[b]p1.[/b] What is the maximal number of pieces of two shapes, [img]https://cdn.artofproblemsolving.com/attachments/a/5/6c567cf6a04b0aa9e998dbae3803b6eeb24a35.png[/img] and [img]https://cdn.artofproblemsolving.com/attachments/8/a/7a7754d0f2517c93c5bb931fb7b5ae8f5e3217.png[/img], that can be used to tile a $7\times 7$ square? [b]p2.[/b] Six shooters participate in a shooting competition. Every participant has $5$ shots. Each shot adds from $1$ to $10$ points to shooter’s score. Every person can score totally for all five shots from $5$ to $50$ points. Each participant gets $7$ points for at least one of his shots. The scores of all participants are different. We enumerate the shooters $1$ to $6$ according to their scores, the person with maximal score obtains number $1$, the next one obtains number $2$, the person with minimal score obtains number $6$. What score does obtain the participant number $3$? The total number of all obtained points is $264$. [b]p2.[/b] There are exactly $n$ students in a high school. Girls send messages to boys. The first girl sent messages to $5$ boys, the second to $7$ boys, the third to $6$ boys, the fourth to $8$ boys, the fifth to $7$ boys, the sixth to $9$ boys, the seventh to $8$, etc. The last girl sent messages to all the boys. Prove that $n$ is divisible by $3$. [b]p4.[/b] In what minimal number of triangles can one cut a $25 \times 12$ rectangle in such a way that one can tile by these triangles a $20 \times 15$ rectangle. [b]p5.[/b] There are $2014$ stones in a pile. Two players play the following game. First, player $A$ takes some number of stones (from $1$ to $30$) from the pile, then player B takes $1$ or $2$ stones, then player $A$ takes $2$ or $3$ stones, then player $B$ takes $3$ or $4$ stones, then player A takes $4$ or $5$ stones, etc. The player who gets the last stone is the winner. If no player gets the last stone (there is at least one stone in the pile but the next move is not allowed) then the game results in a draw. Who wins the game using the right strategy? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a_1 = \sqrt 7$ and $b_i = \lfloor a_i \rfloor$, $a_{i+1} = \dfrac{1}{b_i - \lfloor b_i \rfloor}$ for each $i\geq i$. What is the smallest integer $n$ greater than $2004$ such that $b_n$ is divisible by $4$? ($\lfloor x \rfloor$ denotes the largest integer less than or equal to $x$) $ \textbf{(A)}\ 2005 \qquad\textbf{(B)}\ 2006 \qquad\textbf{(C)}\ 2007 \qquad\textbf{(D)}\ 2008 \qquad\textbf{(E)}\ \text{None of above} $

Let $n$ be a positive integer that is not divisible by either $2$ or $5$. In the decimal expansion of $\frac{1}{n}= 0.a_1a_2a_3\cdots$ a finite number of digits after the decimal point are chosen arbitrarily to be deleted. Clearly the decimal number obtained by this procedure is also rational, so it's equal to $\frac{a}{b}$ for some integers $a,b$. Prove that $b$ is divisible by $n$.

There are 15 cities, and there is a train line between each pair operated by either the Carnegie Rail Corporation or the Mellon Transportation Company. A tourist wants to visit exactly three cities by travelling in a loop, all by travelling on one line. What is the minimum number of such 3-city loops?

Given a sequence $a_1,a_2,a_3,\ldots $ of positive integers in which every positive integer occurs exactly once. Prove that there exist integers $\ell $ and $m,\ 1<\ell <m$, such that $a_1+a_m=2a_{\ell}$.

The vertices of a triangle with sides $a\ge b\ge c$ are centers of three circles, such that no two of the circles have common interior points and none contains any other vertex of the triangle. Determine the maximum possible total area of these three circles.

Do there exist irrational numbers $a,b$ both greater than $1$, such that $\lfloor{a^m}\rfloor\neq \lfloor{b^n}\rfloor$ for all $m,n\in\mathbb{N}$ ?

Given a solid $R$ contained in a semi cylinder with the hight $1$ which has a semicircle with radius $1$ as the base. The cross section at the hight $x\ (0\leq x\leq 1)$ is the form combined with two right-angled triangles as attached figure as below. Answer the following questions. (1) Find the cross-sectional area $S(x)$ at the hight $x$. (2) Find the volume of $R$. If necessary, when you integrate, set $x=\sin t.$

In a certain language there are only two letters, $A$ and $B$. The words of this language obey the following rules: (i) The only word of length $1$ is $A$; (ii) A sequence of letters $X_1X_2...X_{n+1}$, where $X_i\in \{A,B\}$ for each $i$, forms a word of length $n+1$ if and only if it contains at least one letter $A$ and is not of the form $WA$ for a word $W$ of length $n$. Show that the number of words consisting of $1998 A$’s and $1998 B$’s and not beginning with $AA$ equals $\binom{3995}{1997}-1$

Calculate $ \lim_{n\to\infty }\left(\lim_{x\to 0} \left( -\frac{n}{x}+1+\frac{1}{x}\sum_{r=2}^{n+1}\sqrt[r!]{1+\sin rx}\right)\right) . $ [i]Constantin Rusu[/i]

$ ABC $ is a triangle with $ \angle BCA= 90^{\circ } $ and $ D,E $ on sides $ BC,CA, $ rspectively, so that $ \frac{BD}{AC}=\frac{AE}{CD}=k. $ The line $ BE $ meets $ AD $ at $ O. $ Show that $ \angle BOD =60^{\circ } $ if and only if $ k=\sqrt 3. $

