p1. Known points $A (-1.2)$, $B (0,2)$, $C (3,0)$, and $D (3, -1)$ as seen in the following picture. Determine the measure of the angle $AOD$ . [img]https://cdn.artofproblemsolving.com/attachments/f/2/ca857aaf54c803db34d8d52505ef9a80e7130f.png[/img] p2. Determine all prime numbers $p> 2$ until $p$ divides $71^2 - 37^2 - 51$. p3. A ball if dropped perpendicular to the ground from a height then it will bounce back perpendicular along the high third again, down back upright and bouncing back a third of its height, and next. If the distance traveled by the ball when it touches the ground the fourth time is equal to $106$ meters. From what height is the ball was dropped? p4. The beam $ABCD.EFGH$ is obtained by pasting two unit cubes $ABCD.PQRS$ and $PQRS.EFGH$. The point K is the midpoint of the edge $AB$, while the point $L$ is the midpoint of the edge $SH$. What is the length of the line segment $KL$? p5. How many integer numbers are no greater than $2004$, with remainder $1$ when divided by $2$, with remainder $2$ when divided by $3$, with remainder $3$ when divided by $4$, and with remainder $4$ when divided by $5$?

The inscribed circle of the triangle $ABC$ touches its sides $AB, BC, CA$, at points $K,N, M$ respectively. It is known that $\angle ANM = \angle CKM$. Prove that the triangle $ABC$ is isosceles. (Vyacheslav Yasinsky)

$$F(x)=x^{6}+15x^{5}+85x^{4}+225x^{3}+274x^{2}+120x+1$$

Find all natural numbers $a>1$, with the property that every prime divisor of $a^6-1$ divides also at least one of the numbers $a^3-1$, $a^2-1$. [i]K. Dochev[/i]

We have considered an arbitrary segment from each line in a plane. Show that the set of points of these segments have a subset such that the points of this subset form a triangle in the plane.

For every $x$, $y$, $z>{3\over 2}$ prove the inequality $$ x^{24} + \root 5\of {y^{60}+z^{40}} \geq \left(x^4 y^3 + {1\over 3} y^2 z^2 + {1\over 9} x^3 z^3 \right)^2. $$

During a day $2016$ customers visited the store. Every customer has been only once at the store(a customer enters the store,spends some time, and leaves the store). Find the greatest integer $k$ that makes the following statement always true. We can find $k$ customers such that either all of them have been at the store at the same time, or any two of them have not been at the same store at the same time.

A pair of 8-sided dice have sides numbered 1 through 8. Each side has the same probability (chance) of landing face up. The probability that the product of the two numbers that land face-up exceeds 36 is $\textbf{(A)}\ \dfrac{5}{32} \qquad \textbf{(B)}\ \dfrac{11}{64} \qquad \textbf{(C)}\ \dfrac{3}{16} \qquad \textbf{(D)}\ \dfrac{1}{4} \qquad \textbf{(E)}\ \dfrac{1}{2}$

in a right-angled triangle $ABC$ with $\angle C=90$,$a,b,c$ are the corresponding sides.Circles $K.L$ have their centers on $a,b$ and are tangent to $b,c$;$a,c$ respectively,with radii $r,t$.find the greatest real number $p$ such that the inequality $\frac{1}{r}+\frac{1}{t}\ge p(\frac{1}{a}+\frac{1}{b})$ always holds.

Let $O$ be the circumcentre of triangle $ABC$. Lines $AO$ and $BC$ intersect at point $D$. Let $S$ be a point on line $BO$ such that $DS \parallel AB$ and lines $AS$ and $BC$ intersect at point $T$. Prove that if $O, D, S$ and $T$ lie on the same circle, then $ABC$ is an isosceles triangle.

Consider an infinite strip, divided into unit squares. A finite number of nuts is placed in some of these squares. In a step, we choose a square $A$ which has more than one nut and take one of them and put it on the square on the right, take another nut (from $A$) and put it on the square on the left. The procedure ends when all squares has at most one nut. Prove that, given the initial configuration, any procedure one takes will end after the same number of steps and with the same final configuration.

Find all four-digit numbers $\overline{abcd}$ such that they are multiples of $3$ and that $\overline{ab}-\overline{cd}=11$. ($\overline{abcd}$ is a four-digit number; $\overline{ab}$ is a two digit-number as $\overline{cd}$ is).

Given the polynomial $P(x)=(x^2-7x+6)^{2n}+13$ where $n$ is a positive integer. Prove that $P(x)$ can't be written as a product of $n+1$ non-constant polynomials with integer coefficients.

Let $ABC$ be an acute triangle with orthocenter $H$. The circumcircle of the triangle $BHC$ intersects $AC$ a second time in point $P$ and $AB$ a second time in point $Q$. Prove that $H$ is the circumcenter of the triangle $APQ$. [i](Karl Czakler)[/i]

Let $ A_n $ be the set of partitions of the sequence $ 1,2,..., n $ into several subsequences such that every two neighbouring terms of each subsequence have different parity,and $ B_n $ the set of partitions of the sequence $ 1,2,..., n $ into several subsequences such that all the terms of each subsequence have the same parity ( for example,the partition $ {(1,4,5,8),(2,3),(6,9),(7)} $ is an element of $ A_9 $,and the partition $ {(1,3,5),(2,4),(6)} $ is an element of $ B_6 $ ). Prove that for every positive integer $ n $ the sets $ A_n $ and $ B_{n+1} $ contain the same number of elements.

