Two numbers have a sum of $32$. If one of the numbers is $ – 36$, what is the other number? $\textbf{(A)}\ 68 \qquad \textbf{(B)}\ -4 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 72 \qquad \textbf{(E)}\ -68$

The number $4567$ is tripled. The ones digit (units digit) in the resulting number is $\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 1$

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}$.

Find the greatest $n$ such that $(z+1)^n = z^n + 1$ has all its non-zero roots in the unitary circumference, e.g. $(\alpha+1)^n = \alpha^n + 1, \alpha \neq 0$ implies $|\alpha| = 1.$

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? $\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 20$

Suppose that \[\sum_{i=1}^{982} 7^{i^2}\] can be expressed in the form $983q + r$, where $q$ and $r$ are integers and $0 \leq r \leq 492$. Find $r$. [i]Author: Alex Zhu[/i]

Calculate the Gauss integral \[\oint\frac{(d\overrightarrow{A},d\overrightarrow{B},\overrightarrow{A}-\overrightarrow{B})}{|\overrightarrow{A}-\overrightarrow{B}|^3}\] where $\overrightarrow{A}$ runs along the curve $x=\cos\alpha$, $y=\sin\alpha$, $z=0$, and $\overrightarrow{B}$ along the curve $x=2\cos^2\beta$, $y=\frac12\sin\beta$, $z=\sin2\beta$. Note: that $\oint$ was supposed to be oiint (i.e. $\iint$ with a circle) but the command does not work on AoPS.

For all integers $n\geq 1$ we define $x_{n+1}=x_1^2+x_2^2+\cdots +x_n^2$, where $x_1$ is a positive integer. Find the least $x_1$ such that 2006 divides $x_{2006}$.

A cube measures $10 \text{cm} \times 10 \text{cm} \times10 \text{cm}$ . Three cuts are made parallel to the faces of the cube as shown creating eight separate solids which are then separated. What is the increase in the total surface area? $\textbf{(A)}\ 300 \text{cm}^2 \qquad \textbf{(B)}\ 800 \text{cm}^2 \qquad \textbf{(C)}\ 1200 \text{cm}^2 \qquad \textbf{(D)}\ 600 \text{cm}^2 \qquad \textbf{(E)}\ 0 \text{cm}^2$

On a large piece of paper, Dana creates a “rectangular spiral” by drawing line segments of lengths, in cm, of 1, 1, 2, 2, 3, 3, 4, 4, ... as shown. Dana’s pen runs out of ink after the total of all the lengths he has drawn is 3000 cm. What is the length of the longest line segment that Dana draws? $\textbf{(A)}\ 38 \qquad \textbf{(B)}\ 39 \qquad \textbf{(C)}\ 54 \qquad \textbf{(D)}\ 55 \qquad \textbf{(E)}\ 30$

Let $n$ be a positive integer, and $x$ be a positive real number. Prove that $$\sum_{k=1}^{n} \left( x \left[\frac{k}{x}\right] - (x+1)\left[\frac{k}{x+1}\right]\right) \leq n,$$ where $[x]$ denotes the largest integer not exceeding $x$.

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\% $

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.

Let $\overline{a_{n}a_{n-1}\ldots a_{1}a_{0}}$ be the decimal representation of a prime positive integer such that $n>1$ and $a_{n}>1$. Prove that the polynomial $P(x)=a_{n}x^{n}+\ldots +a_{1}x+a_{0}$ cannot be written as a product of two non-constant integer polynomials.

Find the number of positive integer n, which follows the following $ \bigstar $ $ n=[\frac{m^3}{2024}] $ $n$ has a positive integer $m$ that follows this equation ($ m \le 1000$)

Let $D$ be a closed disc in the complex plane. Prove that for all positive integers $n$, and for all complex numbers $z_1,z_2,\ldots,z_n\in D$ there exists a $z\in D$ such that $z^n = z_1\cdot z_2\cdots z_n$.