Let $x_1,x_2,\dots,x_n$ be a finite sequence of real numbersm mwhere $0<x_i<1$ for all $i=1,2,\dots,n$. Put $P=x_1x_2\cdots x_n$, $S=x_1+x_2+\cdots+x_n$ and $T=\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n}$. Prove that \[\frac{T-S}{1-P}>2.\]

Let $ABC$ be an isosceles triangle such that $AB=AC$. Find angles of triangle $ABC$ if $\frac{AB}{BC}=1+2\cos{\frac{2\pi}{7}}$

We say that a triangle $ABC$ is great if the following holds: for any point $D$ on the side $BC$, if $P$ and $Q$ are the feet of the perpendiculars from $D$ to the lines $AB$ and $AC$, respectively, then the reflection of $D$ in the line $PQ$ lies on the circumcircle of the triangle $ABC$. Prove that triangle $ABC$ is great if and only if $\angle A = 90^{\circ}$ and $AB = AC$. [i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]

Zandrew Hao has $n^2$ dollars, where $n$ is an integer. He is a massive fan of the singer Pachary Zerry, and he wants to buy many copies of his $3$ albums, which cost $\$8$, $\$623$, and $\$835$ (two of them are very rare). Find the sum of the $3$ greatest values of $n$ such that Zandrew can't spend all of his money on albums.

Prove that a nontrivial finite ring is not a skew field if and only if the equation $ x^n+y^n=z^n $ has nontrivial solutions in this ring for any natural number $ n. $

$(BEL 5)$ Let $G$ be the centroid of the triangle $OAB.$ $(a)$ Prove that all conics passing through the points $O,A,B,G$ are hyperbolas. $(b)$ Find the locus of the centers of these hyperbolas.

Let $a, b, c, d$ be positive integers such that $ab = cd$. Prove that $w = a^{2006} + b^{2006} + c^{2006} + d^{2006}$ is composite.

In the plane we have points $P,Q,A,B,C$ such triangles $APQ,QBP$ and $PQC$ are similar accordantly (same direction). Then let $A'$ ($B',C'$ respectively) be the intersection of lines $BP$ and $CQ$ ($CP$ and $AQ;$ $AP$ and $BQ,$ respectively.) Show that the points $A,B,C,A',B',C'$ lie on a circle.

The triangle $ABC$ has sides lengths $AB = 39$, $BC = 57$, and $CA = 70$ as shown. Median $\overline{AD}$ is divided into three congruent segments by points $E$ and $F$. Lines $BE$ and $BF$ intersect side $\overline{AC}$ at points $G$ and $H$, respectively. Find the distance from $G$ to $H$. [asy] import graph; size(7cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); pair A = (-2,0), B = (3,0), D = (0,4), C = (0.5,0), F = (0.17,2.66), G = (0.6,3.2), H = (1.5,2); draw(A--B); draw(D--B); draw(D--A); draw(D--C); draw(G--A); draw(A--H); label("$ A $",(-0.16,4.6),SE*labelscalefactor); label("$ B $",(-2.66,0.3),SE*labelscalefactor); label("$ C $",(3.34,0.18),SE*labelscalefactor); label("$ D $",(0.28,-0.1),SE*labelscalefactor); label("$ E $",(0.44,1.4),SE*labelscalefactor); label("$ F $",(-0.24,3.15),SE*labelscalefactor); label("$ H $",(0.82,3.6),SE*labelscalefactor); label("$G$",(1.84,2.25),SE*labelscalefactor); label("39",(-1.68,2.5),SE*labelscalefactor); label("57",(0.3,-0.74),SE*labelscalefactor); label("70",(2,2.98),SE*labelscalefactor); dot(A); dot(B); dot(D); dot(C); dot((0.33,1.33)); dot(F); dot(G); dot(H); [/asy]

$n$ people sitting at a round table. In the beginning, everyone writes down a positive number $n$ on piece of paper in front of them. From now on, in every minute, they write down the number that they get if they subtract the number of their right-hand neighbour from their own number. They write down the new number and erase the original. Give those number $n$ that there exists an integer $k$ in a way that regardless of the starting numbers, after $k$ minutes, everyone will have a number that is divisible by $n$.

Suppose that $a, b,$ and $c$ are positive real numbers such that $$a + b + c \ge \frac{1}{a} + \frac{1}{b} + \frac{1}{c}.$$ Find the largest possible value of the expression $$\frac{a + b - c}{a^3 + b^3 + abc} + \frac{b + c - a}{b^3 + c^3 + abc} + \frac{c + a - b}{c^3 + a^3 + abc}.$$

for acute triangle $\triangle ABC$ with orthocenter $H$, and $E,F$ the feet of altitudes for $B,C$, we have $P$ on $EF$ such as that $HO \perp HP$. $Q$ is on segment $AH$ so $HM \perp PQ$. prove $QA=3QH$

The positive real numbers $a$ and $b$ ($a> b$) are written on the board. At every step, with numbers written on the board, one of the following operations can be performed: a) choose one of the numbers and write its square or its inverse. b) choose two numbers written on the board ¸and write their sum or their positive difference. Show how the product $a \cdot b$ can be obtained with the help of the defined operations.

Given a triangle $ABC$ satisfying $AC+BC=3\cdot AB$. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Let $K$ and $L$ be the reflections of the points $D$ and $E$ with respect to $I$. Prove that the points $A$, $B$, $K$, $L$ lie on one circle. [i]Proposed by Dimitris Kontogiannis, Greece[/i]