Let $p$ and $q$ be two positive integers that have no common divisor. The set of integers shall be partioned into three subsets $A$, $B$, $C$ such that for each integer $z$ in each of the sets $A$, $B$, $C$ there is exactly one of the numbers $z$, $z+p$ and $z+q$. a) Prove that such a decomposition is possible if and only if $p+q$ is divisible by $3$. b) In the case we omit the restriction that $p$, $q$ may not have a common divisor, prove that for $p \neq q$ the number $\frac{p+q}{\gcd(p,q)}$ is divisible by 3.

Determine all triplets of prime numbers $p<q<r$, such that $p+q=r$ and $(r-p)(q-p)-27p$ is a square.

Triangle $ABC$ is given in a plane. Draw the bisectors of all three of its angles. Then draw the line that connects the points where the bisectors of angles $ABC$ and $ACB$ meet the opposite sides of the triangle. Through the point of intersection of this line and the bisector of angle $BAC$, draw another line parallel to $BC$. Let this line intersect $AB$ in $M$ and $AC$ in $N$. Prove that $2MN = BM+CN$.

Let $ ABC$ be an obtuse inscribed in a circle of radius $ 1$. Prove that $ \triangle ABC$ can be covered by an isosceles right-angled triangle with hypotenuse of length $ \sqrt {2} \plus{} 1$.

[u]Part 1[/u] [b]p1.[/b] Two kids $A$ and $B$ play a game as follows: From a box containing $n$ marbles ($n > 1$), they alternately take some marbles for themselves, such that: 1. $A$ goes first. 2. The number of marbles taken by $A$ in his first turn, denoted by $k$, must be between $1$ and $n$, inclusive. 3. The number of marbles taken in a turn by any player must be between $1$ and $k$, inclusive. The winner is the one who takes the last marble. What is the sum of all $n$ for which $B$ has a winning strategy? [b]p2.[/b] How many ways can your rearrange the letters of "Alejandro" such that it contains exactly one pair of adjacent vowels? [b]p3.[/b] Assuming real values for $p, q, r$, and $s$, the equation $$x^4 + px^3 + qx^2 + rx + s$$ has four non-real roots. The sum of two of these roots is $q + 6i$, and the product of the other two roots is $3 - 4i$. Find the smallest value of $q$. [b]p4.[/b] Lisa has a $3$D box that is $48$ units long, $140$ units high, and $126$ units wide. She shines a laser beam into the box through one of the corners, at a $45^o$ angle with respect to all of the sides of the box. Whenever the laser beam hits a side of the box, it is reflected perfectly, again at a $45^o$ angle. Compute the distance the laser beam travels until it hits one of the eight corners of the box. [u]Part 2[/u] [b]p5.[/b] How many ways can you divide a heptagon into five non-overlapping triangles such that the vertices of the triangles are vertices of the heptagon? [b]p6.[/b] Let $a$ be the greatest root of $y = x^3 + 7x^2 - 14x - 48$. Let $b$ be the number of ways to pick a group of $a$ people out of a collection of $a^2$ people. Find $\frac{b}{2}$ . [b]p7.[/b] Consider the equation $$1 -\frac{1}{d}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c},$$ with $a, b, c$, and $d$ being positive integers. What is the largest value for $d$? [b]p8.[/b] The number of non-negative integers $x_1, x_2,..., x_{12}$ such that $$x_1 + x_2 + ... + x_{12} \le 17$$ can be expressed in the form ${a \choose b}$ , where $2b \le a$. Find $a + b$. [u]Part 3[/u] [b]p9.[/b] In the diagram below, $AB$ is tangent to circle $O$. Given that $AC = 15$, $AB = 27/2$, and $BD = 243/34$, compute the area of $\vartriangle ABC$. [img]https://cdn.artofproblemsolving.com/attachments/b/f/b403e5e188916ac4fb1b0ba74adb7f1e50e86a.png[/img] [b]p10.[/b] If $$\left[2^{\log x}\right]^{[x^{\log 2}]^{[2^{\log x}]...}}= 2, $$ where $\log x$ is the base-$10$ logarithm of $x$, then it follows that $x =\sqrt{n}$. Compute $n^2$. [b]p11.[/b] [b]p12.[/b] Find $n$ in the equation $$133^5 + 110^5 + 84^5 + 27^5 = n^5, $$ where $n$ is an integer less than $170$. [u]Part 4[/u] [b]p13.[/b] Let $x$ be the answer to number $14$, and $z$ be the answer to number $16$. Define $f(n)$ as the number of distinct two-digit integers that can be formed from digits in $n$. For example, $f(15) = 4$ because the integers $11$, $15$, $51$, $55$ can be formed from digits of $15$. Let $w$ be such that $f(3xz - w) = w$. Find $w$. [b]p14.[/b] Let $w$ be the answer to number $13$ and $z$ be the answer to number $16$. Let $x$ be such that the coefficient of $a^xb^x$ in $(a + b)^{2x}$ is $5z^2 + 2w - 1$. Find $x$. [b]p15.[/b] Let $w$ be the answer to number $13$, $x$ be the answer to number $14$, and $z$ be the answer to number $16$. Let $A$, $B$, $C$, $D$ be points on a circle, in that order, such that $\overline{AD}$ is a diameter of the circle. Let $E$ be the intersection of $\overleftrightarrow{AB}$ and $\overleftrightarrow{DC}$, let $F$ be the intersection of $\overleftrightarrow{AC}$ and $\overleftrightarrow{BD}$, and let $G$ be the intersection of $\overleftrightarrow{EF}$ and $\overleftrightarrow{AD}$. Now, let $AE = 3x$, $ED = w^2 - w + 1$, and $AD = 2z$. If $FG = y$, find $y$. [b]p16.[/b] Let $w$ be the answer to number $13$, and $x$ be the answer to number $16$. Let $z$ be the number of integers $n$ in the set $S = \{w,w + 1, ... ,16x - 1, 16x\}$ such that $n^2 + n^3$ is a perfect square. Find $z$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Define sequence $a_{0}, a_{1}, a_{2}, \ldots, a_{2018}, a_{2019}$ as below: $ a_{0}=1 $ $a_{n+1}=a_{n}-\frac{a_{n}^{2}}{2019}$, $n=0,1,2, \ldots, 2018$ Prove $a_{2019} < \frac{1}{2} < a_{2018}$

