A sequence of three real numbers forms an arithmetic progression with a first term of $ 9$. If $ 2$ is added to the second term and $ 20$ is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term of the geometric progression? $ \textbf{(A)}\ 1\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 36\qquad \textbf{(D)}\ 49\qquad \textbf{(E)}\ 81$

Call admissible a set $A$ of integers that has the following property: If $x,y \in A$ (possibly $x=y$) then $x^2+kxy+y^2 \in A$ for every integer $k$. Determine all pairs $m,n$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is the set of all integers. [i]Proposed by Warut Suksompong, Thailand[/i]

$n$ teams take part in a tournament, in which every two teams compete exactly once, and that no draws are possible. It is known that for any two teams, there exists another team which defeated both of the two teams. Find all $n$ for which this is possible.

Jenny and Kenny are walking in the same direction, Kenny at 3 feet per second and Jenny at 1 foot per second, on parallel paths that are 200 feet apart. A tall circular building 100 feet in diameter is centered midway between the paths. At the instant when the building first blocks the line of sight between Jenny and Kenny, they are 200 feet apart. Let $t$ be the amount of time, in seconds, before Jenny and Kenny can see each other again. If $t$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?

Let $\displaystyle{z}$ be a complex number with $\displaystyle{\left| {z + 1} \right| > 2}$. Prove that $\displaystyle{\left| {{z^3} + 1} \right| > 1}$. [i]Proposed by Walther Janous and Gerhard Kirchner, Innsbruck.[/i]

Suppose that for a prime number $p$ and integers $a,b,c$ the following holds: \[6\mid p+1,\quad p\mid a+b+c,\quad p\mid a^4+b^4+c^4.\] Prove that $p\mid a,b,c$.

Let $k$ be any fixed positive integer. Let's look at integer pairs $(a, b)$, for which the quadratic equations $x^2 - 2ax + b = 0$ and $y^2 + 2ay + b = 0$ are real solutions (not necessarily different), which can be denoted by $x_1, x_2$ and $y_1, y_2$, respectively, in such an order that the equation $x_1 y_1 - x_2 y_2 = 4k$. a) Find the largest possible value of the second component $b$ of such a pair of numbers ($a, b)$. b) Find the sum of the other components of all such pairs of numbers.

Find all quadruples of real numbers $(a,b,c,d)$ such that the equalities \[X^2 + a X + b = (X-a)(X-c) \text{ and } X^2 + c X + d = (X-b)(X-d)\] hold for all real numbers $X$. [i]Morteza Saghafian, Iran[/i]

When rolling a certain unfair six-sided die with faces numbered $1, 2, 3, 4, 5$, and $6$, the probability of obtaining face $F$ is greater than $\frac{1}{6}$, the probability of obtaining the face opposite is less than $\frac{1}{6}$, the probability of obtaining any one of the other four faces is $\frac{1}{6}$, and the sum of the numbers on opposite faces is $7$. When two such dice are rolled, the probability of obtaining a sum of $7$ is $\frac{47}{288}$. Given that the probability of obtaining face $F$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

Let $a \neq b a,b \in \mathbb{R}$ such that $(x^2+20ax+10b)(x^2+20bx+10a)=0$ has no roots for $x$. Prove that $20(b-a)$ is not an integer.

Three travel companies provide transportation between $n$ cities, such that each connection between a pair of cities is covered by one company only. Prove that, for $n \geq 11$, there must exist a round-trip through some four cities, using the services of a same company, while for $n < 11$ this is not anymore necessarily true. [i]Dan Schwarz[/i]

Let $a, b, c, d$ three strictly positive real numbers such that \[a^{2}+b^{2}+c^{2}=d^{2}+e^{2},\] \[a^{4}+b^{4}+c^{4}=d^{4}+e^{4}.\] Compare \[a^{3}+b^{3}+c^{3}\] with \[d^{3}+e^{3},\]

The remainder on dividing the polynomial $p(x)$ by $x^2 - (a+b)x + ab$ (where $a \not = b$) is $mx + n$. Find the coefficients $m, n$ in terms of $a, b$. Find $m, n$ for the case $p(x) = x^{200}$ divided by $x^2 - x - 2$ and show that they are integral.

If the arithmetic mean of $a$ and $b$ is double their geometric mean, with $a>b>0$, then a possible value for the ratio $\frac{a}{b}$, to the nearest integer, is $\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 14 \qquad\textbf{(E)}\ \text{none of these} $

Find the smallest odd integer $ k$ such that: for every $ 3\minus{}$degree polynomials $ f$ with integer coefficients, if there exist $ k$ integer $ n$ such that $ |f(n)|$ is a prime number, then $ f$ is irreducible in $ \mathbb{Z}[n]$.

