Let $P(x) = x^{16}-x^{15}+·...-x+ 1$, and let p be a prime such that $p-1$ is divisible by $34$ ($p = 103$ is an example). How many integers a between $1$ and $ p-1$ inclusive satisfy the property that $P(a)$ is divisible by $p$?

Alice and Bob play the following game. They alternate selecting distinct nonzero digits (from $1$ to $9$) until they have chosen seven such digits, and then consider the resulting seven-digit number by concatenating the digits in the order selected, with the seventh digit appearing last (i.e. $\overline{A_1B_2A_3B_4A_6B_6A_7}$). Alice wins if and only if the resulting number is the last seven decimal digits of some perfect seventh power. Please determine which player has the winning strategy.

Determine all positive integers $n$ for which there exists a polynomial $f(x)$ with real coefficients, with the following properties: (1) for each integer $k$, the number $f(k)$ is an integer if and only if $k$ is not divisible by $n$; (2) the degree of $f$ is less than $n$. [i](Hungary) Géza Kós[/i]

In the country of Postonia, one wants to have only two values of stamps. These values should be integers greater than $1$ with the difference $2$, and should have the property that one can combine the stamps for any postage which is greater than or equal to the sum of these two values. What values can be chosen?

Consider the equation $x^3 + y^3 + z^3 = 2$. a) Prove that it has infinitely many integer solutions $x,y,z$. b) Determine all integer solutions $x, y, z$ with $|x|, |y|, |z| \leq 28$.

Let $p > 10$ be prime. Prove that there are positive integers $m$ and $n$ with $m + n < p$ exist for which $p$ is a divisor of $5^m7^n-1$.

Let $k$ be a positive integer and $a_1,a_2,\dots$ be an infinite sequence of positive integers such that \[a_ia_{i+1} \mid k-a_i^2\] for all integers $i \ge 1$. Prove that there exists a positive integer $M$ such that $a_n=a_{n+1}$ for all integers $n \ge M$.

[b]p1.[/b] Each point in the interior and on the boundary of a square of side $2$ inches is colored either red or blue. Prove that there exists at least one pair of points of the same color whose distance apart is not less than $-\sqrt5$ inches. [b]p2.[/b] $ABC$ is an equilateral triangle of altitude $h$. A circle with center $0$ and radius $h$ is tangent to side $AB$ at $Z$ and intersects side $AC$ in point $X$ and side $BC$ in point $Y$. Prove that the circular arc $XZY$ has measure $60^o$. [img]https://cdn.artofproblemsolving.com/attachments/b/e/ac70942f7a14cd0759ac682c3af3551687dd69.png[/img] [b]p3.[/b] Find all of the real and complex solutions (if any exist) of the equation $x^7 + 7^7 = (x + 7)^7$ [b]p4.[/b] The four points $A, B, C$, and $D$ are not in the same plane. Given that the three angles, angle $ABC$, angle $BCD$, and angle $CDA$, are all right angles, prove that the fourth angle, angle $DAB$, of this skew quadrilateral is acute. [b]p5.[/b] $A, B, C$ and $D$ are four positive whole numbers with the following properties: (i) each is less than the sum of the other three, and (ii) each is a factor of the sum of the other three. Prove that at least two of the numbers must be equal. (An example of four such numbers: $A = 4$, $B = 4$, $C = 2$, $D = 2$.) [b]p6.[/b] $S$ is a set of six points and $L$ is a set of straight line segments connecting certain pairs of points in $S$ so that each point of $S$ is connected with at least four of the other points. Let $A$ and $B$ denote two arbitrary points of $S$. Show that among the triangles having sides in $L$ and vertices in $S$ there are two with the properties: (i) The two triangles have no common vertex. (ii) $A$ is a vertex of one of the triangles, and $B$ is a vertex of the other. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ x \in \mathbb{Q}^+$. Prove that there exits $\alpha_1,\alpha_2,...,\alpha_k \in \mathbb{N}$ and pairwe distinct such that \[x= \sum_{i=1}^{k} \frac{1}{\alpha_i}\]

Every integer is to be coloured blue, green, red, or yellow. Can this be done in such a way that if $a, b, c, d$ are not all $0$ and have the same colour, then $3a-2b \neq 2c-3d$? [size=85][color=#0000FF][Mod edit: Question fixed][/color][/size]

If $\frac{1}{a} + \frac{1}{b} = \frac{1}{c}$, where $a,b,c$ are positive integers with no common factor, prove that $(a +b)$ is a square.

