Found problems: 85335
A point $ P$ is performing a random walk on the $ X$-axis. At the instant $ t\equal{}0$, $ P$ is at a point $ x_0$ ($ |x_0|\le N$, where $ x_0$ and $ N$ denote integers, $ N>0$). If at an instant $ t$ ($ t$ being a nonnegative integer), $ P$ is at a point of $ x$ integer abscissa and $ |x|<N$, then by the instant $ t\plus{}1$ it reaches either the point $ x\plus{}1$ or the point $ x\minus{}1$, each with probability $ \frac12$. If at the instant $ t$, $ P$ is at the point $ x\equal{}N$ [$ x\equal{}\minus{}N$], then by the instant $ t\plus{}1$ it is certain to reach the point $ N\minus{}1$ [$ \minus{}N\plus{}1$]. Denote by $ P_k(t)$ the probability of $ P$ being at $ x\equal{}k$ at instant $ t$ ($ k$ is an integer). Find $ \lim_{t\to \infty}P_{k}(2t)$ and $ \lim_{t\to \infty}P_k(2t\plus{}1)$ for every fixed $ k$.
Show that for each natural number $k$ there exist only finitely many triples $(p, q, r)$ of distinct primes for which $p$ divides $qr-k$, $q$ divides $pr-k$, and $r$ divides $pq - k$.
Prove that for some positive integer $n$ there exist positive integers $a$,$b$ and $c$ such that $a^2-n=xy$, $b^2-n=yz$ and $c^2-n=xz$ where $x,y$ and $z$ - some pairwise different positive integers.
Five marbles of various sizes are placed in a conical funnel. Each marble is in contact with the adjacent marble(s). Also, each marble is in contact all around the funnel wall. The smallest marble has a radius of $8$, and the largest marble has a radius of $18$. What is the radius of the middle marble?
A convex quadrilateral $ABCD$ with $AC \neq BD$ is inscribed in a circle with center $O$. Let $E$ be the intersection of diagonals $AC$ and $BD$. If $P$ is a point inside $ABCD$ such that $\angle PAB+\angle PCB=\angle PBC+\angle PDC=90^\circ$, prove that $O$, $P$ and $E$ are collinear.
The sum of two prime numbers is $85$. What is the product of these two prime numbers?
$\textbf{(A) }85\qquad\textbf{(B) }91\qquad\textbf{(C) }115\qquad\textbf{(D) }133\qquad \textbf{(E) }166$
A polynomial $P(x)$ has unity as the coefficient of its highest power, and has the property that with natural number arguments, it can take all values of form $2^M$ , where $M$ is a natural number. Prove that the polynomial is of degree $1$.
Consider a board consisting of $n\times n$ unit squares where $n \ge 2$. Two cells are called neighbors if they share a horizontal or vertical border. In the beginning, all cells together contain $k$ tokens. Each cell may contain one or several tokens or none. In each turn, choose one of the cells that contains at least one token for each of its neighbors and move one of those to each of its neighbors. The game ends if no such cell exists.
(a) Find the minimal $k$ such that the game does not end for any starting configuration and choice of cells during the game.
(b) Find the maximal $k$ such that the game ends for any starting configuration and choice of cells during the game.
Proposed by Theresia Eisenkölbl
Let $R $ be the circumradius of a circumscribed quadrilateral $ABCD $. Let $h_1$ and $h_2$ be the altitudes from $A $ to $BC $ and $CD $ respectively. Similarly $h_3$ and $h_4$ are the altitudes from $C $ to $AB $ and $AD$. Prove that $$\frac {h_1+h_2- 2R}{h_1h_2}=\frac {h_3+h_4-2R}{h_3h_4} $$
On complex plane, if $A,B$ are two angles of acute triangle $ABC$, then the point $z=(\cos B-\sin A)+\text{i}(\sin B-\cos A)$ corresponding to is in
$\text{(A)}$Quadrant I
$\text{(B)}$Quadrant II
$\text{(C)}$Quadrant III
$\text{(D)}$Quadrant IV
We color each number in the set $S = \{1, 2, ..., 61\}$ with one of $25$ given colors, where it is not necessary that every color gets used. Let $m$ be the number of non-empty subsets of $S$ such that every number in the subset has the same color. What is the minimum possible value of $m$?
