Found problems: 85335
Let $x,y,z$ be positive real numbers.Prove that
$(xy^2+yz^2+zx^2)(x^2y+y^2z+z^2x)(xy+yz+zx)\geq 3(x+y+z)^2(xyz)^2.$
Let $ABCD$ be a cyclic quadrilateral with center $O$.
Suppose the circumcircles of triangles $AOB$ and $COD$ meet again at $G$, while the circumcircles of triangles $AOD$ and $BOC$ meet again at $H$.
Let $\omega_1$ denote the circle passing through $G$ as well as the feet of the perpendiculars from $G$ to $AB$ and $CD$.
Define $\omega_2$ analogously as the circle passing through $H$ and the feet of the perpendiculars from $H$ to $BC$ and $DA$.
Show that the midpoint of $GH$ lies on the radical axis of $\omega_1$ and $\omega_2$.
[i]Proposed by Yang Liu[/i]
Let $P(x)$ denote the product of all (decimal) digits of a natural number $x$. For any positive integer $x_1$, define the sequence $(x_n)$ recursively by $x_{n+1} = x_n + P(x_n)$. Prove or disprove that the sequence $(x_n)$ is necessarily bounded.
How many primes between $2$ and $2^{30}$ are $1$ more than a multiple of $2017$? If $C$ is the correct answer and $A$ is your answer, then your score will be rounded up from $\max(0,25-15|\ln\tfrac{A}{C}|)$.
Is there is a convex quadrilateral which the diagonals divide into four triangles with areas of distinct primes?
Let $f:A^3\to A$ where $A$ is a nonempty set and $f$ satisfies:
(a) for all $x,y\in A$, $f(x,y,y)=x=f(y,y,x)$ and
(b) for all $x_1,x_2,x_3,y_1,y_2,y_3,z_1,z_2,z_3\in A$,
$$f(f(x_1,x_2,x_3),f(y_1,y_2,y_3),f(z_1,z_2,z_3))=f(f(x_1,y_1,z_1),f(x_2,y_2,z_2),f(x_3,y_3,z_3)).$$
Prove that for an arbitrary fixed $a\in A$, the operation $x+y=f(x,a,y)$ is an Abelian group addition.
Let $ABC$ be an isosceles triangle with $AC=BC$, let $M$ be the midpoint of its side $AC$, and let $Z$ be the line through $C$ perpendicular to $AB$. The circle through the points $B$, $C$, and $M$ intersects the line $Z$ at the points $C$ and $Q$. Find the radius of the circumcircle of the triangle $ABC$ in terms of $m = CQ$.
What value of $x$ satisfies
$$x- \frac{3}{4} = \frac{5}{12} - \frac{1}{3}?$$
$\textbf{(A)}\ -\frac{2}{3}\qquad\textbf{(B)}\ \frac{7}{36}\qquad\textbf{(C)}\ \frac{7}{12}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{5}{6}$
Let $ S(n)$ denote the sum of the digits of a natural number $ n$ (in base $ 10$). Prove that for every $ n$, $ S(2n) \le 2S(n) \le 10S(2n)$. Prove also that there is a positive integer $ n$ with $ S(n)\equal{}1996S(3n)$.
Triangle $ABC$ with $\angle BAC=90^{\circ}$ is the base of the pyramid $ABCD$. Moreover, $AD=BD$ and $AB=CD$. Prove that $\angle ACD\ge 30^{\circ}$.
Let $BB_1$ and $CC_1$ be altitudes of triangle $ABC$. The tangents to the circumcircle of $AB_1C_1$ at $B_1$ and $C_1$ meet AB and $AC$ at points $M$ and $N$ respectively. Prove that the common point of circles $AMN$ and $AB_1C_1$ distinct from $A$ lies on the Euler line of $ABC$.
Let $N$ be the midpoint of side $BC$ of triangle $ABC$. Right isosceles triangles $ABM$ and $ACP$ are constructed outside the triangle, with bases $AB$ and $AC$. Prove that $\triangle MNP$ is also a right isosceles triangle.
Different points $A, B, C, D$ lie on a circle with a center at the point $O$ at such way that $\angle AOB$ $= \angle BOC =$ $\angle COD =$ $60^o$. Point $P$ lies on the shorter arc $BC$ of this circle. Points $K, L, M$ are projections of $P$ on lines $AO, BO, CO$ respectively . Show that
(a) the triangle $KLM$ is equilateral,
(b) the area of triangle $KLM$ does not depend on the choice of the position of point $P$ on the shorter arc $BC$
Determine all positive integers in the form $a<b<c<d$ with the property that each of them divides the sum of the other three.
