You are given a right triangle $ABC$ with $\angle ACB = 90^\circ$. Let $W_A , W_B$ respectively be the midpoints of the smaller arcs $BC$ and $AC$ of the circumcircle of $\triangle ABC$, and $N_A , N_B$ respectively be the midpoints of the larger arcs $BC$ and $AC$ of this circle. Denote by $P$ and $Q$ the points of intersection of segment $AB$ with the lines $N_AW_B$ and $N_BW_A$, respectively. Prove that $AP = BQ$. [i]Proposed by Oleksiy Masalitin[/i]

[b]p1.[/b] What is the maximum number of points of intersection between a square and a triangle, assuming that no side of the triangle is parallel to any side of the square? [b]p2.[/b] Two angles of an isosceles triangle measure $80^o$ and $x^o$. What is the sum of all the possible values of $x$? [b]p3.[/b] Let $p$ and $q$ be prime numbers such that $p + q$ and p + $7q$ are both perfect squares. Find the value of $pq$. [b]p4.[/b] Anna, Betty, Carly, and Danielle are four pit bulls, each of which is either wearing or not wearing lipstick. The following three facts are true: (1) Anna is wearing lipstick if Betty is wearing lipstick. (2) Betty is wearing lipstick only if Carly is also wearing lipstick. (3) Carly is wearing lipstick if and only if Danielle is wearing lipstick The following five statements are each assigned a certain number of points: (a) Danielle is wearing lipstick if and only if Carly is wearing lipstick. (This statement is assigned $1$ point.) (b) If Anna is wearing lipstick, then Betty is wearing lipstick. (This statement is assigned $6$ points.) (c) If Betty is wearing lipstick, then both Anna and Danielle must be wearing lipstick. (This statement is assigned $10$ points.) (d) If Danielle is wearing lipstick, then Anna is wearing lipstick. (This statement is assigned $12$ points.) (e) If Betty is wearing lipstick, then Danielle is wearing lipstick. (This statement is assigned $14$ points.) What is the sum of the points assigned to the statements that must be true? (For example, if only statements (a) and (d) are true, then the answer would be $1 + 12 = 13$.) [b]p5.[/b] Let $f(x)$ and $g(x)$ be functions such that $f(x) = 4x + 3$ and $g(x) = \frac{x + 1}{4}$. Evaluate $g(f(g(f(42))))$. [b]p6.[/b] Let $A,B,C$, and $D$ be consecutive vertices of a regular polygon. If $\angle ACD = 120^o$, how many sides does the polygon have? [b]p7.[/b] Fred and George have a fair $8$-sided die with the numbers $0, 1, 2, 9, 2, 0, 1, 1$ written on the sides. If Fred and George each roll the die once, what is the probability that Fred rolls a larger number than George? [b]p8.[/b] Find the smallest positive integer $t$ such that $(23t)^3 - (20t)^3 - (3t)^3$ is a perfect square. [b]p9.[/b] In triangle $ABC$, $AC = 8$ and $AC < AB$. Point $D$ lies on side BC with $\angle BAD = \angle CAD$. Let $M$ be the midpoint of $BC$. The line passing through $M$ parallel to $AD$ intersects lines $AB$ and $AC$ at $F$ and $E$, respectively. If $EF =\sqrt2$ and $AF = 1$, what is the length of segment $BC$? (See the following diagram.) [img]https://cdn.artofproblemsolving.com/attachments/2/3/4b5dd0ae28b09f5289fb0e6c72c7cbf421d025.png[/img] [b]p10.[/b] There are $2011$ evenly spaced points marked on a circular table. Three segments are randomly drawn between pairs of these points such that no two segments share an endpoint on the circle. What is the probability that each of these segments intersects the other two? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The isosceles triangle $\triangle ABC$, with $AB=AC$, is inscribed in the circle $\omega$. Let $P$ be a variable point on the arc $\stackrel{\frown}{BC}$ that does not contain $A$, and let $I_B$ and $I_C$ denote the incenters of triangles $\triangle ABP$ and $\triangle ACP$, respectively. Prove that as $P$ varies, the circumcircle of triangle $\triangle PI_BI_C$ passes through a fixed point.

Let $m$ and $n$ be distinct positive integers. Prove that $2^{2^m} + 1$ and $2^{2^n} + 1$ have no common divisor greater than $1$.

