Found problems: 85335
A particle starts at $(0,0,0)$ in three-dimensional space. Each second, it randomly selects one of the eight lattice points a distance of $\sqrt{3}$ from its current location and moves to that point. What is the probability that, after two seconds, the particle is a distance of $2\sqrt{2}$ from its original location?
[i]Proposed by Connor Gordon[/i]
Let $n$ be a positive integer, $x_1,x_2,\ldots,x_{2n}$ be non-negative real numbers with sum $4$. Prove that there exist integer $p$ and $q$, with $0 \le q \le n-1$, such that
\[ \sum_{i=1}^q x_{p+2i-1} \le 1 \mbox{ and } \sum_{i=q+1}^{n-1} x_{p+2i} \le 1, \]
where the indices are take modulo $2n$.
[i]Note:[/i] If $q=0$, then $\sum_{i=1}^q x_{p+2i-1}=0$; if $q=n-1$, then $\sum_{i=q+1}^{n-1} x_{p+2i}=0$.
Given a positive prime number $p$. Prove that there exist a positive integer $\alpha$ such that $p|\alpha(\alpha-1)+3$, if and only if there exist a positive integer $\beta$ such that $p|\beta(\beta-1)+25$.
Denote by $ S$ the set of all positive integers. Find all functions $ f: S \rightarrow S$ such that
\[ f (f^2(m) \plus{} 2f^2(n)) \equal{} m^2 \plus{} 2 n^2\]
for all $ m,n \in S$.
[i]Bulgaria[/i]
Let $I$ be the incentre of acute triangle $ABC$ with $AB\neq AC$. The incircle $\omega$ of $ABC$ is tangent to sides $BC, CA$, and $AB$ at $D, E,$ and $F$, respectively. The line through $D$ perpendicular to $EF$ meets $\omega$ at $R$. Line $AR$ meets $\omega$ again at $P$. The circumcircles of triangle $PCE$ and $PBF$ meet again at $Q$.
Prove that lines $DI$ and $PQ$ meet on the line through $A$ perpendicular to $AI$.
[i]Proposed by Anant Mudgal, India[/i]
Find all triples of prime numbers $(p, q, r)$ such that $p^q + p^r$ is a perfect square.
Prove, for angles $\alpha$, $\beta$ and $\gamma$ of a right triangle, the following relation: $$\text{sin } \alpha \text{ sin } \beta \text{ sin } (\alpha-\beta) \text{ } + \text{ sin } \beta \text{ sin } \gamma \text{ sin } (\beta-\gamma) \text{ }+ \text{ sin } \gamma \text{ sin } \alpha \text{ sin } (\gamma-\alpha) \text{ }+ \text{ sin } (\alpha-\beta) \text{ sin } (\beta-\gamma) \text{ sin } (\gamma-\alpha) = 0.$$
Let $\omega$ be the semicircle with diameter $PQ$. A circle $k$ is tangent internally to $\omega$ and to segment $PQ$ at $C$. Let $AB$ be the tangent to $K$ perpendicular to $PQ$, with $A$ on $\omega$ and $B$ on the segment $CQ$. Show that $AC$ bisects angle $\angle PAB$
Is there a positive integer $N>10^{20}$ such that all its decimal digits are odd,
the numbers of digits 1, 3, 5, 7, 9 in its decimal representation are equal,
and it is divisible by each 20-digit number obtained from it by deleting
digits? (Neither deleted nor remaining digits must be consecutive.)
A rectangle $ABCD$ has side lengths $AB = m$, $AD = n$, with $m$ and $n$ relatively prime and both odd. It is divided into unit squares and the diagonal AC intersects the sides of the unit squares at the points $A_1 = A, A_2, A_3, \ldots , A_k = C$. Show that \[ A_1A_2 - A_2A_3 + A_3A_4 - \cdots + A_{k-1}A_k = {\sqrt{m^2+n^2}\over mn}. \]
Peeter, Juri, Kati and Mari are standing at the entrance of a dark tunnel. They have one torch and none of them dares to be in the tunnel without it, but the tunnel is so narrow that at most two people can move together. It takes $1$ minute for Peeter, $2$ minutes for Juri, $5$ for Kati and $10$ for Mari to pass the tunnel. Find the minimum time in which they can all pass through the tunnel.
Let $ABC$ be an acute triangle where $AC > BC$. Let $T$ denote the foot of the altitude from vertex $C$, denote the circumcentre of the triangle by $O$. Show that quadrilaterals $ATOC$ and $BTOC$ have equal area.
Reconstruct an acute-angled triangle given the orthocenter and midpoints of two sides.
