Found problems: 85335
Let $N_9$ be the answer to problem 9.
In a rainforest, there is a row of nine rocks labeled from $1$ to $N_9$. A gecko is standing on rock $1$. The gecko jumps according to following rules:
[list]
[*] if it is on rock $1$, then it will jump with equal probability to any of the other rocks.
[*] if it is on rock $R$ and $R$ is prime, then it will jump to rock $N_9$.
[*] if it is on rock $4$, then it will jump with equal probability to rock $1$ or rock $6$.
[*] if it is on rock $R$ and $R$ is composite with $4<R<N_9$, then it will jump with equal probability to rock $Q$ or $S$, where $Q$ is the greatest composite number less than $R$, and $S$ is the smallest composite number greater than $R$.
[*] if it is on rock $N_9$, then it stops jumping.
[/list]
The expected number of jumps the gecko will take to reach rock $N_9$ is $\tfrac{p}{q}$, where $p$ are $q$ relatively prime positive integers. Compute $p+q$.
In a game similar to three card monte, the dealer places three cards on the table: the queen of spades and two red cards. The cards are placed in a row, and the queen starts in the center; the card configuration is thus RQR. The dealer proceeds to move. With each move, the dealer randomly switches the center card with one of the two edge cards (so the configuration after the first move is either RRQ or QRR). What is the probability that, after 2004 moves, the center card is the queen?
There are eight different symbols designed on $n\geq 2$ different T-shirts. Each shirt contains at least one symbol, and no two shirts contain all the same symbols. Suppose that for any $k$ symbols $(1\leq k\leq 7)$ the number of shirts containing at least one of the $k$ symbols is even. Determine the value of $n$.
Let $ x,y$ be reals s.t. $ x^2\plus{}y^2\leq1$ and $ n$ a natural number.Prove that:
$ (x^n\plus{}y)^2\plus{}y^2\geq\dfrac{1}{n\plus{}2}(x^2\plus{}y^2)^n$
Define $f_{0}(x)=e^x$ and $f_{n+1}(x)=x f_{n}'(x)$. Show that $\sum_{n=0}^{\infty} \frac{f_{n}(1)}{n!}=e^e$.
Find all perfect powers whose last $ 4$ digits are $ 2,0,0,8$, in that order.
Determine all functions $ f$ defined in the set of rational numbers and taking their values in the same set such that the equation $ f(x + y) + f(x - y) = 2f(x) + 2f(y)$ holds for all rational numbers $x$ and $y$.
The $25$ integers from $-10$ to $14,$ inclusive, can be arranged to form a $5$-by-$5$ square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum?
$\textbf{(A) }2 \qquad\textbf{(B) } 5\qquad\textbf{(C) } 10\qquad\textbf{(D) } 25\qquad\textbf{(E) } 50$
$O$ and $I$ are the circumcentre and incentre of $\vartriangle ABC$ respectively. Suppose $O$ lies in the interior of $\vartriangle ABC$ and $I$ lies on the circle passing through $B, O$, and $C$. What is the magnitude of $\angle B AC$ in degrees?
Let $A{}$ be a point in the Cartesian plane. At each step, Ann tells Bob a number $0\leqslant a\leqslant 1$ and he then moves $A{}$ in one of the four cardinal directions, at his choice, by a distance of $a{}.$ This process cotinues as long as Ann wishes. Amongst every 100 consecutive moves, each of the four possible moves should have been made at least once. Ann's goal is to force Bob to eventually choose a point at a distance greater than 100 from the initial position of $A.{}$ Can Ann achieve her goal?
[i]Selected from an Argentine Olympiad[/i]
The diagram below shows a regular hexagon with an inscribed square where two sides of the square are parallel to two sides of the hexagon. There are positive integers $m$, $n$, and $p$ such that the ratio of the area of the hexagon to the area of the square can be written as $\tfrac{m+\sqrt{n}}{p}$ where $m$ and $p$ are relatively prime. Find $m + n + p$.
[asy]
import graph; size(4cm);
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
draw((0,1)--(1,1)--(1.5,1.87)--(1,2.73)--(0,2.73)--(-0.5,1.87)--cycle);
filldraw((1.13,2.5)--(-0.13,2.5)--(-0.13,1.23)--(1.13,1.23)--cycle,grey);
draw((0,1)--(1,1));
draw((1,1)--(1.5,1.87));
draw((1.5,1.87)--(1,2.73));
draw((1,2.73)--(0,2.73));
draw((0,2.73)--(-0.5,1.87));
draw((-0.5,1.87)--(0,1));
draw((1.13,2.5)--(-0.13,2.5));
draw((-0.13,2.5)--(-0.13,1.23));
draw((-0.13,1.23)--(1.13,1.23));
draw((1.13,1.23)--(1.13,2.5)); [/asy]
Set $M$ contains $n \ge 2$ positive integers. It's known that for any two different $a, b \in M$, $a^2+1$ is divisible by $b$. What is the largest possible value of $n$?
