The real factors of $ x^2\plus{}4$ are: $\textbf{(A)}\ (x^2+2)(x^2+2) \qquad \textbf{(B)}\ (x^2+2)(x^2-2) \qquad \textbf{(C)}\ x^2(x^2+4) \qquad\\ \textbf{(D)}\ (x^2-2x+2)(x^2+2x+2) \qquad \textbf{(E)}\ \text{Non-existent}$

Let's write p,q, and r as three distinct prime numbers, where 1 is not a prime. Which of the following is the smallest positive perfect cube leaving $ n \equal{} pq^2r^4$ as a divisor? $ \textbf{(A)}\ p^8q^8r^8\qquad \textbf{(B)}\ (pq^2r^2)^3\qquad \textbf{(C)}\ (p^2q^2r^2)^3\qquad \textbf{(D)}\ (pqr^2)^3\qquad \textbf{(E)}\ 4p^3q^3r^3$

Let $O$ be the center of the circle circumscribed to triangle $ABC$ and let $ S_ {A} $, $ S_ {B} $, $ S_ {C} $ be the circles centered on $O$ that are tangent to the sides $BC, CA, AB$ respectively. Show that the sum of the angle between the two tangents $ S_ {A} $ from $A$ plus the angle between the two tangents $ S_ {B} $ from $B$ plus the angle between the two tangents $ S_ {C} $ from $C$ is $180$ degrees.

Regular tetrahedron $P_1P_2P_3P_4$ has side length $1$. Define $P_i$ for $i > 4$ to be the centroid of tetrahedron $P_{i-1}P_{i-2}P_{i-3}P_{i-4}$, and $P_{ \infty} = \lim_{n\to \infty} P_n$. What is the length of $P_5P_{ \infty}$?

Let $m,n\geq 2$ be integers. Let $f(x_1,\dots, x_n)$ be a polynomial with real coefficients such that $$f(x_1,\dots, x_n)=\left\lfloor \frac{x_1+\dots + x_n}{m} \right\rfloor\text{ for every } x_1,\dots, x_n\in \{0,1,\dots, m-1\}.$$ Prove that the total degree of $f$ is at least $n$.

Let us consider a first-order language $L$ with a ternary predicate $\operatorname{Plus}$. Hence (well-formed) formulas of $L$ are built of symbols for variables, logical connectives, quantifiers, brackets, and the predicate symbol $\operatorname{Plus}$. $$(\exists x_1)(\forall x_2):\operatorname{Plus}(x_2,x_1,x_2)\wedge(\forall x_3):\neg\operatorname{Plus}(x_1,x_3,x_3)$$ is an example of such a formula. Recall that a formula is [i]closed[/i] iff each variable symbol occurs within the scope of a quantifier. Show that there exists an algorithm which decides whether or not a given closed formula of $L$ is true for the set $\mathbb N$ of natural numbers ($\{0,1,2,\ldots\}$) where $\operatorname{Plus}(x,y,z)$ is interpreted as $x+y=z$.

Let points $P$ and $Q$ be isogonally conjugated with respect to a triangle $ABC$. The line $PQ$ meets the circumcircle of $ABC$ at point $X$. The reflection of $BC$ about $PQ$ meets $AX$ at point $E$. Prove that $A, P, Q, E$ are concyclic.

A convex quadrilateral $ABCD$ is given. $\omega_1$ is a circle with diameter $BC$, $\omega_2$ is a circle with diameter $AD$. $AC$ meets $\omega_1$ and $\omega_2$ for the second time at $B_1$ and $D_1$. $BD$ meets $\omega_1$ and $\omega_2$ for the second time at $C_1$ and $A_1$. $AA_1$ meets $DD_1$ at $X$, $BB_1$ meets $CC_1$ at $Y$. $\omega_1$ intersects $\omega_2$ at $P$ and $Q$. $XY$ meets $PQ$ at $N$. Prove that $XN=NY$.

Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.

What is the probability of having no $B$ before the first $A$ in a random permutation of the word $\text{ABRAKADABRA}$? $ \textbf{(A)}\ \dfrac 23 \qquad\textbf{(B)}\ \dfrac 57 \qquad\textbf{(C)}\ \dfrac 56 \qquad\textbf{(D)}\ \dfrac 67 \qquad\textbf{(E)}\ \text{None of the preceding} $

Let $a,b$ be two natural numbers. When we divide $a^2+b^2$ by $a+b$, we the the remainder $r$ and the quotient $q.$ Determine all pairs $(a, b)$ for which $q^2 + r = 1977.$

Given any integer $m>1$ prove that there exist infinitely many positive integers $n$ such that the last $m$ digits of $5^n$ are a sequence $a_m,a_{m-1},\ldots ,a_1=5\ (0\le a_j<10)$ in which each digit except the last is of opposite parity to its successor (i.e., if $a_i$ is even, then $a_{i-1}$ is odd, and if $a_i$ is odd, then $a_{i-1}$ is even).

