Given the equations $x^2+kx+6=0$ and $x^2-kx+6=0$. If, when the roots of the equation are suitably listed, each root of the second equation is $5$ more than the corresponding root of the first equation, then $k$ equals: $\textbf{(A)}\ 5 \qquad \textbf{(B)}\ -5 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ -7 \qquad \textbf{(E)}\ \text{none of these}$

The sequence of Fibonacci numbers $F_0, F_1, F_2, . . .$ is defined by $F_0 = F_1 = 1 $ and $F_{n+2} = F_n+F_{n+1}$ for all $n > 0$. For example, we have $F_2 = F_0 + F_1 = 2, F_3 = F_1 + F_2 = 3, F_4 = F_2 + F_3 = 5$, and $F_5 = F_3 + F_4 = 8$. The sequence $a_0, a_1, a_2, ...$ is defined by $a_n =\frac{1}{F_nF_{n+2}}$ for all $n \ge 0$. Prove that for all $m \ge 0$ we have: $a_0 + a_1 + a_2 + ... + a_m < 1$.

Determine the least integer $n$ such that for any set of $n$ lines in the 2D plane, there exists either a subset of $1001$ lines that are all parallel, or a subset of $1001$ lines that are pairwise nonparallel. [i]Proposed by Samuel Wang[/i] [hide=Solution][i]Solution.[/i] $\boxed{1000001}$ Since being parallel is a transitive property, we note that in order for this to not exist, there must exist at most $1001$ groups of lines, all pairwise intersecting, with each group containing at most $1001$ lines. Thus, $n = 1000^2 + 1 = \boxed{1000001}$.[/hide]

Determine all natural numbers $n \geq 2$ with at most four natural divisors, which have the property that for any two distinct proper divisors $d_1$ and $d_2$ of $n$, the positive integer $d_1-d_2$ divides $n$.

The lengths of two altitudes of a triangles are $8$ and $12$. Which of the following cannot be the third altitude? $ \textbf{a)}\ 4 \qquad\textbf{b)}\ 7 \qquad\textbf{c)}\ 8 \qquad\textbf{d)}\ 12 \qquad\textbf{e)}\ 23 $

Let $ K$ be the circumscribed circle of the trapezoid $ ABCD$ . In this trapezoid the diagonals $ AC$ and $ BD$ are perpendicular. The parallel sides $ AB\equal{}a$ and $ CD\equal{}c$ are diameters of the circles $ K_{a}$ and $ K_{b}$ respectively. Find the perimeter and the area of the part inside the circle $ K$, that is outside circles $ K_{a}$ and $ K_{b}$.

You use a lock with four dials, each of which is set to a number between 0 and 9 (inclusive). You can never remember your code, so normally you just leave the lock with each dial one higher than the correct value. Unfortunately, last night someone changed all the values to 5. All you remember about your code is that none of the digits are prime, 0, or 1, and that the average value of the digits is 5. How many combinations will you have to try?

Let $ABC$ be an isosceles triangle with $AC=BC$, whose incentre is $I$. Let $P$ be a point on the circumcircle of the triangle $AIB$ lying inside the triangle $ABC$. The lines through $P$ parallel to $CA$ and $CB$ meet $AB$ at $D$ and $E$, respectively. The line through $P$ parallel to $AB$ meets $CA$ and $CB$ at $F$ and $G$, respectively. Prove that the lines $DF$ and $EG$ intersect on the circumcircle of the triangle $ABC$. [i]Proposed by Hojoo Lee, Korea[/i]

Show that:\[\alpha = \pm \frac{\pi}{12} + k\cdot \frac{\pi}2 (k\in \mathbb{Z}) \Longleftrightarrow\ |{\tan \alpha}| + |{\cot \alpha}| = 4\]

A regular pentagon is drawn in the plane, along with all its diagonals. All its sides and diagonals are extended infinitely in both directions, dividing the plane into regions, some of which are unbounded. An ant starts in the center of the pentagon, and every second, the ant randomly chooses one of the edges of the region it's in, with an equal probability of choosing each edge, and crosses that edge into another region. If the ant enters an unbounded region, it explodes. After first leaving the central region of the pentagon, let $x$ be the expected number of times the ant re-enters the central region before it explodes. Find the closest integer to $100x$.

$101$ people, sitting at a round table in any order, had $1,2,... , 101$ cards, respectively. A transfer is someone give one card to one of the two people adjacent to him. Find the smallest positive integer $k$ such that there always can through no more than $ k $ times transfer, each person hold cards of the same number, regardless of the sitting order.

Find the number of quadruples $(a, b, c, d)$ of integers with absolute value at most $5$ such that $$(a^2 + b^2 + c^2 + d^2)^2 = (a + b + c + d)(a - b + c -d)((a - c)^2 + (b - d)^2).$$

The product of three positive numbers is $1$ and their sum is greater than the sum of their inverses. Prove that one of these numbers is greater than $1$, while the other two are smaller than $1$.

Evaluate $\int_{\sqrt{2}}^{\sqrt{3}} (x^2+\sqrt{x^4-1})(\frac{1}{\sqrt{x^2+1}}+{\frac{1}{\sqrt{x^2-1}})dx.}$ [i]Proposed by kunny[/i]

In triangle $ABC$, let $I$ be the incentre, $O$ be the circumcentre, and $H$ be the orthocentre. It is given that $IO = IH$. Show that one of the angles of triangle $ABC$ must be equal to $60$ degrees.

A four-digit number $n$ is said to be [i]literally 1434[/i] if, when every digit is replaced by its remainder when divided by $5$, the result is $1434$. For example, $1984$ is [i]literally 1434[/i] because $1$ mod $5$ is $1$, $9$ mod $5$ is $4$, $8$ mod $5$ is $3$, and $4$ mod $5$ is $4$. Find the sum of all four-digit positive integers that are [i]literally 1434[/i]. [i]Proposed by Evin Liang[/i] [hide=Solution] [i]Solution.[/i] $\boxed{67384}$ The possible numbers are $\overline{abcd}$ where $a$ is $1$ or $6$, $b$ is $4$ or $9$, $c$ is $3$ or $8$, and $d$ is $4$ or $9$. There are $16$ such numbers and the average is $\dfrac{8423}{2}$, so the total in this case is $\boxed{67384}$. [/hide]