Find all prime numbers $p, q, r$ satisfying the equation \[ p^4+2p+q^4+q^2=r^2+4q^3+1 \]

Let $a,b$ be two positive integers, such that $ab\neq 1$. Find all the integer values that $f(a,b)$ can take, where \[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]

Let $\pi(n)$ denote the number of primes less than or equal to $n$. A subset of $S=\{1,2,\ldots, n\}$ is called [i]primitive[/i] if there are no two elements in it with one of them dividing the other. Prove that for $n\geq 5$ and $1\leq k\leq \pi(n)/2,$ the number of primitive subsets of $S$ with $k+1$ elements is greater or equal to the number of primitive subsets of $S$ with $k$ elements. [i]Proposed by Cs. Sándor, Budapest[/i]

Let $S=\{x^2+2y^2\mid x,y\in\mathbb Z\}$. If $a$ is an integer with the property that $3a$ belongs to $S$, prove that then $a$ belongs to $S$ as well.

Let $ a_{1}, a_{2}...a_{n}$ be non-negative reals, not all zero. Show that that (a) The polynomial $ p(x) \equal{} x^{n} \minus{} a_{1}x^{n \minus{} 1} \plus{} ... \minus{} a_{n \minus{} 1}x \minus{} a_{n}$ has preceisely 1 positive real root $ R$. (b) let $ A \equal{} \sum_{i \equal{} 1}^n a_{i}$ and $ B \equal{} \sum_{i \equal{} 1}^n ia_{i}$. Show that $ A^{A} \leq R^{B}$.

Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{ 1,2,\ldots , n \}$ such that the sums of the different pairs are different integers not exceeding $n$?

Find the greatest possible distance between any two points inside or along the perimeter of an equilateral triangle with side length $2$. [i]Proposed by Alex Li[/i]

Robin is playing notes on an 88-key piano. He starts by playing middle C, which is actually the 40th lowest note on the piano (i.e. there are 39 notes lower than middle C). After playing a note, Robin plays with probability $\tfrac12$ the lowest note that is higher than the note he just played, and with probability $\tfrac12$ the highest note that is lower than the note he just played. What is the probability that he plays the highest note on the piano before playing the lowest note?

Let $ABC$ be a triangle and $O$ be its circumcircle. Let $A', B', C'$ be the midpoints of minor arcs $AB$, $BC$ and $CA$ respectively. Let $I$ be the center of incircle of $ABC$. If $AB = 13$, $BC = 14$ and $AC = 15$, what is the area of the hexagon $AA'BB'CC'$? Suppose $m \angle BAC = \alpha$ , $m \angle CBA = \beta$, and $m \angle ACB = \gamma$. [b]p10.[/b] Let the incircle of $ABC$ be tangent to $AB, BC$, and $AC$ at $J, K, L$, respectively. Compute the angles of triangles $JKL$ and $A'B'C'$ in terms of $\alpha$, $\beta$, and $\gamma$, and conclude that these two triangles are similar. [b]p11.[/b] Show that triangle $AA'C'$ is congruent to triangle $IA'C'$. Show that $AA'BB'CC'$ has twice the area of $A'B'C'$. [b]p12.[/b] Let $r = JL/A'C'$ and the area of triangle $JKL$ be $S$. Using the previous parts, determine the area of hexagon $AA'BB'CC'$ in terms of $ r$ and $S$. [b]p13.[/b] Given that the circumradius of triangle $ABC$ is $65/8$ and that $S = 1344/65$, compute $ r$ and the exact value of the area of hexagon $AA'BB'CC'$. PS. You had better use hide for answers.

Suppose an integer-valued function $f$ satisfies $$\sum_{k=1}^{2n+1}f(k)=\ln|2n+1|-4\ln|2n-1|\enspace\text{and}\enspace\sum_{k=0}^{2n}f(k)=4e^n-e^{n-1}$$ for all non-negative integers $n$. Determine $\sum_{n=0}^\infty\frac{f(n)}{2^n}$.

A doubly-indexed sequence $a_{m,n}$, for $m$ and $n$ nonnegative integers, is defined as follows: [list] [*]$a_{m,0}=0$ for all $m>0$ and $a_{0,0}=1$. [*]$a_{m,1}=0$ for all $m>1$, $a_{1,1}=1$, and $a_{0,1}=0$. [*]$a_{0,n}=a_{0,n-1}+a_{0,n-2}$ for all $n\geq 2$. [*]$a_{m,n}=a_{m,n-1}+a_{m,n-2}+a_{m-1,n-1}-a_{m-1,n-2}$ for all $m>0$, $n\geq 2$. [/list] Then there exists a unique value of $x$ so $\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\frac{a_{m,n}x^m}{3^{n-m}}=1$. Find $\lfloor 1000x^2 \rfloor$.

