Found problems: 85335
Let $k$ be a positive integer, and $n$ a positive integer greater than $2$. Define
$$f_{1}(n)=n,\;\; f_{2}(n)=n^{f_{1}(n)},\;\ldots\;, f_{j+1}(n)=n^{f_{j}(n)}.$$
Prove either part of the inequality
$$f_{k}(n) < n!! \cdots ! < f_{k+1}(n),$$
where the middle term has $k$ factorial symbols.
Let $\Gamma$ be a circunference and $O$ its center. $AE$ is a diameter of $\Gamma$ and $B$ the midpoint of one of the arcs $AE$ of $\Gamma$. The point $D \ne E$ in on the segment $OE$. The point $C$ is such that the quadrilateral $ABCD$ is a parallelogram, with $AB$ parallel to $CD$ and $BC$ parallel to $AD$. The lines $EB$ and $CD$ meets at point $F$. The line $OF$ cuts the minor arc $EB$ of $\Gamma$ at $I$.
Prove that the line $EI$ is the angle bissector of $\angle BEC$.
We are given a line $ g$ with four successive points $ P$, $ Q$, $ R$, $ S$, reading from left to right. Describe a straightedge and compass construction yielding a square $ ABCD$ such that $ P$ lies on the line $ AD$, $ Q$ on the line $ BC$, $ R$ on the line $ AB$ and $ S$ on the line $ CD$.
Compute the sum $$\frac{\varphi(50!)}{\varphi(49!)}+ \frac{\varphi(51!)}{\varphi(50!)} + \dots + \frac{\varphi(100!)}{\varphi(99!)}$$ where $\varphi(n)$ returns the number of positive integers less than $n$ that are relatively prime to $n$.
[i]Proposed by Andy Xu[/i]
An integer $n$ is called $k$-special, with $k$ a positive integer, if it's the sum of the squares of $k$ consecutive integers. For example, $13$ is $2$-special, since $13=2^2+3^2$, and $2$ is $3$-special, since $2=(-1)^2+0^2+1^2$.
a) Prove that there's no perfect square that is $4$-special.
b) Find a perfect square that is $I^2$-special, for some odd positive integer $I$ with $I\ge 3$.
Find minimum value of $F(x,y)=\sqrt{(x+1)^{2}+(y-1)^{2}}+\sqrt{(x-1)^{2}+(y+1)^{2}}+\sqrt{(x+2)^{2}+(y+2)^{2}}$, where $x,y\in\mathbb{R}$.
Find all integer solutions to the equation $$\frac{xyz}{w}+\frac{yzw}{x}+\frac{zwx}{y}+\frac{wxy}{z}=4$$
Let $A$ and $B$ be rectangles in the plane and $f : A \rightarrow B$ be a mapping which is uniform on the interior of $A$, maps the boundary of $A$ homeomorphically to the boundary of $B$ by mapping the sides of $A$ to corresponding sides in $B$. Prove that $f$ is an affine transformation.
Find all positive integers $n$ that have distinct positive divisors $d_1, d_2, \dots, d_k$, where $k>1$, that are in arithmetic progression and $$n=d_1+d_2+\cdots+d_k.$$ Note that $d_1, d_2, \dots, d_k$ do not have to be all the divisors of $n$.
Let $ \triangle{XOY}$ be a right-angled triangle with $ m\angle{XOY}\equal{}90^\circ$. Let $ M$ and $ N$ be the midpoints of legs $ OX$ and $ OY$, respectively. Given that $ XN\equal{}19$ and $ YM\equal{}22$, find $ XY$.
$ \textbf{(A)}\ 24 \qquad
\textbf{(B)}\ 26 \qquad
\textbf{(C)}\ 28 \qquad
\textbf{(D)}\ 30 \qquad
\textbf{(E)}\ 32$
You are given $n \ge 2$ distinct positive integers. For every pair $a<b$ of them, Vlada writes on the board the largest power of $2$ that divides $b-a$. At most how many distinct powers of $2$ could Vlada have written?
[i]Proposed by Oleksiy Masalitin[/i]
The sum of two positive numbers is $5$ times their difference. What is the ratio of the larger number to the smaller?
$\textbf{(A) }\dfrac54\qquad\textbf{(B) }\dfrac32\qquad\textbf{(C) }\dfrac95\qquad\textbf{(D) }2\qquad\textbf{(E) }\dfrac52$
Let $m$ and $n$ be positive integers such that $m^4 - n^4 = 3439$. What is the value of $mn$?
If $a,b,c$ are three real numbers and \[ a + \dfrac{1}{b} = b + \dfrac{1}{c} = c + \dfrac{1}{a} = t \] for some real number $t$, prove that $abc + t = 0 .$
Let $X$ be a set of $n$ elements. Prove that the sum of the numbers of elements of sets $A\cap B$, where $A$ and $B$ run over all subsets of $X$, is equal to $n4^{n-1}$.
