$ABCD$ is a parallelogram. Take $E$ on the line $AB$ so that $BE = BC$ and $B$ lies between $A$ and $E$. Let the line through $C$ perpendicular to $BD$ and the line through $E$ perpendicular to $AB$ meet at $F$. Show that $\angle DAF = \angle BAF$.

Let $180^o < \theta_1 < \theta_2 <...< \theta_n = 360^o$. For $i = 1,2,..., n$, $P_i = (\cos \theta_i^o, \sin \theta_i^o)$ is a point on the circle $C$ with centre $(0,0)$ and radius $1$. Let $P$ be any point on the upper half of $C$. Find the coordinates of $P$ such that the sum of areas $[PP_1P_2] + [PP_2P_3] + ...+ [PP_{n-1}P_n]$ attains its maximum.

Find the lowest odd positive integer with an odd number of divisors and is divisible by $d^2$ and $a+b+c+d+e+f$, where $a, b, c, d, e, f$ are consecutive prime numbers.

Does there exist a positive integer, $x$, such that $(x+2)^{2023}-x^{2023}$ has exactly $2023^{2023}$ factors? [i]Proposed by Wong Jer Ren[/i]

$V$ is a real vector space and $ f, f_{i}: V \rightarrow \mathbb{R} $ are linear for $i = 1, 2, ... , k.$ Also $f $ is zero at all points for which all of $ f_{i }$ are zero. Show that $ f $ is a linear combination of the $f_{i}$.

If $n$ is an integer, then find all values of $n$ for which $\sqrt{n}+\sqrt{n+2005}$ is an integer as well.

\[\int_0^1 \frac{1-x}{x^{5/2}+x^{3/2}+x^{1/2}}\mathrm dx\] [i]Proposed by Connor Gordon[/i]

Suppose $f(x)$ is a function satisfying $\left | f(m+n)-f(m) \right | \leq \frac{n}{m}$ for all positive integers $m$,$n$. Show that for all positive integers $k$: \[\sum_{i=1}^{k}\left |f(2^k)-f(2^i) \right |\leq \frac{k(k-1)}{2}\].

In the following triangular lattice distance of two vertices is length of the shortest path between them. Let $ A_{1},A_{2},\dots,A_{n}$ be constant vertices of the lattice. We want to find a vertex in the lattice whose sum of distances from vertices is minimum. We start from an arbitrary vertex. At each step we check all six neighbors and if sum of distances from vertices of one of the neighbors is less than sum of distances from vertices at the moment we go to that neighbor. If we have more than one choice we choose arbitrarily, as seen in the attached picture. Obviusly the algorithm finishes a) Prove that when we can not make any move we have reached to the problem's answer. b) Does this algorithm reach to answer for each connected graph?

A board is called ''Guapo'' if it can be totally covered by pieces equal to that shown in the picture, without gaps and without overlaps or pieces that cover areas outside the board. Is possible to rotate or reflect the pieces. [img]https://imgur.com/o6jX1JO.jpeg[/img] Find all positive integers $n$ such that a board $n \times (n+1)$ is ''Guapo''.

Solve : $y(x+y)^2=9 $ ; $y(x^3-y^3)=7$

An object is thrown with a fixed initial speed $v_\text{0}$ at various angles $\alpha$ relative to the horizon. At some constant height $h$ above the launch point the speed $v$ of the object is measured as a function of the initial angle $\alpha$. Which of the following best describes the dependence of $v$ on $\alpha$? (Assume that the height h is achieved, and assume that there is no air resistance.) (A) $v$ will increase monotonically with $\alpha$. (B) $v$ will increase to some critical value $v_{max}$ and then decrease. (C) $v$ will remain constant, independent of $\alpha$. (D) $v$ will decrease to some critical value $v_{min}$ and then increase. (E) None of the above.

The elements of the set $F$ are some subsets of $\left\{1,2,\ldots ,n\right\}$ and satisfy the conditions: i) if $A$ belongs to $F$, then $A$ has three elements; ii)if $A$ and $B$ are distinct elements of $F$ , then $A$ and $B$ have at most one common element. Let $f(n)$ be the greatest possible number of elements of $F$. Prove that $\frac{n^{2}-4n}{6}\leq f(n) \leq \frac{n^{2}-n}{6}$

