Found problems: 85335
Find all functions $f: \mathbb N \to \mathbb N$ such that for all positive integers $m,n$,
[b](i)[/b] $mf(f(m))=\left( f(m) \right)^2$,
[b](ii)[/b] If $\gcd(m,n)=d$, then $f(mn) \cdot f(d)=d \cdot f(m) \cdot f(n)$,
[b](iii)[/b] $f(m)=m$ if and only if $m=1$.
Let $ ABC$ be an equilateral triangle with side length 2, and let $ \Gamma$ be a circle with radius $ \frac {1}{2}$ centered at the center of the equilateral triangle. Determine the length of the shortest path that starts somewhere on $ \Gamma$, visits all three sides of $ ABC$, and ends somewhere on $ \Gamma$ (not necessarily at the starting point). Express your answer in the form of $ \sqrt p \minus{} q$, where $ p$ and $ q$ are rational numbers written as reduced fractions.
a) Let $n$ be a positive integer $G$ be a a group with $|G|<\frac{4n^2}{n-\varphi(n)}$. Suppose that $Z(G)$ contains at least $\varphi(n)+1$ elements of order $n$. Prove that $G$ is abelian.
b) Find a noncommutative group $G$ with $16$ elements such that $Z(G)$ contains two elements of order two.
[i]Robert Rogozsan & Filip Munteanu[/i]
Let $ T$ denote the set of all ordered triples $ (p,q,r)$ of nonnegative integers. Find all functions $ f: T \rightarrow \mathbb{R}$ satisfying
\[ f(p,q,r) = \begin{cases} 0 & \text{if} \; pqr = 0, \\
1 + \frac{1}{6}(f(p + 1,q - 1,r) + f(p - 1,q + 1,r) & \\
+ f(p - 1,q,r + 1) + f(p + 1,q,r - 1) & \\
+ f(p,q + 1,r - 1) + f(p,q - 1,r + 1)) & \text{otherwise} \end{cases}
\]
for all nonnegative integers $ p$, $ q$, $ r$.
The $KLMN$ tetrahedron (triangle pyramid) vertices are situated inside or on the faces or on the edges of the $ABCD$ tetrahedron. Prove that perimeter of $KLMN$ is less than $4/3$ perimeter of $ABCD$.
Let $x_1, x_2, x_3, x_4$ be the distances from an arbitrary point inside the tetrahedron to the planes of its faces, and let $h_1, h_2, h_3, h_4$ be the corresponding heights of the tetrahedron. Prove that $$\sqrt{h_1+h_2+h_3+h_4} \ge \sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}+\sqrt{x_4}$$
(Dmitry Nomirovsky)
There are $ n$ websites $ 1,2,\ldots,n$ ($ n \geq 2$). If there is a link from website $ i$ to $ j$, we can use this link so we can move website $ i$ to $ j$.
For all $ i \in \left\{1,2,\ldots,n - 1 \right\}$, there is a link from website $ i$ to $ i+1$.
Prove that we can add less or equal than $ 3(n - 1)\log_{2}(\log_{2} n)$ links so that for all integers $ 1 \leq i < j \leq n$, starting with website $ i$, and using at most three links to website $ j$. (If we use a link, website's number should increase. For example, No.7 to 4 is impossible).
Sorry for my bad English.
Suppose we have an (infinite) cone $ \mathcal C$ with apex $ A$ and a plane $ \pi$. The intersection of $ \pi$ and $ \mathcal C$ is an ellipse $ \mathcal E$ with major axis $ BC$, such that $ B$ is closer to $ A$ than $ C$, and $ BC \equal{} 4$, $ AC \equal{} 5$, $ AB \equal{} 3$. Suppose we inscribe a sphere in each part of $ \mathcal C$ cut up by $ \mathcal E$ with both spheres tangent to $ \mathcal E$. What is the ratio of the radii of the spheres (smaller to larger)?
Evaluate
\[\frac{1/2}{1+\sqrt2}+\frac{1/4}{1+\sqrt[4]2}+\frac{1/8}{1+\sqrt[8]2}+\frac{1/16}{1+\sqrt[16]2}+\cdots\]
[i]Proposed by Ethan Tan[/i]
Place points $A$, $B$, $C$, $D$, $E$, and $F$ evenly spaced on a unit circle. Compute the area of the shaded $12$-sided region, where the region is bounded by line segments $AD$, $DF$, $FB$, $BE$, $EC$, and $CA$.
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The real numbers $x_1,\ldots ,x_{2011}$ satisfy
\[x_1+x_2=2x_1',\ x_2+x_3=2x_2', \ \ldots, \ x_{2011}+x_1=2x_{2011}'\]
where $x_1',x_2',\ldots,x_{2011}'$ is a permutation of $x_1,x_2,\ldots,x_{2011}$. Prove that $x_1=x_2=\ldots =x_{2011}$ .
In triangle $ABC$, side $BC$ is equal to $a$, and the angles of the triangle adjacent to it are equal to $\alpha$ and $\beta$. A circle passing through points $A $and $B$ intersects lines $CA$ and $CB$ for second time at points $ P$ and $M$. It is known that straight line $RM$ passes through the center of the circle circumscribed around $ABC$. Find the length of the segment $PM$
Inside the convex polyhedron, the point $P$ and several lines $\ell_1,\ell_2, ..., \ell_n$ passing through $P$ and not lying in the same plane. To each face of the polyhedron we associate one of the lines $l_1, l_2, ..., l_n$ that forms the largest angle with the plane of this face (if there are there are several direct ones, we will choose any of them). Prove that there is a face that intersects with its corresponding line.
