Let $A$ be the sum of the first $2k+1$ positive odd integers, and let $B$ be the sum of the first $2k+1$ positive even integers. Show that $A+B$ is a multiple of $4k+3$.

Let $n$ be a positive integer. Does $n^2$ has more positive divisors of the form $4k+1$ or of the form $4k-1$?

For which digits $a$ do exist integers $n \geq 4$ such that each digit of $\frac{n(n+1)}{2}$ equals $a \ ?$

For any number $n\in\mathbb{N},n\ge 2$, denote by $P(n)$ the number of pairs $(a,b)$ whose elements are of positive integers such that \[\frac{n}{a}\in (0,1),\quad \frac{a}{b}\in (1,2)\quad \text{and}\quad \frac{b}{n}\in (2,3). \] $a)$ Calculate $P(3)$. $b)$ Find $n$ such that $P(n)=2002$.

For every positive integers $n,m$, show that there exist two sets $A,B$ which satisfy the following. [list] [*]$A$ is a set of $n$ successive positive integers, and $B$ is a set of $m$ successive positive integers. [*]$A\cup B = \phi$ [*]For every $a\in A$ and $b\in B$, $a$ and $b$ are not relatively prime. [/list]

At the wedding of two Bulgarian nationals in mathematics, every guest who gave a positive integer \(n\), not yet given by another guest, which divides \(3^n-3\) but does not divide \(2^n-2\), received a prize. If there were an infinite number of guests, would the newlyweds theoretically need an infinite number of gifts?

$S=\sin{64x}+\sin{65x}$ and $C=\cos{64x}+\cos{65x}$ are both rational for some $x$. Prove, that for one of these sums both summands are rational too.

(a) Let $x$ and $y$ be two positive integers such that $\sqrt{x} +\sqrt{y}$ is an integer. Show that $\sqrt{x}$ and $\sqrt{y}$ are both integers. (b) Find all positive integers $x$ and $y$ such that $\sqrt{x} +\sqrt{y}=\sqrt{2007}$.

Finite set $A$ has such property: every six its distinct elements’ sum isn’t divisible by $6$. Does there exist such set $A$ consisting of $11$ distinct natural numbers?

Let $T_n$ denotes the least natural such that $$n\mid 1+2+3+\cdots +T_n=\sum_{i=1}^{T_n} i$$ Find all naturals $m$ such that $m\ge T_m$. [i]Proposed by Nicolás De la Hoz [/i]

Find all prime numbers $p$ for which the number $p^2+11$ has less than $11$ divisors.

Is there a number $n$ such that one can write $n$ as the sum of $2017$ perfect squares and (with at least) $2017$ distinct ways?

Let $X$ be the sum of all divisors of the number $(3\cdot 2009)^{((2\cdot 2009)^{2009}-1)}$ . Find the last digit of $X$.

Let $n+1, n \geq 1$ positive integers be formed by taking the product of $n$ given prime numbers (a prime number can appear several times or also not appear at all in a product formed in this way.) Prove that among these $n+1$ one can find some numbers whose product is a perfect square.

A program works as follows. If the input is given a natural number $n$ ($n \ge 2$), then the program consecutively performs the following procedure: it determines the greatest proper divisor of the number $ n$ (that is, different from $1$ and $n$) and subtracts it from the number $n$, then applies again the same procedure to the obtained result and so on. If the program cannot find any proper divisor of the given number at a step, then it stops and outputs the total number $m$ of procedures performed (this number can be equal to $0$). The input was given the number $n = 13^{13}$. Determine the respective number $m$ at the output.

Find all positive integers $k>1$, such that there exist positive integer $n$, such that the number $A=17^{18n}+4.17^{2n}+7.19^{5n}$ is product of $k$ consecutive positive integers.

Determine the maximum integer $ n $ such that for each positive integer $ k \le \frac{n}{2} $ there are two positive divisors of $ n $ with difference $ k $.

For any natural number $m$, we denote by $\phi (m)$ the number of integers $k$ relatively prime to $m$ and satisfying $1 \le k \le m$. Determine all positive integers $n$ such that for every integer $k > n^2$, we have $n | \phi (nk + 1)$. (Daniel Harrer)

[b]p1.[/b] A matrix is a rectangular array of numbers. For example, $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $\begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix}$ are $2 \times 2$ matrices. A [i]saddle [/i] point in a matrix is an entry which is simultaneously the smallest number in its row and the largest number in its column. a. Write down a $2 \times 2$ matrix which has a saddle point, and indicate which entry is the saddle point. b. Write down a $2 \times 2$ matrix which has no saddle point c. Prove that a matrix of any size, all of whose entries are distinct, can have at most one saddle point. [b]p2.[/b] a. Find four different pairs of positive integers satisfying the equation $\frac{7}{m}+\frac{11}{n}=1$. b. Prove that the solutions you have found in part (a) are all possible pairs of positive integers satisfying the equation $\frac{7}{m}+\frac{11}{n}=1$. [b]p3.[/b] Let $ABCD$ be a quadrilateral, and let points $M, N, O, P$ be the respective midpoints of sides $AB$, $BC$, $CD$, $DA$. a. Show, by example, that it is possible that $ABCD$ is not a parallelogram, but $MNOP$ is a square. Be sure to prove that your construction satisfies all given conditions. b. Suppose that $MO$ is perpendicular to $NP$. Prove that $AC = BD$. [b]p4.[/b] A [i]Pythagorean triple[/i] is an ordered collection of three positive integers $(a, b, c)$ satisfying the relation $a^2 + b^2 = c^2$. We say that $(a, b, c)$ is a [i]primitive [/i] Pythagorean triple if $1$ is the only common factor of $a, b$, and $c$. a. Decide, with proof, if there are infinitely many Pythagorean triples. b. Decide, with proof, if there are infinitely many primitive Pythagorean triples of the form $(a, b, c)$ where $c = b + 2$. c. Decide, with proof, if there are infinitely many primitive Pythagorean triples of the form $(a, b, c)$ where $c = b + 3$. [b]p5.[/b] Let $x$ and $y$ be positive real numbers and let $s$ be the smallest among the numbers $\frac{3x}{2}$,$\frac{y}{x}+\frac{1}{x}$ and $\frac{3}{y}$. a. Find an example giving $s > 1$. b. Prove that for any positive $x$ and $y,s <2$. c. Find, with proof, the largest possible value of $s$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Define the Fibanocci sequence recursively by $F_1=1$, $F_2=1$ and $F_{i+2} = F_i + F_{i+1}$ for all $i$. Prove that for all integers $b,c>1$, there exists an integer $n$ such that the sum of the digits of $F_n$ when written in base $b$ is greater than $c$. [i]Proposed by Ryan Alweiss[/i]

