Found problems: 85335
Find the area of the equilateral triangle that includes vertices at $\left(-3,5\right)$ and $\left(-5,9\right)$.
$\text{(A) }3\sqrt{3}\qquad\text{(B) }10\sqrt{3}\qquad\text{(C) }\sqrt{30}\qquad\text{(D) }2\sqrt{15}\qquad\text{(E) }5\sqrt{3}$
$n$ is a natural number. for every positive real numbers $x_{1},x_{2},...,x_{n+1}$ such that $x_{1}x_{2}...x_{n+1}=1$ prove that:
$\sqrt[x_{1}]{n}+...+\sqrt[x_{n+1}]{n} \geq n^{\sqrt[n]{x_{1}}}+...+n^{\sqrt[n]{x_{n+1}}}$
For any point $(x, y)$ with $0\le x < 1$ and $0 \le y < 1$, Jenny can perform a shuffle on that point, which takes the point to $(\{3x + y\} ,\{x + 2y\})$ where $\{a\}$ denotes the fractional or decimal part of $a$ (so for example,$\{\pi\} = \pi - 3 = 0.1415...$). How many points $p$ are there such that after $3$ shuffles on $p$, $p$ ends up in its original position?
Let $f(x) = 3x + 2.$ Prove that there exists $m \in \mathbb{N}$ such that $f^{100}(m)$ is divisible by $1988$.
let $p$and $q=p+2$ be twin primes. consider the diophantine equation $(+)$ given by
$n!+pq^2=(mp)^2$ $m\geq1$, $n\geq1$
i. if $m=p$,find the value of $p$.
ii. how many solution quadruple $(p,q,m,n)$ does $(+)$ have ?
Define polynomials $f_{n}(x)$ for $n \geq 0$ by $f_{0}(x)=1, f_{n}(0)=0$ for $n \geq 1,$ and
$$
\frac{d}{d x} f_{n+1}(x)=(n+1) f_{n}(x+1)
$$
for $n \geq 0 .$ Find, with proof, the explicit factorization of $f_{100}(1)$ into powers of distinct primes.
An arbitrary point $D$ is marked on the hypotenuse $AB$ of a right triangle $ABC$. The circle circumscribed around the triangle $ACD$ intersects the line $BC$ at the point $E$ for the second time, and the circle circumscribed around the triangle $BCD$ intersects the line $AC$ for the second time at the point $F$. Prove that the line $EF$ passes through the point $D$.
Eighteen students participate in a team selection test with three problems, each worth up to seven points. All scores are nonnegative integers. After the competition, the results are posted by Evan in a table with 3 columns: the student's name, score, and rank (allowing ties), respectively. Here, a student's rank is one greater than the number of students with strictly higher scores (for example, if seven students score $0, 0, 7, 8, 8, 14, 21$ then their ranks would be $6, 6, 5, 3, 3, 2, 1$ respectively).
When Richard comes by to read the results, he accidentally reads the rank column as the score column and vice versa. Coincidentally, the results still made sense! If the scores of the students were $x_1 \le x_2 \le \dots \le x_{18}$, determine the number of possible values of the $18$-tuple $(x_1, x_2, \dots, x_{18})$. In other words, determine the number of possible multisets (sets with repetition) of scores.
[i]Proposed by Yang Liu[/i]
Let $\alpha$, $\beta$ and $\gamma$ be the roots of the polynomial $2023x^3-2023x^2-1$. Find $$\frac{1}{\alpha^3}+\frac{1}{\beta^3}+\frac{1}{\gamma^3}$$.
[i]Proposed by Andy Xu[/i]
Abolf is on the second step of a stairway to heaven in every step of this stairway except the first one which is the hell there is a devil who is either a human, an elf or a demon and tempts Abolf. The devil in the second step is Satan himself as one of three forms. Whenever an elf or a demon tries to tempt Abolf he resists and walks one step up but when a human tempts Abolf he is deceived and hence he walks one step down. However if Abolf is deceived by Satan for the first time he resists and does not fall down to hell but the second time he falls down to eternal hell. Every time a devil makes a temptation it changes its form from a human, an elf, a demon to an elf, a demon, a human respectively. Prove that Abolf passes each step after some time.
[i]Proposed by Yaser Ahmadi Fouladi[/i]
Let $n > 3$ be a natural number. Prove that $4^n + 1$ has a prime divisor $> 20$.
Determine the smallest positive integer $a$ for which there exist a prime number $p$ and a positive integer $b \ge 2$ such that $$\frac{a^p -a}{p}=b^2.$$
The numbers $1, 2, 3, ..., 1000$ are written on the board. Two people take turns erasing one number at a time. The game ends when two numbers remain on the board. If their sum is divisible by three, then the one who made the first move wins. if not, then his partner. Which one will win if played correctly?
