This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 95

1998 Gauss, 6
Tags: gauss
In the multiplication question, the sum of the digits in the four boxes is [img]http://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNy83L2NmMTU0MzczY2FhMGZhM2FjMjMwZDcwYzhmN2ViZjdmYjM4M2RmLnBuZw==&rn=U2NyZWVuc2hvdCAyMDE3LTAyLTI1IGF0IDUuMzguMjYgUE0ucG5n[/img] $\textbf{(A)}\ 13 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 27 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 22$

1991 Arnold's Trivium, 39
Tags: calculus , trigonometry , gauss , integration
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.

1999 Gauss, 4
Tags: gauss
$1+\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}$ is equal to $\textbf{(A)}\ \dfrac{15}{8} \qquad \textbf{(B)}\ 1\dfrac{3}{14} \qquad \textbf{(C)}\ \dfrac{11}{8} \qquad \textbf{(D)}\ 1\dfrac{3}{4} \qquad \textbf{(E)}\ \dfrac{7}{8}$

1999 Gauss, 25
Tags: gauss
In a softball league, after each team has played every other team 4 times, the total accumulated points are: Lions 22, Tigers 19, Mounties 14, and Royals 12. If each team received 3 points for a win, 1 point for a tie and no points for a loss, how many games ended in a tie? $\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 10$

1951 Miklós Schweitzer, 16
Tags: calculus , advanced fields , gauss , integration
Let $ \mathcal{F}$ be a surface which is simply covered by two systems of geodesics such that any two lines belonging to different systems form angles of the same opening. Prove that $ \mathcal{F}$ can be developed (that is, isometrically mapped) into the plane.

2003 Iran MO (3rd Round), 1
Tags: number theory , gauss , relatively prime
suppose this equation: x <sup>2</sup> +y <sup>2</sup> +z <sup>2</sup> =w <sup>2</sup> . show that the solution of this equation ( if w,z have same parity) are in this form: x=2d(XZ-YW), y=2d(XW+YZ),z=d(X <sup>2</sup> +Y <sup>2</sup> -Z <sup>2</sup> -W <sup>2</sup> ),w=d(X <sup>2</sup> +Y <sup>2</sup> +Z <sup>2</sup> +W <sup>2</sup> )

1989 China Team Selection Test, 3
Tags: combinatorial geometry , calculus , algebra , gauss , derivative , polynomial
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 $|\alpha| = 1.$

1998 Gauss, 15
Tags: gauss
The diagram shows a magic square in which the sums of the numbers in any row, column or diagonal are equal. What is the value of $n$? $\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 11$

1994 China Team Selection Test, 2
Tags: algebra , polynomial , calculus , integration , gauss , prime number , number theory
Given distinct prime numbers $p$ and $q$ and a natural number $n \geq 3$, find all $a \in \mathbb{Z}$ such that the polynomial $f(x) = x^n + ax^{n-1} + pq$ can be factored into 2 integral polynomials of degree at least 1.

1998 Gauss, 5
Tags: gauss
If a machine produces $150$ items in one minute, how many would it produce in $10$ seconds? $\textbf{(A)}\ 10 \qquad \textbf{(B)}\ 15 \qquad \textbf{(C)}\ 20 \qquad \textbf{(D)}\ 25 \qquad \textbf{(E)}\ 30$

2006 Turkey Team Selection Test, 1
Tags: induction , modular arithmetic , quadratic , gauss , prime number , number theory
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 $x_{2006}$.

1999 Gauss, 3
Tags: gauss
Susan wants to place 35.5 kg of sugar in small bags. If each bag holds 0.5 kg, how many bags are needed? $\textbf{(A)}\ 36 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 53 \qquad \textbf{(D)}\ 70 \qquad \textbf{(E)}\ 71$

1998 Gauss, 1
Tags: gauss
The value of $\frac{1998- 998}{1000}$ is $\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 1000 \qquad \textbf{(C)}\ 0.1 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 0.001$

1999 Gauss, 14
Tags: gauss
Which of the following numbers is an odd integer, contains the digit 5, is divisible by 3, and lies between $12^2$ and $13^2$? $\textbf{(A)}\ 105 \qquad \textbf{(B)}\ 147 \qquad \textbf{(C)}\ 156 \qquad \textbf{(D)}\ 165 \qquad \textbf{(E)}\ 175$

1998 Gauss, 23
Tags: gauss
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? $\textbf{(A)}\ 300 \text{cm}^2 \qquad \textbf{(B)}\ 800 \text{cm}^2 \qquad \textbf{(C)}\ 1200 \text{cm}^2 \qquad \textbf{(D)}\ 600 \text{cm}^2 \qquad \textbf{(E)}\ 0 \text{cm}^2$

2010 Sharygin Geometry Olympiad, 2
Tags: geometry , circumcircle , trigonometry , gauss , euler , homothety , geometric transformation
Bisectors $AA_1$ and $BB_1$ of a right triangle $ABC \ (\angle C=90^\circ )$ meet at a point $I.$ Let $O$ be the circumcenter of triangle $CA_1B_1.$ Prove that $OI \perp AB.$

2008 China Team Selection Test, 3
Tags: polynomial , combinatorial geometry , geometry , algebra , gauss , complex number
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.

PEN A Problems, 23
Tags: number theory , quadratic , gauss , modular arithmetic , relatively prime , divisibility theory
(Wolstenholme's Theorem) Prove that if \[1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{p-1}\] is expressed as a fraction, where $p \ge 5$ is a prime, then $p^{2}$ divides the numerator.

PEN Q Problems, 12
Tags: polynomial , algebra , gauss
Prove that if the integers $a_{1}$, $a_{2}$, $\cdots$, $a_{n}$ are all distinct, then the polynomial \[(x-a_{1})^{2}(x-a_{2})^{2}\cdots (x-a_{n})^{2}+1\] cannot be expressed as the product of two nonconstant polynomials with integer coefficients.

PEN E Problems, 14
Tags: algebra , function , number theory , polynomial , logarithm , limit , gauss
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}$.