Find all quadruples $(p, q, r, n)$ of prime numbers $p, q, r$ and positive integer numbers $n$, such that $$p^2 = q^2 + r^n$$ (Walther Janous)

Let $D$ be a point on the side $AB$ of the triangle $ABC$ such that $BD = CD$, and let the points $E$ on the side $BC$ and $F$ on the extension $AC$ beyond the point $C$ be such that $EF\parallel CD$. The lines $AE$ and $CD$ intersect at the point $G$. Prove that $BC$ is the bisector of the angle $FBG$.

Let $a, b, c$ be positive reals such that $a + b + c = 1$ and $P(x) = 3^{2005}x^{2007 }- 3^{2005}x^{2006} - x^2$. Prove that $P(a) + P(b) + P(c) \le -1$.

Two circles with different radius $O_1$ and $O_2$ are both tangent to a larger circle $O$, tangent points are $S,T$. Note that intersections of $O_1$ and $O_2$ are $M,N$, prove that the sufficient and necessary condition of $OM\perp MN$ is $S,N,T$ are colinear.

Suppose that the roots $p,q$ of the equation $x^2-x+c=0$ where $c \in \mathbb{R}$, are rational numbers. Prove that the roots of the equation $x^2+px-q=0$ are also rational numbers.

A wheel consists of a fixed circular disk and a mobile circular ring. On the disk the numbers $1, 2, 3, \ldots ,N$ are marked, and on the ring $N$ integers $a_1,a_2,\ldots ,a_N$ of sum $1$ are marked. The ring can be turned into $N$ different positions in which the numbers on the disk and on the ring match each other. Multiply every number on the ring with the corresponding number on the disk and form the sum of $N$ products. In this way a sum is obtained for every position of the ring. Prove that the $N$ sums are different.

We are given an equilateral triangle ABC with the length of its side equal to $1$. There are $n-1$ points on each side of the triangle $ABC$ that equally divide the side into $n$ segments. We draw all possible lines that pass through any two of all those $3(n-1)$ points such that they are parallel to one of three sides of triangle $ABC$. All such lines divide triangle $ABC$ into some lesser triangles whose vertices are called [i]nodes[/i]. We assign a real number for each [i]node[/i] such that the following conditions are satisfied: (I) real numbers $a,b,c$ are assigned to $A,B,C$ respectively; (II) for any rhombus that is consisted of two lesser triangles that share a common side, the sum of the numbers of vertices on its one diagonal is equal to that of vertices on the other diagonal. 1) Find the minimum distance between the [i]node[/i] with the maximal number to the [i]node[/i] with the minimal number; 2) Denote by $S$ the sum of the numbers of all [i]nodes[/i], find $S$.

[b]2.[/b] Show that the series $\sum_{n=1}^{\infty}\frac{1}{n}sin(asin(\frac{2n\pi}{N}))e^{bcos(\frac{2n\pi}{N})}$ is convergent for every positive integer N and any real numbers a and b. [b](S. 25)[/b]

Let $19$ integer numbers are given. Let Hamza writes on the paper the greatest common divisor for each pair of numbers. It occurs that the difference between the biggest and smallest numbers written on the paper is less than $180$. Prove that not all numbers on the paper are different.

Let $a_0, a_1, a_2, ...$ be a sequence such that: $a_0 = 2$; each $a_n = 2$ or $3; a_n =$the number of $3$s between the $n$th and $n+1$th $2$ in the sequence. So the sequence starts: $233233323332332 ...$ . Show that we can find $\alpha$ such that $a_n = 2$ iff $n = [\alpha m]$ for some integer $m \geq 0$.

Sam dumps tea for $6$ hours at a constant rate of $60$ tea crates per hour. Eddie takes $4$ hours to dump the same amount of tea at a different constant rate. How many tea crates does Eddie dump per hour? [i]Proposed by Samuel Tsui[/i] [hide=Solution] [i]Solution.[/i] $\boxed{90}$ Sam dumps a total of $6 \cdot 60 = 360$ tea crates and since it takes Eddie $4$ hours to dump that many he dumps at a rate of $\dfrac{360}{4}= \boxed{90}$ tea crates per hour. [/hide]

$\triangle ABC$ has $AB=5,BC=6,$ and $AC=7.$ Let $M$ be the midpoint of $BC,$ and let the circumcircle of $\triangle ABM$ intersect $AC$ at $N.$ If the length of segment $MN$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a,b$ find $a+b.$ [i]Proposed by Alex Li[/i]

Find all positive integers $n$ for which there exists an integer multiple of $2022$ such that the sum of the squares of its digits is equal to $n$.

Suppose that ${x_1, x_2, \dots , x_n}$ are positive integers for which $x_1 + x_2 + \cdots+ x_n = 2(n + 1)$. Show that there exists an integer $r$ with $0 \leq r \leq n - 1$ for which the following $n - 1$ inequalities hold: \[x_{r+1} + \cdots + x_{r+i} \leq 2i+ 1, \qquad \qquad \forall i, 1 \leq i \leq n - r; \] \[x_{r+1} + \cdots + x_n + x_1 + \cdots+ x_i \leq 2(n - r + i) + 1, \qquad \qquad \forall i, 1 \leq i \leq r - 1.\] Prove that if all the inequalities are strict, then $r$ is unique and that otherwise there are exactly two such $r.$

