Let $ z_{1},z_{2},z_{3}$ be three complex numbers of moduli less than or equal to $ 1$. $ w_{1},w_{2}$ are two roots of the equation $ (z \minus{} z_{1})(z \minus{} z_{2}) \plus{} (z \minus{} z_{2})(z \minus{} z_{3}) \plus{} (z \minus{} z_{3})(z \minus{} z_{1}) \equal{} 0$. Prove that, for $ j \equal{} 1,2,3$, $\min\{|z_{j} \minus{} w_{1}|,|z_{j} \minus{} w_{2}|\}\leq 1$ holds.

In a “Fibonacci” sequence of numbers, each term beginning with the third, is the sum of the previous two terms. The first number in such a sequence is 2 and the third is 9. What is the eighth term in the sequence? $\textbf{(A)}\ 34 \qquad \textbf{(B)}\ 36 \qquad \textbf{(C)}\ 107 \qquad \textbf{(D)}\ 152 \qquad \textbf{(E)}\ 245$

The integer 287 is exactly divisible by $\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 6$

The letters of the word ‘GAUSS’ and the digits in the number ‘1998’ are each cycled separately and then numbered as shown. 1. AUSSG 9981 2. USSGA 9819 3. SSGAU 8199 etc. If the pattern continues in this way, what number will appear in front of GAUSS 1998? $\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 20$

When the celebrated German mathematician Karl Gauss (1777-1855) was nine years old, he was asked to add all the integers from 1 through 100. He quickly added 1 and 100, 2 and 99, and so on for 50 pairs of numbers each adding in 101. His answer was 50 · 101=5,050. Now find the sum of all the digits in the integers from 1 through 1,000,000 (i.e. all the digits in those numbers, not the numbers themselves).

Each of the digits 3, 5, 6, 7, and 8 is placed one to a box in the diagram. If the two digit number is subtracted from the three digit number, what is the smallest difference? $\textbf{(A)}\ 269 \qquad \textbf{(B)}\ 278 \qquad \textbf{(C)}\ 484 \qquad \textbf{(D)}\ 271 \qquad \textbf{(E)}\ 261$

Which one of the following gives an odd integer? $\textbf{(A)}\ 6^2 \qquad \textbf{(B)}\ 23-17 \qquad \textbf{(C)}\ 9\times24 \qquad \textbf{(D)}\ 96\div8 \qquad \textbf{(E)}\ 9\times41$

Five students named Fred, Gail, Henry, Iggy, and Joan are seated around a circular table in that order. To decide who goes first in a game, they play “countdown”. Henry starts by saying ‘34’, with Iggy saying ‘33’. If they continue to count down in their circular order, who will eventually say ‘1’? $\textbf{(A)}\ \text{Fred} \qquad \textbf{(B)}\ \text{Gail} \qquad \textbf{(C)}\ \text{Henry} \qquad \textbf{(D)}\ \text{Iggy} \qquad \textbf{(E)}\ \text{Joan}$

If two real numbers $ x$ and $ y$ satisfy the equation $ \frac{x}{y} \equal{} x \minus{} y$, then: $ \textbf{(A)}\ {x \ge 4}\text{ and }{x \le 0}\qquad \\ \textbf{(B)}\ {y}\text{ can equal }{1}\qquad \\ \textbf{(C)}\ \text{both }{x}\text{ and }{y}\text{ must be irrational}\qquad \\ \textbf{(D)}\ {x}\text{ and }{y}\text{ cannot both be integers}\qquad \\ \textbf{(E)}\ \text{both }{x}\text{ and }{y}\text{ must be rational}$

Given an acute triangle $ PBC$ with $ PB\neq PC.$ Points $ A,D$ lie on $ PB,PC,$ respectively. $ AC$ intersects $ BD$ at point $ O.$ Let $ E,F$ be the feet of perpendiculars from $ O$ to $ AB,CD,$ respectively. Denote by $ M,N$ the midpoints of $ BC,AD.$ $ (1)$: If four points $ A,B,C,D$ lie on one circle, then $ EM\cdot FN \equal{} EN\cdot FM.$ $ (2)$: Determine whether the converse of $ (1)$ is true or not, justify your answer.

Through a point $P$ within a triangle $ABC$ the lines $l, m$, and $n$ perpendicular respectively to $AP,BP,CP$ are drawn. Prove that if $l$ intersects the line $BC$ in $Q$, $m$ intersects $AC$ in $R$, and $n$ intersects $AB$ in $S$, then the points $Q, R$, and $S$ are collinear.

Consider a sequence $\{a_n\}_{n\geq 0}$ such that $a_{n+1}=a_n-\lfloor{\sqrt{a_n}}\rfloor\ (n\geq 0),\ a_0\geq 0$. (1) If $a_0=24$, then find the smallest $n$ such that $a_n=0$. (2) If $a_0=m^2\ (m=2,\ 3,\ \cdots)$, then for $j$ with $1\leq j\leq m$, express $a_{2j-1},\ a_{2j}$ in terms of $j,\ m$. (3) Let $m\geq 2$ be integer and for integer $p$ with $1\leq p\leq m-1$, let $a\0=m^2-p$. Find $k$ such that $a_k=(m-p)^2$, then find the smallest $n$ such that $a_n=0$.

