Found problems: 85335
Among the positive integers less than $100$, each of whose digits is a prime number, one is selected at random. What is the probablility that the selected number is prime?
$\textbf{(A) } \dfrac{8}{99}
\qquad\textbf{(B) } \dfrac{2}{5}
\qquad\textbf{(C) } \dfrac{9}{20}
\qquad\textbf{(D) } \dfrac{1}{2}
\qquad\textbf{(E) } \dfrac{9}{16}
$
Andy, Bess, Charley and Dick play on a $1000 \times 1000$ board. They make moves in turn: Andy first, then Bess, then Charley and finally Dick, after that Andy moves again and so on. At each move a player must paint several unpainted squares forming $2 \times 1, 1 \times 2, 1 \times 3$, or $3 \times 1$ rectangle. The player that cannot move loses. Prove that some three players can cooperate to make the fourth player lose.
A sequence of real numbers $ \{a_{n}\}$ is defineds by $ a_{0}\neq 0,1$, $ a_1\equal{}1\minus{}a_0$,$ a_{n\plus{}1}\equal{}1\minus{}a_n(1\minus{}a_n)$, $ n\equal{}1,2,...$.
Prove that for any positive integer $ n$, we have
$ a_{0}a_{1}...a_{n}(\frac{1}{a_0}\plus{}\frac{1}{a_1}\plus{}...\plus{}\frac{1}{a_n})\equal{}1$
Which of these numbers is the largest?
$\textbf{(A)} \sqrt{\sqrt[3]{5\cdot 6}}\qquad
\textbf{(B)} \sqrt{6\sqrt[3]{5}}\qquad
\textbf{(C)} \sqrt{5\sqrt[3]{6}}\qquad
\textbf{(D)} \sqrt[3]{5\sqrt{6}}\qquad
\textbf{(E)} \sqrt[3]{6\sqrt{5}}$
Find all non-negative integers $a, b, c, d$ such that $7^a = 4^b + 5^c + 6^d$
For $n$ an odd positive integer, the unit squares of an $n\times n$ chessboard are coloured alternately black and white, with the four corners coloured black. A it tromino is an $L$-shape formed by three connected unit squares. For which values of $n$ is it possible to cover all the black squares with non-overlapping trominos? When it is possible, what is the minimum number of trominos needed?
Prove that \[ 1 < \frac{1}{1001} + \frac{1}{1002} + \frac{1}{1003} + \cdots + \frac{1}{3001} < 1 \frac{1}{3}. \]
Determine all functions $f: \mathbb{R} \to \mathbb{R}$, such that
$$f(xf(y)+1)=y+f(f(x)f(y))$$
for all $x, y \in \mathbb{R}$.
(Theresia Eisenkölbl)
In acute triangle $\triangle ABC$, $\overline{AB} = 20$ and $\overline{AC} = 21$. Let the feet of the perpendiculars from $A$ to the angle bisectors of $\angle ACB$ and $\angle ABC$ be $X$ and $Y$, respectively. Let $M$ be the midpoint of $\overline{XY}$. Suppose $P$ is the point on side $BC$ such that $MP$ is parallel to the angle bisector of $\angle BAC$. If given that $\overline{BP} = 11$, then the length of side $BC$ can be expressed as a common fraction in the form $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
The numbers from $1$ through $100$ are written in some order on a circle.
We call a pair of numbers on the circle [i]good [/i] if the two numbers are not neighbors on the circle and if at least one of the two arcs they determine on the circle only contains numbers smaller then both of them. What may be the total number of good pairs on the circle.
7. There are 100 chess bishops on white squares of a $100 \times 100$ chess board. Some of them are white and some of them are black. They can move in any order and capture the bishops of opposing color. What number of moves is sufficient for sure to retain only one bishop on the chess board?
At a recent math contest, Evan was asked to find $2^{2016} \pmod{p}$ for a given prime number $p$ with $100 < p < 500$. Evan has forgotten what the prime $p$ was, but still remembers how he solved it:
[list]
[*] Evan first tried taking $2016$ modulo $p - 1$, but got a value $e$ larger than $100$.
[*] However, Evan noted that $e - \frac{1}{2}(p - 1) = 21$, and then realized the answer was $-2^{21} \pmod{p}$.