Let $ k > 1$ be an integer, and consider the infinite array given by the integer lattice in the first quadrant of the plane, filled with real numbers. The array is said to be constant if all its elements are equal in value. The array is said to be $ k$-balanced if it is non-constant, and the sums of the elements of any $ k\times k$ sub-square have a constant value $ v_k$. An array which is both $ p$-balanced and $ q$-balanced will be said to be $ (p, q)$-balanced, or just doubly-balanced, if there is no confusion as to which $ p$ and $ q$ are meant. If $p, q$ are relatively prime, the array is said to be co-prime. We will call $ (M\times N)$-seed a $ M \times N$ array, anchored with its lower left corner in the origin of the plane, which extended through periodicity in both dimensions in the plane results into a $ (p, q)$-balanced array; more precisely, if we denote the numbers in the array by $ a_{ij}$ , where $ i, j$ are the coordinates of the lower left corner of the unit square they lie in, we have, for all non-negative integers $ i, j$ \[ a_{i \plus{} M,j} \equal{} a_{i,j} \equal{} a_{i,j \plus{} N}\] (a) Prove that $ q^2v_p \equal{} p^2v_q$ for a $ (p, q)$-balanced array. (b) Prove that more than two different values are used in a co-prime $ (p,q)$-balanced array. Show that this is no longer true if $ (p, q) > 1$. (c) Prove that any co-prime $ (p, q)$-balanced array originates from a seed. (d) Show there exist $ (p, q)$-balanced arrays (using only three different values) for arbitrary values $ p, q$. (e) Show that neither a $ k$-balanced array, nor a $ (p, q)$-balanced array if $ (p, q) > 1$, need originate from a seed. (f) Determine the minimal possible value $ T$ for a square $ (T\times T)$-seed resulting in a co-prime $ (p, q)$-balanced array, when $p,q$ are both prime. (g) Show that for any relatively prime $ p, q$ there must exist a co-prime $ (p, q)$-balanced array originating from a square $ (T\times T)$-seed, with no lesser $ (M\times N)$-seed available ($ M\leq T, N\leq T$ and $MN< T^2$). [i]Dan Schwarz[/i]

A line $ x \equal{} k$ intersects the graph of $ y \equal{} \log_5{x}$ and the graph of $ y \equal{} \log_5{(x \plus{} 4)}$. The distance between the points of intersection is $ 0.5$. Given that $ k \equal{} a \plus{} \sqrt{b}$, where $ a$ and $ b$ are integers, what is $ a \plus{} b$? $ \textbf{(A)}\ 6\qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 9\qquad \textbf{(E)}\ 10$

Do there exist infinitely many triples of positive integers $(a, b, c)$ such that $a$, $b$, $c$ are pairwise coprime, and $a! + b! + c!$ is divisible by $a^2 + b^2 + c^2$? [i]Proposed by Anzo Teh Zhao Yang[/i]

Mr. Squash bought a large parking lot in Utah, which has an area of $600$ square meters. A car needs $6$ square meters of parking space while a bus needs $30$ square meters of parking space. Mr. Squash charges $\$2.50$ per car and $\$7.50$ per bus, but Mr. Squash can only handle at most $60$ vehicles at a time. Find the ordered pair $(a,b)$ where $a$ is the number of cars and $b$ is the number of buses that maximizes the amount of money Mr. Squash makes. [i]Proposed by Nathan Cho[/i]

Let $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$, $(x_4, y_4)$, and $(x_5, y_5)$ be the vertices of a regular pentagon centered at $(0, 0)$. Compute the product of all positive integers k such that the equality $x_1^k+x_2^k+x_3^k+x_4^k+x_5^k=y_1^k+y_2^k+y_3^k+y_4^k+y_5^k$ must hold for all possible choices of the pentagon.

For which integers $k$ does there exist a function $f : N \to Z$ such that $f(1995) =1996$ and $f(xy) = f(x)+ f(y)+k f(gcd(x,y))$ for all $x,y \in N$?

Prove that there exist natural numbers $n_k, m_k, k=0,1,2,\ldots$, such that the numbers $n_k+m_k, k=1,2,\ldots$ are pairwise distinct primes and the set of linear combination of the polynomials $x^{n_k}y^{m_k}$ is dense in $C([0,1] \times [0,1])$ under the supremum norm. (translated by Miklós Maróti)

Given is a finite set of points $M$ and an equilateral triangle $\Delta$ in the plane. It is known that for any subset $M' \subset M$, which has no more than $9$ points, can be covered by two translations of the triangle $\Delta$. Prove that the entire set $M$ can be covered by two translations of $\Delta$.

Let $ABCD$ be a convex quadrilateral. The lines parallel to $AD$ and $CD$ through the orthocentre $H$ of $ABC$ intersect $AB$ and $BC$ Crespectively at $P$ and $Q$. prove that the perpendicular through $H$ to th eline $PQ$ passes through th eorthocentre of triangle $ACD$

Represent $\frac{1}{2021}$ as a difference of two irreducible fractions with smaller denominators. [i](Proposed by Bogdan Rublov)[/i]

For a particular peculiar pair of dice, the probabilities of rolling 1, 2, 3, 4, 5 and 6 on each die are in the ratio $ 1: 2: 3: 4: 5: 6$. What is the probability of rolling a total of 7 on the two dice? $ \textbf{(A) } \frac 4{63} \qquad \textbf{(B) } \frac 18 \qquad \textbf{(C) } \frac 8{63} \qquad \textbf{(D) } \frac 16 \qquad \textbf{(E) } \frac 27$