Found problems: 85335
Find all triples $(a, b, c)$ of positive integers such that
$a^2 + b^2 = n\cdot lcm(a, b) + n^2$
$b^2 + c^2 = n \cdot lcm(b, c) + n^2$
$c^2 + a^2 = n \cdot lcm(c, a) + n^2$
for some positive integer $n$.
Let $\Gamma_1$ be a circle with center $O$ and $A$ a point on it. Consider the circle $\Gamma_2$ with center at $A$ and radius $AO$. Let $P$ and $Q$ be the intersection points of $\Gamma_1$and $\Gamma_2$. Consider the circle $\Gamma_3$ with center at $P$ and radius $PQ$. Let $C$ be the second intersection point of $\Gamma_3$ and $\Gamma_1$. The line $OP$ cuts $\Gamma_3$ at $R$ and $S$, with $R$ outside $\Gamma_1$. $RC$ cuts $\Gamma_1$ into $B$. $CS$ cuts $\Gamma_1$ into $D$. Show that $ABCD$ is a square.
The function $f: Z \to Z$ satisfies the following conditions:
1) $f(f(n))=n$ for all integers $n$
2) $f(f(n+2)+2) = n$ for all integers $n$
3) $f(0)=1$.
Find the value of $f(1995)$ and $f(-1994)$.
In a scalene triangle $ABC$ with centroid $G$ and circumcircle $\omega$ centred at $O$, the extension of $AG$ meets $\omega$ at $M$; lines $AB$ and $CM$ intersect at $P$; and lines $AC$ and $BM$ intersect at $Q$. Suppose the circumcentre $S$ of the triangle $APQ$ lies on $\omega$ and $A, O, S$ are collinear. Prove that $\angle AGO = 90^{o}$.
Prove that any identity that holds for every finite $ n$-distributive lattice also holds for the lattice of all convex subsets of the $ (n\minus{}1)$-dimensional Euclidean space. (For convex subsets, the lattice operations are the set-theoretic intersection and the convex hull of the set-theoretic union. We call a lattice $ n$-$ \textit{distributive}$ if \[ x \wedge (\bigvee_{i\equal{}0}^n y_i)\equal{}\bigvee_{j\equal{}0}^n(x \wedge (\bigvee_{0\leq i \leq n, \;i \not\equal{} j\ }y_i))\] holds for all elements of the lattice.)
[i]A. Huhn[/i]
Compute the value of the infinite series \[\sum_{n=2}^\infty \dfrac{n^4+3n^2+10n+10}{2^n\cdot(n^4+4)}\]
Let $S$ be the square consisting of all pints $(x,y)$ in the plane with $0\le x,y\le 1$. For each real number $t$ with $0<t<1$, let $C_t$ denote the set of all points $(x,y)\in S$ such that $(x,y)$ is on or above the line joining $(t,0)$ to $(0,1-t)$.
Prove that the points common to all $C_t$ are those points in $S$ that are on or above the curve $\sqrt{x}+\sqrt{y}=1$.
Suppose $a$ and $b$ are two variables that satisfy $\textstyle\int_0^2(-ax^2+b)dx=0$. What is $\tfrac{a}{b}$?
$n \geq k$ -two natural numbers. $S$ -such natural, that have not less than $n$ divisors. All divisors of $S$ are written
in descending order. What minimal number of divisors can have number from $k$-th place ?
Let $ABC$ be a triangle with incenter $I$, and suppose the incircle is tangent to $CA$ and $AB$ at $E$ and $F$. Denote by $G$ and $H$ the reflections of $E$ and $F$ over $I$. Let $Q$ be the intersection of $BC$ with $GH$, and let $M$ be the midpoint of $BC$. Prove that $IQ$ and $IM$ are perpendicular.
A $ 3\times3\times3$ cube composed of $ 27$ unit cubes rests on a horizontal plane. Determine the number of ways of selecting two distinct unit cubes from a $ 3\times3\times1$ block (the order is irrelevant) with the property that the line joining the centers of the two cubes makes a $ 45^\circ$ angle with the horizontal plane.
Determine all values of $a \in R$ such that there exists a function $f : [0, 1] \to R$ fulfilling the following inequality for all $x \ne y$: $$|f(x) - f(y)| \ge a.$$
One of the receipts for a math tournament showed that $72$ identical trophies were purchased for $\$$-$99.9$-, where the first and last digits were illegible. How much did each trophy cost?
