Found problems: 85335
Let $A$ be the set $\{k^{19}-k: 1<k<20, k\in N\}$. Let $G$ be the GCD of all elements of $A$.
Then the value of $G$ is?
Consider two lines $\ell$ and $\ell ' $ and a fixed point $P$ equidistant from these lines. What is the locus of projections $M$ of $P$ on $AB$, where $A$ is on $\ell $, $B$ on $\ell ' $, and angle $\angle APB$ is right?
Given
$$x_1x_2 \cdots x_{2022} = 1,$$
$$(x_1 +1)(x_2 +1)\cdots (x_{2022} +1)=2,$$
$$\text{and so on},$$
$$(x_1 + 2021) (x_2 + 2021) \cdots (x_{2022} + 2021) = 2^{2021},$$
compute
$$(x_1 +2022)(x_2 +2022) \cdots (x_{2022} +2022).$$
Two congruent rectangles are located as shown in the figure. Find the area of the shaded part.
[img]https://cdn.artofproblemsolving.com/attachments/2/e/10b164535ab5b3a3b98ce1a0b84892cd11d76f.png[/img]
Find all functions $f : R \to R$ such that for all $x, y \in R$ holds $$yf(2x) - xf(2y) = 8xy(x^2 - y^2).$$
Let $ABC$ be an obtuse triangle with obtuse angle $\angle B$ and altitudes $AD, BE, CF$. Let $T$ and $S$ be the midpoints of $AD$ and $CF$, respectively. Let $M$ and $N$ and be the symmetric images of $T$ with respect to lines $BE$ and $BD$, respectively.
Prove that $S$ lies on the circumcircle of triangle $BMN$.
In the scalene triangle $ABC$, the bisector of angle A cuts side $BC$ at point $D$.
The tangent lines to the circumscribed circunferences of triangles $ABD$ and $ACD$ on point D, cut lines $AC$ and $AB$ on points $E$ and $F$ respectively. Let $G$ be the intersection point of lines $BE$ and $CF$.
Prove that angles $EDG$ and $ADF$ are equal.
Find the sum of the coefficients of the polynomial $P(x) = x^4- 29x^3 + ax^2 + bx + c$, given that $P(5) = 11$, $P(11) = 17$, and $P(17) = 23$.
Let $n > 1$ be a fixed integer. On an infinite row of squares, there are $n$ stones on square $1$ and no stones on squares $2$, $3$, $4$, $\ldots$. Curious George plays a game in which a [i]move[/i] consists of taking two adjacent piles of sizes $a$ and $b$, where $a-b$ is a nonzero even integer, and transferring stones to equalize the piles (so that both piles have $\frac{a+b}{2}$ stones). The game ends when no more moves can be made. George wants to analyze the number of moves it takes to end the game.
(a) Suppose George wants to end the game as quickly as possible. How many moves will it take him?
(b) Suppose George wants to end the game as slowly as possible. Show that for all $n > 2$, the game will end after at most $\frac{3}{16}n^2$ moves.
[i]Scoring note:[/i] For part (b), partial credit will be awarded for correct proofs of weaker bounds, eg. $\frac{1}{4}n^2$, $n^k$, or $k^n$ (for some $k \geq 2$).
Inside a hoop of radius $ 2r $ a disk of radius $ r $ rolls on the hoop without slipping. What line is traced by a point arbitrarily chosen on the edge of the disk?
Let $S=\{1,2,\ldots ,98\}$. Find the least natural number $n$ such that we can pick out $10$ numbers in any $n$-element subset of $S$ satisfying the following condition: no matter how we equally divide the $10$ numbers into two groups, there exists a number in one group such that it is coprime to the other numbers in that group, meanwhile there also exists a number in the other group such that it is not coprime to any of the other numbers in the same group.
All faces of a tetrahedron are right-angled triangles. It is known that three of its edges have the same length $s$. Find the volume of the tetrahedron.
