Found problems: 85335
Let $n,k$ be integers such that $n\geq k\geq3$. Consider $n+1$ points in a plane (there is no three collinear points) and $k$ different colors, then, we color all the segments that connect every two points. We say that an angle is good if its vertex is one of the initial set, and its two sides aren't the same color. Show that there exist a coloration such that the \\ total number of good angles is greater than $n \binom{k}{2} \lfloor(\frac{n}{k})\rfloor^2$
Let $O,A,B,C$ be variable points in the plane such that $OA=4$, $OB=2\sqrt3$ and $OC=\sqrt {22}$. Find the maximum value of the area $ABC$.
[i]Mihai Baluna[/i]
Find the number of cubic polynomials $p(x) = x^3 + ax^2 + bx + c$, where $a$, $b$, and $c$ are integers in $\{-20, -19,-18, \dots , 18, 19, 20\}$, such that there is a unique integer $m \neq 2$ with $p(m) = p(2)$.
For a regular $n$-gon, let $M$ be the set of the lengths of the segments joining its vertices. Show that the sum of the squares of the elements of $M$ is greater than twice the area of the polygon.
Let $ABC$ be an acute triangle such that $\angle BAC = 45^o$. Let $D$ a point on $AB$ such that $CD \perp AB$. Let $P$ be an internal point of the segment $CD$. Prove that $AP\perp BC$ if and only if $|AP| = |BC|$.
Find maximal value of positive integer $n$ such that there exists subset of $S=\{1,2,...,2001\}$ with $n$ elements, such that equation $y=2x$ does not have solutions in set $S \times S$
Prove that for all positive integers $n$, all complex roots $r$ of the polynomial
\[P(x) = (2n)x^{2n} + (2n-1)x^{2n-1} + \dots + (n+1)x^{n+1} + nx^n + (n+1)x^{n-1} + \dots + (2n-1)x + 2n\]
lie on the unit circle (i.e. $|r| = 1$).
Let $\mathbb{N}$ denote the set of natural numbers, and let $\left( a_i, b_i \right)$, $1 \leq i \leq 9$, be nine distinct tuples in $\mathbb{N} \times \mathbb{N}$. Show that there are three distinct elements in the set $\{ 2^{a_i} 3^{b_i} \colon 1 \leq i \leq 9 \}$ whose product is a perfect cube.
All angles of $ABC$ are in $(30,90)$. Circumcenter of $ABC$ is $O$ and circumradius is $R$. Point $K$ is projection of $O$ to angle bisector of $\angle B$, point $M$ is midpoint $AC$. It is known, that $2KM=R$. Find $\angle B$
Is it true that for $\forall$ prime number $p$, there exist non-constant polynomials $P$ and $Q$ with $P,Q\in \mathbb{Z} [x]$ for which the remainder modulo $p$ of the coefficient in front of $x^n$ in the product $PQ$ is 1 for $n=0$ and $n=4$; $p-2$ for $n=2$ and is 0 for all other $n\geq 0$?
Let $T = \{9^k : k \ \text{is an integer}, 0 \le k \le 4000\}$. Given that $9^{4000}$ has 3817 digits and that its first (leftmost) digit is 9, how many elements of $T$ have 9 as their leftmost digit?
Find all the positive integers $n$ with the property:
there exists an integer $k > 2$ and the positive rational numbers $a_1, a_2, ..., a_k$
such that $a_1 + a_2 + .. + a_k = a_1a_2 . . . a_k = n$.
Given a positive integer $ n$, find the number of $ n$-digit natural numbers consisting of digits 1, 2, 3 in which any two adjacent digits are either distinct or both equal to 3.
Berk tries to guess the two-digit number that Ayca picks. After each guess, Ayca gives a hint indicating the number of digits which match the number picked. If Berk can guarantee to guess Ayca's number in $n$ guesses, what is the smallest possible value of $n$?
$
\textbf{(A)}\ 9
\qquad\textbf{(B)}\ 10
\qquad\textbf{(C)}\ 11
\qquad\textbf{(D)}\ 15
\qquad\textbf{(E)}\ 20
$
Let $ a,b,c$ be positive numbers, satisfying $ abc\geq 1$. Prove that
\[ a^{3} \plus{} b^{3} \plus{} c^{3} \geq ab \plus{} bc \plus{} ca.\]
Let triangle $ABC$ with incenter $I$ satisfy $AB = 3$, $AC = 4$, and $BC = 5$. Suppose that $D$ and $E$ lie on $AB$ and $AC$, respectively, such that $D$, $I$, and $E$ are collinear and $DE \perp AI$. Points $P$ and $Q$ lie on side $BC$ such that $IP = BP$ and $IQ = CQ$, and lines $DP$ and $EQ$ meet at $S$. Compute $SI^2$.
Let $x$ be a real number between 0 and $\tfrac{\pi}{2}$ for which the function $3\sin^2 x + 8\sin x \cos x + 9\cos^2 x$ obtains its maximum value, $M$. Find the value of $M + 100\cos^2x$.
Let $m$ and $n$ be positive integers. Let $S$ be the set of all points $(x, y)$ with integer
coordinates such that $1 \leq x,y \leq m+n-1$ and $m+1 \leq x +y \leq 2m+n-1.$ Let $L$ be the
set of the $3m+3n-3$ lines parallel to one of $x = 0, y = 0,$ or $x + y = 0$ and passing through
at least one point in $S$. For which pairs $(m, n)$ does there exist a subset $T$ of $S$ such that
every line in $L$ intersects an odd number of elements of $T$?
In my isosceles triangle $\vartriangle ABC$ with $AB = CA$, we draw $D$ the midpoint of $BC$. Let $E$ be a point on $AC$ such that $\angle CDE = 60^o$ and $M$ the midpoint of $DE$. Prove that $\angle AME = \angle BMD$.
Consider a solid hemisphere of radius $1$. Find the distance from its center of mass to the base.
Prove that if the circumcircles of the faces of a tetrahedron $ABCD$ have equal radii, then $AB=CD$, $AC=BD$ and $AD=BC$.
Find positive reals $a, b, c$ which maximizes the value of $a+ 2b+ 3c$ subject to the constraint that $9a^2 + 4b^2 + c^2 = 91$
Suppose that $a$, $b$ and $x$ are positive real numbers. Prove that $\log_{ab} x =\dfrac{\log_a x\log_b x}{\log_ax+\log_bx}$.
Let $P(z)=a_nz^n+a_{n-1}z^{n-1}+\ldots+a_mz^m$ be a polynomial with complex coefficients such that $a_m\neq 0, a_n\neq 0$ and $n>m$. Prove that
\[\text{max}_{|z|=1}\{|P(z)|\}\ge\sqrt{2|a_ma_n|+\sum_{k=m}^{n} |a_k|^2}\]
Rational numbers $a$ and $b$ are chosen at random among all rational numbers in the interval $[0,2)$ that can be written as fractions $\tfrac nd$ where $n$ and $d$ are integers with $1\leq d\leq 5$. What is the probability that \[(\cos(a\pi)+i\sin(b\pi))^4\] is a real number?
$\textbf{(A) }\dfrac3{50}\qquad\textbf{(B) }\dfrac4{25}\qquad\textbf{(C) }\dfrac{41}{200}\qquad\textbf{(D) }\dfrac6{25}\qquad\textbf{(E) }\dfrac{13}{50}$