Let $ABCD$ be a square and consider the points $K\in AB, L\in BC,$ and $M\in CD$ such that $\Delta KLM$ is a right isosceles triangle, with the right angle at $L$. Prove that the lines $AL$ and $DK$ are perpendicular to each other.

[b](a)[/b] For every positive integer $n$ prove that \[1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2} <2\] [b](b)[/b] Let $X=\{1, 2, 3 ,\ldots, n\} \ ( n \geq 1)$ and let $A_k$ be non-empty subsets of $X \ (k=1,2,3, \ldots , 2^n -1).$ If $a_k$ be the product of all elements of the set $A_k,$ prove that \[\sum_{i=1}^{m} \sum_{j=1}^m \frac{1}{a_i \cdot j^2} <2n+1\]

Determine all $6$-tuples $(p,q,r,x,y,z)$ where $p,q,r$ are prime, and $x,y,z$ natural numbers such that $p^{2x}=q^yr^z+1$.

[b]p1.[/b] Two trains, $A$ and $B$, travel between cities $P$ and $Q$. On one occasion $A$ started from $P$ and $B$ from $Q$ at the same time and when they met $A$ had travelled $120$ miles more than $B$. It took $A$ four $(4)$ hours to complete the trip to $Q$ and B nine $(9)$ hours to reach $P$. Assuming each train travels at a constant speed, what is the distance from $P$ to $Q$? [b]p2.[/b] If $a$ and $b$ are integers, $b$ odd, prove that $x^2 + 2ax + 2b = 0$ has no rational roots. [b]p3.[/b] A diameter segment of a set of points in a plane is a segment joining two points of the set which is at least as long as any other segment joining two points of the set. Prove that any two diameter segments of a set of points in the plane must have a point in common. [b]p4.[/b] Find all positive integers $n$ for which $\frac{n(n^2 + n + 1) (n^2 + 2n + 2)}{2n + 1}$ is an integer. Prove that the set you exhibit is complete. [b]p5.[/b] $A, B, C, D$ are four points on a semicircle with diameter $AB = 1$. If the distances $\overline{AC}$, $\overline{BC}$, $\overline{AD}$, $\overline{BD}$ are all rational numbers, prove that $\overline{CD}$ is also rational. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n \ge 3$ be an integer. Prove that there exist positive integers $\ge 2$, $a_1,a_2,..,a_n$, such that $a_1 a_2 ... \widehat{a_i}... a_n \equiv 1$ (mod $a_i$), for $i = 1,..., n$. Here $\widehat{a_i}$ means the term $a_i$ is omitted.

Let $S = \{a_1, \ldots, a_n \}$ be a finite set of positive integers of size $n \ge 1$, and let $T$ be the set of all positive integers that can be expressed as sums of perfect powers (including $1$) of distinct numbers in $S$, meaning \[ T = \left\{ \sum_{i=1}^n a_i^{e_i} \mid e_1, e_2, \dots, e_n \ge 0 \right\}. \] Show that there is a positive integer $N$ (only depending on $n$) such that $T$ contains no arithmetic progression of length $N$. [i]Yang Liu[/i]

Let $p$ be a prime and $n$ a positive integer below $100$. What’s the probability that $p$ divides $n$?

Let $2=p_1<p_2<\ldots<p_n<\ldots$ be all prime numbers. Prove that for any positive integer $n \geq 3$ there exist at least $p_n+n-1$ prime numbers, that do not exceed $p_1p_2\ldots p_n$ [i]I. Voronovich[/i]

Find all right triangles with integer sides and inradius 6.

Prove that if the function $ ax^2 + bx + c $ takes an integer value for every integer value of the variable $ x $, then $ 2a $, $ a + b $, $ c $ are integers and vice versa.

Let $P(a)$ be the largest prime positive divisor of $a^2 + 1$. Prove that exist infinitely many positive integers $a, b, c$ such that $P(a)=P(b)=P(c)$. [i]A. Golovanov[/i]

The arithmetic mean of two different positive integers $x,y$ is a two digit integer. If one interchanges the digits, the geometric mean of these numbers is archieved. a) Find $x,y$. b) Show that a)'s solution is unique up to permutation if we work in base $g=10$, but that there is no solution in base $g=12$. c) Give more numbers $g$ such that a) can be solved; give more of them such that a) can't be solved, too.

