cuts to be made to achieve this minimum value.

Let $c$ be a complex number. Suppose there exist distinct complex numbers $r$, $s$, and $t$ such that for every complex number $z$, we have \[ (z - r)(z - s)(z - t) = (z - cr)(z - cs)(z - ct). \] Compute the number of distinct possible values of $c$.

This is a very classical problem. Let the $ABCD$ be a cyclic quadrilateral with circumcircle $\omega_1$.Lines $AC$ and $BD$ intersect at point $E$,and lines $AD$,$BC$ intersect at point $F$.Circle $\omega_2$ is tangent to segments $EB,EC$ at points $M,N$ respectively,and intersects with circle $\omega_1$ at points $Q,R$.Lines $BC,AD$ intersect line $MN$ at $S,T$ respectively.Show that $Q,R,S,T$ are concyclic.

Consider the set $X$ = $\{ 1,2 \ldots 10 \}$ . Find two disjoint nonempty sunsets $A$ and $B$ of $X$ such that a) $A \cup B = X$; b) $\prod_{x\in A}x$ is divisible by $\prod_{x\in B}x$, where $\prod_{x\in C}x$ is the product of all numbers in $C$; c) $\frac{ \prod\limits_{x\in A}x}{ \prod\limits_{x\in B}x}$ is as small as possible.

$ABCD$ is a rectangular sheet of paper that has been folded so that corner $B$ is matched with point $B'$ on edge $AD$. The crease is $EF$, where $E$ is on $AB$ and $F$is on $CD$. The dimensions $AE=8$, $BE=17$, and $CF=3$ are given. The perimeter of rectangle $ABCD$ is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, B=(25,0), C=(25,70/3), D=(0,70/3), E=(8,0), F=(22,70/3), Bp=reflect(E,F)*B, Cp=reflect(E,F)*C; draw(F--D--A--E); draw(E--B--C--F, linetype("4 4")); filldraw(E--F--Cp--Bp--cycle, white, black); pair point=( 12.5, 35/3 ); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); label("$B^\prime$", Bp, dir(point--Bp)); label("$C^\prime$", Cp, dir(point--Cp));[/asy]

Find all positive integers $x$ and $y$ such that $${x \choose y} = 1432$$

In a football tournament of one round (each team plays each other once, $2$ points for win , $1$ point for draw and $0$ points for loss), $28$ teams compete. During the tournament more than $75\%$ of the matches finished in a draw . Prove that there were two teams who finished with the same number of points. (M . Vora, gymnasium student , Hungary)

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 $x+y\leq m.$

In triangle $ABC, \angle A= 45^o, BH$ is the altitude, the point $K$ lies on the $AC$ side, and $BC = CK$. Prove that the center of the circumscribed circle of triangle $ABK$ coincides with the center of an excircle of triangle $BCH$.

Let $ABCD$ a cyclique quadrilateral. We consider the Following points: $A'$ the orthogonal projection of $A$ over $(BD)$, $B'$ the orthogonal projection of $B$ over $(AC)$, $C'$ the orthgonal projection of $C$ over $(BD)$ and $D'$ being the orthogonal projection of $D$ over $(AC)$ Prove that $A', B', C'$ and $D'$.

There are 7 pieces of paper in the hat. On the $ n $th piece of paper there is written the number $ 2^n-1 $ ($ n = 1, 2, \ldots, 7 $). We draw cards randomly until the sum exceeds 124. What is the most probable value of this sum?

$B$ is the center of unit circle. $A,C$ are points on the circle (the order of $A,B,C$ is clockwise), and $\angle ABC=2\alpha(0<\alpha<\frac{\pi}{3})$. Then we will rotate $\triangle ABC$ anticlockwise. In the first rotation, $A$ is the center of rotation, the result is that $B$ is on the circle. In the second rotation, $B$ is the center of rotation, the result is that $C$ is on the circle. In the third rotation, $C$ is the center of rotation, the result is that $A$ is on the circle. ... After we rotate for $100$ times, the distance $A$ travelled is $\text{(A)}22\pi(1+\sin\alpha)-66\alpha\qquad\text{(B)}\frac{67}{3}\pi\qquad\text{(C)}22\pi+\frac{68}{3}\pi\sin\alpha-66\alpha\qquad\text{(D)}33\pi-66\alpha$

$ABC$ is triangle with $AB=BC$. $X,Y$ are midpoints of $AC$ and $AB$. $Z$ is base of perpendicular from $B$ to $CY$. Prove, that circumcenter of $XYZ$ lies on $AC$

The edges of $K_{2017}$ are each labeled with $1,2,$ or $3$ such that any triangle has sum of labels at least $5.$ Determine the minimum possible average of all $\dbinom{2017}{2}$ labels. (Here $K_{2017}$ is defined as the complete graph on 2017 vertices, with an edge between every pair of vertices.) [i]Proposed by Michael Ma[/i]

Let $ a_0,a_1,\dots,a_{n \plus{} 1}$ be natural numbers such that $ a_0 \equal{} a_{n \plus{} 1} \equal{} 1$, $ a_i>1$ for all $ 1\leq i \leq n$, and for each $ 1\leq j\leq n$, $ a_i|a_{i \minus{} 1} \plus{} a_{i \plus{} 1}$. Prove that there exist one $ 2$ in the sequence.

Let $ABC$ be a right triangle with $\angle A= 90^{\circ}$. A circle $\omega$ centered on $BC$ is tangent to $AB$ at $D$ and $AC$ at $E$. Let $F$ and $G$ be the intersections of $\omega$ and $BC$ so that $F$ lies between $B$ and $G$. If lines $DG$ and $EF$ intersect at $X$, show that $AX=AD.$