Found problems: 85335
Suppose two regular pyramids with the same base $ABC$: $P-ABC$ and $Q-ABC$ are circumscribed by the same sphere. If the angle formed by one of the lateral face and the base of pyramid $P-ABC$ is $\frac{\pi}{4}$, find the tangent value of the angle formed by one of the lateral face and the base of the pyramid $Q-ABC$.
There are $1001$ points in the plane such that no three are collinear. The points are joined by $1001$ line segments such that each point is an endpoint of exactly two of the line segments.
Prove that there does not exist a straight line in the plane that intersects each of the $1001$ segments in an interior point.
[i]An interior point of a line segment is a point of the line segment that is not one of the two endpoints.[/i]
Determine the greatest common divisor of the numbers:
$$5^5-5, 7^7-7, 9^9-9 ,..., 2017^{2017}-2017,$$
If $x=\frac{1}{3}$, what is the value, rounded to $100$ decimal digits, of $\sum_{n=0}^{7}\frac{2^n}{1+x^{2^n}}$?
Let $ABC$ be a triangle with $AB = 3$, $BC = 5$, $AC = 7$, and let $ P$ be a point in its interior. If $G_A$, $G_B$, $G_C$ are the centroids of $\vartriangle PBC$, $\vartriangle PAC$, $\vartriangle PAB$, respectively, find the maximum possible area of $\vartriangle G_AG_BG_C$.
Members of the State Duma formed factions in such a way that for any two fractions $A $ and $B$ (not necessarily different), $\overline{A \cup B}$ is also faction ($\overline{C}$ denotes the set of all members of the Duma, not including in $C$). Prove that for any two factions $A$ and $B$, $A \cup B$ is also a faction.
Let $a$, $b$, $c$ be positive integers such that $abc + bc + c = 2014$. Find the minimum possible value of $a + b + c$.
During a break, $n$ children at school sit in a circle around their teacher to play a game. The teacher walks clockwise close to the children and hands out candies to some of them according to the following rule:
He selects one child and gives him a candy, then he skips the next child and gives a candy to the next one, then he skips 2 and gives a candy to the next one, then he skips 3, and so on.
Determine the values of $n$ for which eventually, perhaps after many rounds, all children will have at least one candy each.
Let $k,n>1$ be integers such that the number $p=2k-1$ is prime. Prove that, if the number $\binom{n}{2}-\binom{k}{2}$ is divisible by $p$, then it is divisible by $p^2$.
Let $ABCD$ be rhombus $ABCD$ where the triangles $ABD$ and $BCD$ are equilateral. Let $M$ and $N$ be points on the sides $BC$ and $CD$, respectively, such that $\angle MAN = 30^o$. Let $X$ be the intersection point of the diagonals $AC$ and $BD$. Prove that $\angle XMN = \angle\ DAM$ and $\angle XNM = \angle BAN$.
A triangle is constructed with the lengths of the sides chosen from the set $\{2, 3, 5, 8, 13, 21, 34, 55, 89, 144\}$. Show that this triangle must be isosceles.
(A triangle is isosceles if it has at least two sides the same length.)
Determine $n \in \Bbb{N}$ such that $n^2 + 2$ divides $2 + 2001n.$
In the interior of a circle centred at $O$ consider the $1200$ points $A_1,A_2,\ldots ,A_{1200}$, where for every $i,j$ with $1\le i\le j\le 1200$, the points $O,A_i$ and $A_j$ are not collinear. Prove that there exist the points $M$ and $N$ on the circle, with $\angle MON=30^{\circ}$, such that in the interior of the angle $\angle MON$ lie exactly $100$ points.
Postive real numbers $a, b, c$ satisfy $abc=1$. Show that $$\frac{a^3+a^2}{1+bc}+\frac{b^3+b^2}{1+ca}+\frac{c^3+c^2}{1+ab}\geq3.$$
Determine the integers $a, b, c$ for which
$$\frac{a+1}{3}=\frac{b+2}{4}=\frac{5}{c+3}$$
Points $K$ and $L$ are taken on the sides $BC$ and $CD$ of a square $ABCD$ so that $\angle AKB = \angle AKL$. Find $\angle KAL$.
The area of the smallest square that will contain a circle of radius 4 is
$\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 64 \qquad \textbf{(E)}\ 128$
Determine all functions $ f: \mathbb{R} \mapsto \mathbb{R}$ with $ x,y \in \mathbb{R}$ such that
\[ f(x \minus{} f(y)) \equal{} f(x\plus{}y) \plus{} f(y)\]
Let $A_1,A_2,\dots, A_n$ be finite, nonempty sets. Define the function
\[f(t)=\sum_{k=1}^n \sum_{1\leq i_1<i_2<\dots<i_k\leq n} (-1)^{k-1}t^{|A_{i_1}\cup A_{i_2}\cup \dots\cup A_{i_k}|}.\]
Prove that $f$ is nondecreasing on $[0,1].$
($|A|$ denotes the number of elements in $A.$)
Let $A$ be a set of functions $f : R\to R$.
For all $f_1, f_2 \in A$ there exists a $f_3 \in A$ such that $f_1(f_2(y) - x)+ 2x = f_3(x + y)$ for all $x, y \in R$.
Prove that for all $f \in A$, we have $f(x - f(x))= 0$ for all $x \in R$.
Positive integers $ a,b,$ and $ c$ are randomly and independently selected with replacement from the set $ \{ 1,2,3,\dots,2010 \}.$ What is the probability that $ abc \plus{} ab \plus{} a$ is divisible by $ 3$?
$ \textbf{(A)}\ \dfrac{1}{3} \qquad\textbf{(B)}\ \dfrac{29}{81} \qquad\textbf{(C)}\ \dfrac{31}{81} \qquad\textbf{(D)}\ \dfrac{11}{27} \qquad\textbf{(E)}\ \dfrac{13}{27}$
Let $a_{0,1}, a_{0,2}, . . . , a_{0, 2016}$ be positive real numbers. For $n\geq 0$ and $1 \leq k < 2016$ set $$a_{n+1,k} = a_{n,k} +\frac{1}{2a_{n,k+1}} \ \ \text{and} \ \ a_{n+1,2016} = a_{n,2016} +\frac{1}{2a_{n,1}}.$$
Show that $\max_{1\leq k \leq 2016} a_{2016,k} > 44.$
$ABCD$ is a square such that $AB$ lies on the line $y=x+4$ and points $C$ and $D$ lie on the graph of parabola $y^2=x$. Compute the sum of all possible areas of $ABCD$.
A line through the incenter $I$ of triangle $ABC$, perpendicular to $AI$, intersects $AB$ at $P$ and $AC$ at $Q$. Prove that the circle tangent to $AB$ at $P$ and to $AC$ at $Q$ is also tangent to the circumcircle of triangle $ABC$.
Let $a$, $b$, $c$ and $d$ be positive real numbers with $a+b+c+d = 4$. Prove that $\frac{a}{b^2 + 1} + \frac{b}{c^2+1} + \frac{c}{d^2+1} + \frac{d}{a^2+1} \geq 2$.