[/list]
What was the prime $p$?
Find all prime numbers $p$ for which one can find a positive integer $m$ and nonnegative integers $a_0,a_1,...,a_m$ less than $p$ such that $$\begin{cases} a_0+a_1p+...+a_{m-1}p^{m-1}+a_{m}p^{m} = 2013 \\
a_0+a_1+...+a_{m-1}+a_{m} = 11\end{cases}$$
Evaluate $\int_{-1}^{0}\sqrt{\frac{1+x}{1-x}}dx.$
How many $10$-digit positive integers containing only the numbers $1,2,3$ can be written such that the first and the last digits are same, and no two consecutive digits are same?
$
\textbf{(A)}\ 768
\qquad\textbf{(B)}\ 642
\qquad\textbf{(C)}\ 564
\qquad\textbf{(D)}\ 510
\qquad\textbf{(E)}\ 456
$
The Lindelöf number $L(X)$ of a topological space $X$ is the least infinite cardinal $\lambda$ with the property that every open covering of $X$ has a subcovering of cardinality at most $\lambda$. Prove that if evert non-countably infinite subset of a first countable space $X$ has a point of condensation, then $L(X)=\sup L(A)$, where $A$ runs over the separable closed subspaces of $X$.
(A point of condensation of a subset $H\subseteq X$ is a point $x\in X$ such that any neighbourhood of $x$ intersects $H$ in a non-countably infinite set.)
By default, iPhone passcodes consist of four base-$10$ digits. However, Freya decided to be unconventional and use hexadecimal (base-$16$) digits instead of base-$10$ digits! (Recall that $10_{16} = 16_{10}$.) She sets her passcode such that exactly two of the hexadecimal digits are prime. How many possible passcodes could she have set?
Prove that the plane dividing in equal proportions the surface area and volume of the circumscribed polyhedron passes through the center of the sphere inscribed in this polyhedron.
Don Miguel places a token in one of the $(n+1)^2$ vertices determined by an $n \times n$ board. A [i]move[/i] consists of moving the token from the vertex on which it is placed to an adjacent vertex which is at most $\sqrt2$ away, as long as it stays on the board. A [i]path[/i] is a sequence of moves such that the token was in each one of the $(n+1)^2$ vertices exactly once. What is the maximum number of diagonal moves (those of length $\sqrt2$) that a path can have in total?
A positive integer is called a repunit, if it is written only by ones. The repunit with $n$ digits will be denoted as $\underbrace{{11\cdots1}}_{n}$ . Prove that:
α) the repunit $\underbrace{{11\cdots1}}_{n}$is divisible by $37$ if and only if $n$ is divisible by $3$
b) there exists a positive integer $k$ such that the repunit $\underbrace{{11\cdots1}}_{n}$ is divisible by $41$ if $n$ is divisible by $k$
A [i]zigzag [/i] is a polyline on a plane formed from two parallel rays and a segment connecting the origins of these rays. What is the maximum number of parts a plane can be split into using $ n $ zigzags?
A rectangular sheet of paper whose dimensions are $12 \times 18$ is folded along a diagonal, creating the $M$-shaped region drawn in the picture (see below). Find the area of the shaded region.
[img]https://cdn.artofproblemsolving.com/attachments/4/7/d82cde3e91ab83fa14cd6cefe9bba28462dde1.png[/img]
Show that there are no positive real numbers $x, y, z$ such $(12x^2+yz)(12y^2+xz)(12z^2+xy)= 2014x^2y^2z^2$ .
Let \(n\ge3\) be a fixed integer, and let \(\alpha\) be a fixed positive real number. There are \(n\) numbers written around a circle such that there is exactly one \(1\) and the rest are \(0\)'s. An [i]operation[/i] consists of picking a number \(a\) in the circle, subtracting some positive real \(x\le a\) from it, and adding \(\alpha x\) to each of its neighbors.
Find all pairs \((n,\alpha)\) such that all the numbers in the circle can be made equal after a finite number of operations.
[i]Proposed by Anthony Wang[/i]
A convex hexagon $ABCDEF$ with $AB = BC = a, CD = DE = b, EF = FA = c$ is inscribed in a circle. Show that this hexagon has three (pairwise disjoint) pairs of mutually perpendicular diagonals.