Found problems: 85335
Prove that if $ n $ is a natural number greater than $ 2 $, then $$\sqrt[n + 1]{n+1} < \sqrt[n]{n}.$$
Let $\Omega$ be a circle in a plane. $2022$ pink points and $2565$ blue points are placed inside $\Omega$ such that no point has two colors and no two points are collinear with the center of $\Omega$. Prove that there exists a sector of $\Omega$ such that the angle at the center is acute and the number of blue points inside the sector is greater than the number of pink points by exactly $100$. (Note: such sector may contain no pink points.)
If $a,b,c$ are positive real numbers, prove that $\frac{a^2b^2}{a+b}+\frac{b^2c^2}{b+c}+\frac{c^2a^2}{c+a}\le \frac{a^3+b^3+c^3}{2}$
$ABC$ is an arbitrary triangle. $A',B',C'$ are midpoints of arcs $BC, AC, AB$. Sides of triangle $ABC$, intersect sides of triangle $A'B'C'$ at points $P,Q,R,S,T,F$. Prove that \[\frac{S_{PQRSTF}}{S_{ABC}}=1-\frac{ab+ac+bc}{(a+b+c)^{2}}\]
Let $S$ be a union of finitely many disjoint subintervals of $[0,1]$ such that no two points in $S$ have distance $1/10$. Show that the total length of the intervals comprising $S$ is at most $1/2$.
Find the exponent of $37$ in the representation of the number $111...... 11$ with $3\cdot 37^{2000}$ digits equals to $1$, as product of prime powers
Let $I$ denote the center of the circle inscribed in the right triangle $ABC$ with right angle at the vertex $A$. Next, denote by $M$ and $N$ the midpoints of the lines $AB$ and $BI$. Prove that the line $CI$ is tangent to the circumscribed circle of triangle $BMN$.
(Patrik Bak, Josef Tkadlec)
The diagram below shows rectangle $ABDE$ where $C$ is the midpoint of side $\overline{BD}$, and $F$ is the midpoint of side $\overline{AE}$. If $AB=10$ and $BD=24$, find the area of the shaded region.
[asy]
size(300);
defaultpen(linewidth(0.8));
pair A = (0,10),B=origin,C=(12,0),D=(24,0),E=(24,10),F=(12,10),G=extension(C,E,D,F);
filldraw(A--C--G--F--cycle,gray(0.7));
draw(A--B--D--E--F^^E--G--D);
label("$A$",A,NW);
label("$B$",B,SW);
label("$C$",C,S);
label("$D$",D,SE);
label("$E$",E,NE);
label("$F$",F,N);
[/asy]
Prove that for all $a,b,c$ positive real numbers
$\dfrac{a^4+1}{b^3+b^2+b}+\dfrac{b^4+1}{c^3+c^2+c}+\dfrac{c^4+1}{a^3+a^2+a} \ge 2$
Let $n\geq 2$ be a positive integer. We call a [i]vertex[/i] every point in the coordinate plane, whose $x$ and $y$ coordinates both are from the set $\{1,2,3,...,n\}$. We call a segment between two vertices an [i]edge[/i], if its length if $1$. We've colored some edges red, such that between any two vertices, there is a unique path of red edges (a path may contain each edge at most once). The red edge $f$ is [i]vital[/i] for an edge $e$, if the path of red edges connecting the two endpoints of $e$ contain $f$. Prove that there is a red edge, such that it is vital for at least $n$ edges.
Let $n > 2$ be a natural number. A subset $A$ of $R$ is said $n$-[i]small [/i]if there exist $n$ real numbers $t_1 , t_2 , ..., t_n$ such that the sets $t_1 + A , t_2 + A ,... , t_n + A$ are different . Show that $R$ can not be represented as a union of $ n - 1$ $n$-[i]small [/i] sets .
Notation : if $r \in R$ and $B \subset R$ , then $r + B = \{ r + b | b \in B\}$ .
