Found problems: 260
[color=darkred]A circle $ \Omega_1$ is tangent outwardly to the circle $ \Omega_2$ of bigger radius. Line $ t_1$ is tangent at points $ A$ and $ D$ to the circles $ \Omega_1$ and $ \Omega_2$ respectively. Line $ t_2$, parallel to $ t_1$, is tangent to the circle $ \Omega_1$ and cuts $ \Omega_2$ at points $ E$ and $ F$. Point $ C$ belongs to the circle $ \Omega_2$ such that $ D$ and $ C$ are separated by the line $ EF$. Denote $ B$ the intersection of $ EF$ and $ CD$. Prove that circumcircle of $ ABC$ is tangent to the line $ AD$.[/color]
Let $A$ be the largest subset of $\{1,\dots,n\}$ such that for each $x\in A$, $x$ divides at most one other element in $A$. Prove that \[\frac{2n}3\leq |A|\leq \left\lceil \frac{3n}4\right\rceil. \]
Let $ A$ and $ B$ be the endpoints of a semicircular arc of radius $ 2$. The arc is divided into seven congruent arcs by six equally spaced points $ C_1,C_2,\ldots,C_6$. All chords of the form $ \overline{AC_i}$ or $ \overline{BC_i}$ are drawn. Let $ n$ be the product of the lengths of these twelve chords. Find the remainder when $ n$ is divided by $ 1000$.
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]
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}$
Can the cells of a $2002 \times 2002$ table be filled with the numbers from $1$ to $2002^2$ (one per cell) so that for any cell we can find three numbers $a, b, c$ in the same row or column (or the cell itself) with $a = bc$?
Bernardo chooses a three-digit positive integer $N$ and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer $S$. For example, if $N=749$, Bernardo writes the numbers 10,444 and 3,245, and LeRoy obtains the sum $S=13,689$. For how many choices of $N$ are the two rightmost digits of $S$, in order, the same as those of $2N$?
${ \textbf{(A)}\ 5\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 15\qquad\textbf{(D}}\ 20\qquad\textbf{(E)}\ 25 $
Let $ P(x)$ be a polynomial with a non-negative coefficients. Prove that if the inequality $ P\left(\frac {1}{x}\right)P(x)\geq 1$ holds for $ x \equal{} 1$, then this inequality holds for each positive $ x$.
Find all natural numbers $ x,y$ such that $ y| (x^{2}+1)$ and $ x^{2}| (y^{3}+1)$.
Prove that there exist infinitely many integers $n$ such that $n, n+1, n+2$ are each the sum of the squares of two integers.
The altitude $CH$ of a right triangle $ABC$, with $\angle{C}=90$, cut the angles bisectors $AM$ and $BN$ at $P$ and $Q$, and let $R$ and $S$ be the midpoints of $PM$ and $QN$. Prove that $RS$ is parallel to the hypotenuse of $ABC$
Prove that for any non-zero real numbers $a, b, c,$
\[\frac{(b+c-a)^2}{(b+c)^2+a^2} + \frac{(c+a-b)^2}{(c+a)^2+b^2} + \frac{(a+b-c)^2}{(a+b)^2+c^2} \geq \frac 35.\]
Let $ n \ge 1$ and $ k \ge 3$ be integers. A circle is divided into $ n$ sectors $ a_1,a_2,\dots,a_n$. We will color the $ n$ sectors with $ k$ different colors such that $ a_i$ and $ a_{i \plus{} 1}$ have different color for each $ i \equal{} 1,2,\dots,n$ where $ a_{n \plus{} 1}\equal{}a_1$. Find the number of ways to do such coloring.
In acute-angled triangle $ABC$, a semicircle with radius $r_a$ is constructed with its base on $BC$ and tangent to the other two sides. $r_b$ and $r_c$ are defined similarly. $r$ is the radius of the incircle of $ABC$. Show that \[ \frac{2}{r} = \frac{1}{r_a} + \frac{1}{r_b} + \frac{1}{r_c}. \]
Let $ T$ be a bounded linear operator on a Hilbert space $ H$, and assume that $ \|T^n \| \leq 1$ for some natural number $ n$. Prove the existence of an invertible linear operator $ A$ on $ H$ such that $ \| ATA^{\minus{}1} \| \leq 1$.
