The real numbers $c, b, a$ form an arithmetic sequence with $a\ge b\ge c\ge 0$. The quadratic $ax^2+bx+c$ has exactly one root. What is this root? $\textbf{(A)}\ -7-4\sqrt{3}\qquad\textbf{(B)}\ -2-\sqrt{3}\qquad\textbf{(C)}\ -1\qquad\textbf{(D)}\ -2+\sqrt{3}\qquad\textbf{(E)}\ -7+4\sqrt{3} $

Find eight positive integers $n_1, n_2, \dots , n_8$ with the following property: For every integer $k$, $-1985 \leq k \leq 1985$, there are eight integers $a_1, a_2, \dots, a_8$, each belonging to the set $\{-1, 0, 1\}$, such that $k=\sum_{i=1}^{8} a_i n_i .$

Which one satisfies the equation $\sqrt[3]{6+\sqrt{x}} + \sqrt[3]{6-\sqrt{x}} = \sqrt[3]{3}$ ? $ \textbf{(A)}\ 27 \qquad \textbf{(B)}\ 32 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 52 \qquad \textbf{(E)}\ 63$

The squares of a chessboard are numbered in the following way. The upper left corner is numbered 1. The two squares on the next diagonal from top-right to bottom-left are numbered 2 and 3. The three squares on the next diagonal are numbered 4, 5 and 6, and so on. The two squares on the second-to-last diagonal are numbered 62 and 63, and the lower right corner is numbered 64. Peter puts eight pebbles on the squares of the chessboard in such a way that there is exactly one pebble in each column and each row. Then he moves each pebble to a square with a number greater than that of the original square. Can it happen that there is still exactly one pebble in each column and each row? [i](8 points)[/i]

Find the value of $$(2^{40}+12^{41}+23^{42}+67^{43}+87^{44})^{45!+46}\mod11$$ (variation but same answer) [hide=Answer]3[/hide]

Let $A_1,A_2,\cdots , A_n$ be arithmetic progressions of integers, each of $k$ terms, such that any two of these arithmetic progressions have at least two common elements. Suppose $b$ of these arithmetic progressions have common difference $d_1$ and the remaining arithmetic progressions have common difference $d_2$ where $0<b<n$. Prove that \[b \le 2\left(k-\frac{d_2}{gcd(d_1,d_2)}\right)-1.\]

Points $M$, $N$, $P$ and $Q$ are midpoints of the sides $AB$, $BC$, $CD$ and $DA$ of the inscribed quadrilateral $ABCD$ with intersection point $O$ of its diagonals. Let $K$ be the second intersection point of the circumscribed circles of $MOQ$ and $NOP$. Prove that $OK\perp AC$.

Determine all values of $x$ in the interval $0 \leq x \leq 2\pi$ which satisfy the inequality \[ 2 \cos{x} \leq \sqrt{1+\sin{2x}}-\sqrt{1-\sin{2x}} \leq \sqrt{2}. \]

On the basis of VSEPR theory, what geometry is predicted for the central sulfur atom in $\ce{SOCl2}$? $ \textbf{(A) }\text{bent}\qquad\textbf{(B) }\text{T-shaped}\qquad\textbf{(C) }\text{trigonal planar} \qquad\textbf{(D) }\text{trigonal pyramidal} \qquad$

Find all real solutions of the equation $(x -4) (x^2 - 8x + 14)^2 = (x - 4)^3$.

The angle $\angle XOY =\alpha $ and the points $A$ and $B$ on OY are given such that $OA = a$ and $OB = b$ with $a > b$. A circle passes through the points $A$ and $B$ and is tangent to $OX$. a) Calculate the radius of that circle in terms of $a, b$ and $\alpha $. b) If $a$ and $b$ are constants and $\alpha $ varies, show that the minimum value of the radius of the circle is $\frac{a-b}{2}$.

Let there be given a set of $1996$ equal circles in the plane, no two of them having common interior points. Prove that there exists a circle touching at most three other circles.

Suppose $P(x,y)$ is a homogenous non-constant polynomial with real coefficients such that $P(\sin t, \cos t) = 1$ for all real $t$. Prove that $P(x,y) = (x^2+y^2)^k$ for some positive integer $k$. (A polynomial $A(x,y)$ with real coefficients and having a degree of $n$ is homogenous if it is the sum of $a_ix^iy^{n-i}$ for some real number $a_i$, for all integer $0 \le i \le n$.)