Let $k$ be a fixed integer greater than 1, and let ${m=4k^2-5}$. Show that there exist positive integers $a$ and $b$ such that the sequence $(x_n)$ defined by \[x_0=a,\quad x_1=b,\quad x_{n+2}=x_{n+1}+x_n\quad\text{for}\quad n=0,1,2,\dots,\] has all of its terms relatively prime to $m$. [i]Proposed by Jaroslaw Wroblewski, Poland[/i]

Classify up to homeomorphism the topological spaces of the support of functions that are real quadratic polynoms of three variables and and irreducible over the set of real numbers.

Let $S$ be the set of all rational numbers expressible in the form \[\frac{(a_1^2+a_1-1)(a_2^2+a_2-1)\ldots (a_n^2+a_n-1)}{(b_1^2+b_1-1)(b_2^2+b_2-1)\ldots (b_n^2+b_n-1)}\] for some positive integers $n, a_1, a_2 ,\ldots, a_n, b_1, b_2, \ldots, b_n$. Prove that there is an infinite number of primes in $S$.

Let $n$ be a positive integer. Show that there are infinitely many primes $p$ such that the smallest positive primitive root of $p$ is greater than $n$.

If the sum of all the angles except one of a convex polygon is $ 2190^{\circ}$, then the number of sides of the polygon must be $ \textbf{(A)}\ 13 \qquad \textbf{(B)}\ 15 \qquad \textbf{(C)}\ 17 \qquad \textbf{(D)}\ 19 \qquad \textbf{(E)}\ 21$

Consider the sequence $(a^n + 1)_{n\geq 1}$, with $a>1$ a fixed integer. i) Prove there exist infinitely many primes, each dividing some term of the sequence. ii) Prove there exist infinitely many primes, none dividing any term of the sequence. [i](Dan Schwarz)[/i]

Two parallel chords in a circle have lengths $10$ and $14$, and the distance between them is $6$. The chord parallel to these chords and midway between them is of length $\sqrt{a}$ where $a$ is [asy] // note: diagram deliberately not to scale -- azjps void htick(pair A, pair B, real r){ D(A--B); D(A-(r,0)--A+(r,0)); D(B-(r,0)--B+(r,0)); } size(120); pathpen = linewidth(0.7); pointpen = black+linewidth(3); real min = -0.6, step = 0.5; pair[] A, B; D(unitcircle); for(int i = 0; i < 3; ++i) { A.push(intersectionpoints((-9,min+i*step)--(9,min+i*step),unitcircle)[0]); B.push(intersectionpoints((-9,min+i*step)--(9,min+i*step),unitcircle)[1]); D(D(A[i])--D(B[i])); } MP("10",(A[0]+B[0])/2,N); MP("\sqrt{a}",(A[1]+B[1])/2,N); MP("14",(A[2]+B[2])/2,N); htick((B[1].x+0.1,B[0].y),(B[1].x+0.1,B[2].y),0.06); MP("6",(B[1].x+0.1,B[0].y/2+B[2].y/2),E);[/asy] $\textbf{(A)}\ 144 \qquad \textbf{(B)}\ 156 \qquad \textbf{(C)}\ 168 \qquad \textbf{(D)}\ 176 \qquad \textbf{(E)}\ 184$

Prove that: there exists a positive constant $K$, and an integer series $\{a_n\}$, satisfying: $(1)$ $0<a_1<a_2<\cdots <a_n<\cdots $; $(2)$ For any positive integer $n$, $a_n<1.01^n K$; $(3)$ For any finite number of distinct terms in $\{a_n\}$, their sum is not a perfect square.

Let $A, B, C, D, E$ be integers, $B \neq 0$ and $F=AD^{2}-BCD+B^{2}E \neq 0$. Prove that the number $N$ of pairs of integers $(x, y)$ such that \[Ax^{2}+Bxy+Cx+Dy+E=0,\] satisfies $N \le 2 d( \vert F \vert )$, where $d(n)$ denotes the number of positive divisors of positive integer $n$.

Let a quadratic polynomial $g(x) = ax^2 + bx + c$ be given and an integer $n \ge 1$. Prove that there exists at most one polynomial $f(x)$ of $n$th degree such that $f(g(x)) = g(f(x)).$