Found problems: 85335
Find all pairs $(a,b)$ of positive integers such that $a^{2017}+b$ is a multiple of $ab$.
Let $S$ be the set of positive integers $x$ for which there exist positive integers $y$ and $m$ such that $y^2-2^m=x^2$.
(a) Find all of the elements of $S$.
(b) Find all $x$ such that both $x$ and $x+1$ are in $S$.
Suppose that the positive divisors of a positive integer $n$ are $1=d_1<d_2<\ldots<d_k=n$, where $k \geq 5$. Given that $k \leq 1000$ and $n={d_2}^{d_3}{d_4}^{d_5}$, compute, with proof, all possible values of $k$.
Three groups $A, B$ and $C$ of mathematicians from different countries have invited to a ceremony. We have formed meetings such that three mathematicians participate in every meeting and there is exactly one mathematician from each group in every meeting. Also every two mathematicians have participated in exactly one meeting with each other.
[b](a)[/b] Prove that if this is possible, then number of mathematicians of the groups is equal.
[b](b)[/b] Prove that if there exist $3$ mathematicians in each group, then that work is possible.
[b](c)[/b] Prove that if number mathematicians of the groups be equal, then that work is possible.
Real numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $[-1,1].$ Find the probability that $$|x|+|y|+1 \le 3\min\{|x+y+1|, |x+y-1|\}.$$
Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect?
$ \textbf{(A)}\ 8 \qquad
\textbf{(B)}\ 9 \qquad
\textbf{(C)}\ 10 \qquad
\textbf{(D)}\ 12 \qquad
\textbf{(E)}\ 16$
Let $\mathcal P$ be a finite set of squares on an infinite chessboard. Kelvin the Frog notes that $\mathcal P$ may be tiled with only $1 \times 2$ dominoes, while Alex the Kat notes that $\mathcal P$ may be tiled with only $2 \times 1$ dominoes. The dominoes cannot be rotated in each tiling. Prove that the area of $\mathcal P$ is a multiple of 4.
Barbara and Jenna play the following game, in which they take turns. A number of coins lie on a table. When it is Barbara's turn, she must remove $2$ or $4$ coins, unless only one coin remains, in which case she loses her turn. When it is Jenna's turn, she must remove $1$ or $3$ coins. A coin flip determines who goes first. Whoever removes the last coin wins the game. Assume both players use their best strategy. Who will win when the game starts with $2013$ coins and when the game starts with $2014$ coins?
$\textbf{(A)}$ Barbara will win with $2013$ coins, and Jenna will win with $2014$ coins.
$\textbf{(B)}$ Jenna will win with $2013$ coins, and whoever goes first will win with $2014$ coins.
$\textbf{(C)}$ Barbara will win with $2013$ coins, and whoever goes second will win with $2014$ coins.
$\textbf{(D)}$ Jenna will win with $2013$ coins, and Barbara will win with $2014$ coins.
$\textbf{(E)}$ Whoever goes first will win with $2013$ coins, and whoever goes second will win with $2014$ coins.
Find all real numbers $k$ such that $r^4+kr^3+r^2+4kr+16=0$ is true for exactly one real number $r$.
[u]Down the In finite Corridor[/u]
Consider an isosceles triangle $T$ with base $10$ and height $12$. Defi ne a sequence $\omega_1$, $\omega_2$,$...$of circles such that $\omega_1$ is the incircle of $T$ and $\omega_{i+1}$ is tangent to $\omega_i$ and both legs of the isosceles triangle for $i > 1$.
[b]p1.[/b] Find the radius of $\omega_1$.
[b]p2.[/b] Find the ratio of the radius of $\omega_{i+1}$ to the radius of $\omega_i$.
[b]p3.[/b] Find the total area contained in all the circles.
Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$
[i]Israel[/i]
An evil witch is making a potion to poison the people of PUMAClandia. In order for the potion to work, the number of poison dart frogs cannot exceed $5$, the number of wolves’ teeth must be an even number, and the number of dragon scales has to be a multiple of $6$. She can also put in any number of tiger nails. Given that the stew has exactly $2021$ ingredients, in how many ways can she add ingredients for her potion to work?
Find the area of the region bounded by $C: y=-x^4+8x^3-18x^2+11$ and the tangent line which touches $C$ at distinct two points.