Point $O$ is the origin of a space. Points $A_1, A_2,\dots, A_n$ have nonnegative coordinates. Prove the following inequality: $$|\overrightarrow{OA_1}|+|\overrightarrow {OA_2}|+\dots+|\overrightarrow {OA_n}|\leq \sqrt{3}|\overrightarrow {OA_1}+\overrightarrow{OA_2}+\dots+\overrightarrow{OA_n}|$$ [I]Proposed by A. Khrabrov[/i]

Show that the fraction $ \frac{n^2+n-1}{n^2+2n}$ is irreducible for every positive integer n.

$ 15 $ teams participate in a soccer league. Each team plays each of the remaining teams exactly once. If a team beats another team in a match they receive $ 3 $ points, while the loser receives $ 1 $ point. In the event of a tie, both teams receive $ 2 $ points. When all possible league matches are held, the following can be observed: $\bullet$ No two teams have finished with the same amount of points. $\bullet$ Each team finished the league with at least $ 21 $ points. Let $W$ be the team that finished the league with the highest score. Determine how many points $W$ scored and show that there were at least four ties in the league.

Let $ABCD$ be an isosceles trapezoid inscribed in circle $\omega$, such that $AD \| BC$. Point $E$ is chosen on the arc $BC$ of $\omega$ not containing $A$. Let $BC$ and $DE$ intersect at $F$. Show that if $E$ is chosen such that $EB = EC$, the area of $AEF$ is maximized.

Let $O$ be the center of the concentric circles $C_1,C_2$ of radii $3$ and $5$ respectively. Let $A\in C_1, B\in C_2$ and $C$ point so that triangle $ABC$ is equilateral. Find the maximum length of $ [OC] $.

Real numbers $a$ and $b$ are chosen with $1<a<b$ such that no triangle with positive area has side lengths $1,a,$ and $b$ or $\tfrac{1}{b}, \tfrac{1}{a},$ and $1$. What is the smallest possible value of $b$? ${ \textbf{(A)}\ \dfrac{3+\sqrt{3}}{2}\qquad\textbf{(B)}\ \dfrac52\qquad\textbf{(C)}\ \dfrac{3+\sqrt{5}}{2}\qquad\textbf{(D)}}\ \dfrac{3+\sqrt{6}}{2}\qquad\textbf{(E)}\ 3 $

Prove that if equations $$x^2 + mx + n = 0 \,\,\,\, and\,\, \,\, x^2 + px + q = 0$$ have a common root, there is a relationship between the coefficients of these equations $$ (n - q)^2 - (m - p) (np - mq) = 0.$$

Evaluate $\sum^{15}_{k=0}\left(2^{560}(-1)^k \cos^{560}\left( \frac{k\pi}{16}\right)\right) \pmod{17}.$

Define a [i]small[/i] prime to be a prime under $1$ billion. Find the sum of all [i]small[/i] primes of the form $20^n + 1$, given that the answer is greater than $1000$. [i]Lightning 3.3[/i]

Let $P$ be a point chosen on the interior of side $\overline{BC}$ of triangle $\triangle ABC$ with side lengths $\overline{AB} = 10, \overline{BC} = 10, \overline{AC} = 12$. If $X$ and $Y$ are the feet of the perpendiculars from $P$ to the sides $AB$ and $AC$, then the minimum possible value of $PX^2 + PY^2$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by Andrew Wen[/i]

Let $n>1$ be an integer and let $a_0,a_1,\ldots,a_n$ be non-negative real numbers. Definite $S_k=\sum_{i\equal{}0}^k \binom{k}{i}a_i$ for $k=0,1,\ldots,n$. Prove that\[\frac{1}{n} \sum_{k\equal{}0}^{n-1} S_k^2-\frac{1}{n^2}\left(\sum_{k\equal{}0}^{n} S_k\right)^2\le \frac{4}{45} (S_n-S_0)^2.\]

Let $a,b,c$ be real numbers such that $a+b+c = 4$ and $a,b,c > 1$. Prove that: \[\frac 1{a-1} + \frac 1{b-1} + \frac 1{c-1} \ge \frac 8{a+b} + \frac 8{b+c} + \frac 8{c+a}\]