Prove that there do not exist polynomials $ P$ and $ Q$ such that \[ \pi(x)\equal{}\frac{P(x)}{Q(x)}\] for all $ x\in\mathbb{N}$.

Given a complex number $z$ such that $z^{13}=1$, find all possible value of $z+z^3+z^4+z^9+z^{10}+z^{12}$.

Calculate the integral of the Gaussian curvature of the surface \[z^4+(x^2+y^2-1)(2x^2+3y^2-1)=0\]

The vertical axis indicates the number of employees, but the scale was accidentally omitted from this graph. What percent of the employees at the Gauss company have worked there for $5$ years or more? [asy] for(int a=1; a<11; ++a) { draw((a,0)--(a,-.5)); } draw((0,10.5)--(0,0)--(10.5,0)); label("$1$",(1,-.5),S); label("$2$",(2,-.5),S); label("$3$",(3,-.5),S); label("$4$",(4,-.5),S); label("$5$",(5,-.5),S); label("$6$",(6,-.5),S); label("$7$",(7,-.5),S); label("$8$",(8,-.5),S); label("$9$",(9,-.5),S); label("$10$",(10,-.5),S); label("Number of years with company",(5.5,-2),S); label("X",(1,0),N); label("X",(1,1),N); label("X",(1,2),N); label("X",(1,3),N); label("X",(1,4),N); label("X",(2,0),N); label("X",(2,1),N); label("X",(2,2),N); label("X",(2,3),N); label("X",(2,4),N); label("X",(3,0),N); label("X",(3,1),N); label("X",(3,2),N); label("X",(3,3),N); label("X",(3,4),N); label("X",(3,5),N); label("X",(3,6),N); label("X",(3,7),N); label("X",(4,0),N); label("X",(4,1),N); label("X",(4,2),N); label("X",(5,0),N); label("X",(5,1),N); label("X",(6,0),N); label("X",(6,1),N); label("X",(7,0),N); label("X",(7,1),N); label("X",(8,0),N); label("X",(9,0),N); label("X",(10,0),N); label("Gauss Company",(5.5,10),N); [/asy] $\text{(A)}\ 9\% \qquad \text{(B)}\ 23\frac{1}{3}\% \qquad \text{(C)}\ 30\% \qquad \text{(D)}\ 42\frac{6}{7}\% \qquad \text{(E)}\ 50\% $

Susan wants to place 35.5 kg of sugar in small bags. If each bag holds 0.5 kg, how many bags are needed? $\textbf{(A)}\ 36 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 53 \qquad \textbf{(D)}\ 70 \qquad \textbf{(E)}\ 71$

The average of 10, 4, 8, 7, and 6 is $\textbf{(A)}\ 33 \qquad \textbf{(B)}\ 13 \qquad \textbf{(C)}\ 35 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 7$

Let $ U\subset\mathbb R^n$ be an open set, and let $ L: U\times\mathbb R^n\to\mathbb R$ be a continuous, in its second variable first order positive homogeneous, positive over $ U\times (\mathbb R^n\setminus\{0\})$ and of $ C^2$-class Langrange function, such that for all $ p\in U$ the Gauss-curvature of the hyper surface \[ \{ v\in\mathbb R^n \mid L(p,v) \equal{} 1 \}\] is nowhere zero. Determine the extremals of $ L$ if it satisfies the following system \[ \sum_{k \equal{} 1}^n y^k\partial_k\partial_{n \plus{} i}L \equal{} \sum_{k \equal{} 1}^n y^k\partial_i\partial_{n \plus{} k} L \qquad (i\in\{1,\dots,n\})\] of partial differetial equations, where $ y^k(u,v) : \equal{} v^k$ for $ (u,v)\in U\times\mathbb R^k$, $ v \equal{} (v^1,\dots,v^k)$.

Each time a bar of soap is used, its volume decreases by $10\%$. What is the minimum number of times a new bar would have to be used so that less than one-half its volume remains? $\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$

The first 9 positive odd integers are placed in the magic square so that the sum of the numbers in each row, column and diagonal are equal. Find the value of $A + E$. \[ \begin{tabular}{|c|c|c|}\hline A & 1 & B \\ \hline 5 & C & 13\\ \hline D & E & 3 \\ \hline\end{tabular} \] $\textbf{(A)}\ 32 \qquad \textbf{(B)}\ 28 \qquad \textbf{(C)}\ 26 \qquad \textbf{(D)}\ 24 \qquad \textbf{(E)}\ 16$

The positive integers $a$ and $b$ are such that the numbers $15a+16b$ and $16a-15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?

What is \[ \sum^{100}_{i=1} \sum^{100}_{j=1} (i+j) ? \] $ \textbf{(A) }100,100 \qquad \textbf{(B) }500,500\qquad \textbf{(C) }505,000 \qquad \textbf{(D) }1,001,000 \qquad \textbf{(E) }1,010,000 \qquad $

Prove that for every prime $p>100$ and every integer $r$, there exist two integers $a$ and $b$ such that $p$ divides $a^2+b^5-r$.

Calculate \[\int\cdots\int \exp\left(-\sum_{1\le i\le j\le n}x_ix_j\right)dx_1\cdots dx_n\]