[b]p1.[/b] There are $32$ balls in a box: $6$ are blue, $8$ are red, $4$ are yellow, and $14$ are brown. If I pull out three balls at once, what is the probability that none of them are brown?
[b]p2.[/b] Circles $A$ and $B$ are concentric, and the area of circle $A$ is exactly $20\%$ of the area of circle $B$. The circumference of circle $B$ is $10$. A square is inscribed in circle $A$. What is the area of that square?
[b]p3.[/b] If $x^2 +y^2 = 1$ and $x, y \in R$, let $q$ be the largest possible value of $x+y$ and $p$ be the smallest possible value of $x + y$. Compute $pq$.
[b]p4.[/b] Yizheng and Jennifer are playing a game of ping-pong. Ping-pong is played in a series of consecutive matches, where the winner of a match is given one point. In the scoring system that Yizheng and Jennifer use, if one person reaches $11$ points before the other person can reach $10$ points, then the person who reached $11$ points wins. If instead the score ends up being tied $10$-to-$10$, then the game will continue indefinitely until one person’s score is two more than the other person’s score, at which point the person with the higher score wins. The probability that Jennifer wins any one match is $70\%$ and the score is currently at $9$-to-$9$. What is the probability that Yizheng wins the game?
[b]p5.[/b] The squares on an $8\times 8$ chessboard are numbered left-to-right and then from top-to-bottom (so that the top-left square is $\#1$, the top-right square is $\#8$, and the bottom-right square is $\#64$). $1$ grain of wheat is placed on square $\#1$, $2$ grains on square $\#2$, $4$ grains on square $\#3$, and so on, doubling each time until every square of the chessboard has some number of grains of wheat on it. What fraction of the grains of wheat on the chessboard are on the rightmost column?
[b]p6.[/b] Let $f$ be any function that has the following property: For all real numbers $x$ other than $0$ and $1$, $$f \left( 1 - \frac{1}{x} \right) + 2f \left( \frac{1}{1 - x}\right)+ 3f(x) = x^2.$$ Compute $f(2)$.
[b]p7.[/b] Find all solutions of: $$(x^2 + 7x + 6)^2 + 7(x^2 + 7x + 6)+ 6 = x.$$
[b]p8.[/b] Let $\vartriangle ABC$ be a triangle where $AB = 25$ and $AC = 29$. $C_1$ is a circle that has $AB$ as a diameter and $C_2$ is a circle that has $BC$ as a diameter. $D$ is a point on $C_1$ so that $BD = 15$ and $CD = 21$. $C_1$ and $C_2$ clearly intersect at $B$; let $E$ be the other point where $C_1$ and $C_2$ intersect. Find all possible values of $ED$.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Ali Baba and the $40$ thieves want to cross Bosporus strait. They made a line so that any two people standing next to each other are friends. Ali Baba is the first, he is also a friend with the thief next to his neighbour. There is a single
boat that can carry $2$ or $3$ people and these people must be friends. Can Ali Baba and the $40$ thieves always cross the strait if a single person cannot sail?
Prove that for any positive integers $a_1, a_2, \ldots , a_n$ the following inequality holds true:
\[\left\lfloor\frac{a_1^2}{a_2}\right\rfloor+\left\lfloor\frac{a_2^2}{a_3}\right\rfloor+\cdots+\left\lfloor\frac{a_n^2}{a_1}\right\rfloor\geqslant a_1+a_2+\cdots+a_n.\]
[i]Maxim Didin[/i]
p1. Bahri lives quite close to the clock gadang in the city of Bukit Tinggi West Sumatra. Bahri has an antique clock. On Monday $4$th March $2013$ at $10.00$ am, Bahri antique clock is two minutes late in comparison with Clock Tower. A day later, the antique clock was four minutes late compared to the Clock Tower. March $6$, $2013$ the clock is late six minutes compared to Jam Gadang. The following days Bahri observed that his antique clock exhibited the same pattern of delay. On what day and what date in $2014$ the antique Bahri clock (hand short and long hands) point to the same number as the Clock Tower?
p2. In one season, the Indonesian Football League is participated by $20$ teams football. Each team competes with every other team twice. The result of each match is $3$ if you win, $ 1$ if you draw, and $0$ if you lose. Every week there are $10$ matches involving all teams. The winner of the competition is the team that gets the highest total score. At the end what week is the fastest possible, the winner of the competition on is the season certain?
p3. Look at the following picture. The quadrilateral $ABCD$ is a cyclic. Given that $CF$ is perpendicular to $AF$, $CE$ is perpendicular to $BD$, and $CG$ is perpendicular to $AB$. Is the following statements true?
