Found problems: 85335
There are seven rods erected at the vertices of a regular heptagonal area. The top of each rod is connected to the top of its second neighbor by a straight piece of wire so that, looking from above, one sees each wire crossing exactly two others. Is it possible to set the respective heights of the rods in such a way that no four tops of the rods are coplanar and each wire passes one of the crossings from above and the other one from below?
The audience arranges $n$ coins in a row. The sequence of heads and tails is chosen arbitrarily. The audience also chooses a number between $1$ and $n$ inclusive. Then the assistant turns one of the coins over, and the magician is brought in to examine the resulting sequence. By an agreement with the assistant beforehand, the magician tries to determine the number chosen by the audience.
[list][b](a)[/b] Prove that if this is possible for some $n$, then it is also possible for $2n$.
[b](b)[/b] Determine all $n$ for which this is possible.[/list]
Every positive integer is marked with a number from the set $\{ 0,1,2\}$, according to the following rule:
$$\text{if a positive integer }k\text{ is marked with }j,\text{ then the integer }k+j\text{ is marked with }0.$$
Let $S$ denote the sum of marks of the first $2019$ positive integers. Determine the maximum possible value of $S$.
[i]Proposed by Ivan Novak[/i]
Tristan is eating his favorite cereal, Tiger Crunch, which has marshmallows of two colors, black and orange. He eats the marshmallows by randomly choosing from those remaining one at a time, and he starts out with $17$ orange and $5$ black marshmallows. If $\frac{p}{q}$ is the expected number of marshmallows remaining the instant that there is only one color left, and $p$ and $q$ are relatively prime positive integers, find $p + q$.
A continuous function $f : R \to R$ satisfies the conditions $f(1000) = 999$ and $f(x)f(f(x)) = 1$ for all real $x$. Determine $f(500)$.
Let $\Omega$ be an $n$-gon inscribed in the unit circle, with vertices $P_1, P_2, ..., P_n$.
(a) Show that there exists a point $P$ on the unit circle such that $PP_1 \cdot PP_2\cdot ... \cdot PP_n \ge 2$.
(b) Suppose for each $P$ on the unit circle, the inequality $PP_1 \cdot PP_2\cdot ... \cdot PP_n \le 2$ holds. Prove that $\Omega$ is regular.
Let $X$ be a subset of $R_{+}^{*}$ with $m$ elements.
Find $X$ such that the number of subsets with the same sum is maximum.
Find the greatest possible sum of integers $a$ and $b$ such that $\frac{2021!}{20^a\cdot 21^b}$ is a positive integer.
[i]Proposed by Aidan Duncan[/i]
Given are the integers $a , b , c, \alpha, \beta$ and the sequence $(u_n)$ is defined by $u_1=\alpha, u_2=\beta, u_{n+2}=au_{n+1}+bu_n+c$ for all $n \geq 1$.
a) Prove that if $a = 3 , b= -2 , c = -1$ then there are infinitely many pairs of integers $(\alpha ; \beta)$ so that $u_{2023}=2^{2022}$.
b) Prove that there exists a positive integer $n_0$ such that only one of the following two statements is true:
i) There are infinitely many positive integers $m$, such that $u_{n_0}u_{n_0+1}\ldots u_{n_0+m}$ is divisible by $7^{2023}$ or $17^{2023}$
ii) There are infinitely many positive integers $k$ so that $u_{n_0}u_{n_0+1}\ldots u_{n_0+k}-1$ is divisible by $2023$
If $ a$ and $ b$ are positive and $ a\not\equal{} 1,\,b\not\equal{} 1$, then the value of $ b^{\log_b{a}}$ is:
$ \textbf{(A)}\ \text{dependent upon }{b} \qquad
\textbf{(B)}\ \text{dependent upon }{a}\qquad
\textbf{(C)}\ \text{dependent upon }{a}\text{ and }{b}\qquad
\textbf{(D)}\ \text{zero}\qquad
\textbf{(E)}\ \text{one}$
The regular octagon of the following figure is inscribed in a circle of radius $1$ and $P$ is a arbitrary point of this circle. Calculate the value of $PA^2 + PB^2 +...+ PH^2$.
[img]https://cdn.artofproblemsolving.com/attachments/4/c/85e8e48c45970556077ac09c843193959b0e5a.png[/img]
The vertices of a regular $n + 1$-gon are denoted by $P_0,P_1,...,P_n$ in some order ($n \ge 2$). Each side of the polygon is assigned a natural number as follows: if the endpoints of the side are $P_i$ and $P_j$, then the assigned number equals $|i - j |$. Let S be the sum of all $n + 1$ assigned numbers.
(a) Given $n$, what is the smallest possible value of $S$?
(b) If $P_0$ is fixed, how many different assignments are there for which $S$ attains the smallest value?
Find all functions $f,g$:$R \to R$ such that $f(x+yg(x))=g(x)+xf(y)$ for $x,y \in R$.
