Found problems: 85335
Prove that among $81$ natural numbers whose prime divisors are in the set $\{2, 3, 5\}$ there exist four numbers whose product is the fourth power of an integer.
Point $T$ lies on the bisector of $\angle B$ of acuteangled $\triangle ABC$. Circle $S$ with diameter $BT$ intersects $AB$ and $BC$ at points $P$ and $Q$. Circle, that goes through point $A$ and tangent to $S$ at $P$ intersects line $AC$ at $X$. Circle, that goes through point $C$ and tangent to $S$ at $Q$ intersects line $AC$ at $Y$. Prove, that $TX=TY$
Let $AD$ and $AE$ be the altitude and median of triangle $ABC$, in with $\angle B = 2\angle C$. Prove that $AB = 2DE$.
Let $I$ be the incenter of a triangle $ABC$ with $AB = 20$, $BC = 15$, and $BI = 12$. Let $CI$ intersect the circumcircle $\omega_1$ of $ABC$ at $D \neq C $. Alice draws a line $l$ through $D$ that intersects $\omega_1$ on the minor arc $AC$ at $X$ and the circumcircle $\omega_2$ of $AIC$ at $Y$ outside $\omega_1$. She notices that she can construct a right triangle with side lengths $ID$, $DX$, and $XY$. Determine, with proof, the length of $IY$.
Find all triples $(a,b,c)$ of real numbers such that $|2a + b| \ge 4$ and $|ax^2 + bx + c| \le 1$ $ \forall x \in [-1, 1]$.
Right triangle $\triangle{DEF}$ with $\angle{D}=90^\circ$ and $\angle{F}=30^\circ$ is inscribed in equilateral triangle $\triangle{ABC}$ such that $D, E,$ and $F$ lie on segments $\overline{BC}, \overline{CA},$ and $\overline{AB},$ respectively. Given that $BD=7$ and $DC=4,$ compute $DE.$
$S_1,S_2,\ldots,S_n$ are subsets of $\{1,2,\ldots,10000\}$ which satisfy that, whenever $|S_i| > |S_j|$, the sum of all elements in $S_i$ is less than the sum of all elements in $S_j$. Let $m$ be the maximum number of distinct values among $|S_1|,\ldots,|S_n|$. Find $\left\lfloor\frac{m}{100}\right\rfloor$.
For which angles $ \theta$, with $ \theta$ a rational number of degrees, is $ {\tan}^{2}\theta\plus{}{\tan}^{2}2\theta$ is irrational?
Let $\mathbb{N}$ be the set of all positive integers. $f,g:\mathbb{N} \to \mathbb{N}$ be funtions such that $f$ is onto and $g$ is one-one and $f(n)\geq g(n)$ for all positive integers $n$. Prove that $f=g$.
$a_{n}$ is a sequence that $a_{1}=1,a_{2}=2,a_{3}=3$, and \[a_{n+1}=a_{n}-a_{n-1}+\frac{a_{n}^{2}}{a_{n-2}}\] Prove that for each natural $n$, $a_{n}$ is integer.
Let $D \subseteq \mathbb{C}$ be an open set containing the closed unit disk $\{z : |z| \leq 1\}$. Let $f : D \rightarrow \mathbb{C}$ be a holomorphic function, and let $p(z)$ be a monic polynomial. Prove that
$$
|f(0)| \leq \max_{|z|=1} |f(z)p(z)|
$$
Let $k$ a positive integer number such that $\frac{k(k+1)}{3}$ is a perfect square. Show that $\frac{k}{3}$ and $k+1$ are both perfect squares.