Write down your reasons. $$\frac{BD}{CE}=\frac{AB}{CG}+ \frac{AD}{CF}$$
[img]https://cdn.artofproblemsolving.com/attachments/b/0/dbd97b4c72bc4ebd45ed6fa213610d62f29459.png[/img]
p4. Suppose $M=2014^{2014}$. If the sum of all the numbers (digits) that make up the number $M$ equals $A$ and the sum of all the digits that make up the number $A$ equals $B$, then find the sum of all the numbers that make up $B$.
p5. Find all positive integers $n < 200$ so that $n^2 + (n + 1)^2$ is square of an integer.
Let $\mathcal{K}$ be a circle centered in $A$. Let $p$ be a line tangent to $\mathcal{K}$ in $B$ and let a line parallel to $p$ intersect $\mathcal{K}$ in $C$ and $D$. Let the line $AD$ intersect $p$ in $E$ and let $F$ be the intersection of the lines $CE$ and $AB$. Prove that the line through $D$, parallel to the tangent through $A$ to the circumcircle of $AFD$ intersects the line $CF$ on $\mathcal{K}$.
Find the extremum of $f(t)=\int_1^t \frac{\ln x}{x+t}dx\ (t>0)$.
Let ${c\equiv c\left(O, R\right)}$ be a circle with center ${O}$ and radius ${R}$ and ${A, B}$ be two points on it, not belonging to the same diameter. The bisector of angle${\angle{ABO}}$ intersects the circle ${c}$ at point ${C}$, the circumcircle of the triangle $AOB$ , say ${c_1}$ at point ${K}$ and the circumcircle of the triangle $AOC$ , say ${{c}_{2}}$ at point ${L}$. Prove that point ${K}$ is the circumcircle of the triangle $AOC$ and that point ${L}$ is the incenter of the triangle $AOB$.
Evangelos Psychas (Greece)
Let $ABC$ be a non-isosceles and acute triangle. $X$ is a point on arc $BC$ not containing $A$ such that $BA-CA = CX-BX$. The incircle of $\triangle ABC$ touches $AC$ and $AB$ at $E$ and $F$ respectively. The $X$-excircle of $\triangle XBC$ touches $XC$ and $XB$ at $Y$ and $Z$ respectively. Let $T$ be such that $TA$ and $TX$ bisects $\angle BAC$ and $\angle BXC$ respectively.
Prove that $T$ lies on the radical axis of circles $(BFZ)$ and $(CEY)$.
[i](Proposed by Chuah Jia Herng)[/i]
There is three-dimensional space. For every integer $n$ we build planes $ x \pm y\pm z = n$. All space is divided on octahedrons and tetrahedrons.
Point $(x_0,y_0,z_0)$ has rational coordinates but not lies on any plane. Prove, that there is such natural $k$ , that point $(kx_0,ky_0,kz_0)$ lies strictly inside the octahedron of partition.
Show that for each natural number $n > 1$
$1 \cdot \sqrt{{n \choose 1}}+ 2 \cdot \sqrt{{n \choose 2}}+...+n \cdot \sqrt{{n \choose n}} <\sqrt{2^{n-1}n^3}$
Let $k>1$ be an integer. A function $f:\mathbb{N^*}\to\mathbb{N^*}$ is called $k$-[i]tastrophic[/i] when for every integer $n>0$, we have $f_k(n)=n^k$ where $f_k$ is the $k$-th iteration of $f$:
\[f_k(n)=\underbrace{f\circ f\circ\cdots \circ f}_{k\text{ times}}(n)\]
For which $k$ does there exist a $k$-tastrophic function?
Find all positive integers $a, b,c$ greater than $1$, such that $ab + 1$ is divisible by $c, bc + 1$ is divisible by $a$ and $ca + 1$ is divisible by $b$.