On a real number line, the points $1, 2, 3, \dots, 11$ are marked. A grasshopper starts at point $1$, then jumps to each of the other $10$ marked points in some order so that no point is visited twice, before returning to point $1$. The maximal length that he could have jumped in total is $L$, and there are $N$ possible ways to achieve this maximum. Compute $L+N$.
[i]Proposed by Yannick Yao[/i]
Given an equilateral triangle with sidelength $k$ cm. With lines parallel to it's sides, we split it into $k^2$ small equilateral triangles with sidelength $1$ cm. This way, a triangular grid is created. In every small triangle of sidelength $1$ cm, we place exactly one integer from $1$ to $k^2$ (included), such that there are no such triangles having the same numbers. With vertices the points of the grid, regular hexagons are defined of sidelengths $1$ cm. We shall name as [i]value [/i] of the hexagon, the sum of the numbers that lie on the $6$ small equilateral triangles that the hexagon consists of . Find (in terms of the integer $k>4$) the maximum and the minimum value of the sum of the values of all hexagons .
Find all primes that can be written both as a sum of two primes and as a difference of two primes.
Show that $1 \le n^{1/n} \le 2$ for all positive integers $n$.
Find the smallest $k$ such that $1 \le n ^{1/n} \le k$ for all positive integers $n$.
Convex polygons $P_1$ and $P_2$ are drawn in the same plane with $n_1$ and $n_2$ sides, respectively, $n_1 \le n_2$. If $P_1$ and $P_2$ do not have any line segment in common, then the maximum number of intersections of $P_1$ and $P_2$ is:
$\textbf{(A)}\ 2n_1 \qquad\textbf{(B)}\ 2n_2 \qquad\textbf{(C)}\ n_1n_2 \qquad\textbf{(D)}\ n_1+n_2 \qquad\textbf{(E)}\ \text{none of these} $
Two circles $ \left(\Omega_1\right),\left(\Omega_2\right) $ touch internally on the point $ T $. Let $ M,N $ be two points on the circle $ \left(\Omega_1\right) $ which are different from $ T $ and $ A,B,C,D $ be four points on $ \left(\Omega_2\right) $ such that the chords $ AB, CD $ pass through $ M,N $, respectively. Prove that if $ AC,BD,MN $ have a common point $ K $, then $ TK $ is the angle bisector of $ \angle MTN $.
* $ \left(\Omega_2\right) $ is bigger than $ \left(\Omega_1\right) $
There are $n$ points in the plane, some of which are connected by segments.
Some of the segments are colored in white, while the others are colored black in such a way that there exist a completely white as well as a completely black closed broken line of segments, each of them passing through every one of the $n$ points exactly once.
It is known that the segments $AB$ and $BC$ are white. Prove that it is possible to recolor the segments in red and blue in such a way that $AB$ and $BC$ are recolored as red, [hide=not all of which segments are recolored red]meaning that recoloring every white as red and every black as blue is not acceptable[/hide], and that there exist a completely red as well as a completely blue closed broken line of segments, each of them passing through every one of the $n$ points exactly once.
Find the greatest prime factor of the sum of the two largest two-digit prime numbers.
Let $a,b \in R^+$ with $ab = 1$, prove that
$$\frac{1}{a^3 + 3b}+\frac{1}{b^3 + 3a}\le \frac12.$$
For any nonempty, finite set $B$ and real $x$, define
$$d_B(x) = \min_{b\in B} |x-b|$$
(1) Given positive integer $m$. Find the smallest real number $\lambda$ (possibly depending on $m$) such that for any positive integer $n$ and any reals $x_1,\cdots,x_n \in [0,1]$, there exists an $m$-element set $B$ of real numbers satisfying $$d_B(x_1)+\cdots+d_B(x_n) \le \lambda n$$
(2) Given positive integer $m$ and positive real $\epsilon$. Prove that there exists a positive integer $n$ and nonnegative reals $x_1,\cdots,x_n$, satisfying for any $m$-element set $B$ of real numbers, we have
$$d_B(x_1)+\cdots+d_B(x_n) > (1-\epsilon)(x_1+\cdots+x_n)$$
Given a table $m\times n$, in each cell of which a number $+1$ or $-1$ is written. It is known that initially exactly one $-1$ is in the table, all the other numbers being $+1$. During a move, it is allowed to chose any cell containing $-1$, replace this $-1$ by $0$, and simultaneously multiply all the numbers in the neighbouring cells by $-1$ (we say that two cells are neighbouring if they have a common side). Find all $(m,n)$ for which using such moves one can obtain the table containing zeros only, regardless of the cell in which the initial $-1$ stands.
If $A=\{x|\sqrt{x-2}\leq0\},B=\{x|10^{x^2-2}=10^{x}\}$, then $A\cap(\mathbb{R}\backslash B)$ is
$\text{(A)}\{2\}\qquad\text{(B)}\{-1\}\qquad\text{(C)}\{x|x\leq2\}\qquad\text{(D)}\varnothing$