Let $ABCD$ be a trapezoid such that $AB \parallel CD, AD=13, BC=15, AB=20,$ and $CD=34.$ Point $X$ lies inside the trapezoid such that $\angle{XAB}=2\angle{XBA}$ and $\angle{XDC}=2\angle{XCD}.$ Compute $XD-XA.$
We have an isosceles triangle $ABC$ with base $BC$. Through vertex $A$ draw a line $r$ parallel to $BC$. The points $P, Q$ are located on the perpendicular bisectors of $AB$ and $AC$ respectively, such that $PQ\perp BC$. They are points $M$ and $N$ on the line $r$ such that $\angle APM = \angle AQN = 90^o$. Prove that $$\frac{1}{AM} + \frac{1}{AN}\le \frac{2}{ AB}$$
Let $a$ and $b$ be fixed positive integers. We say that a prime $p$ is [i]fun[/i] if there exists a positive integer $n$ satisfying the following conditions:
[list]
[*]$p$ divides $a^{n!} + b$.
[*]$p$ divides $a^{(n + 1)!} + b$.
[*]$p < 2n^2 + 1$.
[/list]
Show that there are finitely many fun primes.
For arbitrary positive integers $a, b$, denote $a @ b =\frac{a-b}{gcd(a,b)}$
Let $n$ be a positive integer. Prove that the following conditions are equivalent:
(i) $gcd(n, n @ m) = 1$ for every positive integer $m < n$,
(ii) $n = p^k$ where $p$ is a prime number and $k$ is a non-negative integer.
Let $\triangle ABC$ be an equilateral triangle of side length 1. Let $D,E,F$ be points on $BC,AC,AB$ respectively, such that $\frac{DE}{20} = \frac{EF}{22} = \frac{FD}{38}$. Let $X,Y,Z$ be on lines $BC,CA,AB$ respectively, such that $XY\perp DE, YZ\perp EF, ZX\perp FD$. Find all possible values of $\frac{1}{[DEF]} + \frac{1}{[XYZ]}$.
The positive integers $1,2,...,121$ are arranged in the squares of a $11 \times 11$ table. Dima found the product of numbers in each row and Sasha found the product of the numbers in each column. Could they get the same set of $11$ numbers?
[i]Proposed by S. Berlov[/i]
When $x$ is added to both the numerator and the denominator of the fraction $a/b, a\neq b, b\neq 0$, the value of the fraction is changed to $c/d$. Then $x$ equals:
$\textbf{(A) }\dfrac1{c-d}\qquad
\textbf{(B) }\dfrac{ad-bc}{c-d}\qquad
\textbf{(C) }\dfrac{ad-bc}{c+d}\qquad
\textbf{(D) }\dfrac{bc-ad}{c-d}\qquad
\textbf{(E) }\dfrac{bc-ad}{c+d}$
Alex, Bill, and Charlie want to play a game of DotA. They each come online at a uniformly random time between $8:00$ and $8:05\text{ }\text{PM}$, and each person queues for $2$ minutes. However, if any of them sees any other of them online while queuing, they merge parties and restart the queue, again waiting for $2$ minutes starting from the merger time.
For example, suppose that Alex logs in at $8:00\text{ PM}$, Bill logs in at $8:01\text{ PM}$, and Charlie logs in at $8:02:30\text{ PM}$ ($30$ seconds past $8:02\text{ PM}$). At $8:01\text{ PM}$, Alex and Bill would merge parties and queue for $2$ minutes starting at $8:01\text{ PM}$. At $8:02:30\text{ PM}$, Charlie would merge with Alex and Bill’s party, since Alex and Bill have waited together for only $1.5$ minutes.
What is the probability that they will play as a party of $3$?
Evaluate $\sum_{n=1}^\infty \frac{n}{(2^n-2^{-n})^2}+\frac{(-1)^nn}{(2^n-2^{-n})^2}$
For each odd prime number $p$, prove that the integer
$$1!+2!+3!+\cdots +p!-\left\lfloor \frac{(p-1)!}{e}\right\rfloor$$is divisible by $p$
(Here, $e$ denotes the base of the natural logarithm and $\lfloor x\rfloor$ denotes the largest integer that is less than or equal to $x$.)
Consider a semicircle with center $M$ and diameter $AB$. Let $P$ be a point in the semicircle, different from $A$ and $B$, and let $Q$ be the midpoint of the arc $AP$. The line parallel to $QP$ through $M$ intersects $PB$ at the point $S$. Prove that the triangle $PMS$ is isosceles.
Let be a sequence $ \left( x_n \right)_{n\ge 0} $ with $ x_0\in (0,1) $ and defined as
$$ 2x_n=x_{n-1}+\sqrt{3-3x_{n-1}^2} . $$
Prove that this sequence is bounded and periodic. Moreover, find $ x_0 $ for which this sequence is convergent.
[i]Ovidiu Țâțan[/i]