Let $\mathcal{P}_0=A_0A_1\ldots A_{n-1}$ be a convex polygon such that $A_iA_{i+1}=2^{[i/2]}$ for $i=0, 1,\ldots ,n-1$ (where $A_n=A_0$). Define the sequence of polygons $\mathcal{P}_k=A_0^kA_1^k\ldots A_{n-1}^k$ as follows: $A_i^1$ is symmetric to $A_i$ with respect to $A_0$, $A_i^2$ is symmetric to $A_i^1$ with respect to $A_1^1$, $A_i^3$ is symmetric to $A_i^2$ with respect to $A_2^2$ and so on. Find the values of $n$ for which infinitely many polygons $\mathcal{P}_k$ coincide with $\mathcal{P}_0$.

Let $ABCDEF$ be a convex hexagon such that $AB=CD=EF=20, \ BC=DE=FA=21,$ and $\angle A=\angle C=\angle E=90^{\circ}.$ The area of $ABCDEF$ can then be expressed in the form $a+\tfrac{b\sqrt{c}}{d},$ where $a,\ b,\ c,$ and $d$ are positive integers, $b$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d.$

A finite sequence is said to be [i]disorderly[/i], if no two terms of the sequence have their average in between them. For example, $(0, 2, 1)$ is disorderly, for $1 = \frac{0+2}{2}$ is not in between $0$ and $2$, and the other averages $\frac{0+1}{2} = \frac{1}{2}$ and $\frac{2+1}{2} = 1\frac{1}{2}$ do not even occur in the sequence. Prove that for every $n \in \Bbb{N}$ there is a disorderly sequence enumerating the numbers $0, 1,\ldots , n$ without repetitions.

In a group $G$, we have two elements $x,y$ such that $x^{n}=e,y^2=e,yxy=x^{-1}$, $n\ge 1$. Prove that for any $k\in\mathbb{N}$ holds $(x^ky)^2=e$. Note : e=group's identity .

A tetrahedron is a polyhedron composed of four triangular faces. Faces $ABC$ and $BCD$ of a tetrahedron $ABCD$ meet at an angle of $\pi/6$. The area of triangle $\triangle ABC$ is $120$. The area of triangle $\triangle BCD$ is $80$, and $BC = 10$. What is the volume of the tetrahedron? We call the volume of a tetrahedron as one-third the area of it's base times it's height.

The graph of $r=2+\cos2\theta$ and its reflection over the line $y=x$ bound five regions in the plane. Find the area of the region containing the origin.

Find minimum value of $F(x,y)=\sqrt{(x+1)^{2}+(y-1)^{2}}+\sqrt{(x-1)^{2}+(y+1)^{2}}+\sqrt{(x+2)^{2}+(y+2)^{2}}$, where $x,y\in\mathbb{R}$.