[i]E. Druszt[/i]
Let $T$ be a non-empty finite subset of positive integers $\ge 1$. A subset $S$ of $T$ is called [b]good [/b] if for every integer $t\in T$ there exists an $s$ in $S$ such that $gcd(t,s) >1$. Let
\[A={(X,Y)\mid X\subseteq T,Y\subseteq T,gcd(x,y)=1 \text{for all} x\in X, y\in Y}\]
Prove that :
$a)$ If $X_0$ is not [b]good[/b] then the number of pairs $(X_0,Y)$ in $A$ is [b]even[/b].
$b)$ the number of good subsets of $T$ is [b]odd[/b].
For each positive integer $n\leq 49$ we define the numbers $a_n = 3n+\sqrt{n^2-1}$ and $b_n=2(\sqrt{n^2+n}+\sqrt{n^2-n})$. Prove that there exist two integers $A,B$ such that
\[ \sqrt{a_1-b_1}+\sqrt{a_2-b_2} + \cdots + \sqrt{a_{49}-b_{49}} = A+B\sqrt2. \]
Here is a problem given at the mathematical test at some school:
[i]The hypotenuse of the right triangle is 12 inches. The height (distance from the opposite vertex to the hypotenuse) is 12 inches. Find the area of the triangle[/i]
Everybody in the class got the answer $42$ square inches, except for the two best students. Can you explain why the two best students could not get the same answer as the majority?
Consider a circle $(O)$ and two fixed points $B,C$ on $(O)$ such that $BC$ is not the diameter of $(O)$. $A$ is an arbitrary point on $(O)$, distinct from $B,C$. Let $D,J,K$ be the midpoints of $BC,CA,AB$, respectively, $E,M,N$ be the feet of perpendiculars from $A$ to $BC$, $B$ to $DJ$, $C$ to $DK$, respectively. The two tangents at $M,N$ to the circumcircle of triangle $EMN$ meet at $T$. Prove that $T$ is a fixed point (as $A$ moves on $(O)$).
Two circles are tangent to each other internally at a point $\ T $. Let the chord $\ AB $ of the larger circle be tangent to the smaller circle at a point $\ P $. Prove that the line $\ TP $ bisects $\ \angle ATB $.
Let $ABCD$ be a convex quadrilateral. Lines $AB$ and $CD$ meet at $P$, and lines $AD$ and $BC$ meet at $Q$. Let $O$ be a point in
the interior of $ABCD$ such that $\angle BOP = \angle DOQ$. Prove that
$\angle AOB +\angle COD = 180$.
Find a $n\in\mathbb{N}$ such that for all primes $p$, $n$ is divisible by $p$ if and only if $n$ is divisible by $p-1$.
Prove that there are infinitely many positive integers $ n$ such that $ n(n\plus{}1)$ can be represented as a sum of two positive squares in at least two different ways. (Here $ a^{2}\plus{}b^{2}$ and $ b^{2}\plus{}a^{2}$ are considered as the same representation.)
Find all monic polynomials $f(x)$ in $\mathbb Z[x]$ such that $f(\mathbb Z)$ is closed under multiplication.
[i]By Mohsen Jamali[/i]
In an acute-angled triangle $ABC$, we consider the feet $H_a$ and $H_b$ of the altitudes from $A$ and $B$, and the intersections $W_a$ and $W_b$ of the angle bisectors from $A$ and $B$ with the opposite sides $BC$ and $CA$ respectively. Show that the centre of the incircle $I$ of triangle $ABC$ lies on the segment $H_aH_b$ if and only if the centre of the circumcircle $O$ of triangle $ABC$ lies on the segment $W_aW_b$.