Dots are spaced one unit apart, horizontally and vertically. The number of square units enclosed by the polygon is [asy] for(int a=0; a<4; ++a) { for(int b=0; b<4; ++b) { dot((a,b)); } } draw((0,0)--(0,2)--(1,2)--(2,3)--(2,2)--(3,2)--(3,0)--(2,0)--(2,1)--(1,0)--cycle);[/asy] $ \text{(A)}\ 5\qquad\text{(B)}\ 6\qquad\text{(C)}\ 7\qquad\text{(D)}\ 8\qquad\text{(E)}\ 9 $

How many distinct lines pass through the point $(0, 2016)$ and intersect the parabola $y = x^2$ at two lattice points? (A lattice point is a point whose coordinates are integers.)

Suppose one is given $n$ real numbers, not all zero, but such that their sum is zero. Prove that one can label these numbers $a_1, a_2, ..., a_n$ in such a manner that $a_1a_2 + a_2a_3 +...+a_{n-1}a_n + a_na_1 < 0$.

If $x$ is such that $\dfrac{1}{x}<2$ and $\dfrac{1}{x}>-3$, then: $\textbf{(A)}\ -\dfrac{1}{3}<x<\dfrac{1}{2} \qquad \textbf{(B)}\ -\dfrac{1}{2}<x<3 \qquad \textbf{(C)}\ x>\dfrac{1}{2} \qquad\\ \textbf{(D)}\ x>\dfrac{1}{2}\text{ or }-\dfrac{1}{3}<x<0 \qquad \textbf{(E)}\ x>\dfrac{1}{2}\text{ or }x<-\dfrac{1}{3}$

For all rational $x$ satisfying $0 \leq x < 1$, the functions $f$ is defined by \[f(x)=\begin{cases}\frac{f(2x)}{4},&\mbox{for }0 \leq x < \frac 12,\\ \frac 34+ \frac{f(2x - 1)}{4}, & \mbox{for } \frac 12 \leq x < 1.\end{cases}\] Given that $x = 0.b_1b_2b_3 \cdots $ is the binary representation of $x$, find, with proof, $f(x)$.

[b]p1.[/b] The clock is $2$ hours $20$ minutes ahead of the correct time each week. The clock is set to the correct time at midnight Sunday to Monday. What time does this clock show at 6pm correct time on Thursday? [b]p2.[/b] Five cities $A,B,C,D$, and $E$ are located along the straight road in the alphabetical order. The sum of distances from $B$ to $A,C,D$ and $E$ is $20$ miles. The sum of distances from $C$ to the other four cities is $18$ miles. Find the distance between $B$ and $C$. [b]p3.[/b] Does there exist distinct digits $a, b, c$, and $d$ such that $\overline{abc}+\overline{c} = \overline{bda}$? Here $\overline{abc}$ means the three digit number with digits $a, b$, and $c$. [b]p4.[/b] Kuzya, Fyokla, Dunya, and Senya participated in a mathematical competition. Kuzya solved $8$ problems, more than anybody else. Senya solved $5$ problem, less than anybody else. Each problem was solved by exactly $3$ participants. How many problems were there? [b]p5.[/b] Mr Mouse got to the cellar where he noticed three heads of cheese weighing $50$ grams, $80$ grams, and $120$ grams. Mr. Mouse is allowed to cut simultaneously $10$ grams from any two of the heads and eat them. He can repeat this procedure as many times as he wants. Can he make the weights of all three pieces equal? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $(A,+,\cdot)$ be an unit ring with the property: for all $x\in A$, \[ x+x^{2}+x^{3}=x^{4}+x^{5}+x^{6} \] [list=a] [*]Let $x\in A$ and let $n\geq2$ be an integer such that $x^{n}=0$. Prove that $x=0$. [*]Prove that $x^{4}=x$, for all $x\in A$.[/list]

Let $S$ be a non-empty set of positive integers such that for any $n \in S$, all positive divisors of $2^n+1$ are also in $S$. Prove that $S$ contains an integer of the form $(p_1p_2 \ldots p_{2023})^{2023}$, where $p_1, p_2, \ldots, p_{2023}$ are distinct prime numbers, all greater than $2023$.

Any positive integer $n$ can be written in the form $n = 2^aq$, where $a \ge 0$ and $q$ is odd. We call $q$ the [i]odd part[/i] of $n$. Define the sequence $a_0,a_1,...$ as follows: $a_0 = 2^{2011}-1$ and for $m > 0, a_{m+i}$ is the odd part of $3a_m + 1$. Find $a_{2011}$.

Let $x,y$ be positive real numbers .Find the minimum of $x+y+\frac{|x-1|}{y}+\frac{|y-1|}{x}$.