Found problems: 85335
Alien Connor starts at $(0,0)$ and walks around on the integer lattice. Specifically, he takes one step of length one in a uniformly random cardinal direction every minute, unless his previous four steps were all in the same directionin which case he randomly picks a new direction to step in. Every time he takes a step, he leaves toxic air on the lattice point he just left, and the toxic cloud remains there for $150$ seconds. After taking $5$ steps total, the probability that he has not encountered his own toxic waste canb be written as $\tfrac{a}{b}$ for relatively prime positive integers $a,b.$ Find $a+b.$
Consider the sequence: $x_1=19,x_2=95,x_{n+2}=\text{lcm} (x_{n+1},x_n)+x_n$, for $n>1$, where $\text{lcm} (a,b)$ means the least common multiple of $a$ and $b$. Find the greatest common divisor of $x_{1995}$ and $x_{1996}$.
A $10\times10\times10$ cube has $1000$ unit cubes. $500$ of them are coloured black and $500$ of them are coloured white. Show that there are at least $100$ unit squares, being the common face of a white and a black unit cube.
Let $ a$ and $ b$ be two positive integers such that $ a \cdot b \plus{} 1$ divides $ a^{2} \plus{} b^{2}$. Show that $ \frac {a^{2} \plus{} b^{2}}{a \cdot b \plus{} 1}$ is a perfect square.
Suppose that at the end of any year, a unit of money has lost $10\%$ of the value it had at the beginning of that year. Find the smallest integer $n$ such that after $n$ years, the money will have lost at least $90\%$ of its value. (To the nearest thousandth $\log_{10}3=.477$.)
$\text{(A)}\ 14 \qquad \text{(B)}\ 16 \qquad \text{(C)}\ 18 \qquad \text{(D)}\ 20 \qquad \text{(E)}\ 22$
Find the side lengths of the triangle $ABC$ with area $S$ and $\angle BAC = x$ such that the side $BC$ is as short as possible.
Find all integers n for which number $ \log_{2n\minus{}1}(n^2\plus{}2)$ is rational.
Consider the sequence defined by $a_1 = 2022$ and $a_{n+1} = a_n + e^{-a_n}$ for $n \geq 1$. Prove that there exists a positive real number $r$ for which the sequence $$\{ra_1\}, \{ra_{10}\}, \{ra_{100}\}, . . . $$converges.
[i]Note[/i]: $\{x \} = x - \lfloor x \rfloor$ denotes the part of $x$ after the decimal point.
[i]Proposed by Ethan Tan[/i]
Let $k > 2$ be an integer. A positive integer $l$ is said to be $k-pable$ if the numbers $1, 3, 5, . . . , 2k - 1$ can be partitioned into two subsets $A$ and $B$ in such a way that the sum of the elements of $A$ is exactly $l$ times as large as the sum of the elements of $B$.
Show that the smallest $k-pable$ integer is coprime to $k$.
Calculate the sum $\sum_{n=1}^{1.000.000}[ \sqrt{n} ]$ .
You may use the formula $\sum_{i=1}^{k} i^2=\frac{k(k +1)(2k +1)}{6}$ without a proof.
Triangles $MA_2B_2$ and $MA_1B_1$ are similar to each other and have the same orientation. Prove that the circles circumcribed around these triangles and the straight lines $A_1A_2$ , $B_1B_2$ have a common point.
Let $n>6$ be a perfect number, and let $n=p_1^{e_1}\cdot\cdot\cdot p_k^{e_k}$ be its prime factorisation with $1<p_1<\dots <p_k$. Prove that $e_1$ is an even number.
A number $n$ is [i]perfect[/i] if $s(n)=2n$, where $s(n)$ is the sum of the divisors of $n$.
(Proposed by Javier Rodrigo, Universidad Pontificia Comillas)
Given positive integers $k, m, n$ such that $1 \leq k \leq m \leq n$. Evaluate
\[\sum^{n}_{i=0} \frac{(-1)^i}{n+k+i} \cdot \frac{(m+n+i)!}{i!(n-i)!(m+i)!}.\]
Do there exist continuous functions $f, g: \mathbb{R} \to \mathbb{R}$, both nowhere differentiable, such that $f \circ g$ is differentiable?