Let $\vartriangle ABC$ be a triangle. The angle bisector of $\angle CAB$ intersects$ BC$ at $L$. On the interior of line segments $AC$ and $AB$, two points, $M$ and $N$, respectively, are chosen in such a way that the lines $AL, BM$ and $CN$ are concurrent, and such that $\angle AMN = \angle ALB$. Prove that $\angle NML = 90^o$.

The numbers from $1$ to $100$ are placed without repetition in each cell of a \(10 \times 10\) board. An increasing path of length \(k\) on this board is a sequence of cells \(c_1, c_2, \ldots, c_k\) such that, for each \(i = 2, 3, \ldots, k\), the following properties are satisfied: • The cells \(c_i\) and \(c_{i-1}\) share a side or a vertex; • The number in \(c_i\) is greater than the number in \(c_{i-1}\). What is the largest positive integer \(k\) for which we can always find an increasing path of length \(k\), regardless of how the numbers from 1 to 100 are arranged on the board?

Points $ A_1$, $ A_2$, $ ...$ are placed on a circle with center $ O$ such that $ \angle OA_n A_{n\plus{}1}\equal{}35^\circ$ and $ A_n\neq A_{n\plus{}2}$ for all positive integers $ n$. What is the smallest $ n>1$ for which $ A_n\equal{}A_1$?

Prove the following theorem: If there are $n$ pairs of different points $P_i$, $i = 1, 2, ..., n$, $n > 2$ in three dimensions space, such that each of them is at a smaller distance from one and the same point $Q$ than any other $P_i$, then $n < 15$.

Sixteen dots are arranged in a four by four grid as shown. The distance between any two dots in the grid is the minimum number of horizontal and vertical steps along the grid lines it takes to get from one dot to the other. For example, two adjacent dots are a distance 1 apart, and two dots at opposite corners of the grid are a distance 6 apart. The mean distance between two distinct dots in the grid is $\frac{m}{n}$, where m and n are relatively prime positive integers. Find $m + n$. [center][img]https://i.snag.gy/c1tB7z.jpg[/img][/center]

Find all real numbers$p, q$ for which the polynomial equation $P(x) = x^4 - \frac{8p^2}{q}x^3 + 4qx^2 - 3px + p^2 = 0$ has four positive roots.

Let $a$ and $b$ be natural numbers with $a > b$ and having the same parity. Prove that the solutions of the equation \[ x^2 - (a^2 - a + 1)(x - b^2 - 1) - (b^2 + 1)^2 = 0 \] are natural numbers, none of which is a perfect square. [i]Albania[/i]

(a) Let $a$ and $b$ be positive integers such that $M(a, b) = a - \frac1b +b(b + \frac3a)$ is an integer. Prove that $M(a,b)$ is a square. (b) Find nonzero integers $a$ and $b$ such that $M(a,b)$ is a positive integer, but not a square.

Let $\triangle ABC$ and $\triangle C'B'A$ be two congruent triangles ( with this order and orient. ). Define point $M$ as the midpoint of segment $AB$ and suppose that the extension of $CB'$ from $B'$ passes trough $M$ , if $F$ be a point on the smaller arc $MC$ of circumcircle of triangle $\triangle BMC$ such that $\angle FB'A=90$ and $\angle C'CB' \neq 90$ , then prove that $\angle B'C'C=\angle CAF$. [i]Proposed by Alireza Dadgarnia[/i]

What figure can the central projection of a triangle be? (The center of the projection does not lie on the plane of the triangle.)

Solve the equation $\sqrt{x} = \log_2 x$.

Let $n$ denote the product of the first $2013$ primes. Find the sum of all primes $p$ with $20 \le p \le 150$ such that (i) $\frac{p+1}{2}$ is even but is not a power of $2$, and (ii) there exist pairwise distinct positive integers $a,b,c$ for which \[ a^n(a-b)(a-c) + b^n(b-c)(b-a) + c^n(c-a)(c-b) \] is divisible by $p$ but not $p^2$. [i]Proposed by Evan Chen[/i]

Among the $n$ inhabitants of an island, every two are either friends or enemies. Some day, the chief of the island orders that each inhabitant (including himself) makes and wears a necklace consisting of marbles, in such a way that the necklaces of two friends have at least one marble of the same type and that the necklaces of two enemies differ at all marbles. (A necklace may also be marbleless). Show that the chief’s order can be achieved by using $\left\lfloor\frac{n^2}4\right\rfloor$ different types of stones, but not necessarily by using fewer types.