Suppose that the function $ g : (0,1) \rightarrow \mathbb{R}$ can be uniformly approximated by polynomials with nonnegative coefficients. Prove that $ g$ must be analytic. Is the statement also true for the interval $ (\minus{}1,0)$ instead of $ (0,1)$?
[i]J. Kalina, L. Lempert[/i]
Prove the following inequality.
\[\int_{-1}^1 \frac{e^x+e^{-x}}{e^{e^{e^x}}}dx<e-\frac{1}{e}\]
Own
Tim and Allen are playing a match of [i]tenus[/i]. In a match of [i]tenus[/i], the two players play a series of games, each of which is won by one of the two players. The match ends when one player has won exactly two more games than the other player, at which point the player who has won more games wins the match. In odd-numbered games, Tim wins with probability $3/4$, and in the even-numbered games, Allen wins with probability $3/4$. What is the expected number of games in a match?
There are three number-transforming machines. We input the pair $(a_1, a_2)$, and the machine returns $(b_1, b_2)$. We denote this transformation as $(a_1, a_2) \to (b_1, b_2)$.
(a) The first machine can perform two transformations:
- $(a, b) \to (a - 1, b - 1)$
- $(a, b) \to (a + 13, b + 5)$
If the input pair is $(25,32)$, is it possible to obtain the pair $(82,98)$ after a series of transformations?
(b) The second machine can perform two transformations:
- $(a, b) \to (a - 1, b - 1)$
- $(a, b) \to (2a, 2b)$
If the input pair is $(34,60)$, is it possible to obtain the pair $(2000, 2008)$ after a series of transformations?
(c) The third machine can perform two transformations:
- $(a, b) \to (a - 2, b + 2)$
- $(a, b) \to (2a - b + 1, 2b - 1 - a)$
If the input pair is $(145,220)$, is it possible to obtain the pair $(363,498)$ after a series of transformations?
Let $\mathcal{S}$ be the set $\{1,2,3,\ldots,10\}.$ Let $n$ be the number of sets of two non-empty disjoint subsets of $\mathcal{S}.$ (Disjoint sets are defined as sets that have no common elements.) Find the remainder obtained when $n$ is divided by $1000.$
Find the maximum value of $\lambda ,$ such that for $\forall x,y\in\mathbb R_+$ satisfying $2x-y=2x^3+y^3,x^2+\lambda y^2\leqslant 1.$
$a$ and $b$ are natural numbers such that $b > a > 1$, and $a$ does not divide $b$. The sequence of natural numbers $\{b_n\}_{n=1}^\infty$ satisfies $b_{n + 1} \geq 2b_n \forall n \in \mathbb{N}$. Does there exist a sequence $\{a_n\}_{n=1}^\infty$ of natural numbers such that for all $n \in \mathbb{N}$, $a_{n + 1} - a_n \in \{a, b\}$, and for all $m, l \in \mathbb{N}$ ($m$ may be equal to $l$), $a_m + a_l \not\in \{b_n\}_{n=1}^\infty$?
Polynomial $P(x)=c_{2006}x^{2006}+c_{2005}x^{2005}+\ldots+c_1x+c_0$ has roots $r_1,r_2,\ldots,r_{2006}$. The coefficients satisfy $2i\tfrac{c_i}{c_{2006}-i}=2j\tfrac{c_j}{c_{2006}-j}$ for all pairs of integers $0\le i,j\le2006$. Given that $\sum_{i\ne j,i=1,j=1}^{2006} \tfrac{r_i}{r_j}=42$, determine $\sum_{i=1}^{2006} (r_1+r_2+\ldots+r_{2006})$.
Ben and Connor are playing a game of wallball. The first player to lead by $2$ points wins the game. Suppose Ben wins each point with probability $\tfrac{3}{4}$ and is gracious enough to allow Connor to start with a $1$ point lead. The probability that Ben wins the game is $\tfrac{m}{n}$ for coprime positive integers $m$ and $n.$ What is $m + n$?
For $135^\circ < x < 180^\circ$, points $P=(\cos x, \cos^2 x), Q=(\cot x, \cot^2 x), R=(\sin x, \sin^2 x)$ and $S =(\tan x, \tan^2 x)$ are the vertices of a trapezoid. What is $\sin(2x)$?
$ \textbf{(A)}\ 2-2\sqrt{2}\qquad\textbf{(B)}\ 3\sqrt{3}-6\qquad\textbf{(C)}\ 3\sqrt{2}-5\qquad\textbf{(D)}\ -\frac{3}{4}\qquad\textbf{(E)}\ 1-\sqrt{3} $
Eight football teams participate in a tournament of one round (each team plays each other team once) . There are no draws. Prove that it is possible at the conclusion of the tournament to be able to find $4$ teams , say $A, B, C$ and $D$ so that $A$ defeated $B, C$ and $D, B$ defeated $C$ and $D$ , and $C$ defeated $D$ .
Let $a, b$ and $n$ be positive integers with $a>b$ such that all of the following hold:
i. $a^{2021}$ divides $n$,
ii. $b^{2021}$ divides $n$,
iii. 2022 divides $a-b$.
Prove that there is a subset $T$ of the set of positive divisors of the number $n$ such that the sum of the elements of $T$ is divisible by 2022 but not divisible by $2022^2$.
[i]Proposed by Silouanos Brazitikos, Greece[/i]