Circles $ \Gamma_1$ and $ \Gamma_2$ are internally tangent to circle $ \Gamma$ at $ P$ and $ Q$, respectively. Let $ P_1$ and $ Q_1$ are on $ \Gamma_1$ and $ \Gamma_2$ respectively such that $ P_1Q_1$ is the common tangent of $ P_1$ and $ Q_1$. Assume that $ \Gamma_1$ and $ \Gamma_2$ intersect at $ R$ and $ R_1$. Define $ O_1,O_2,O_3$ as the intersection of $ PQ$ and $ P_1Q_1$, the intersection of $ PR$ and $ P_1R_1$, and the intersection $ QR$ and $ Q_1R_1$. Prove that the points $ O_1,O_2,O_3$ are collinear.
[i]Rudi Adha Prihandoko, Bandung[/i]
A polynomial $x^n + a_1x^{n-1} + a_2x^{n-2} +... + a_{n-2}x^2 + a_{n-1}x + a_n$ has $n$ distinct real roots $x_1, x_2,...,x_n$, where $n > 1$. The polynomial $nx^{n-1}+ (n - 1)a_1x^{n-2} + (n - 2)a_2x^{n-3} + ...+ 2a_{n-2}x + a_{n-1}$ has roots $y_1, y_2,..., y_{n_1}$.
Prove that $\frac{x^2_1+ x^2_2+ ...+ x^2_n}{n}>\frac{y^2_1 + y^2_2 + ...+ y^2_{n-1}}{n - 1}$
In a mathematical competition, in which $6$ problems were posed to the participants, every two of these problems were solved by more than $\frac 25$ of the contestants. Moreover, no contestant solved all the $6$ problems. Show that there are at least $2$ contestants who solved exactly $5$ problems each.
[i]Radu Gologan and Dan Schwartz[/i]
Let be $\triangle ABC$ .Let $A'$, $B'$, $C'$ be the foot of perpendiculars from $A$, $B$ and $C$ respectively.
The points $E$ and $F$ are on the sides $CB'$ and $BC'$ respectively, such that $B'E\cdot C'F = BF\cdot CE$.
Show that $AEA'F$ is cyclic.
In a football tournament $2n$ teams play in a round. Every round consists of $n$ pairs
of teams that haven’t played with each other yet. Every round’s schedule is determined before the
round is held. Find the minimal positive integer $k$ such that the following situation is possible: after
$k$ rounds it’s impossible to schedule the next round.
Given an acute triangle $ABC$ with circumcenter $O$ and orthocenter $H$. Let $K$ be a point inside $ABC$ which is not $O$ nor $H$. Point $L$ and $M$ are located outside the triangle $ABC$ such that $AKCL$ and $AKBM$ are parallelogram. At last, let $BL$ and $CM$ intersects at $N$, and let $J$ be the midpoint of $HK$. Show that $KONJ$ is also a parallelogram.
[i]Raja Oktovin, Pekanbaru[/i]
$\triangle ABC$ is inscribed to unit circle. Bisector of $\angle A,\angle B,\angle C$ intersect the circle at $A_1,B_1,C_1$ respectively. The value of $\frac{\displaystyle AA_1\cdot\cos\frac{A}{2}+BB_1\cdot\cos\frac{B}{2}+CC_1\cdot\cos\frac{C}{2}}{\sin A+\sin B+\sin C}$ is
$\text{(A)}2\qquad\text{(B)}4\qquad\text{(C)}6\qquad\text{(D)}8$
Let $x_1$, $x_2$,$x_3$, $y_1$, $y_2$, and $y_3 $ be positive real numbers, such that $x_1 + y_2 = x_2 + y_3 = x_3 + y_1 =1$. Prove that $$ x_1y_1 + x_2y_2 + x_3y_3 <1$$
Several zeros, ones and twos are written on the blackboard. An anonymous clean in turn pairs of different numbers, writing, instead of cleaned, the number not equal to each. ($0$ instead of pair $\{1,2\}, 1$ instead of $\{0,2\}, 2$ instead of $\{0,1\}$). Prove that if there remains one number only, it does not depend on the processing order.
Consider a regular $n$-gon with $n>3$, call a line [i]acceptable[/i] if it passes through the interior of this $n$-gon. Draw $m$ different acceptable lines, so that the $n$-gon is divided into several smaller polygons.
(a) Prove that there exists an $m$, depending only on $n$, such that any collection of $m$ acceptable lines results in one of the smaller polygons having $3$ or $4$ sides.
(b) Find the smallest possible $m$ which guarantees that at least one of the smaller polygons will have $3$ or $4$ sides.