A biologist wants to calculate the number of fish in a lake. On May 1 she catches a random sample of 60 fish, tags them, and releases them. On September 1 she catches a random sample of 70 fish and finds that 3 of them are tagged. To calculate the number of fish in the lake on May 1, she assumes that $25\%$ of these fish are no longer in the lake on September 1 (because of death and emigrations), that $40\%$ of the fish were not in the lake May 1 (because of births and immigrations), and that the number of untagged fish and tagged fish in the September 1 sample are representative of the total population. What does the biologist calculate for the number of fish in the lake on May 1?

Let $p{}$ and $q{}$ be two coprime positive integers. A frog hops along the integer line so that on every hop it moves either $p{}$ units to the right or $q{}$ units to the left. Eventually, the frog returns to the initial point. Prove that for every positive integer $d{}$ with $d < p + q$ there are two numbers visited by the frog which differ just by $d{}$. [i]Nikolay Belukhov[/i]

$ABC$ is an acute-angled triangle. The tangents to the circumcircle at $A$ and $C$ meet the tangent at $B$ at $M$ and $N$. The altitude from $B$ meets $AC$ at $P$. Show that $BP$ bisects the angle $MPN$

Consider an isosceles triangle $ABC$ with $AB=AC$. Point $D$ is on the smaller arc $AB$ of its circumcirle. Point $E$ lies on the continuation of segment $AD$ beyond point $D$ so that both $A$ and $E$ lie in the same half-plane relative to $BC$. The circumcirle of triangle $BDE$ intersects side $AB$ at point $F$. Prove that lines $EF$ and $BC$ are parallel. (Author: R. Zhenodarov)

$n$ assistants start simultaneously from one vertex of a cube-shaped planet with edge length $1$. Each assistant moves along the edges of the cube at a constant speed of $2, 4, 8, \cdots, 2^n$, and can only change their direction at the vertices of the cube. The assistants can pass through each other at the vertices, but if they collide at any point that is not a vertex, they will explode. Determine the maximum possible value of $n$ such that the assistants can move infinitely without any collisions.

There are $n$ holes in a circle. The holes are numbered $1,2,3$ and so on to $n$. In the beginning, there is a peg in every hole except for hole $1$. A peg can jump in either direction over one adjacent peg to an empty hole immediately on the other side. After a peg moves, the peg it jumped over is removed. The puzzle will be solved if all pegs disappear except for one. For example, if $n=4$ the puzzle can be solved in two jumps: peg $3$ jumps peg $4$ to hole $1$, then peg $2$ jumps the peg in $1$ to hole $4$. (See illustration below, in which black circles indicate pegs and white circles are holes.) [center][img]http://i.imgur.com/4ggOa8m.png[/img][/center] [list=a] [*]Can the puzzle be solved for $n=5$? [*]Can the puzzle be solved for $n=2014$? [/list] In each part (a) and (b) either describe a sequence of moves to solve the puzzle or explain why it is impossible to solve the puzzle.

Determine all integers $ n > 1$ such that \[ \frac {2^n \plus{} 1}{n^2} \] is an integer.

Let $d(k)$ denote the number of positive divisors of a positive integer $k$. Prove that there exist infinitely many positive integers $M$ that cannot be written as \[M=\left(\frac{2\sqrt{n}}{d(n)}\right)^2\] for any positive integer $n$.

A natural number is written in each box of the $13 \times 13$ grid area. Prove that you can choose $2$ rows and $4$ columns such that the sum of the numbers written at their $8$ intersections is divisible by $8$.

Find the first two values of $40!(\text{mod }1763)$ [hide=Answer]1311, 3074[\hide]

In a geometric progression whose terms are positive, any term is equal to the sum of the next two following terms. then the common ratio is: $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ \text{about }\frac{\sqrt{5}}{2} \qquad\textbf{(C)}\ \frac{\sqrt{5}\minus{}1}{2} \qquad\textbf{(D)}\ \frac{1\minus{}\sqrt{5}}{2} \qquad\textbf{(E)}\ \frac{2}{\sqrt{5}}$

Two circles in space are said to be tangent to each other if they have a corni-non tangent at the same point of tangency. Assume that there are three circles in space which are mutually tangent at three distinct points. Prove that they either alI lie in a plane or all lie on a sphere.

