Found problems: 85335
Let $OABC$ be a trihedral angle such that \[ \angle BOC = \alpha, \quad \angle COA = \beta, \quad \angle AOB = \gamma , \quad \alpha + \beta + \gamma = \pi . \] For any interior point $P$ of the trihedral angle let $P_1$, $P_2$ and $P_3$ be the projections of $P$ on the three faces. Prove that $OP \geq PP_1+PP_2+PP_3$.
[i]Constantin Cocea[/i]
Find the maximal $x$ such that the expression $4^{27} + 4^{1000} + 4^x$ is the exact square.
$ABCDEF$ is a regular hexagon. Let $R$ be the overlap between $\vartriangle ACE$ and $\vartriangle BDF$. What is the area of $R$ divided by the area of $ABCDEF$?
Let $AD$ be a bisector of triangle $ABC$. Points $M$ and $N$ are projections of $B$ and $C$ respectively to $AD$. The circle with diameter $MN$ intersects $BC$ at points $X$ and $Y$. Prove that $\angle BAX = \angle CAY$.
For which digits $a$ do exist integers $n \geq 4$ such that each digit of $\frac{n(n+1)}{2}$ equals $a \ ?$
Find all triples $(a,b,c)$ of real numbers such that $|2a + b| \ge 4$ and $|ax^2 + bx + c| \le 1$ $ \forall x \in [-1, 1]$.
A harmonic progression is a sequence of numbers such that their reciprocals are in arithmetic progression.
Let $S_n$ represent the sum of the first $n$ terms of the harmonic progression; for example $S_3$ represents the sum of the first three terms. If the first three terms of a harmonic progression are $3,4,6$, then:
$ \textbf{(A)}\ S_4=20 \qquad\textbf{(B)}\ S_4=25\qquad\textbf{(C)}\ S_5=49\qquad\textbf{(D)}\ S_6=49\qquad\textbf{(E)}\ S_2=\frac12 S_4 $
Show that any graph with $6$ points has a triangle or three points which are not joined to each other.
Let $x_{1}$ be a positive real number and for every integer $n \geq 1$ let $x_{n+1} = 1 + x_{1}x_{2}\ldots x_{n-1}x_{n}$.
If $x_{5} = 43$, what is the sum of digits of the largest prime factors of $x_{6}$?
A rook is randomly placed on an otherwise empty $8 \times 8$ chessboard. Owen makes moves with the rook by randomly choosing $1$ of the $14$ possible moves. Find the expected value of the number of moves it takes Owen to move the rook to the top left square. Note that a rook can move any number of squares either in the horizontal or vertical direction each move.
Call a positive integer "useful but not optimized " (!), if it can be written as a sum of distinct powers of $3$ and powers of $5$.
Prove that there exist infinitely many positive integers which they are not "useful but not optimized".
(e.g. $37=(3^0+3^1+3^3)+(5^0+5^1)$ is a " useful but not optimized" number)
[i]Proposed by Mohsen Jamali[/i]
What is the largest number of acute angles that a convex hexagon can have?
$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$
Let $a, b, n$ be positive integers, $b < 10$ and $2^n = 10a + b$.
Prove that if $n > 3$, then $6$ divides $ab$.
For a given positive integer $n$, we consider the set $M$ of all intervals of the form $[l, r]$, where the integers $l$ and $r$ satisfy the condition $0 \leq l < r \leq n$. What largest number of elements of $M$ can be chosen so that each chosen interval completely contains at most one other selected interval?
[i]Proposed by Anton Trygub[/i]
a) Compute
$$\lim_{n\to\infty}n\int^1_0\left(\frac{1-x}{1+x}\right)^ndx.$$
b) Let $k\ge1$ be an integer. Compute
$$\lim_{n\to\infty}n^{k+1}\int^1_0\left(\frac{1-x}{1+x}\right)^nx^kdx.$$
Let $M,N,P$ be midpoints of $BC,AC$ and $AB$ of triangle $\triangle ABC$ respectively. $E$ and $F$ are two points on the segment $\overline{BC}$ so that $\angle NEC = \frac{1}{2} \angle AMB$ and $\angle PFB = \frac{1}{2} \angle AMC$. Prove that $AE=AF$.
[i]Proposed by Alireza Dadgarnia[/i]
A group of $100$ students from different countries meet at a mathematics competition. Each student speaks the same number of languages, and, for every pair of students $A$ and $B$, student $A$ speaks some language that student $B$ does not speak, and student $B$ speaks some language that student $A$ does not speak. What is the least possible total number of languages spoken by all the students?
$
\textbf{(A) }9 \qquad
\textbf{(B) }10 \qquad
\textbf{(C) }12 \qquad
\textbf{(D) }51 \qquad
\textbf{(E) }100 \qquad
$
Bricklayer Brenda would take $ 9$ hours to build a chimney alone, and bricklayer Brandon would take $ 10$ hours to build it alone. When they work together they talk a lot, and their combined output is decreased by $ 10$ bricks per hour. Working together, they build the chimney in $ 5$ hours. How many bricks are in the chimney?
$ \textbf{(A)}\ 500 \qquad
\textbf{(B)}\ 900 \qquad
\textbf{(C)}\ 950 \qquad
\textbf{(D)}\ 1000 \qquad
\textbf{(E)}\ 1900$
A $1\times 2$ rectangle is inscribed in a semicircle with longer side on the diameter. What is the area of the semicircle?
$\textbf{(A)}\ \frac\pi2 \qquad \textbf{(B)}\ \frac{2\pi}3 \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ \frac{4\pi}3 \qquad \textbf{(E)}\ \frac{5\pi}3$
Let be a natural number $ n, $ a complex number $ a, $ and two matrices $ \left( a_{pq}\right)_{1\le q\le n}^{1\le p\le n} ,\left( b_{pq}\right)_{1\le q\le n}^{1\le p\le n}\in\mathcal{M}_n(\mathbb{C} ) $ such that
$$ b_{pq} =a^{p-q}\cdot a_{pq},\quad\forall p,q\in\{ 1,2,\ldots ,n\} . $$
Calculate the determinant of $ B $ (in function of $ a $ and the determinant of $ A $ ).
A rectangular piece of paper $ABCD$ has sides of lengths $AB = 1$, $BC = 2$. The rectangle is folded in half such that $AD$ coincides with $BC$ and $EF$ is the folding line. Then fold the paper along a line $BM$ such that the corner $A$ falls on line $EF$. How large, in degrees, is $\angle ABM$?
A circle passing through the vertices $B$ and $D$ of quadrilateral $ABCD$ meets $AB$, $BC$, $CD$, and $DA$ at points $K$, $L$, $M$, and $N$ respectively. A circle passing through $K$ and $M$ meets $AC$ at $P$ and $Q$. Prove that $L$, $N$, $P$, and $Q$ are concyclic.
There is a frog jumping on a $ 2k \times 2k$ chessboard, composed of unit squares. The frog's jumps are $ \sqrt{1 \plus{} k^2}$ long and they carry the frog from the center of a square to the center of another square. Some $ m$ squares of the board are marked with an $ \times$, and all the squares into which the frog can jump from an $ \times$'d square (whether they carry an $ \times$ or not) are marked with an $ \circ$. There are $ n$ $ \circ$'d squares. Prove that $ n \ge m$.
Price of the book increased by $20\%$, and then decreased by $10\%$. How many percents should we decrease current price so we get a price which is $54\%$ percent of an original one?
Find all complex roots (with multiplicities) of the polynomial
$$p(x)=\sum_{n=1}^{2008}(1004-|1004-n|)x^n.$$