Answer both (i) and (ii). Test for convergence the series
(i) \[ \frac 1{\log (2!)} + \frac 1{\log (3!)} + \frac 1{\log (4!)} + \cdots + \frac 1{\log (n!)} +\cdots\]
(ii) \[ \frac 13 + \frac 1{3\sqrt3} + \frac 1{3\sqrt3 \sqrt[3]3} + \cdots + \frac 1{3\sqrt3 \sqrt[3]3 \cdots \sqrt[n]3} + \cdots\]
For every integer $n\geq 3$, let $s_n$ be the sum of all primes (strictly) less than $n$. Are there infinitely many integers $n\geq 3$ such that $s_n$ is coprime to $n$?
[i]Russian Competition[/i]
If $X$, $Y$ and $Z$ are different digits, then the largest possible $3-$digit sum for
$\begin{tabular}{ccc}
X & X & X \\
& Y & X \\
+ & & X \\ \hline
\end{tabular}$
has the form
$\text{(A)}\ XXY \qquad \text{(B)}\ XYZ \qquad \text{(C)}\ YYX \qquad \text{(D)}\ YYZ \qquad \text{(E)}\ ZZY$
A positive integer $n$ is said to be $m$-free if $n \leq m!$ and $\gcd(i,n)=1$ for each $i=1,2,...,m$. Let $\mathcal{S}_k$ denote the sum of the squares of all the $k$-free integers. Find the remainder when $\mathcal{S}_7-\mathcal{S}_6$ is divided by $1000$.
[i]Proposed by [b]FedeX333X[/b][/i]
Find all continuous functions $f\colon \mathbb{R}_{>0}\to \mathbb{R}_{\geq 1}$ for which the following equation holds for all positive reals $x$, $y$:
\[f\left(\frac{f(x)}{y}\right)-f\left(\frac{f(y)}{x}\right)=xy\left(f(x+1)-f(y+1)\right)\]
If Bobby’s age is increased by $6$, it’s a number with an integral (positive) square root. If his age is decreased by $6$, it’s that square root. How old is Bobby?
Given a scalene triangle $ABC$. Its incircle touches $BC,AC,AB$ at $D,E,F$ respectvely. Let $L,M,N$ be the symmetric points of $D$ with $EF$,of $E$ with $FD$,of $F$ with $DE$,respectively. Line $AL$ intersects $BC$ at $P$,line $BM$ intersects $CA$ at $Q$,line $CN$ intersects $AB$ at $R$. Prove that $P,Q,R$ are collinear.
Prove that if the diagonals of a convex quadrilateral are perpendicular, then the feet of perpendiculars dropped from the intersection point of diagonals on the sides of this quadrilateral lie on one circle. Is the converse true?
6. Circles $W_{1}$ and $W_{2}$ with equal radii are given. Let $P$,$Q$ be the intersection of the circles.
points $B$ and $C$ are on $W_{1}$ and $W_{2}$ such that they are inside $W_{2}$ and $W_{1}$ respectively.
Points $X$,$Y$ $\neq$ $P$ are on $W_{1}$ and $W_{2}$ respectively, such that $\angle{BPQ}=\angle{BYQ}$ and $\angle{CPQ}=\angle{CXQ}$.Denote by $S$ as the other intersection of $(YPB)$ and $(XPC)$. Prove that $QS,BC,XY$ are concurrent.
Determine the polynomial
$$f(x) = x^k + a_{k-1} x^{k-1}+\cdots +a_1 x +a_0 $$
of smallest degree such that $a_i \in \{-1,0,1\}$ for $0\leq i \leq k-1$ and $f(n)$ is divisible by $30$ for all positive integers $n$.
Walter rolls four standard six-sided dice and finds that the product of the numbers on the upper face is 144. Which of the following could NOT be on the sum of the upper four faces?
$ \textbf{(A)}\ 14 \qquad \textbf{(B)}\ 15 \qquad \textbf{(C)}\ 16 \qquad \textbf{(D)}\ 17 \qquad \textbf{(E)}\ 18$
How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.)
$\textbf{(A) }3\qquad\textbf{(B) }6\qquad\textbf{(C) }12\qquad\textbf{(D) }18\qquad\textbf{(E) }24$
In a triangle $ABC$ , we have a point $O$ on $BC$ . Now show that there exists a line $l$ such that $l||AO$ and $l$ divides the triangle $ABC$ into two halves of equal area .