Found problems: 85335
Let $ABCD$ and $BCFG$ be two faces of a cube with $AB=12.$ A beam of light emanates from vertex $A$ and reflects off face $BCFG$ at point $P,$ which is 7 units from $\overline{BG}$ and 5 units from $\overline{BC}.$ The beam continues to be reflected off the faces of the cube. The length of the light path from the time it leaves point $A$ until it next reaches a vertex of the cube is given by $m\sqrt{n},$ where $m$ and $n$ are integers and $n$ is not divisible by the square of any prime. Find $m+n.$
What is the volume of the region in three-dimensional space defined by the inequalities $|x|+|y|+|z|\le1$ and $|x|+|y|+|z-1|\le1$?
$\textbf{(A)}\ \frac{1}{6} \qquad
\textbf{(B)}\ \frac{1}{3} \qquad
\textbf{(C)}\ \frac{1}{2} \qquad
\textbf{(D)}\ \frac{2}{3} \qquad
\textbf{(E)}\ 1$
Does there exist a convex pentagon $A_1A_2A_3A_4A_5$ and a point $X$ inside it such that $XA_i=A_{i+2}A_{i+3}$ for all $i=1,...,5$ (all indices are considered modulo $5$) ?
I. Voronovich
Find all triples of natural numbers $(a, b, c)$ for which the number $$2^a + 2^b + 2^c + 3$$ is the square of an integer.
Let $TABCD$ be a pyramid with top vertex $T$, such that its base $ABCD$ is a square of side length 4. It is given that, among the triangles $TAB$, $TBC$, $TCD$ and $TDA$, one can find an isosceles triangle and a right-angled triangle. Find all possible values for the volume of the pyramid.
It is known that some \(d\) distinct divisors of a positive integer number \(n\) form an arithmetic progression. Prove that the number \(n\) has at least \(2d - 2\) divisors.
[i]Proposed by Anton Trygub[/i]
Construct a triangle knowing an angle, the ratio of the sides that form it and the radius of the inscribed circle.
Let $S = \frac{1}{a_1}+\frac{2}{a_2}+ ... +\frac{100}{a_{100}}$ where $a_1, a_2,..., a_{100}$ are positive integers. What are all the possible integer values that $S$ can take ?
Eight points are equally spaced around a circle of radius $r$. If we draw a circle of radius $1$ centered at each of the eight points, then each of these circles will be tangent to two of the other eight circles that are next to it. IF $r^2=a+b\sqrt{2}$, where $a$ and $b$ are integers, then what is $a+b$?
$\text{(A) }3\qquad\text{(B) }4\qquad\text{(C) }5\qquad\text{(D) }6\qquad\text{(E) }7$
$S_{1},S_{2},S_{3}$ are three spheres in $\mathbb R^{3}$ that their centers are not collinear. $k\leq8$ is the number of planes that touch three spheres. $A_{i},B_{i},C_{i}$ is the point that $i$-th plane touch the spheres $S_{1},S_{2},S_{3}$. Let $O_{i}$ be circumcenter of $A_{i}B_{i}C_{i}$. Prove that $O_{i}$ are collinear.
Prove that for every four positive numbers $a,b,c,d$ the following inequality is true:
$$\sqrt{\frac{a^2+b^2+c^2+d^2}4}\ge\sqrt[3]{\frac{abc+abd+acd+bcd}4}.$$
A function $f$ is defined recursively by $f(1)=f(2)=1$ and $$f(n)=f(n-1)-f(n-2)+n$$ for all integers $n \geq 3$. What is $f(2018)$?
$\textbf{(A)} \text{ 2016} \qquad \textbf{(B)} \text{ 2017} \qquad \textbf{(C)} \text{ 2018} \qquad \textbf{(D)} \text{ 2019} \qquad \textbf{(E)} \text{ 2020}$
Big Welk writes the letters of the alphabet in order, and starts again at $A$ each time he gets to $Z.$ What is the $4^3$-rd letter that he writes down?
Which of the following are true?