For a positive integer $n$, let $s(n)$ be the number of ways that $n$ can be written as the sum of strictly increasing perfect $2010^{\text{th}}$ powers. For instance, $s(2) = 0$ and $s(1^{2010} + 2^{2010}) = 1$. Show that for every real number $x$, there exists an integer $N$ such that for all $n > N$,
\[\frac{\max_{1 \leq i \leq n} s(i)}{n} > x.\]
[i]Alex Zhu.[/i]
Suppose that the roots of the polynomial $P(x)=x^3+ax^2+bx+c$ are $\cos \frac{2\pi}7,\cos \frac{4\pi}7,$ and $\cos \frac{6\pi}7$, where angles are in radians. What is $abc$?
$\textbf{(A) }-\frac{3}{49} \qquad \textbf{(B) }-\frac{1}{28} \qquad \textbf{(C) }\frac{^3\sqrt7}{64} \qquad \textbf{(D) }\frac{1}{32}\qquad \textbf{(E) }\frac{1}{28}$
A subset $S \in \{0, 1, 2, \cdots , 2000\}$ satisfies $|S|=401$.
Prove that there exists a positive integer $n$ such that there are at least $70$ positive integers $x$ such that $x, x+n \in S$
Let $f(x)$ be a periodic function with periods $T$ and $1$($0<T<1$).Prove that:
(1)If $T$ is rational,then there exists a prime $p$ such that $\frac{1}{p}$ is also a period of $f$;
(2)If $T$ is irrational,then there exists a strictly decreasing infinite sequence {$a_n$},with $1>a_n>0$ for all positive integer $n$,such that all $a_n$ are periods of $f$.
Given are twenty-two different five-element sets, such that any two of them have exactly two elements in common. Prove that they all have two elements in common.
Let \(n\) be a positive integer. For every positive integer $1 \leq k \leq n$ the sequence ${\displaystyle {\{ a_{i}+ki\}}_{i=1}^{n }}$ is defined, where $a_1,a_2, \dots ,a_n$ are integers. Among these \(n\) sequences, for at most how many of them does all the elements of the sequence give different remainders when divided by \(n\)?
Albatross and Frankinfueter both own a circle. Frankinfueter also has stolen from Prof. Trugweg a ruler. Before that, Trugweg had two points with a distance of $1$ drawn his (infinitely large) board. For a natural number $n$, let A $(n)$ be the number of the construction steps that Albatross needs at least to create two points with a distance of $n$ to construct. Similarly, Frankinfueter needs at least $F(n)$ steps for this.
How big can $\frac{A (n)}{F (n)}$ become?
There are only the following three construction steps:
a) Mark an intersection of two straight lines, two circles or a straight line with one circle.
b) Pierce at a marked point $P$ and draw a circle around $P$ through one marked point .
c) Draw a straight line through two marked points (this implies possession of a ruler ahead!).
Let $ABC$ be an arbitrary triangle and let $S_1, S_2,\cdots, S_7$ be circles satisfying the following conditions:
$S_1$ is tangent to $CA$ and $AB$,
$S_2$ is tangent to $S_1, AB$, and $BC,$
$S_3$ is tangent to $S_2, BC$, and $CA,$
..............................
$S_7$ is tangent to $S_6, CA$ and $AB.$
Prove that the circles $S_1$ and $S_7$ coincide.
Four different points $A,B,C$ and $D$ lie in a plane. No three of these points lie on a single straight line. Describe the construction of a square $PQRS$ such that on each of the sides of $PQRS$, or the extensions , lies one of the points $A, B, C$ and $D$.
Given a triangle with sides of lengths $a,b,c$, prove that $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}< 2$.
Let $f_0$ be the function from $\mathbb{Z}^2$ to $\{0,1\}$ such that $f_0(0,0)=1$ and $f_0(x,y)=0$ otherwise. For each positive integer $m$, let $f_m(x,y)$ be the remainder when \[ f_{m-1}(x,y) + \sum_{j=-1}^{1} \sum_{k=-1}^{1} f_{m-1}(x+j,y+k) \] is divided by $2$.
Finally, for each nonnegative integer $n$, let $a_n$ denote the number of pairs $(x,y)$ such that $f_n(x,y) = 1$.
Find a closed form for $a_n$.
[i]Proposed by Bobby Shen[/i]
For any real $ a$ define $ f_a : \mathbb{R} \rightarrow \mathbb{R}^2$ by the law $ f_a(t) \equal{} \left( \sin(t), \cos(at) \right)$.
a) Prove that $ f_{\pi}$ is not periodic.
b) Determine the values of the parameter $ a$ for which $ f_a$ is periodic.
[b]Remark[/b]. L. Euler proved in $ 1737$ that $ \pi$ is irrational.