Found problems: 85335
Arman, starting from a number, calculates the sum of the cubes of the digits of that number, and again calculates the sum of the cubes of the digits of the resulting number and continues the same process. Arman calls a number $Good$ if it reaches $1$ after performing a number of steps. Prove that there is an arithmetic progression of length $1402$ of good numbers.
[i]Proposed by Navid Safaei [/i]
Let $\alpha,\beta$ be the distinct positive roots of the equation of $2x=\tan x$.
Evaluate
\[\int_0^1 \sin \alpha x\sin \beta x\ dx\]
Prove that it is impossible to draw a convex quadrangle, with one diagonal equal to doubled another, the angle between them $45$ degrees, on the coordinate plane, so, that all the vertices' coordinates would be integers.
Prove that for any natural numbers $n,r$ with $r + 3 \le n $the binomial coefficients $n \choose r$, $n \choose r+1$, $n \choose r+2 $, $n \choose r+3 $ cannot be successive terms of an arithmetic progression.
Let $K$ be the measure of the area bounded by the $x$-axis, the line $x=8$, and the curve defined by \[f=\{(x,y)\,|\, y=x\text{ when }0\leq x\leq 5,\,y=2x-5\text{ when }5\leq x\leq 8\}.\] Then $K$ is:
$\textbf{(A) }21.5\qquad
\textbf{(B) }36.4\qquad
\textbf{(C) }36.5\qquad
\textbf{(D) }44\qquad$
$\textbf{ (E) }\text{less than 44 but arbitrarily close to it.}$
Grasshopper Gerald and his 2020 friends play leapfrog on a plane as follows. At each turn Gerald jumps over a friend so that his original point and his resulting point are symmetric with respect to this friend. Gerald wants to perform a series of jumps such that he jumps over each friend exactly once. Let us say that a point is achievable if Gerald can finish the 2020th jump in it. What is the maximum number $N{}$ such that for some initial placement of the grasshoppers there are just $N{}$ achievable points?
[i]Mikhail Svyatlovskiy[/i]
Given two closed broken lines in the plane with odd numbers of edges. All the lines, containing those edges are different, and not a triple of them intersects in one point. Prove that it is possible to chose one edge from each line such, that the chosen edges will be the opposite sides of a convex quadrangle.
If $ \theta$ is a constant such that $ 0 < \theta < \pi$ and $ x \plus{} \frac{1}{x} \equal{} 2\cos{\theta}$. then for each positive integer $ n$, $ x^n \plus{} \frac{1}{x^n}$ equals
$ \textbf{(A)}\ 2\cos{\theta}\qquad
\textbf{(B)}\ 2^n\cos{\theta}\qquad
\textbf{(C)}\ 2\cos^n{\theta}\qquad
\textbf{(D)}\ 2\cos{n\theta}\qquad
\textbf{(E)}\ 2^n\cos^n{\theta}$
In a $4 \times 6$ grid, all edges and diagonals are drawn (see attachment). Determine the number of parallelograms in the grid that uses only the line segments drawn and none of its four angles are right.
$(SWE 4)$ Let $a_0, a_1, a_2, \cdots$ be determined with $a_0 = 0, a_{n+1} = 2a_n + 2^n$. Prove that if $n$ is power of $2$, then so is $a_n$
There are $26$ students in the class.
They agreed that each of them would either be a liar (liars always lie) or a knight (knights always tell the truth).
When they came to the class and sat down for desks, each of them said: “I am sitting next to a liar.”
Then some students moved for other desks. After that, everyone says: “ I am sitting next to a knight .”
Is this possible?
Every time exactly two students sat at any desk.
A square is drawn inside a rectangle. The ratio of the width of the rectangle to a side of the square is $ 2: 1$. The ratio of the rectangle's length to its width is $ 2: 1$. What percent of the rectangle's area is inside the square?