Find the minimal positive integer $n$ such that no matter what $n$ distinct numbers from $1$ to $1000$ you choose, such that no two are divisible by a square of the same prime, one of the chosen numbers is a square of prime.
[i]D. Zmiaikou[/i]
Let $S=\{1,2,3,4,5,6\}.$ Find the number of bijective functions $f:S\rightarrow S$ for which there exist exactly $6$ bijective functions $g:S\rightarrow S$ such that $f(g(x))=g(f(x))$ for all $x\in S$.
[i]Proposed by Nathan Ramesh
A flat board has a circular hole with radius $1$ and a circular hole with radius $2$ such that the distance between the centers of the two holes is 7. Two spheres with equal radii sit in the two holes such that the spheres are tangent to each other. The square of the radius of the spheres is $\frac{m}n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Two cards are dealt from a deck of four red cards labeled $A$, $B$, $C$, $D$ and four green cards labeled $A$, $B$, $C$, $D$. A winning pair is two of the same color or two of the same letter. What is the probability of drawing a winning pair?
$\textbf{(A)}\ \frac{2}{7} \qquad
\textbf{(B)}\ \frac{3}{8} \qquad
\textbf{(C)}\ \frac{1}{2} \qquad
\textbf{(D)}\ \frac{4}{7} \qquad
\textbf{(E)}\ \frac{5}{8}$
Let $ \triangle ABC $ be an equilateral triangle with height $13$, and let $O$ be its center. Point $X$ is chosen at random from all points inside $ \triangle ABC $. Given that the circle of radius $1$ centered at $X$ lies entirely inside $ \triangle ABC $, what is the probability that this circle contains $O$?
Does there exist a positive integer $ n$ such that $ n$ has exactly 2000 prime divisors and $ n$ divides $ 2^n \plus{} 1$?
In this figure $\angle RFS = \angle FDR$, $FD = 4$ inches, $DR = 6$ inches, $FR = 5$ inches, $FS = 7\dfrac{1}{2}$ inches. The length of $RS$, in inches, is:
[asy]
import olympiad;
pair F,R,S,D;
F=origin;
R=5*dir(aCos(9/16));
S=(7.5,0);
D=4*dir(aCos(9/16)+aCos(1/8));
label("$F$",F,SW);label("$R$",R,N); label("$S$",S,SE); label("$D$",D,W);
label("$7\frac{1}{2}$",(F+S)/2.5,SE);
label("$4$",midpoint(F--D),SW);
label("$5$",midpoint(F--R),W);
label("$6$",midpoint(D--R),N);
draw(F--D--R--F--S--R);
markscalefactor=0.1;
draw(anglemark(S,F,R)); draw(anglemark(F,D,R));
//Credit to throwaway1489 for the diagram[/asy]
$\textbf{(A)}\ \text{undetermined} \qquad
\textbf{(B)}\ 4\qquad
\textbf{(C)}\ 5\dfrac{1}{2} \qquad
\textbf{(D)}\ 6 \qquad
\textbf{(E)}\ 6\dfrac{1}{4}$
[b]p17.[/b] Let the roots of the polynomial $f(x) = 3x^3 + 2x^2 + x + 8 = 0$ be $p, q$, and $r$. What is the sum $\frac{1}{p} +\frac{1}{q} +\frac{1}{r}$ ?
[b]p18.[/b] Two students are playing a game. They take a deck of five cards numbered $1$ through $5$, shuffle them, and then place them in a stack facedown, turning over the top card next to the stack. They then take turns either drawing the card at the top of the stack into their hand, showing the drawn card to the other player, or drawing the card that is faceup, replacing it with the card on the top of the pile. This is repeated until all cards are drawn, and the player with the largest sum for their cards wins. What is the probability that the player who goes second wins, assuming optimal play?