The sequence $ \{x_{n}\}$ is defined by $ x_{1} \equal{} 2,x_{2} \equal{} 12$, and $ x_{n \plus{} 2} \equal{} 6x_{n \plus{} 1} \minus{} x_{n}$, $ (n \equal{} 1,2,\ldots)$. Let $ p$ be an odd prime number, let $ q$ be a prime divisor of $ x_{p}$. Prove that if $ q\neq2,3,$ then $ q\geq 2p \minus{} 1$.

Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.

For a positive integer $n$, let $\sigma (n)$ be the sum of the divisors of $n$ (for example $\sigma (10) = 1 + 2 + 5 + 10 = 18$). For how many $n \in \{1, 2,. .., 100\}$, do we have $\sigma (n) < n+ \sqrt{n}$?

Prove or disprove that there exists a positive real $u$ such that $\lfloor u^n\rfloor-n$ is an even integer for all positive integers $n$.

Let $a,b$ be positive real numbers. For any positive integer $n$, denote by $x_n$ the sum of digits of the number $[an+b]$ in it's decimal representation. Show that the sequence $(x_n)_{n\ge 1}$ contains a constant subsequence. [i]Laurentiu Panaitopol[/i]

The following sentece is written on a board: [center]The equation $x^2-824x+\blacksquare 143=0$ has two integer solutions.[/center] Where $\blacksquare$ represents algarisms of a blurred number on the board. What are the possible equations originally on the board?

Prove that there are infinitely many positive integers $n$ such that $(n!)^{n+2015}$ divides $(n^{2})!$.

Find all pairs of prime numbers $(p,q)$ for which \[2^p = 2^{q-2} + q!.\]

Prove that there is no polynomial $P$ with integer coefficients such that $P\left(\sqrt[3]5+\sqrt[3]{25}\right)=5+\sqrt[3]5$.

Let $f$ be the function of the set of positive integers into itself, defined by $f(1) = 1$, $f(2n) = f(n)$ and $f(2n + 1) = f(n) + f(n + 1)$. Show that, for any positive integer $n$, the number of positive odd integers m such that $f(m) = n$ is equal to the number of positive integers[color=#0000FF][b] less or equal to [/b][/color]$n$ and coprime to $n$. [color=#FF0000][mod: the initial statement said less than $n$, which is wrong.][/color]

Let $ n$ be a positive integer. Show that the numbers \[ \binom{2^n \minus{} 1}{0},\; \binom{2^n \minus{} 1}{1},\; \binom{2^n \minus{} 1}{2},\; \ldots,\; \binom{2^n \minus{} 1}{2^{n \minus{} 1} \minus{} 1}\] are congruent modulo $ 2^n$ to $ 1$, $ 3$, $ 5$, $ \ldots$, $ 2^n \minus{} 1$ in some order. [i]Proposed by Duskan Dukic, Serbia[/i]

(a) Prove that for every positive integer $m$ there exists an integer $n\ge m$ such that $$\left \lfloor \frac{n}{1} \right \rfloor \cdot \left \lfloor \frac{n}{2} \right \rfloor \cdots \left \lfloor \frac{n}{m} \right \rfloor =\binom{n}{m} \\\\\\\\\\\\\\\ (*)$$ (b) Denote by $p(m)$ the smallest integer $n \geq m$ such that the equation $ (*)$ holds. Prove that $p(2018) = p(2019).$ Remark: For a real number $x,$ we denote by $\left \lfloor x \right \rfloor$ the largest integer not larger than $x.$

The sequence $ (a_n)$ is defined by $ a_1\equal{}a_2\equal{}a_3\equal{}1$ and $ a_{n\plus{}1}a_{n\minus{}2}\minus{}a_n a_{n\minus{}1}\equal{}2$ for all $ n \ge 3.$ Prove that $ a_n$ is a positive integer for all $ n \ge 1$.