Found problems: 85335
Let $r(m)$ be the number of positive integers a less than or equal to $m$ where $\gcd(a, m)$ is prime. Find the sum of all positive integers $m < 300$ such that $r(m) = \varphi(m),$ where $\varphi(m)$ denotes the number of positive integers $a$ less than $m$ where $\gcd(a, m) = 1.$
Let $n$ be a positive integer. Set $a_{n,0}=1$. For $k\geq 0$, choose an integer $m_{n,k}$ uniformly at random from the set $\{1,\,\ldots,\,n\}$, and let
\[
a_{n,k+1}=
\begin{cases}
a_{n,k}+1, & \text{if $m_{n,k}>a_{n,k}$;}\\
a_{n,k}, & \text{if $m_{n,k}=a_{n,k}$;}\\
a_{n,k}-1, & \text{if $m_{n,k}<a_{n,k}$.}
\end{cases}
\]
Let $E(n)$ be the expected value of $a_{n,n}$. Determine $\lim_{n\to\infty}E(n)/n$.
Consider the set $M=\{1,2,3,...,2020\}.$ Find the smallest positive integer $k$ such that for any subset $A$ of $M$ with $k$ elements, there exist $3$ distinct numbers $a,b,c$ from $M$ such that $a+b, b+c$ and $c+a$ are all in $A.$
In the set $ S_n \equal{} \{1, 2,\ldots ,n\}$ a new multiplication $ a*b$ is defined with the following properties:
[b](i)[/b] $ c \equal{} a * b$ is in $ S_n$ for any $ a \in S_n, b \in S_n.$
[b](ii)[/b] If the ordinary product $ a \cdot b$ is less than or equal to $ n,$ then $ a*b \equal{} a \cdot b.$
[b](iii)[/b] The ordinary rules of multiplication hold for $ *,$ i.e.:
[b](1)[/b] $ a * b \equal{} b * a$ (commutativity)
[b](2)[/b] $ (a * b) * c \equal{} a * (b * c)$ (associativity)
[b](3)[/b] If $ a * b \equal{} a * c$ then $ b \equal{} c$ (cancellation law).
Find a suitable multiplication table for the new product for $ n \equal{} 11$ and $ n \equal{} 12.$
Compute the remainder when $2^{30}$ is divided by $1000$.
The incircle of triangle $ ABC $ has centre $I$ and touches the sides $ BC $, $ CA $, $ AB $ at points $ A_1 $, $ B_1 $, $ C_1 $, respectively. Let $ I_a $, $ I_b $, $ I_c $ be excentres of triangle $ ABC $, touching the sides $ BC $, $ CA $, $ AB $ respectively. The segments $ I_aB_1 $ and $ I_bA_1 $ intersect at $ C_2 $. Similarly, segments $ I_bC_1 $ and $ I_cB_1 $ intersect at $ A_2 $, and the segments $ I_cA_1 $ and $ I_aC_1 $ at $ B_2 $. Prove that $ I $ is the center of the circumcircle of the triangle $ A_2B_2C_2 $.
[i]L. Emelyanov, A. Polyansky[/i]
Let $ A \equal{} \{1,2,\ldots,2008\}$
a) Find the number of subset of $ A$ which satisfy : the product of its elements is divisible by 7
b) Let $ N(i)$ denotes the number of subset of $ A$ which sum of its elements remains $ i$ when divided by 7. Prove that $ N(0) \minus{} N(1) \plus{} N(2) \minus{} N(3) \plus{} N(4) \minus{} N(5) \plus{} N(6)\minus{}N(7) \equal{} 0$
EDITED : thx for cosinator.. BTW, your statement and my correction give 80% hint of the solution :D
The LHS Math Team wants to play Among Us. There are so many people who want to play that they are going to form several games. Each game has at most 10 people. People are $\textit{happy}$ if they are in a game that has at least 8 people in it. What is the largest possible number of people who would like to play Among Us such that it is impossible to make everyone $\textit{happy}$?
[i]Proposed by Sammy Charney[/i]
Let $Z_1,Z_2,\cdots ,Z_n$ be complex numbers satisfying $|Z_1|+|Z_2|+\cdots +|Z_n|=1$. Show that there exist some among the $n$ complex numbers such that the modulus of the sum of these complex numbers is not less than $1/6$.
Let $ABCD$ be a circumscribed quadrilateral (i. e. a quadrilateral which has an incircle). The exterior angle bisectors of the angles $DAB$ and $ABC$ intersect each other at $K$; the exterior angle bisectors of the angles $ABC$ and $BCD$ intersect each other at $L$; the exterior angle bisectors of the angles $BCD$ and $CDA$ intersect each other at $M$; the exterior angle bisectors of the angles $CDA$ and $DAB$ intersect each other at $N$. Let $K_{1}$, $L_{1}$, $M_{1}$ and $N_{1}$ be the orthocenters of the triangles $ABK$, $BCL$, $CDM$ and $DAN$, respectively. Show that the quadrilateral $K_{1}L_{1}M_{1}N_{1}$ is a parallelogram.
