For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots? $\textbf{(A) }6 \qquad \textbf{(B) }8 \qquad \textbf{(C) }9 \qquad \textbf{(D) }14 \qquad \textbf{(E) }16$

Let $x$ be a real number such that $x^2 -x+1 = 7$ and $x^2 +x+1 = 13$. Compute the value of $x^4$.

In how many ways can the integers from $1$ to $n$ be ordered subject to the condition that, except for the first integer on the left, every integer differs by $1$ from some integer to the left of it?

Let $ABCD$ be a parallelogram and $H$ the Orthocentre of $\triangle{ABC}$. The line parallel to $AB$ through $H$ intersects $BC$ at $P$ and $AD$ at $Q$ while the line parallel to $BC$ through $H$ intersects $AB$ at $R$ and $CD$ at $S$. Show that $P$, $Q$, $R$ and $S$ are concyclic. [i](Swiss Mathematical Olympiad 2011, Final round, problem 8)[/i]

For any positive integer $n$, define $\boxed{n}$ to be the sum of the positive factors of $n$. For example, $\boxed{6} = 1 + 2 + 3 + 6 = 12$. Find $\boxed{\boxed{11}}$. $\textbf{(A)}\ 13 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 28 \qquad \textbf{(E)}\ 30$

Let ${k}$ be a fixed positive integer. A finite sequence of integers ${x_1,x_2, ..., x_n}$ is written on a blackboard. Pepa and Geoff are playing a game that proceeds in rounds as follows. - In each round, Pepa first partitions the sequence that is currently on the blackboard into two or more contiguous subsequences (that is, consisting of numbers appearing consecutively). However, if the number of these subsequences is larger than ${2}$, then the sum of numbers in each of them has to be divisible by ${k}$. - Then Geoff selects one of the subsequences that Pepa has formed and wipes all the other subsequences from the blackboard. The game finishes once there is only one number left on the board. Prove that Pepa may choose his moves so that independently of the moves of Geoff, the game finishes after at most ${3k}$ rounds. (Poland)

Let $b$ be a number with $-2 < b < 0$. Prove that there exists a positive integer $n$ such that all the coefficients of the polynomial $(x + 1)^n(x^2 + bx + 1)$ are positive.

4. Given is an integer $n\geq 1$. Find out the number of possible values of products $k \cdot m$, where $k,m$ are integers satisfying $n^{2}\leq k \leq m \leq (n+1)^{2}$.

1. Baron Munchhausen was told that some polynomial $P(x)=a_{n} x^{n}+\ldots+a_{1} x+a_{0}$ is such that $P(x)+P(-x)$ has exactly 45 distinct real roots. Baron doesn't know the value of $n$. Nevertheless he claims that he can determine one of the coefficients $a_{n}, \ldots, a_{1}, a_{0}$ (indicating its position and value). Isn't Baron mistaken? Boris Frenkin

Ryan stands on the bottom-left square of a 2017 by 2017 grid of squares, where each square is colored either black, gray, or white according to the pattern as depicted to the right. Each second he moves either one square up, one square to the right, or both one up and to the right, selecting between these three options uniformly and independently. Noting that he begins on a black square, find the probability that Ryan is still on a black square after 2017 seconds. [center][img]http://i.imgur.com/WNp59XW.png[/img][/center]

Prove that in any triangle, at most one side can be shorter than the altitude from the opposite vertex.

[b]p1.[/b] Alan leaves home when the clock in his cardboard box says $7:35$ AM and his watch says $7:41$ AM. When he arrives at school, his watch says $7:47$ AM and the $7:45$ AM bell rings. Assuming the school clock, the watch, and the home clock all go at the same rate, how many minutes behind the school clock is the home clock? [b]p2.[/b] Compute $$\left( \frac{2012^{2012-2013} + 2013}{2013} \right) \times 2012.$$ Express your answer as a mixed number. [b]p3.[/b] What is the last digit of $$2^{3^{4^{5^{6^{7^{8^{9^{...^{2013}}}}}}}}} ?$$ [b]p4.[/b] Let $f(x)$ be a function such that $f(ab) = f(a)f(b)$ for all positive integers $a$ and $b$. If $f(2) = 3$ and $f(3) = 4$, find $f(12)$. [b]p5.[/b] Circle $X$ with radius $3$ is internally tangent to circle $O$ with radius $9$. Two distinct points $P_1$ and $P_2$ are chosen on $O$ such that rays $\overrightarrow{OP_1}$ and $\overrightarrow{OP_2}$ are tangent to circle $X$. What is the length of line segment $P_1P_2$? [b]p6.[/b] Zerglings were recently discovered to use the same $24$-hour cycle that we use. However, instead of making $12$-hour analog clocks like humans, Zerglings make $24$-hour analog clocks. On these special analog clocks, how many times during $ 1$ Zergling day will the hour and minute hands be exactly opposite each other? [b]p7.[/b] Three Small Children would like to split up $9$ different flavored Sweet Candies evenly, so that each one of the Small Children gets $3$ Sweet Candies. However, three blind mice steal one of the Sweet Candies, so one of the Small Children can only get two pieces. How many fewer ways are there to split up the candies now than there were before, assuming every Sweet Candy is different? [b]p8.[/b] Ronny has a piece of paper in the shape of a right triangle $ABC$, where $\angle ABC = 90^o$, $\angle BAC = 30^o$, and $AC = 3$. Holding the paper fixed at $A$, Ronny folds the paper twice such that after the first fold, $\overline{BC}$ coincides with $\overline{AC}$, and after the second fold, $C$ coincides with $A$. If Ronny initially marked $P$ at the midpoint of $\overline{BC}$, and then marked $P'$ as the end location of $P$ after the two folds, find the length of $\overline{PP'}$ once Ronny unfolds the paper. [b]p9.[/b] How many positive integers have the same number of digits when expressed in base $3$ as when expressed in base $4$? [b]p10.[/b] On a $2 \times 4$ grid, a bug starts at the top left square and arbitrarily moves north, south, east, or west to an adjacent square that it has not already visited, with an equal probability of moving in any permitted direction. It continues to move in this way until there are no more places for it to go. Find the expected number of squares that it will travel on. Express your answer as a mixed number. PS. You had better use hide for answers.