[b]p19.[/b] Compute the sum of all primes $p$ such that $2^p + p^2$ is also prime.
[b]p20.[/b] In how many ways can one color the $8$ vertices of an octagon each red, black, and white, such that no two adjacent sides are the same color?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Rodrigo has a very large sheet of graph paper. First he draws a line segment connecting point $(0,4)$ to point $(2,0)$ and colors the $4$ cells whose interiors intersect the segment, as shown below. Next Rodrigo draws a line segment connecting point $(2000,3000)$ to point $(5000,8000)$. How many cells will he color this time?
[asy]
filldraw((0,4)--(1,4)--(1,3)--(0,3)--cycle, gray(.75), gray(.5)+linewidth(1));
filldraw((0,3)--(1,3)--(1,2)--(0,2)--cycle, gray(.75), gray(.5)+linewidth(1));
filldraw((1,2)--(2,2)--(2,1)--(1,1)--cycle, gray(.75), gray(.5)+linewidth(1));
filldraw((1,1)--(2,1)--(2,0)--(1,0)--cycle, gray(.75), gray(.5)+linewidth(1));
draw((-1,5)--(-1,-1),gray(.9));
draw((0,5)--(0,-1),gray(.9));
draw((1,5)--(1,-1),gray(.9));
draw((2,5)--(2,-1),gray(.9));
draw((3,5)--(3,-1),gray(.9));
draw((4,5)--(4,-1),gray(.9));
draw((5,5)--(5,-1),gray(.9));
draw((-1,5)--(5, 5),gray(.9));
draw((-1,4)--(5,4),gray(.9));
draw((-1,3)--(5,3),gray(.9));
draw((-1,2)--(5,2),gray(.9));
draw((-1,1)--(5,1),gray(.9));
draw((-1,0)--(5,0),gray(.9));
draw((-1,-1)--(5,-1),gray(.9));
dot((0,4));
label("$(0,4)$",(0,4),NW);
dot((2,0));
label("$(2,0)$",(2,0),SE);
draw((0,4)--(2,0));
draw((-1,0) -- (5,0), arrow=Arrow);
draw((0,-1) -- (0,5), arrow=Arrow);
[/asy]
$\textbf{(A) }6000\qquad\textbf{(B) }6500\qquad\textbf{(C) }7000\qquad\textbf{(D) }7500\qquad\textbf{(E) }8000$
Find, with proof, all polynomials $f$ such that $f$ has nonnegative integer coefficients, $f$($1$) = $8$ and $f$($2$) = $2012$.
Acute triangle $\triangle ABC$ satis es $AB < AC$. Let the circumcircle of this triangle be $O$, and the midpoint of $BC,CA,AB$ be $D,E,F$. Let $P$ be the intersection of the circle with $AB$ as its diameter and line $DF$, which is in the same side of $C$ with respect to $AB$. Let $Q$ be the intersection of the circle with $AC$ as its diameter and the line $DE$, which is in the same side of $B$ with respect to $AC$. Let $PQ \cap BC = R$, and let the line passing through $R$ and perpendicular to $BC$ meet $AO$ at $X$. Prove that $AX = XR$.
On a blackboard lies $50$ magnets in a line numbered from $1$ to $50$, with different magnets containing different numbers. David walks up to the blackboard and rearranges the magnets into some arbitrary order. He then writes underneath each pair of consecutive magnets the positive difference between the numbers on the magnets. If the expected number of times he writes the number $1$ can be written in the form $\tfrac mn$ for relatively prime positive integers $m$ and $n$, compute $100m+n$.
[i] Proposed by David Altizio [/i]
Let $ b_1<b_2<b_3<\dots $ be the sequence of all natural numbers which are sum of squares of two natural numbers.
Prove that there exists infinite natural numbers like $m$ which $b_{m+1}-b_m=2015$ .
If $x$ and $y$ are positive integers, and $x^4+y^4=4721$, find all possible values of $x+y$