An $80 \times 80$ grid is colored orange and black. A square is black if and only if either the square below it or the square to the left of it is black, but not both (If there is no such square, consider it as if it were orange). The only exception is the bottom left square, which is black. Consider the diagonal from the upper left to the lower right. How many black squares does this diagonal have?

Determine the number of ordered pairs $(x,y)$ with $x$ and $y$ integers between $-5$ and $5,$ inclusive, such that $(x+y)(x+3y)=(x+2y)^2.$

Consider $A, B$ and $C$ three collinear points of the plane such that $B$ is between $A$ and $C$. Let $S$ be the circle of diameter $AB$ and $L$ a line that passes through $C$, which does not intersect $S$ and is not perpendicular to line $AC$. The points $M$ and $N$ are, respectively, the feet of the altitudes drawn from $A$ and $B$ on the line $L$. From $C$ draw the two tangent lines to $S$, where $P$ is the closest tangency point to $L$. Prove that the quadrilateral $MPBC$ is cyclic if and only if the lines $MB$ and $AN$ are perpendicular.

Determine all functions $ f$ defined in the set of rational numbers and taking their values in the same set such that the equation $ f(x + y) + f(x - y) = 2f(x) + 2f(y)$ holds for all rational numbers $x$ and $y$.

Let $a$ and $ b$ be integers and $n$ a positive integer. Prove that \[\frac{b^{n-1}a(a + b)(a + 2b) \cdots (a + (n - 1)b)}{n!}\] is an integer.

The positive numbers $a$, $b$, and $c$ satisfy $abc = 1$. Show that: $$ \frac{1}{a^2+a}+\frac{1}{b^2+b}+\frac{1}{c^2+c} \ge \frac{3}{2}. $$

A point $P$ is chosen in the interior of $\triangle ABC$ so that when lines are drawn through $P$ parallel to the sides of $\triangle ABC$, the resulting smaller triangles, $t_1$, $t_2$, and $t_3$ in the figure, have areas 4, 9, and 49, respectively. Find the area of $\triangle ABC$. [asy] size(200); pathpen=black+linewidth(0.65);pointpen=black; pair A=(0,0),B=(12,0),C=(4,5); D(A--B--C--cycle); D(A+(B-A)*3/4--A+(C-A)*3/4); D(B+(C-B)*5/6--B+(A-B)*5/6);D(C+(B-C)*5/12--C+(A-C)*5/12); MP("A",C,N);MP("B",A,SW);MP("C",B,SE); /* sorry mixed up points according to resources diagram. */ MP("t_3",(A+B+(B-A)*3/4+(A-B)*5/6)/2+(-1,0.8),N); MP("t_2",(B+C+(B-C)*5/12+(C-B)*5/6)/2+(-0.3,0.1),WSW); MP("t_1",(A+C+(C-A)*3/4+(A-C)*5/12)/2+(0,0.15),ESE);[/asy]

Let $ABCDEF$ be a convex hexagon. In the hexagon there is a point $K$, such that $ABCK,DEFK$ are both parallelograms. Prove that the three lines connecting $A,B,C$ to the midpoints of segments $CE,DF,EA$ meet at one point.

Let $f(x)$ be a non-constant polynomial with integer coefficients such that $f(1) \neq 1$. For a positive integer $n$, define $\text{divs}(n)$ to be the set of positive divisors of $n$. A positive integer $m$ is $f$-cool if there exists a positive integer $n$ for which $$f[\text{divs}(m)]=\text{divs}(n).$$ Prove that for any such $f$, there are finitely many $f$-cool integers. (The notation $f[S]$ for some set $S$ denotes the set $\{f(s):s \in S\}$.)

Let $m,n,k$ be positive integers, $m> n> k$. An $1 \times m$ strip of paper is divided into the $1 \times 1$ cells. A teacher asks Bill and Pit to place numbers $0$ and $1$ in the cells of the strip so that the sum of the numbers in any $n$ consecutive cells is equal to $k$. After the task was performed it turned out that the sum $S(B)$ of all numbers on the strip of Bill was different from the sum $S(P)$ of Pit. Find the largest possible value of $|S(B) - S(P) |$. (I. Voronovich)

Find the pairs of real numbers $(a,b)$ such that the biggest of the numbers $x=b^2-\frac{a-1}{2}$ and $y=a^2+\frac{b+1}{2}$ is less than or equal to $\frac{7}{16}$

Let $h_a$, $h_b$ and $h_c$ be the heights and $r$ the inradius of a triangle. Prove that the triangle is equilateral if and only if $h_a + h_b + h_c = 9r$.