$ \textbf{(A)}\ 12.5 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 75 \qquad \textbf{(E)}\ 87.5$
At a party, there are $100$ cats. Each pair of cats flips a coin, and they shake paws if and only if the coin comes up heads. It is known that exactly $4900$ pairs of cats shook paws. After the party, each cat is independently assigned a ``happiness index" uniformly at random in the interval $[0,1]$. We say a cat is [i]practical[/i] if it has a happiness index that is strictly greater than the index of every cat with which it shook paws. The expected value of the number of practical cats is $\frac{m}{n}$, where $m$ and $n$ are positive integers with $\gcd(m,n)=1$. Compute $100m + n$.
[i]Proposed by Brandon Wang[/i]
Show that the set of positive integers that cannot be represented as a sum of distinct perfect squares is finite.
The number $1- \frac12 +\frac13-\frac14+...+\frac{1}{2n-1}-\frac{1}{2n}$ is represented as an irreducible fraction. If $3n+1$ is a prime number, prove that the numerator of this fraction is a multiple of $3n + 1$.
There are distinct points $A_1, A_2, \dots, A_{2n}$ with no three collinear. Prove that one can relabel the points with the labels $B_1, \dots, B_{2n}$ so that for each $1 \le i < j \le n$ the segments $B_{2i-1} B_{2i}$ and $B_{2j-1} B_{2j}$ do not intersect and the following inequality holds.
\[ B_1 B_2 + B_3 B_4 + \dots + B_{2n-1} B_{2n} \ge \frac{2}{\pi} (A_1 A_2 + A_3 A_4 + \dots + A_{2n-1} A_{2n}) \]
On the equal $ AC $ and $ BC $ of an isosceles right triangle $ ABC $ , points $ D $ and $ E $ are marked respectively, so that $ CD = CE $. Perpendiculars on the straight line $ AE $, passing through the points $ C $ and $ D $, intersect the side $ AB $ at the points $ P $ and $ Q $.Prove that $ BP = PQ $.
In triangle $ABC$ point $M$ is the midpoint of side $AB$, and point $D$ is the foot of altitude $CD$.
Prove that $\angle A = 2\angle B$ if and only if $AC = 2MD$
A convex $2004$-sided polygon $P$ is given such that no four vertices are cyclic. We call a triangle whose vertices are vertices of $P$ thick if all other $2001$ vertices of $P$ lie inside the circumcircle of the triangle, and thin if they all lie outside its circumcircle. Prove that the number of thick triangles is equal to the number of thin triangles.
Given is the sequence $(a_n)_{n\geq 0}$ which is defined as follows:$a_0=3$ and $a_{n+1}-a_n=n(a_n-1) \ , \ \forall n\geq 0$.
Determine all positive integers $m$ such that $\gcd (m,a_n)=1 \ , \ \forall n\geq 0$.
The following diagram shows an equilateral triangle and two squares that share common edges. The area of each square is $75$. Find the distance from point $A$ to point $B$.
[asy]
size(175);
defaultpen(linewidth(0.8));
pair A=(-3,0),B=(3,0),C=rotate(60,A)*B,D=rotate(270,B)*C,E=rotate(90,C)*B,F=rotate(270,C)*A,G=rotate(90,A)*C;
draw(A--G--F--C--A--B--C--E--D--B);
label("$A$",F,N);
label("$B$",E,N);[/asy]
Prove that : $\ln \frac{11}{27}<\int_{\frac 14}^{\frac 34} \frac{1}{\ln (1-x)}\ dx<\ln \frac{7}{15}.$
Let $S=\{1,2,3,\ldots,280\}$. Find the smallest integer $n$ such that each $n$-element subset of $S$ contains five numbers which are pairwise relatively prime.
Let $A,B,P$ be three points on a circle. Prove that if $a,b$ are the distances from $P$ to the tangents at $A,B$ respectively, and $c$ is the distance from $P$ to the chord $AB$, then $c^2 =ab$.
Prove that there exist infinitely many positive integers $ n$ such that $ n!$ is not divisible by $ n^2\plus{}1$.