(A. Zaslavsky)
Henry invited $2N$ guests to his birthday party. He has $N$ white hats and $N$ black hats. He wants to place hats on his guests and split his guests into one or several dancing circles so that in each circle there would be at least two people and the colors of hats of any two neighbours would be different. Prove that Henry can do this in exactly $(2N)!$ different ways. (All the hats with the same color are identical, all the guests are obviously distinct, $(2N)! = 1 \cdot 2 \cdot . . . \cdot (2N)$.)
Gleb Pogudin
Let $b$ be a positive integer. Determine all $2002$-tuples of non-negative integers $(a_{1}, a_{2}, \cdots, a_{2002})$ satisfying \[\sum^{2002}_{j=1}{a_{j}}^{a_{j}}=2002{b}^{b}.\]
Let $k \leq 2022$ be a positive integer. Alice and Bob play a game on a $2022 \times 2022$ board. Initially, all cells are white. Alice starts and the players alternate. In her turn, Alice can either color one white cell in red or pass her turn. In his turn, Bob can either color a $k \times k$ square of white cells in blue or pass his turn. Once both players pass, the game ends and the person who colored more cells wins (a draw can occur). For each $1 \leq k \leq 2022$, determine which player (if any) has a winning strategy.
Let $P(x)=x^{2016}+2x^{2015}+...+2017,Q(x)=1399x^{1398}+...+2x+1$. Prove that there are strictly increasing sequances $a_i,b_i, i=1,...$ of positive integers such that $gcd(a_i,a_{i+1})=1$ for each $i$. Moreover, for each even $i$, $P(b_i) \nmid a_i, Q(b_i) | a_i$ and for each odd $i$, $P(b_i)|a_i,Q(b_i) \nmid a_i$
Proposed by [i]Shayan Talaei[/i]
For a sequence with some ones and zeros, we count the number of continuous runs of equal digits in it. (For example the sequence $011001010$ has $7$ continuous runs: $0,11,00,1,0,1,0$.) Find the sum of the number of all continuous runs for all possible sequences with $2019$ ones and $2019$ zeros.
Two circles intersect at two points $A$ and $B$. A line $\ell$ which passes through the point $A$ meets the two circles again at the points $C$ and $D$, respectively. Let $M$ and $N$ be the midpoints of the arcs $BC$ and $BD$ (which do not contain the point $A$) on the respective circles. Let $K$ be the midpoint of the segment $CD$. Prove that $\measuredangle MKN = 90^{\circ}$.
Let $A,B$ be two points interior of circle $C(O,R)$ and $M$ a point on the circle. Let $A_1,B_1$ be the intersections of the circle with lines $MA$,$MB$ respectively. Let $G$ be the midpoint of $AB$and $G_1= C\cap MG$. Prove that$$\frac{MA}{AA_1}+ \frac{MB}{BB_1}> 2\frac{MG}{GG_1}$$
Find all functions $f : \mathbb Z \to \mathbb Z$ such that
\[f (n |m|) + f (n(|m| +2)) = 2f (n(|m| +1)) \qquad \forall m,n \in \mathbb Z.\]
[b]Note.[/b] $|x|$ denotes the absolute value of the integer $x.$
Let $a > 3$ be an odd integer. Show that for every positive integer $n$ the number $a^{2^n}- 1$ has at least $n + 1$ distinct prime divisors.
Let $\triangle ABC$ be given, it's $A-$mixtilinear incirlce, $\omega$, and it's excenter $I_A$. Let $H$ be the foot of altitude from $A$ to $BC$, $E$ midpoint of arc $\overarc{BAC}$ and denote by $M$ and $N$, midpoints of $BC$ and $AH$, respectively. Suposse that $MN\cap AE=\{ P \}$ and that line $I_AP$ meet $\omega$ at $S$ and $T$ in this order: $I_A-T-S-P$.
Prove that circumcircle of $\triangle BSC$ and $\omega$ are tangent to each other.
[hide=Diagram]
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[/asy][/hide]
At the vertices of a regular hexagon are written six nonnegative integers whose sum is $2003^{2003}$. Bert is allowed to make moves of the following form: he may pick a vertex and replace the number written there by the absolute value of the difference between the numbers written at the two neighboring vertices. Prove that Bert can make a sequence of moves, after which the number 0 appears at all six vertices.
Prove that there is no function $f:\mathbb{R}^{+} \to \mathbb{R}^{+}$ such that $f(x)^2 \geq f(x+y)(f(x)+y)$ for all $x,y \in \mathbb{R}^{+}$.