$\textbf{(A)}~\exists f:\mathbb N\to\mathbb Z\text{ onto and increasing}$
$\textbf{(B)}~\exists f:\mathbb Z\to\mathbb Q\text{ onto and increasing}$
$\textbf{(C)}~\exists f:\mathbb Q\to\mathbb Z\text{ onto and increasing and bounded}$
$\textbf{(D)}~\text{None of the above}$
Find all functions $f$ from positive integers to themselves such that:
1)$f(mn)=f(m)f(n)$ for all positive integers $m, n$
2)$\{1, 2, ..., n\}=\{f(1), f(2), ... f(n)\}$ is true for infinitely many positive integers $n$.
A cylinder is inscribed within a sphere of radius 10 such that its volume is [i]almost-half[/i] that of the sphere. If [i]almost-half[/i] is defined such that the cylinder has volume $\frac12+\frac{1}{250}$ times the sphere’s volume, find the sum of all possible heights for the cylinder.
Prove the inequality
\[ \frac{a_1+ a_3}{a_1 + a_2} + \frac{a_2 + a_4}{a_2 + a_3} + \frac{a_3 + a_1}{a_3 + a_4} + \frac{a_4 + a_2}{a_4 + a_1} \geq 4, \]
where $a_i > 0, i = 1, 2, 3, 4.$
For positive integer $k,$ we say that it is a [i]Taurus integer[/i] if we can delete one element from the set $M_k=\{1,2,\cdots,k\},$ such that the sum of remaining $k-1$ elements is a positive perfect square. For example, $7$ is a Taurus integer, because if we delete $3$ from $M_7=\{1,2,3,4,5,6,7\},$ the sum of remaining $6$ elements is $25,$ which is a positive perfect square.
$(1)$ Determine whether $2021$ is a Taurus integer.
$(2)$ For positive integer $n,$ determine the number of Taurus integers in $\{1,2,\cdots,n\}.$
Let $ABC$ be an acute-angled triangle inscribed in a circle $\mathcal C (O, R)$ and $F$ a point on the side $AB$ such that $AF < AB/2$. The circle $c_1(F, FA)$ intersects the line $OA$ at the point $A'$ and the circle $\mathcal C$ at $K$. Prove that the quadrilateral $BKFA'$ is cyclic and its circumcircle contains point $O$.
Let triangle \( ABC \) be an acute-angled triangle. Square \( AEFB \) and \( ADGC \) lie outside triangle \( ABC \). \( BD \) intersects \( CE \) at point \( H \), and \( BG \) intersects \( CF \) at point \( I \). The circumcircle of triangle \( BFI \) intersects the circumcircle of triangle \( CGI \) again at point \( K \). Prove that line segment \( HK \) bisects \( BC \).
Let $a_1,a_2, \cdots$ be a sequence of integers that satisfies: $a_1=1$ and $a_{n+1}=a_n+a_{\lfloor \sqrt{n} \rfloor} , \forall n\geq 1 $. Prove that for all positive $k$, there is $m \geq 1$ such that $k \mid a_m$.
Two players in turn paint cells of the $7\times7$ table each using own color. A player can't paint a cell if its row or its column contains a cell painted by the other player. The game stops when one of the players can't make his turn. What
maximal number of the cells can remain unpainted when the game stops?
Team $ A$ and team $ B$ play a series. The first team to win three games wins the series. Each team is equally likely to win each game, there are no ties, and the outcomes of the individual games are independent. If team $ B$ wins the second game and team $ A$ wins the series, what is the probability that team $ B$ wins the first game?
$ \textbf{(A)}\ \frac{1}{5}\qquad
\textbf{(B)}\ \frac{1}{4}\qquad
\textbf{(C)}\ \frac{1}{3}\qquad
\textbf{(D)}\ \frac{1}{2}\qquad
\textbf{(E)}\ \frac{2}{3}$
Find the maximum number $ C$ such that for any nonnegative $ x,y,z$ the inequality
$ x^3 \plus{} y^3 \plus{} z^3 \plus{} C(xy^2 \plus{} yz^2 \plus{} zx^2) \ge (C \plus{} 1)(x^2 y \plus{} y^2 z \plus{} z^2 x)$ holds.
The polynomial $$x^{10} + ?x^9 + ?x^8 + ... + ?x + 1$$ is written on the blackboard. Two players substitute (real) numbers instead of one of the question marks in turn. ($9$ turns total.) The first wins if the polynomial will have no real roots. Who wins?