In triangle $ABC$, $\angle BAC= 60^o$. Point $P$ is taken inside the triangle so that angles $\angle APB=\angle BPC= \angle CP A=120^o$. It is known that $AP = a$. Find the area of triangle $BPC$.
Let $f$ be a function with the following properties:
1) $f(n)$ is defined for every positive integer $n$;
2) $f(n)$ is an integer;
3) $f(2)=2$;
4) $f(mn)=f(m)f(n)$ for all $m$ and $n$;
5) $f(m)>f(n)$ whenever $m>n$.
Prove that $f(n)=n$.
Let $a_1,a_2,\dots, a_{17}$ be a permutation of $1,2,\dots, 17$ such that $(a_1-a_2)(a_2-a_3)\dots(a_{17}-a_1)=n^{17}$ .Find the maximum possible value of $n$ .
On the football training there was $n$ footballers - forwards and goalkeepers. They made $k$ goals. Prove that main trainer can give for every footballer squad number from $1$ to $n$ such, that for every goal the difference between squad number of forward and squad number of goalkeeper is more than $n-k$.
[i](S. Berlov)[/i]
Let $ f:\mathbb{R}\longrightarrow\mathbb{R} ,f(x)=\left\{\begin{matrix} \sin x , & x\not\in\mathbb{Q} \\ 0, & x\in\mathbb{Q}\end{matrix}\right. . $
[b]a)[/b] Determine the maximum length of an interval $ I\subset\mathbb{R} $ such that $ f|_I $ is discontinuous everywhere, yet has the intermediate value property.
[b]b)[/b] Study the convergence of the sequence $ \left( x_n\right)_{n\in\mathbb{N}\cup\{ 0\}} $ defined by $ x_0\in (0,\pi /2),x_{n+1}=f\left( x_n\right),\forall n\ge 0. $
If $a, b, c$ are non-negative real numbers with $ a^2 \plus{} b^2 \plus{} c^2 \equal{} 1$, prove that:
\[ \frac {a}{b^2 \plus{} 1} \plus{} \frac {b}{c^2 \plus{} 1} \plus{} \frac {c}{a^2 \plus{} 1} \geq \frac {3}{4}(a\sqrt {a} \plus{} b\sqrt {b} \plus{} c\sqrt {c})^2\]
Let $ABCD$ be a cyclic quadrilateral. Let lines $AB$ and $CD$ intersect in $P,$ and lines $BC$ and $DA$ intersect in $Q.$ The feet of the perpendiculars from $P$ to $BC$ and $DA$ are $K$ and $L,$ and the feet of the perpendiculars from $Q$ to $AB$ and $CD$ are $M$ and $N.$ The midpoint of diagonal $AC$ is $F.$
Prove that the circumcircles of triangles $FKN$ and $FLM,$ and the line $PQ$ are concurrent.
[i]Based on a problem by Ádám Péter Balogh, Szeged[/i]
Let $ c > 2,$ and let $ a(1), a(2), \ldots$ be a sequence of nonnegative real numbers such that
\[ a(m \plus{} n) \leq 2 \cdot a(m) \plus{} 2 \cdot a(n) \text{ for all } m,n \geq 1,
\]
and $ a\left(2^k \right) \leq \frac {1}{(k \plus{} 1)^c} \text{ for all } k \geq 0.$ Prove that the sequence $ a(n)$ is bounded.
[i]Author: Vjekoslav Kovač, Croatia[/i]
$n$ is a positive integer, $F_n=2^{2^{n}}+1$. Prove that for $n \geq 3$, there exists a prime factor of $F_n$ which is larger than $2^{n+2}(n+1)$.
Let $ f:[0,1]\longrightarrow\mathbb{R} $ a continuous function. Prove that
$$ \int_0^1 f^2\left( x^2 \right) dx\ge \frac{3}{4}\left( \int_0^1 f(x)dx \right)^2 , $$
and find the circumstances under which equality happens.
Let $ABCD$ be a square, and let $\Gamma$ be the circle that is inscribed in square $ABCD$. Let $E$ and $F$ be points on line segments $AB$ and $AD$, respectively, so that $EF$ is tangent to $\Gamma$. Find the ratio of the area of triangle $CEF$ to the area of square $ABCD$.