There is a point source of light in an empty universe. What is the minimum number of solid balls (of any size) one must place in space so that any light ray emanating from the light source intersects at least one ball?

Prove that if the diagonals of a convex quadrilateral are perpendicular, then the feet of perpendiculars dropped from the intersection point of diagonals on the sides of this quadrilateral lie on one circle. Is the converse true?

On side $AB$ of triangle $ABC$ is marked point $K$ such that $AB = CK$. Points $N$ and $M$ are the midpoints of $AK$ and $BC$, respectively. The segments $NM$ and $CK$ intersect in point $P$. Prove that $KN = KP$.

Prove that solution of equation $y=x^2+ax+b$ and $x=y^2+cy+d$ it belong a circle.

Minivan chooses a prime number. Then every second, he adds either the digit $1$ or the digit $3$ to the right end of his number (after the unit digit), such that the new number is also a prime. Can he continue indefinitely? [i](Proposed by Wong Jer Ren)[/i]

Given a $10 \times 10$ chessboard colored as black-and-white alternately. Prove that for any $46$ unit squares without common edges, there are at least $30$ unit squares with the same color.

If the four sides of a quadrilateral are $2, 3, 6$, and $x$, find the sum of all possible integral values for $x$.

The quadrilateral $ ABCD$ inscribed in a circle wich has diameter $ BD$. Let $ A',B'$ are symmetric to $ A,B$ with respect to the line $ BD$ and $ AC$ respectively. If $ A'C \cap BD \equal{} P$ and $ AC\cap B'D \equal{} Q$ then prove that $ PQ \perp AC$

i) Let $a,b$ be real numbers such that $b\leq 0$ and $1+ax+bx^{2} \geq 0$ for every $x\in [0,1]$. Prove that $$\lim_{n\to \infty} n \int_{0}^{1}(1+ax+bx^{2})^{n}dx= \begin{cases} -\frac{1}{a} &\text{if}\; a<0,\\ \infty & \text{if}\; a \geq 0. \end{cases}$$ ii) Let $f:[0,1]\rightarrow[0,\infty)$ be a function with a continuous second derivative and let $f''(x)\leq0$ for every $x\in [0,1]$. Suppose that $L=\lim_{n\to \infty} n \int_{0}^{1}(f(x))^{n}dx$ exists and $0<L<\infty$. Prove that $f'$ has a constant sign and $\min_{x\in [0,1]}|f'(x)|=L^{-1}$.

Compute the sum of all positive integers $n$ such that $n^2-3000$ is a perfect square.

he sequence of real number $ (x_n)$ is defined by $ x_1 \equal{} 0,$ $ x_2 \equal{} 2$ and $ x_{n\plus{}2} \equal{} 2^{\minus{}x_n} \plus{} \frac{1}{2}$ $ \forall n \equal{} 1,2,3 \ldots$ Prove that the sequence has a limit as $ n$ approaches $ \plus{}\infty.$ Determine the limit.

Given a positive integer $n (n>2004)$, we put 1, 2, 3, …,$n^2$ into squares of an $n\times n$ chessboard with one number in a square. A square is called a “good square” if the square satisfies following conditions: 1) There are at least 2004 squares that are in the same row with the square such that any number within these 2004 squares is less than the number within the square. 2) There are at least 2004 squares that are in the same column with the square such that any number within these 2004 squares is less than the number within the square. Find the maximum value of the number of the “good square”.

An ant climbs either two inches or three inches each day. In how many ways can the ant climb twelve inches, if the order of its climbing sequence matters? $\textbf{(A) }8\qquad\textbf{(B) }9\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }14$