Chloe and Zoe are both students in Ms. Demeanor's math class. Last night they each solved half of the problems in their homework assignment alone and then solved the other half together. Chloe had correct answers to only $80\%$ of the problems she solved alone, but overall $88\%$ of her answers were correct. Zoe had correct answers to $90\%$ of the problems she solved alone. What was Zoe's overall percentage of correct answers? $\textbf{(A) }89\qquad\textbf{(B) }92\qquad\textbf{(C) }93\qquad\textbf{(D) }96\qquad\textbf{(E) }98$

Let $ABC$ be a triangle. Suppose $P,Q$ are on lines $AB, AC$ (on the same side of A) with $AP=AC$ and $AB=AQ$. Now suppose points $X,Y$ move along the sides $AB, AC$ of $ABC$ so that $XY || PQ$. Determine the locus of the circumcenters of the variable triangle $AXY$.

Prove that for any positive integers $m>n$, there is infinitely many positive integers $a,b$ such that set of prime divisors of $a^m+b^n$ is equal to set of prime divisors of $a^{2019}+b^{1398}$.

Paula the painter had just enough paint for $ 30$ identically sized rooms. Unfortunately, on the way to work, three cans of paint fell of her truck, so she had only enough paint for $ 25$ rooms. How many cans of paint did she use for the $ 25$ rooms? $ \textbf{(A)}\ 10 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 18 \qquad \textbf{(E)}\ 25$

Consider the real function defined by $f(x) =\frac{1}{|x + 3| + |x + 1| + |x - 2| + |x -5|}$ for all $x \in R$. a) Determine its maximum. b) Graphic representation.

Find all non-negative integer solutions of the equation \[2^x + 3^y = z^2 .\]

Sally the snail sits on the $3 \times 24$ lattice of points $(i, j)$ for all $1 \le i \le 3$ and $1 \le j \le 24$. She wants to visit every point in the lattice exactly once. In a move, Sally can move to a point in the lattice exactly one unit away. Given that Sally starts at $(2, 1)$, compute the number of possible paths Sally can take.

For some polynomial there is an infinite set its values, each of which takes at least at two integer points. Prove that there is at most one the integer value that a polynomial takes at exactly one integer point.

Some math team members decide to study at Cary Library after school. They walk $6$ blocks north, then $8$ blocks west to get there. If they walk $n$ blocks east from the library, they can buy boba from CoCo's. If CoCo's is the same distance from school as it is from the library, find $n$.

Let $ p $ be a natural number and let two partitions $ \mathcal{A} =\left\{ A_1,A_2,...,A_p\right\} ,\mathcal{B}=\left\{ B_1,B_2,...B_p\right\} $ of a finite set $ \mathcal{M} . $ Knowing that, whenever an element of $ \mathcal{A} $ doesn´t have any elements in common with another of $ \mathcal{B} , $ it holds that the number of elements of these two is greater than $ p, $ prove that $ \big| \mathcal{M}\big|\ge\frac{1}{2}\left( 1+p^2\right) . $ Can equality hold?

Let $O$ be a point interior to triangle $ABC$ such that $\angle BAO = 30^o$, $\angle CBO = 20^o$ and $\angle ABO = \angle ACO = 40^o$ . Knowing that triangle $ABC$ is not equilateral, find the measures of its interior angles.

Every $x, 0 \leq x \leq 1$, admits a unique representation $x = \sum_{j=0}^{\infty} a_j 2^{-j}$, where all the $a_j$ belong to $\{0, 1\}$ and infinitely many of them are $0$. If $b(0) = \frac{1+c}{2+c}, b(1) =\frac{1}{2+c},c > 0$, and \[f(x)=a_0 + \sum_{j=0}^{\infty}b(a_0) \cdots b(a_j) a_{j+1}\] show that $0 < f(x) -x < c$ for every $x, 0 < x < 1.$

How many ways are there to cover a $3\times 8$ rectangle with $12$ identical dominoes?

In terms of real parameter $a$ solve inequality: $\log _{a} {x} + \mid a+\log _{a} {x} \mid \cdot \log _{\sqrt{x}} {a} \geq a\log _{x} {a}$ in set of real numbers