[i]Proposed by Oleksiy Masalitin[/i]
Ana and Beto play the following game with a stick of length $15$. Ana starts, and on her first turn, she cuts the stick into two pieces with integer lengths. Then, on each player's turn, they must cut one of the pieces, of their choice, into two new pieces with integer lengths. The player who, on their turn, leaves at least one piece with length equal to $1$ loses. Determine which of the two players has a winning strategy.
Prove that for every convex polygon can be found such three sequential vertices for which a circle that they lie on covers the polygon.
[i]Jordan Tabov[/i]
A positive integer $n$ is $inverosimil$ if there exists $n$ integers not necessarily distinct such that the sum and the product of this integers are equal to $n$. How many positive integers less than or equal to $2022$ are $inverosimils$?
Consider the graph $G$ with $100$ vertices, and the minimum odd cycle goes through $13$ vertices. Prove that the vertices of the graph can be colored in $6$ colors in a way that no two adjacent vertices have the same color.
Suppose that a sequence $(a_n)_{n=1}^{\infty}$ of integers has the following property: For all $n$ large enough (i.e. $n \ge N$ for some $N$ ), $a_n$ equals the number of indices $i$, $1 \le i < n$, such that $a_i + i \ge n$. Find the maximum possible number of integers which occur infinitely many times in the sequence.
Let $ABC$ be an equilateral triangle with the side length equals $a+ b+ c$. On the side $AB{}$ of the triangle $ABC$ points $C_1$ and $C_2$ are chosen, on the side $BC$ points $A_1$ and $A_2$, arc chosen, and on the side $CA$ points $B_1$ and $B_2$ are chosen such that $A_1A_2 = CB_1 = BC_2 = a, B_1B_2 = AC_1 = CA_2 = b,
C_1C_2 = BA_1 = AB_2 = c$. Let the point $A^{’}$ be such that the triangle $A^{'} B_2C_1$ is equilateral, and the points $A$ and $A^{'}$ lie on different sides of the line $B_2C_1$. Similarly, the points $B^{’}$ and $C^{'}$ are constructed (the triangle $B^{'} C_2A_1$ is equilateral, and the points $B$ and $B^{’}$ lie on different sides of the line $C_2A_1$; the triangle $C^{'} A_2B_1$ is equilateral, and the points $C$ and $C^{'}$ lie on different sides of the line $A_2B_1$). Prove that the triangle $A^{'}B^{'}C^{'}$ is equilateral.
Let $ABC$ be an isosceles triangle with $BC=CA$, and let $D$ be a point inside side $AB$ such that $AD< DB$. Let $P$ and $Q$ be two points inside sides $BC$ and $CA$, respectively, such that $\angle DPB = \angle DQA = 90^{\circ}$. Let the perpendicular bisector of $PQ$ meet line segment $CQ$ at $E$, and let the circumcircles of triangles $ABC$ and $CPQ$ meet again at point $F$, different from $C$.
Suppose that $P$, $E$, $F$ are collinear. Prove that $\angle ACB = 90^{\circ}$.
Do there exist real numbers $a,b,c$ such that the system of equations
\begin{align*}
x+y+z&=a\\
x^2+y^2+z^2&=b\\
x^4+y^4+z^4&=c
\end{align*}
has infinitely many real solutions $(x,y,z)$?
Quadrilateral $ABCD$ is inscribed in a circle with $\angle ABC=60^\circ$ and $BC=CD$. Prove that $AB=AD+DC$.
A positive integer $n$ is called [i]Olympic[/i], if there exists a quadratic trinomial with integer coeffecients $f(x)$ satisfying $f(f(\sqrt{n}))=0$. Determine, with proof, the largest Olympic number not exceeding $2015$.
[i]A. Khrabrov[/i]
S is the set of all ($a$, $b$, $c$, $d$, $e$, $f$) where $a$, $b$, $c$, $d$, $e$, $f$ are integers such that $a^2 + b^2 + c^2 + d^2 + e^2 = f^2$. Find the largest $k$ which divides abcdef for all members of $S$.
Find all triples of primes $(p,q,r)$ such that $p^q=2021+r^3$
Find the number of integer solutions of $\left[\frac{x}{100} \left[\frac{x}{100}\right]\right]= 5$