Four circles of radius $3$ are arranged as shown. Their centers are the vertices of a square. The area of the shaded region is closest to
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fill((3,3)--(3,-3)--(-3,-3)--(-3,3)--cycle,lightgray);
fill(arc((3,3),(0,3),(3,0),CCW)--(3,3)--cycle,white);
fill(arc((3,-3),(3,0),(0,-3),CCW)--(3,-3)--cycle,white);
fill(arc((-3,-3),(0,-3),(-3,0),CCW)--(-3,-3)--cycle,white);
fill(arc((-3,3),(-3,0),(0,3),CCW)--(-3,3)--cycle,white);
draw(circle((3,3),3));
draw(circle((3,-3),3));
draw(circle((-3,-3),3));
draw(circle((-3,3),3));
draw((3,3)--(3,-3)--(-3,-3)--(-3,3)--cycle);
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$\text{(A)}\ 7.7 \qquad \text{(B)}\ 12.1 \qquad \text{(C)}\ 17.2 \qquad \text{(D)}\ 18 \qquad \text{(E)}\ 27$
Let $ n$ be an integer,$ n \geq 3.$ Let $ x_1, x_2, \ldots, x_n$ be real numbers such that $ x_i < x_{i\plus{}1}$ for $ 1 \leq i \leq n \minus{} 1$. Prove that
\[ \frac{n(n\minus{}1)}{2} \sum_{i < j} x_ix_j > \left(\sum^{n\minus{}1}_{i\equal{}1} (n\minus{}i)\cdot x_i \right) \cdot \left(\sum^{n}_{j\equal{}2} (j\minus{}1) \cdot x_j \right)\]
Let $f:S\to\mathbb{R}$ be the function from the set of all right triangles into the set of real numbers, defined by $f(\Delta ABC)=\frac{h}{r}$, where $h$ is the height with respect to the hypotenuse and $r$ is the inscribed circle's radius. Find the image, $Im(f)$, of the function.
Let $ R$ be a parallelopiped. Let us assume that areas of all intersections of $ R$ with planes containing centers of three edges of $ R$ pairwisely not parallel and having no common points, are equal. Show that $ R$ is a cuboid.
Using each of the ten digits exactly once, construct two five-digit numbers such that their difference is minimized. Determine this minimal difference.
[i]Proposed by Giorgi Arabidze, Georgia [/i]
Find all natural number which can be expressed in
$ \frac{a\plus{}b}{c}\plus{}\frac{b\plus{}c}{a}\plus{}\frac{c\plus{}a}{b}$
where $ a,b,c\in \mathbb{N}$ satisfy
$ \gcd(a,b)\equal{}\gcd(b,c)\equal{}\gcd(c,a)\equal{}1$
For the nonzero numbers $ a$, $ b$, $ c$, define
\[(a,b,c)\equal{}\frac{a}{b}\plus{}\frac{b}{c}\plus{}\frac{c}{a}.\]
Find $ (2,12,9)$.
$ \textbf{(A)}\ 4 \qquad
\textbf{(B)}\ 5 \qquad
\textbf{(C)}\ 6 \qquad
\textbf{(D)}\ 7 \qquad
\textbf{(E)}\ 8$
Quadrilateral $ABCD$ is inscribed in a circle, $M$ is the intersection point of the lines $AB$ and $CD$ and $N$ is the intersection point of the lines $BC$ and $AD$. It is known that $BM = DN$. Prove that $CM = CN$.
(F Nazarov)
Here $G_{n}$ denotes a simple undirected graph with $n$ vertices, $K_{n}$ denotes the complete graph with $n$ vertices, $K_{n,m}$ the complete bipartite graph whose components have $m$ and $n$ vertices, and $C_{n}$ a circuit with $n$ vertices. The number of edges in the graph $G_{n}$ is denoted $e(G_{n})$.
The edges of $K_{n}(n \geq 3)$ are colored with $n$ colors, and every color is used.
Show that there is a triangle whose sides have different colors.
Nair and Yuli play the following game:
$1.$ There is a coin to be moved along a horizontal array with $203$ cells.
$2.$ At the beginning, the coin is at the first cell, counting from left to right.
$3.$ Nair plays first.
$4.$ Each of the players, in their turns, can move the coin $1$, $2$, or $3$ cells to the right.
$5.$ The winner is the one who reaches the last cell first.
What strategy does Nair need to use in order to always win the game?
Decide whether there is an integer $n > 1$ with the following properties:
(a) $n$ is not a prime number.
(b) For all integers $a$, $a^n - a$ is divisible by $n$
$a$ is a $2000$ digit natural number of the form
$$a=2(A)99…99(B)(C)$$
expressed in base $10$. $a$ is not a multiple of $10$, and $2(A)+(B)(C)=99$. $a=2899..9971$ is a possible example of $a$. $b$ is a number you earn when you write the digits of $a$ in a reverse order(Writing the digits of some number in a reverse order means like reordering $1234$ into $4321$). Find every positive integer $a$ that makes $ab$ a square number.
Determine all functions $f: \mathbb R \to \mathbb R$ with the property
\[f(f(x)+y)=2x+f(f(y)-x), \quad \forall x,y \in \mathbb R.\]