Square $ABCD$ has sides of length 3. Segments $CM$ and $CN$ divide the square's area into three equal parts. How long is segment $CM$ ? [asy] pair A,B,C,D,M,N; A = (0,0); B = (0,3); C = (3,3); D = (3,0); M = (0,1); N = (1,0); draw(A--B--C--D--cycle); draw(M--C--N); label("$A$",A,SW); label("$M$",M,W); label("$B$",B,NW); label("$C$",C,NE); label("$D$",D,SE); label("$N$",N,S);[/asy] $ \text{(A)}\ \sqrt{10}\qquad\text{(B)}\ \sqrt{12}\qquad\text{(C)}\ \sqrt{13}\qquad\text{(D)}\ \sqrt{14}\qquad\text{(E)}\ \sqrt{15} $

2. Find all natural $n>1$ for which value of the sum $2^2+3^2+...+n^2$ equals $p^k$ where p is prime and k is natural

Let $A$ be the set of all permutations $a = (a_1, a_2, \ldots, a_{2003})$ of the 2003 first positive integers such that each permutation satisfies the condition: there is no proper subset $S$ of the set $\{1, 2, \ldots, 2003\}$ such that $\{a_k | k \in S\} = S.$ For each $a = (a_1, a_2, \ldots, a_{2003}) \in A$, let $d(a) = \sum^{2003}_{k=1} \left(a_k - k \right)^2.$ [b]I.[/b] Find the least value of $d(a)$. Denote this least value by $d_0$. [b]II.[/b] Find all permutations $a \in A$ such that $d(a) = d_0$.

Let $ ABCD$ a cuadrilateral inscribed in a circumference. Suppose that there is a semicircle with its center on $ AB$, that is tangent to the other three sides of the cuadrilateral. (i) Show that $ AB \equal{} AD \plus{} BC$. (ii) Calculate, in term of $ x \equal{} AB$ and $ y \equal{} CD$, the maximal area that can be reached for such quadrilateral.

An integer $ n$ is called good if there are $ n \geq 3$ lattice points $ P_1, P_2, \ldots, P_n$ in the coordinate plane satisfying the following conditions: If line segment $ P_iP_j$ has a rational length, then there is $ P_k$ such that both line segments $ P_iP_k$ and $ P_jP_k$ have irrational lengths; and if line segment $ P_iP_j$ has an irrational length, then there is $ P_k$ such that both line segments $ P_iP_k$ and $ P_jP_k$ have rational lengths. (1) Determine the minimum good number. (2) Determine if 2005 is a good number. (A point in the coordinate plane is a lattice point if both of its coordinate are integers.)

Is it possible to cover a rectangle of dimensions $m \times n$ with bricks that have the trimino angular shape (an arrangement of three unit squares forming the letter $\text L$) if: [b](a)[/b] $m \times n = 1985 \times 1987;$ [b](b)[/b] $m \times n = 1987 \times 1989 \quad ?$

How many $0$s are there at the end of the decimal representation of $2000!$? $ \textbf{(A)}\ 222 \qquad\textbf{(B)}\ 499 \qquad\textbf{(C)}\ 625 \qquad\textbf{(D)}\ 999 \qquad\textbf{(E)}\ \text{None of the preceding} $

A school has $450$ students. Each student has at least $100$ friends among the others and among any $200$ students, there are always two that are friends. Prove that $302$ students can be sent on a kayak trip such that each of the $151$ two seater kayaks contain people who are friends. [i]D. Karpov[/i]

Let $p$ be a prime number and $\mathbf{W}\subseteq \mathbb{F}_p[x]$ be the smallest set satisfying the following : [list] (a) $x+1\in \mathbf{W}$ and $x^{p-2}+x^{p-3}+\cdots +x^2+2x+1\in \mathbf{W}$ (b) For $\gamma_1,\gamma_2$ in $\mathbf{W}$, we also have $\gamma(x)\in \mathbf{W}$, where $\gamma(x)$ is the remainder $(\gamma_1\circ \gamma_2)(x)\pmod {x^p-x}$.[/list] How many polynomials are in $\mathbf{W}?$

In the $9$-gon $ABCDEFGHI$, all sides have equal lengths and all angles are equal. Prove that $|AB| + |AC| = |AE|$. [img]https://cdn.artofproblemsolving.com/attachments/6/2/8c82e8a87bf8a557baaf6ac72b3d18d2ba3965.png[/img]

How many distinct sums can be made from adding together exactly 8 numbers that are chosen from the set $\{ 1,4,7,10 \}$, where each number in the set is chosen at least once? (For example, one possible sum is $1+1+1+4+7+7+10+10=41$.)

We are given $13$ apparently equal balls, all but one having the same weight (the remaining one has a different weight). Is it posible to determine the ball with the different weight in $3$ weighings?