Jerry places at most one rook in each cell of a $2025 \times 2025$ grid of cells. A rook [i]attacks[/i] another rook if the two rooks are in the same row or column and there are no other rooks between them.
Determine, with proof, the maximum number of rooks Jerry can place on the grid such that no rook attacks $4$ other rooks.
Let $x_0(t)=1$, $x_{k+1}(t)=(1+t^{k+1})x_k(t)$ for all $k\geq 0$; $y_{n,0}(t)=1$, $y_{n,k}(t)=\frac{t^{n-k+1}-1}{t^k-1}y_{n,k-1}(t)$ for all $n\geq 0$, $1\leq k \leq n$.
Prove that $\sum_{j=0}^{n-1}(-1)^j x_{n-j-1}(t)y_{n,j}(t)=\frac{1-(-1)^n}{2}$ for all $n\geq 1$.
If $x,y,z$ are real numbers such that $xyz=1$, evaluate
$$\frac{x+1}{xy+x+1}+\frac{y+1}{yz+y+1}+\frac{z+1}{zx+z+1}.$$
Is there an infinite sequence of positive real numbers $x_1,x_2,...,x_n$ satisfying for all $n\ge 1$ the relation $x_{n+2}= \sqrt{x_{n+1}}-\sqrt{x_n}$?
From a $n\times (n-1)$ rectangle divided into unit squares, we cut the [i]corner[/i], which consists of the first row and the first column. (that is, the corner has $2n-2$ unit squares). For the following, when we say [i]corner[/i] we reffer to the above definition, along with rotations and symmetry. Consider an infinite lattice of unit squares. We will color the squares with $k$ colors, such that for any corner, the squares in that corner are coloured differently (that means that there are no squares coloured with the same colour). Find out the minimum of $k$.
[i]Proposed by S. Berlov[/i]
Let $n \ge 2018$ be an integer, and let $a_1, a_2, \dots, a_n, b_1, b_2, \dots, b_n$ be pairwise distinct positive integers not exceeding $5n$. Suppose that the sequence
\[ \frac{a_1}{b_1}, \frac{a_2}{b_2}, \dots, \frac{a_n}{b_n} \]
forms an arithmetic progression. Prove that the terms of the sequence are equal.
Find all functions $f(x)$ from nonnegative reals to nonnegative reals such that $f(f(x))=x^4$ and $f(x)\leq Cx^2$ for some constant $C$.
The sequence $a_n$ is defined by $a_0=a_1=1$ and $a_{n+1}=14a_n-a_{n-1}-4$,for all positive integers $n$.
Prove that all terms of this sequence are perfect squares.
Let $ABC$ be an isosceles triangle with $BC=CA$, and let $D$ be a point inside side $AB$ such that $AD< DB$. Let $P$ and $Q$ be two points inside sides $BC$ and $CA$, respectively, such that $\angle DPB = \angle DQA = 90^{\circ}$. Let the perpendicular bisector of $PQ$ meet line segment $CQ$ at $E$, and let the circumcircles of triangles $ABC$ and $CPQ$ meet again at point $F$, different from $C$.
Suppose that $P$, $E$, $F$ are collinear. Prove that $\angle ACB = 90^{\circ}$.
Find the area of the smallest region bounded by the graphs of $y=|x|$ and $x^2+y^2=4$.
$\textbf{(A) }\frac{\pi}{4}\qquad\textbf{(B) }\frac{3\pi}{4}\qquad\textbf{(C) }\pi\qquad\textbf{(D) }\frac{3\pi}{2}\qquad\textbf{(E) }2\pi$
Let $a$, $b$, $c$ be positive real numbers such that $\min(ab,bc,ca) \ge 1$. Prove that $$\sqrt[3]{(a^2+1)(b^2+1)(c^2+1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.$$
[i]Proposed by Tigran Margaryan, Armenia[/i]
Let $\mathcal{C}$ be a circle in the $xy$ plane with radius $1$ and center $(0, 0, 0)$, and let $P$ be a point in space with coordinates $(3, 4, 8)$. Find the largest possible radius of a sphere that is contained entirely in the slanted cone with base $\mathcal{C}$ and vertex $P$.
There is an Equilateral trapezoid $ ABCD. $
$ \bar{AB} =60, \bar{BC}=\bar{DA}= 36, \bar{CD}=108. $
$ M $ is the middle point of $ \bar {AB} $, and point $P$ on $ \bar{AM} $ follows that $ \bar {AP} $ =10.
The foot of perpendicular dropped from $P$ to $ \bar {BD} $ is $E$.
$ \bar{AC} \cap \bar{BD} $ is $ F $.
Point $X$ is on $ \bar {AF} $ which follows $ \bar{MX}=\bar{ME} $
Find $ \bar{AX} \times \bar{AF} $
Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that
\[36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.\]
[i]Proposed by Nazar Serdyuk, Ukraine[/i]
Let $x, y, z$ be positive real numbers such that $xyz = 1$. Prove that:
$$\frac{x^3 + y^3}{x^2 + xy + y^2} +\frac{ y^3 + z^3}{y^2 + yz + z^2} + \frac{z^3 + x^3}{z^2 + zx + x^2}
\ge 2.$$