Given an acute triangle $ABC$ with $AB <AC$. Points $P$ and $Q$ lie on the angle bisector of $\angle BAC$ so that $BP$ and $CQ$ are perpendicular on that angle bisector. Suppose that point $E, F$ lie respectively at sides $AB$ and $AC$ respectively, in such a way that $AEPF$ is a kite. Prove that the lines $BC, PF$, and $QE$ intersect at one point.

Given a circle with radius 1 and 2 points C, D given on it. Given a constant l with $0<l\le 2$. Moving chord of the circle AB=l and ABCD is a non-degenerated convex quadrilateral. AC and BD intersects at P. Find the loci of the circumcenters of triangles ABP and BCP.

Find the smallest positive integer $n$ for which there exist $n$ different positive integers $a_{1}, a_{2}, \cdots, a_{n}$ satisfying [list] [*] $\text{lcm}(a_1,a_2,\cdots,a_n)=1985$,[*] for each $i, j \in \{1, 2, \cdots, n \}$, $gcd(a_i,a_j)\not=1$, [*] the product $a_{1}a_{2} \cdots a_{n}$ is a perfect square and is divisible by $243$, [/list] and find all such $n$-tuples $(a_{1}, \cdots, a_{n})$.

Let O be the circumcenter of triangle ABC. Let H be the orthocenter of triangle ABC. the perpendicular bisector of AB meet AC,BC at D,E. the circumcircle of triangle DEH meet AC,BC,OH again at F,G,L. CH meet FG at T,and ABCT is concyclic. Prove that LHBC is concyclic. graph: https://cdn.luogu.com.cn/upload/image_hosting/w6z6mvm4.png

$ABCD$ is a convex quadrilateral. $P$ is the intersection of external bisectors of $\angle DAC$ and $\angle DBC$. Prove that $\angle APD = \angle BPC$ if and only if $AD+AC=BC+BD$

Determine all sequences $(x_1,x_2,\ldots,x_{2011})$ of positive integers, such that for every positive integer $n$ there exists an integer $a$ with \[\sum^{2011}_{j=1} j x^n_j = a^{n+1} + 1\] [i]Proposed by Warut Suksompong, Thailand[/i]

Prove that there exist positive constants $c_1$ and $c_2$ with the following properties: a) For all real $k>1$, $$\left|\int^1_0\sqrt{1-x^2}\cos(kx)\text dx\right|<\frac{c_1}{k^{3/2}}.$$b) For all real $k>1$, $$\left|\int^1_0\sqrt{1-x^2}\sin(kx)\text dx\right|<\frac{c_2}k.$$

Given a triangle, its midpoint triangle is obtained by joining the midpoints of its sides. A sequence of polyhedra $P_{i}$ is defined recursively as follows: $P_{0}$ is a regular tetrahedron whose volume is 1. To obtain $P_{i+1}$, replace the midpoint triangle of every face of $P_{i}$ by an outward-pointing regular tetrahedron that has the midpoint triangle as a face. The volume of $P_{3}$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Define $d(n)$ as the number of positive divisors of $n\in\mathbb{Z_+^*}$. Let $a$ and $b$ be positive integers satisfying the equality $$a + d(a) = b^2 + 2$$ Prove that $a+b$ is even.

Let $n$ be a positive integer, and let $a_1,a_2,...,a_n$ be $n$ positive real numbers such that $a_1+a_2+...+a_n = 1$. Is it true that $\frac{a_1^4}{a_1^2+a_2^2}+\frac{a_2^4}{a_2^2+a_3^2}+\frac{a_3^4}{a_3^2+a_4^2}+...+\frac{a_{n-1}^4}{a_{n-1}^2+a_n^2}+\frac{a_n^4}{a_n^2+a_1^2}\ge \frac{1}{2n}$ ? Justify your answer.

A directed graph has each vertex with outdegree 2. Prove that it is possible to split the vertices into 3 sets so that for each vertex $v$, $v$ is not simultaneously in the same set with both of the vertices that it points to. [i]David Yang.[/i] [hide="Stronger Version"]See [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=42&t=492100]here[/url].[/hide]

Let $a, b, c$ positive reals such that $a+b+c=1$. Prove that $$\min\{a(1-b),b(1-c),c(1-a)\}\leq \frac{1}{4}$$ $$\max\{a(1-b),b(1-c),c(1-a)\}\geq \frac{2}{9}$$

Let $X$ be an arbitrary set and $f$ a bijection from $X$ to $X$. Show that there exist bijections $g,\ g':X\to X$ s.t. $f=g\circ g',\ g\circ g=g'\circ g'=1_X$.