Show that the number of ordered pairs $(a, b)$ of positive integers with lowest common multiple $n$ is the same as the number of positive divisors of $n^2$.

Show that the equation $14x^2 +15y^2 = 7^{2000}$ has no integer solutions.

[b]p1.[/b] Compute the sum of sharp angles at all five nodes of the star below. ( [url=http://www.math.msu.edu/~mshapiro/NewOlympiad/Olymp2009/10_12_2009.pdf]figure missing[/url] ) [b]p2.[/b] Arrange the integers from $1$ to $15$ in a row so that the sum of any two consecutive numbers is a perfect square. In how many ways this can be done? [b]p3.[/b] Prove that if $p$ and $q$ are prime numbers which are greater than $3$ then $p^2 -q^2$ is divisible by $ 24$. [b]p4.[/b] A city in a country is called Large Northern if comparing to any other city of the country it is either larger or farther to the North (or both). Similarly, a city is called Small Southern. We know that in the country all cities are Large Northern city. Show that all the cities in this country are simultaneously Small Southern. [b]p5.[/b] You have four tall and thin glasses of cylindrical form. Place on the flat table these four glasses in such a way that all distances between any pair of centers of the glasses' bottoms are equal. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the number of ordered $2012$-tuples of integers $(x_1, x_2, . . . , x_{2012})$, with each integer between $0$ and $2011$ inclusive, such that the sum $x_1 + 2x_2 + 3x_3 + · · · + 2012x_{2012}$ is divisible by $2012$.

[u]Round 5[/u] [b]p13.[/b] Let $\{a\} _{n\ge 1}$ be an arithmetic sequence, with $a_ 1 = 0$, such that for some positive integers $k$ and $x$ we have $a_{k+1} = {k \choose x}$, $a_{2k+1} ={k \choose {x+1}}$ , and $a_{3k+1} ={k \choose {x+2}}$. Let $\{b\}_{n\ge 1}$ be an arithmetic sequence of integers with $b_1 = 0$. Given that there is some integer $m$ such that $b_m ={k \choose x}$, what is the number of possible values of $b_2$? [b]p14.[/b] Let $A = arcsin \left( \frac14 \right)$ and $B = arcsin \left( \frac17 \right)$. Find $\sin(A + B) \sin(A - B)$. [b]p15.[/b] Let $\{f_i\}^{9}_{i=1}$ be a sequence of continuous functions such that $f_i : R \to Z$ is continuous (i.e. each $f_i$ maps from the real numbers to the integers). Also, for all $i$, $f_i(i) = 3^i$. Compute $\sum^{9}_{k=1} f_k \circ f_{k-1} \circ ... \circ f_1(3^{-k})$. [u]Round 6[/u] [b]p16.[/b] If $x$ and $y$ are integers for which $\frac{10x^3 + 10x^2y + xy^3 + y^4}{203}= 1134341$ and $x - y = 1$, then compute $x + y$. [b]p17.[/b] Let $T_n$ be the number of ways that n letters from the set $\{a, b, c, d\}$ can be arranged in a line (some letters may be repeated, and not every letter must be used) so that the letter a occurs an odd number of times. Compute the sum $T_5 + T_6$. [b]p18.[/b] McDonald plays a game with a standard deck of $52$ cards and a collection of chips numbered $1$ to $52$. He picks $1$ card from a fully shuffled deck and $1$ chip from a bucket, and his score is the product of the numbers on card and on the chip. In order to win, McDonald must obtain a score that is a positive multiple of $6$. If he wins, the game ends; if he loses, he eats a burger, replaces the card and chip, shuffles the deck, mixes the chips, and replays his turn. The probability that he wins on his third turn can be written in the form $\frac{x^2 \cdot y}{z^3}$ such that $x, y$, and $z$ are relatively prime positive integers. What is $x + y + z$? (NOTE: Use Ace as $1$, Jack as $11$, Queen as $12$, and King as $13$) [u]Round 7[/u] [b]p19.[/b] Let $f_n(x) = ln(1 + x^{2^n}+ x^{2^{n+1}}+ x^{3\cdot 2^n})$. Compute $\sum^{\infty}_{k=0} f_{2k} \left( \frac12 \right)$. [b]p20.[/b] $ABCD$ is a quadrilateral with $AB = 183$, $BC = 300$, $CD = 55$, $DA = 244$, and $BD = 305$. Find $AC$. [b]p21.[/b] Define $\overline{xyz(t + 1)} = 1000x + 100y + 10z + t + 1$ as the decimal representation of a four digit integer. You are given that $3^x5^y7^z2^t = \overline{xyz(t + 1)}$ where $x, y, z$, and t are non-negative integers such that $t$ is odd and $0 \le x, y, z,(t + 1) \le 9$. Compute$3^x5^y7^z$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2782822p24445934]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].