Found problems: 260
2012 Kosovo Team Selection Test, 4
Each term in a sequence $1,0,1,0,1,0...$starting with the seventh is the sum of the last 6 terms mod 10 .Prove that the sequence $...,0,1,0,1,0,1...$ never occurs
PEN B Problems, 5
Let $p$ be an odd prime. If $g_{1}, \cdots, g_{\phi(p-1)}$ are the primitive roots $\pmod{p}$ in the range $1<g \le p-1$, prove that \[\sum_{i=1}^{\phi(p-1)}g_{i}\equiv \mu(p-1) \pmod{p}.\]
1990 India National Olympiad, 5
Let $ a$, $ b$, $ c$ denote the sides of a triangle. Show that the quantity
\[ \frac{a}{b\plus{}c}\plus{}\frac{b}{c\plus{}a}\plus{}\frac{c}{a\plus{}b}\]
must lie between the limits $ 3/2$ and 2. Can equality hold at either limits?
2009 AMC 10, 14
On Monday, Millie puts a quart of seeds, $ 25\%$ of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix of seeds without removing any seeds that are left. Each day the birds eat only $ 25\%$ of the millet in the feeder, but they eat all of the other seeds. On which day, just after Millie has placed the seeds, will the birds find that more than half the seeds in the feeder are millet?
$ \textbf{(A)}\ \text{Tuesday}\qquad \textbf{(B)}\ \text{Wednesday}\qquad \textbf{(C)}\ \text{Thursday} \qquad \textbf{(D)}\ \text{Friday}\qquad \textbf{(E)}\ \text{Saturday}$
2010 Postal Coaching, 5
Let $a, b, c$ be integers such that \[\frac ab+\frac bc+\frac ca= 3\] Prove that $abc$ is a cube of an integer.
1983 Miklós Schweitzer, 4
For which cardinalities $ \kappa$ do antimetric spaces of cardinality $ \kappa$ exist?
$ (X,\varrho)$ is called an $ \textit{antimetric space}$ if $ X$ is a nonempty set, $ \varrho : X^2 \rightarrow [0,\infty)$ is a symmetric map, $ \varrho(x,y)\equal{}0$ holds iff $ x\equal{}y$, and for any three-element subset $ \{a_1,a_2,a_3 \}$ of $ X$ \[ \varrho(a_{1f},a_{2f})\plus{}\varrho(a_{2f},a_{3f}) < \varrho(a_{1f},a_{3f})\] holds for some permutation $ f$ of $ \{1,2,3 \}$.
[i]V. Totik[/i]
2001 AMC 8, 25
There are 24 four-digit whole numbers that use each of the four digits 2, 4, 5 and 7 exactly once. Only one of these four-digit numbers is a multiple of another one. Which of the following is it?
$ \text{(A)}\ 5724\qquad\text{(B)}\ 7245\qquad\text{(C)}\ 7254\qquad\text{(D)}\ 7425\qquad\text{(E)}\ 7542 $
1997 India Regional Mathematical Olympiad, 6
Find the number of unordered pairs $\{ A,B \}$ of subsets of an n-element set $X$ that satisfies the following:
(a) $A \not= B$
(b) $A \cup B = X$
2007 IMS, 3
Prove that $\mathbb R^{2}$ has a dense subset such that has no three collinear points.
2010 Iran MO (3rd Round), 2
$a,b,c$ are positive real numbers. prove the following inequality:
$\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{(a+b+c)^2}\ge \frac{7}{25}(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{a+b+c})^2$
(20 points)