Given an acute-angled triangle $ABC$. Let $CB$ be the altitude and $E$ a random point on the line $CD$. Finally, let $P, Q, R$ and $S$ are the projections of $D$ on the straight lines $AC, AE, BE$ and $BC$. Prove that the points $P, Q, R$ and $S$ lie either on a circle or on one straight line.

Suppose that $m$ and $k$ are non-negative integers, and $p = 2^{2^m}+1$ is a prime number. Prove that [b](a)[/b] $2^{2^{m+1}p^k} \equiv 1$ $(\text{mod } p^{k+1})$; [b](b)[/b] $2^{m+1}p^k$ is the smallest positive integer $n$ satisfying the congruence equation $2^n \equiv 1$ $(\text{mod } p^{k+1})$.

An $n\times n$ matrix $A$ with integer entries is called [i]representative[/i] if, for any integer vector $\mathbf{v}$, there is a finite sequence $0=\mathbf{v}_0,\mathbf{v}_1,\dots,\mathbf{v}_{\ell}=\mathbf{v}$ of integer vectors such that for each $0\leq i <\ell$, either $\mathbf{v}_{i+1}=A\mathbf{v}_{i}$ or $\mathbf{v}_{i+1}-\mathbf{v}_i$ is an element of the standard basis (i.e. one of its entries is $1$, the rest are all equal to $0$). Show that $A$ is not representative if and only if $A^T$ has a real eigenvector with all non-negative entries and non-negative eigenvalue.

A string having a small loop in one end is set over a horizontal pipe so that the ends hang loosely. After that, the other end is put through the loop, pulled as far as possible from the pipe and fixed in that position whereby this end of the string is farther from the pipe than the loop. Let $\alpha$ be the angle by which the string turns at the point where it passes through the loop (see picture). Find $\alpha$. [img]https://cdn.artofproblemsolving.com/attachments/2/1/018bb16d80956699e11c641bad9bb3d0083770.png[/img]

A square and an equilateral triangle have equal perimeters. The area of the triangle is $ 9 \sqrt{3}$ square inches. Expressed in inches the diagonal of the square is: $ \textbf{(A)}\ \frac{9}{2} \qquad \textbf{(B)}\ 2 \sqrt{5} \qquad \textbf{(C)}\ 4 \sqrt{2} \qquad \textbf{(D)}\ \frac{9 \sqrt{2}}{2} \qquad \textbf{(E)}\ \text{none of these}$

For which commutative finite groups is the product of all elements equal to the unit element?

Find all positive integers m, n such that $m^3 - n^3 = 999$.

Any positive integer $n$ can be written in the form $n=2^{k}(2l+1)$ with $k,l$ positive integers. Let $a_n =e^{-k}$ and $b_n = a_1 a_2 a_3 \cdots a_n.$ Prove that $$\sum_{n=1}^{\infty} b_n$$ converges.

Let $ AH_1, BH_2, CH_3$ be the altitudes of an acute angled triangle $ ABC$. Its incircle touches the sides $ BC, AC$ and $ AB$ at $ T_1, T_2$ and $ T_3$ respectively. Consider the symmetric images of the lines $ H_1H_2, H_2H_3$ and $ H_3H_1$ with respect to the lines $ T_1T_2, T_2T_3$ and $ T_3T_1$. Prove that these images form a triangle whose vertices lie on the incircle of $ ABC$.

Let $ABCD$ be a quadrilateral with $\angle A + \angle C = 60^o$. If $AB \cdot CD = BC \cdot AD$, prove that $AB \cdot CD = AC \cdot BD$. Leonard Giugiuc

For each $1\leq i\leq 9$ and $T\in\mathbb N$, define $d_i(T)$ to be the total number of times the digit $i$ appears when all the multiples of $1829$ between $1$ and $T$ inclusive are written out in base $10$. Show that there are infinitely many $T\in\mathbb N$ such that there are precisely two distinct values among $d_1(T)$, $d_2(T)$, $\dots$, $d_9(T)$.

If a person A is taller or heavier than another peoson B, then we note that A is [i]not worse than[/i] B. In 100 persons, if someone is [i]not worse than[/i] other 99 people, we call him [i]excellent boy[/i]. What's the maximum value of the number of [i]excellent boys[/i]? $\text{(A)}1\qquad\text{(B)}2\qquad\text{(C)}50\qquad\text{(D)}100$