If there is a natural number $n$ such that the number $n!$ has exactly $11$ zeros at the end? (With $n!$ is denoted the number $1\cdot 2\cdot 3 \cdot ... (n - )1 \cdot n$).

Check if there exists positive integers $ a, b$ and prime number $p$ such that $a^3-b^3=4p^2$

Three mutually tangent spheres each with radius $5$ sit on a horizontal plane. A triangular pyramid has a base that is an equilateral triangle with side length $6$, has three congruent isosceles triangles for vertical faces, and has height $12$. The base of the pyramid is parallel to the plane, and the vertex of the pyramid is pointing downward so that it is between the base and the plane. Each of the three vertical faces of the pyramid is tangent to one of the spheres at a point on the triangular face along its altitude from the vertex of the pyramid to the side of length $6$. The distance that these points of tangency are from the base of the pyramid is $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [asy] size(200); defaultpen(linewidth(0.8)); pair X=(-.6,.4),A=(-.4,2),B=(-.7,1.85),C=(-1.1,2.05); picture spherex; filldraw(spherex,unitcircle,white); draw(spherex,(-1,0)..(-.2,-.2)..(1,0)^^(0,1)..(-.2,-.2)..(0,-1)); add(shift(-0.5,0.6)*spherex); filldraw(X--A--C--cycle,gray); draw(A--B--C^^X--B); add(shift(-1.5,0.2)*spherex); add(spherex); [/asy]

For a fixed integer $k$, determine all polynomials $f(x)$ with integer coefficients such that $f(n)$ divides $(n!)^k$ for every positive integer $n$.

Let $p$ and $q$ be given prime numbers and $S$ be a subset of ${1,2,3,\dots ,p-2,p-1}$. Prove that the number of elements in the set $A=\{ (x_1,x_2,…,x_q ):x_i\in S,\sum_{i=1}^q x_i \equiv 0(mod\: p)\}$ is multiple of $q$.

Is it possible to find $p,q,r\in\mathbb Q$ such that $p+q+r=0$ and $pqr=1$? [i]Proposed by Máté Weisz, Cambridge[/i]

Let us denote by $b(n)$ the number of ways to represent $n$ in the form $$n = a_0+a_1 \cdot 2 +a_2 \cdot 2^2+...+ a_k \cdot 2^k,$$ where the coefficients at, $r = 1$,$2$,$...$, $k$ can be equal to $0$, $1$ or $2$. Find $b(1996)$.

Suppose $p,q,r$ are three distinct primes such that $rp^3+p^2+p=2rq^2+q^2+q$. Find all possible values of $pqr$.

The remainder of $1991^{2000}$ module $10^6$ is________.

Find all positive integer solutions $ x, y, z$ of the equation $ 3^x \plus{} 4^y \equal{} 5^z.$

Find all pairs of positive integers $(x, y)$, such that $x^3+9x^2-11x-11=2^y$.

Calculate the product of all positive integers less than $100$ and having exactly three positive divisors. Show that this product is a square.

Assume that $ \alpha$ and $ \beta$ are two roots of the equation: $ x^2\minus{}x\minus{}1\equal{}0$. Let $ a_n\equal{}\frac{\alpha^n\minus{}\beta^n}{\alpha\minus{}\beta}$, $ n\equal{}1, 2, \cdots$. (1) Prove that for any positive integer $ n$, we have $ a_{n\plus{}2}\equal{}a_{n\plus{}1}\plus{}a_n$. (2) Find all positive integers $ a$ and $ b$, $ a<b$, satisfying $ b \mid a_n\minus{}2na^n$ for any positive integer $ n$.

Let $p$ be a prime number. For integers $r, s$ such that $rs(r^2 - s^2)$ is not divisible by $p$, let $f(r, s)$ denote the number of integers $n \in \{1, 2, \ldots, p - 1\}$ such that $\{rn/p\}$ and $\{sn/p\}$ are either both less than $1/2$ or both greater than $1/2$. Prove that there exists $N > 0$ such that for $p \geq N$ and all $r, s$, \[ \left\lceil \frac{p-1}{3} \right\rceil \le f(r, s) \le \left\lfloor \frac{2(p-1)}{3} \right\rfloor. \]