Found problems: 85335
The centers of the faces of a cube form a regular octahedron of volume $V$. Through each vertex of the cube we may take the plane perpendicular to the long diagonal from the vertex. These planes also form a regular octahedron. Show that its volume is $27V$.
Inside the convex quadrilateral $ABCD$ lies the point $'M$. Reflect it symmetrically with respect to the midpoints of the sides of the quadrilateral and connect the obtained points so that they form a convex quadrilateral. Prove that the area of this quadrilateral does not depend on the choice of the point $M$.
Tamika selects two different numbers at random from the set $ \{ 8,9,10\} $ and adds them. Carlos takes two different numbers at random from the set $ \{ 3,5,6\} $ and multiplies them. What is the probability that Tamika's result is greater than Carlos' result?
$ \text{(A)}\ \frac{4}{9}\qquad\text{(B)}\ \frac{5}{9}\qquad\text{(C)}\ \frac{1}{2}\qquad\text{(D)}\ \frac{1}{3}\qquad\text{(E)}\ \frac{2}{3} $
Prove the inequality $$(x^3+2y^2+3z)(4y^3+5z^2+6x)(7z^3+8x^2+9y)\geq720(xy+yz+xz)$$ for $x, y, z \geq 1$.
[i]Proposed by K. Kokhas[/i]
Let $V$ be a rectangular prism with integer side lengths. The largest face has area $240$ and the smallest face has area $48$. A third face has area $x$, where $x$ is not equal to $48$ or $240$. What is the sum of all possible values of $x$?
Find all functions $f: \mathbb R^+ \to \mathbb R^+$ such that \[ f \left( x+\frac{f(xy)}{x} \right) = f(xy) f \left( y + \frac 1y \right) \] holds for all $x,y\in\mathbb R^+.$
Prove whether or not there exist natural numbers $n,k$ where $1\le k\le n-2$ such that
\[\binom{n}{k}^2+\binom{n}{k+1}^2=\binom{n}{k+2}^4 \]
Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers with the property that
$$(a_{n+1})^2 + a_na_{n+2} \leq a_n + a_{n+2}$$
for all positive integers $n$. Show that $a_{2022}\leq 1$.
Let $(F_{n})_{n\geq1}$ the Fibonacci sequence. Find all $n \in \mathbb{N}$ such that for every $k=0,1,...,F_{n}$
\[ {F_{n}\choose k} \equiv (-1)^{k} \ (mod \ F_{n}+1) \]
Inside a circle $k$ of radius $R$ some round spots are made. The area of each spot is $1$. Every radius of circle $k$, as well as every circle concentric with $k$, meets in no more than one spot. Prove that the total area of all the spots is less than
$$\pi\sqrt R+\frac12R\sqrt R.$$
Let $\triangle ABC$ be a triangle and let $X$ be a point in the interior of the triangle. The second intersection points of the lines $XA,XB$ and $XC$ with the circumcircle of $\triangle ABC$ are $P,Q$ and $R$. Let $U$ be a point on the ray $XP$ (these are the points on the line $XP$ such that $P$ and $U$ lie on the same side of $X$). The line through $U$ parallel to $AB$ intersects $BQ$ in $V$ . The line through $U$ parallel to $AC$ intersects $CR$ in $W$. Prove that $Q, R, V$ , and $W$ lie on a circle.
Given $ n$ countries with three representatives each, $ m$ committees $ A(1),A(2), \ldots, A(m)$ are called a cycle if
[i](i)[/i] each committee has $ n$ members, one from each country;
[i](ii)[/i] no two committees have the same membership;
[i](iii)[/i] for $ i \equal{} 1, 2, \ldots,m$, committee $ A(i)$ and committee $ A(i \plus{} 1)$ have no member in common, where $ A(m \plus{} 1)$ denotes $ A(1);$
[i](iv)[/i] if $ 1 < |i \minus{} j| < m \minus{} 1,$ then committees $ A(i)$ and $ A(j)$ have at least one member in common.
Is it possible to have a cycle of 1990 committees with 11 countries?
Jeffrey stands on a straight horizontal bridge that measures $20000$ meters across. He wishes to place a pole vertically at the center of the bridge so that the sum of the distances from the top of the pole to the two ends of the bridge is $20001$ meters. To the nearest meter, how long of a pole does Jeffrey need?
$ABCD$ is a convex quadrilateral. We draw its diagnals to divide the quadrilateral to four triabgles. $P$ is the intersection of diagnals. $I_{1},I_{2},I_{3},I_{4}$ are excenters of $PAD,PAB,PBC,PCD$(excenters corresponding vertex $P$). Prove that $I_{1},I_{2},I_{3},I_{4}$ lie on a circle iff $ABCD$ is a tangential quadrilateral.
Determine the largest positive integer $k$ for which there exists a simple graph $G$ of $2014$ vertices that simultaneously satisfies the following conditions:
$a)$ $G$ does not contain triangles
$b)$ For each $i$ between $1$ and $k$, inclusive, at least one vertex of $G$ has degree $i$
$c)$ No vertex of $G$ has a degree greater than $k$
In an exotic country, the National Bank issues coins that can take any value in the interval $[0, 1]$. Find the smallest constant $c > 0$ such that the following holds, no matter the situation in that country:
[i]Any citizen of the exotic country that has a finite number of coins, with a total value of no more than $1000$, can split those coins into $100$ boxes, such that the total value inside each box is at most $c$.[/i]
Find the greatest value of the expression \[ \frac{1}{x^2-4x+9}+\frac{1}{y^2-4y+9}+\frac{1}{z^2-4z+9} \] where $x$, $y$, $z$ are nonnegative real numbers such that $x+y+z=1$.
Find all pairs of prime numbers $p$ and $q$ such that:
\[
p(p + q) = q^p+ 1.
\]
Show that there are infinitely many pairs $(x, y)$ of rational numbers such that $x^3 +y^3 =9$.
Let $ABC$ be a triangle and $\omega$ its incircle. $\omega$ touches $AC,AB$ at $E,F$, respectively. Let $P$ be a point on $EF$. Let $\omega_1=(BFP), \omega_2=(CEP)$. The parallel line through $P$ to $BC$ intersects $\omega_1,\omega_2$ at $X,Y$, respectively. Show that $BX=CY$.
Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.
How many distinct sets are there such that each set contains only non-negative powers of $2$ or $3$ and sum of its elements is $2014$?
$
\textbf{(A)}\ 64
\qquad\textbf{(B)}\ 60
\qquad\textbf{(C)}\ 54
\qquad\textbf{(D)}\ 48
\qquad\textbf{(E)}\ \text{None of the preceding}
$
Let $P,Q,R,S$ be non constant polynomials with real coefficients, such that $P(Q(x))=R(S(x)) $ and the degree of $P$ is multiple of the degree of $R. $ Prove that there exists a polynomial $T$ with real coefficients such that
$$\displaystyle P(x)=R(T(x))$$
Compute all real values of $b$ such that, for $f(x) = x^2+bx-17, f(4)=f'(4)$.
Define $a_k=2^{2^{k-2013}}+k$ for all integers $k$. Simplify $$(a_0+a_1)(a_1-a_0)(a_2-a_1)\cdots(a_{2013}-a_{2012}).$$