Found problems: 95
1998 Gauss, 3
If $S = 6 \times10 000 +5\times 1000+ 4 \times 10+ 3 \times 1$, what is $S$?
$\textbf{(A)}\ 6543 \qquad \textbf{(B)}\ 65043 \qquad \textbf{(C)}\ 65431 \qquad \textbf{(D)}\ 65403 \qquad \textbf{(E)}\ 60541$
1999 Gauss, 1
$1999-999+99$ equals
$\textbf{(A)}\ 901 \qquad \textbf{(B)}\ 1099 \qquad \textbf{(C)}\ 1000 \qquad \textbf{(D)}\ 199 \qquad \textbf{(E)}\ 99$
1998 Gauss, 1
The value of $\frac{1998- 998}{1000}$ is
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 1000 \qquad \textbf{(C)}\ 0.1 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 0.001$
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 $|\alpha| = 1.$
2013 AIME Problems, 3
A large candle is $119$ centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the candle takes $10$ seconds to burn down the first centimeter from the top, $20$ seconds to burn down the second centimeter, and $10k$ seconds to burn down the $k$-th centimeter. Suppose it takes $T$ seconds for the candle to burn down completely. Then $\tfrac{T}{2}$ seconds after it is lit, the candle's height in centimeters will be $h$. Find $10h$.
1991 Arnold's Trivium, 12
Find the flux of the vector field $\overrightarrow{r}/r^3$ through the surface
\[(x-1)^2+y^2+z^2=2\]
2024 Korea Junior Math Olympiad (First Round), 6.
Find the number of $ x $ which follows the following :
$ x-\frac{1}{x}=[x]-[\frac{1}{x}] $
$ ( \frac{1}{100} \le x \le {100} ) $
2005 Romania National Olympiad, 1
Let $ABCD$ be a convex quadrilateral with $AD\not\parallel BC$. Define the points $E=AD \cap BC$ and $I = AC\cap BD$. Prove that the triangles $EDC$ and $IAB$ have the same centroid if and only if $AB \parallel CD$ and $IC^{2}= IA \cdot AC$.
[i]Virgil Nicula[/i]
1998 Gauss, 2
The number $4567$ is tripled. The ones digit (units digit) in the resulting number is
$\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 1$
2001 India Regional Mathematical Olympiad, 2
Find all primes $p$ and $q$ such that $p^2 + 7pq + q^2$ is a perfect square.
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 $A\setminus \{(0,0)\}$
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.
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.
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.
2004 Romania Team Selection Test, 4
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 $z^n = z_1\cdot z_2\cdots z_n$.
1951 Miklós Schweitzer, 16
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.
1998 Gauss, 6
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]
$\textbf{(A)}\ 13 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 27 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 22$
2012 IFYM, Sozopol, 4
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.
1999 Gauss, 7
If the numbers $\dfrac{4}{5},81\%$ and $0.801$ are arranged from smallest to largest, the correct order is
$\textbf{(A)}\ \dfrac{4}{5},81\%,0.801 \qquad \textbf{(B)}\ 81\%,0.801,\dfrac{4}{5} \qquad \textbf{(C)}\ 0.801,\dfrac{4}{5},81\% \qquad \textbf{(D)}\ 81\%,\dfrac{4}{5},0.801 \qquad \textbf{(E)}\ \dfrac{4}{5},0.801,81\%$
2009 USA Team Selection Test, 6
Let $ N > M > 1$ be fixed integers. There are $ N$ people playing in a chess tournament; each pair of players plays each other once, with no draws. It turns out that for each sequence of $ M \plus{} 1$ distinct players $ P_0, P_1, \ldots P_M$ such that $ P_{i \minus{} 1}$ beat $ P_i$ for each $ i \equal{} 1, \ldots, M$, player $ P_0$ also beat $ P_M$. Prove that the players can be numbered $ 1,2, \ldots, N$ in such a way that, whenever $ a \geq b \plus{} M \minus{} 1$, player $ a$ beat player $ b$.
[i]Gabriel Carroll.[/i]