Roy bought a new battery-gasoline hybrid car. On a trip the car ran exclusively on its battery for the first 40 miles, then ran exclusively on gasoline for the rest of the trip, using gasoline at a rate of 0.02 gallons per mile. On the whole trip he averaged 55 miles per gallon. How long was the trip in miles?
$\textbf{(A)}\,140 \qquad\textbf{(B)}\,240 \qquad\textbf{(C)}\,440 \qquad\textbf{(D)}\,640 \qquad\textbf{(E)}\,840$
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$, satisfying \[
f(xy)(f(x) - f(y)) = (x-y)f(x)f(y)
\] for all $x,y$.
Let $ABC$ be a triangle with $AB = 3$, $AC = 8$, $BC = 7$ and let $M$ and $N$ be the midpoints of $\overline{AB}$ and $\overline{AC}$, respectively. Point $T$ is selected on side $BC$ so that $AT = TC$. The circumcircles of triangles $BAT$, $MAN$ intersect at $D$. Compute $DC$.
The sphere inscribed in a tetrahedron $ABCD$ touches face $ABC$ at point $H$. Another sphere touches face $ABC$ at $O$ and the planes containing the other three faces at points exterior to the faces. Prove that if $O$ is the circumcenter of triangle $ABC$, then $H$ is the orthocenter of that triangle.
Given a regular 103-sided polygon. 79 vertices are colored red and the remaining vertices are colored blue. Let $A$ be the number of pairs of adjacent red vertices and $B$ be the number of pairs of adjacent blue vertices.
a) Find all possible values of pair $(A,B).$
b) Determine the number of pairwise non-similar colorings of the polygon satisfying $B=14.$ 2 colorings are called similar if they can be obtained from each other by rotating the circumcircle of the polygon.
There are $n$ coins aligned in a row. In each step, it is allowed to choose a coin with the tail up (but not one of the outermost markers), remove it and reverse the closest coin to the left and the closest coin to the right of it. Initially, all the coins have tails up. Prove that one can achieve the state with only two coins remaining if and only if $n-1$ is not divisible by $3$.
A sequence $(a_1, a_2,\ldots, a_k)$ consisting of pairwise distinct squares of an $n\times n$ chessboard is called a [i]cycle[/i] if $k\geq 4$ and squares $a_i$ and $a_{i+1}$ have a common side for all $i=1,2,\ldots, k$, where $a_{k+1}=a_1$. Subset $X$ of this chessboard's squares is [i]mischievous[/i] if each cycle on it contains at least one square in $X$.
Determine all real numbers $C$ with the following property: for each integer $n\geq 2$, on an $n\times n$ chessboard there exists a mischievous subset consisting of at most $Cn^2$ squares.
A sequence of positive integers $a_1,a_2,a_3,\ldots$ satisfies $$a_{n+1} = n\left \lfloor \frac{a_n}{n} \right \rfloor + 1$$ for all positive integers $n$. If $a_{30}=30$, how many possible values can $a_1$ take? (For a real number $x$, $\lfloor x \rfloor$ denotes the largest integer that is not greater than $x$.)
Calculate
\[\int_{-\infty}^{+\infty}\frac{e^{ikx}}{1+x^2}dx\]
Suppose that $ a, b, c, d$ are real numbers satisfying $ a \geq b \geq c \geq d \geq 0$, $ a^2 \plus{} d^2 \equal{} 1$, $ b^2 \plus{} c^2 \equal{} 1$, and $ ac \plus{} bd \equal{} 1/3$. Find the value of $ ab \minus{} cd$.
Let be a sequence $ \left( a_n \right)_{n\ge 1} $ with $ a_1>0 $ and satisfying the equality
$$ a_n=\sqrt{a_{n+1} -\sqrt{a_{n+1} +a_n}} , $$
for all natural numbers $ n. $
[b]a)[/b] Find a recurrence relation between two consecutive elements of $ \left( a_n \right)_{n\ge 1} . $
[b]b)[/b] Prove that $ \lim_{n\to\infty } \frac{\ln\ln a_n}{n} =\ln 2. $