Found problems: 95
1998 Gauss, 20
Each of the 12 edges of a cube is coloured either red or green. Every face of the cube has at least one
red edge. What is the smallest number of red edges?
$\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$
1994 China Team Selection Test, 2
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.
2003 Brazil National Olympiad, 1
Find the smallest positive prime that divides $n^2 + 5n + 23$ for some integer $n$.
1998 Gauss, 7
A rectangular field is 80 m long and 60 m wide. If fence posts are placed at the corners and are 10 m
apart along the 4 sides of the field, how many posts are needed to completely fence the field?
$\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 26 \qquad \textbf{(C)}\ 28 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 32$
2005 Iran MO (3rd Round), 2
We define a relation between subsets of $\mathbb R ^n$. $A \sim B\Longleftrightarrow$ we can partition $A,B$ in sets $A_1,\dots,A_n$ and $B_1,\dots,B_n$(i.e $\displaystyle A=\bigcup_{i=1} ^n A_i,\ B=\bigcup_{i=1} ^n B_i,
A_i\cap A_j=\emptyset,\ B_i\cap B_j=\emptyset$) and $A_i\simeq B_i$.
Say the the following sets have the relation $\sim$ or not ?
a) Natural numbers and composite numbers.
b) Rational numbers and rational numbers with finite digits in base 10.
c) $\{x\in\mathbb Q|x<\sqrt 2\}$ and $\{x\in\mathbb Q|x<\sqrt 3\}$
d) $A=\{(x,y)\in\mathbb R^2|x^2+y^2<1\}$ and $A\setminus \{(0,0)\}$
2010 Sharygin Geometry Olympiad, 2
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 $OI \perp AB.$
1998 Gauss, 25
Two natural numbers, $p$ and $q$, do not end in zero. The product of any pair, p and q, is a power of 10
(that is, $10, 100, 1000, 10 000$ , ...). If $p >q$, the last digit of $p – q$ cannot be
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 9$
1998 Gauss, 24
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?
$\textbf{(A)}\ 38 \qquad \textbf{(B)}\ 39 \qquad \textbf{(C)}\ 54 \qquad \textbf{(D)}\ 55 \qquad \textbf{(E)}\ 30$
2018 AMC 12/AHSME, 9
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
$
1998 Gauss, 15
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 $n$?
$\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 11$
1957 AMC 12/AHSME, 45
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}$
1999 Gauss, 14
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, 10
At the waterpark, Bonnie and Wendy decided to race each other down a waterslide. Wendy won by
$0.25$ seconds. If Bonnie’s time was exactly $7.80$ seconds, how long did it take for Wendy to go down
the slide?
$\textbf{(A)}\ 7.80~ \text{seconds} \qquad \textbf{(B)}\ 8.05~ \text{seconds} \qquad \textbf{(C)}\ 7.55~ \text{seconds} \qquad \textbf{(D)}\ 7.15~ \text{seconds} \qquad $
$\textbf{(E)}\ 7.50~ \text{seconds}$
1998 Gauss, 16
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?
$\textbf{(A)}\ 269 \qquad \textbf{(B)}\ 278 \qquad \textbf{(C)}\ 484 \qquad \textbf{(D)}\ 271 \qquad \textbf{(E)}\ 261$
2008 Mongolia Team Selection Test, 2
Given positive integers$ m,n$ such that $ m < n$. Integers $ 1,2,...,n^2$ are arranged in $ n \times n$ board. In each row, $ m$ largest number colored red. In each column $ m$ largest number colored blue. Find the minimum number of cells such that colored both red and blue.
1998 Gauss, 1
The value of $\frac{1998- 998}{1000}$ is
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 1000 \qquad \textbf{(C)}\ 0.1 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 0.001$
2013 Stanford Mathematics Tournament, 10
Given a complex number $z$ such that $z^{13}=1$, find all possible value of $z+z^3+z^4+z^9+z^{10}+z^{12}$.
1991 Arnold's Trivium, 38
Calculate the integral of the Gaussian curvature of the surface
\[z^4+(x^2+y^2-1)(2x^2+3y^2-1)=0\]
1999 Gauss, 17
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?
$\textbf{(A)}\ 34 \qquad \textbf{(B)}\ 36 \qquad \textbf{(C)}\ 107 \qquad \textbf{(D)}\ 152 \qquad \textbf{(E)}\ 245$
1999 Gauss, 2
The integer 287 is exactly divisible by
$\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 6$
1989 China Team Selection Test, 3
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 $|\alpha| = 1.$
2012 Online Math Open Problems, 48
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]
1949 Miklós Schweitzer, 7
Find the complex numbers $ z$ for which the series
\[ 1 \plus{} \frac {z}{2!} \plus{} \frac {z(z \plus{} 1)}{3!} \plus{} \frac {z(z \plus{} 1)(z \plus{} 2)}{4!} \plus{} \cdots \plus{} \frac {z(z \plus{} 1)\cdots(z \plus{} n)}{(n \plus{} 2)!} \plus{} \cdots\]
converges and find its sum.
2008 China Team Selection Test, 3
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.
2009 Miklós Schweitzer, 10
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)$.