Let $ABCDEF$ be a cyclic hexagon satisfying $AB\perp BD$ and $BC=EF$.Let $P$ be the intersection of lines $BC$ and $AD$ and let $Q$ be the intersection of lines $EF$ and $AD$.Assume that $P$ and $Q$ are on the same side of $D$ and $A$ is on the opposite side.Let $S$ be the midpoint of $AD$.Let $K$ and $L$ be the incentres of $\triangle BPS$ and $\triangle EQS$ respectively.Prove that $\angle KDL=90^0$.
Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Prove that: \[ \frac 1{a^2}+\frac 1{b^2}+\frac 1{c^2} \geq a^2+b^2+c^2. \]
Michael plays catcher for his school's baseball team. He has always been a great player behind the plate, but this year as a junior, Michael's offense is really improving. His batting average is $.417$ after six games, and the team is $6-0$ (six wins and no losses). They are off to their best start in years.
On the way home from their sixth game, Michael notes to his father that the attendance seems to be increasing due to the team's great start, "There were $181$ people at the first game, then $197$ at the second, $203$ the third, $204$ the fourth, $212$ at the fifth, and there were $227$ at today's game." Just then, Michael's genius younger brother Tony, just seven-years-old, computes the average attendance of the six games. What is their average?
Let $A$, $B$, $C$ be points on a circle whose centre is $O$ and whose radius is $1$, such that $\angle BAC = 45^\circ$. Lines $AC$ and $BO$ (possibly extended) intersect at $D$, and lines $AB$ and $CO$ (possibly extended) intersect at $E$. Prove that $BD \cdot CE = 2$.
Two straight lines intersecting at an angle of $46^o$ are the axes of symmetry of the figure $F$ on the plane. What is the smallest number of axes of symmetry this figure can have?
Let $JHIZ$ be a rectangle, and let $A$ and $C$ be points on sides $ZI$ and $ZJ,$ respectively. The perpendicular from $A$ to $CH$ intersects line $HI$ in $X$ and the perpendicular from $C$ to $AH$ intersects line $HJ$ in $Y.$ Prove that $X,$ $Y,$ and $Z$ are collinear (lie on the same line).
Let ( M , g ) be a Riemannian manifold. Extend the metric tensor g to the set of tangents TM with the following specification: if $a,b\in T_v TM \, (v\in T_p M)$, then $$\tilde g_v (a, b): = g_p (\dot {\alpha} (0), \dot {\beta} (0) ) + g_p (D _ {\alpha} X(0) , D_{\beta} Y(0) )$$ where $\alpha, \beta$ are curves in M such that $\alpha(0) = \beta(0) = p$. X and Y are vector fields along $\alpha,\beta$ respectively, with the condition $\dot X (0) = a,\dot Y(0) = b$. $D _{\alpha}$ and $D _{\beta}$ are the operators of the covariant derivative along the corresponding curves according to the Levi-Civita connection. Is the eigenfunction from the Riemannian manifold (M,g) to the Riemannian manifold $(TM, \tilde g)$ harmonic?
Every $x, 0 \leq x \leq 1$, admits a unique representation $x = \sum_{j=0}^{\infty} a_j 2^{-j}$, where all the $a_j$ belong to $\{0, 1\}$ and infinitely many of them are $0$. If $b(0) = \frac{1+c}{2+c}, b(1) =\frac{1}{2+c},c > 0$, and
\[f(x)=a_0 + \sum_{j=0}^{\infty}b(a_0) \cdots b(a_j) a_{j+1}\]
show that $0 < f(x) -x < c$ for every $x, 0 < x < 1.$
Solve $ x^3-y^3=2xy+7 $ in integers.
How many triples of integers $(a,b,c)$ with $-10\leq a,b,c\leq 10$ satisfy \[a^2+b^2+c^2=(a+b+c)^2?\]
[i]Proposed by David Altizio
Let $S=\{1,2,3,...,12\}$. How many subsets of $S$, excluding the empty set, have an even sum but not an even product?
[i]Proposed by Gabriel Wu[/i]
Let $a, b, c, d$ be four positive integers such that each of them is a difference of two squares of positive integers. Prove that $abcd$ is also a difference of two squares of positive integers.
Determine all real numbers that satisfy the equation $$(x+1)^{21}+(x+1)^{20}(x-1)+(x+1)^{19}(x-1)^2+...+(x-1)^{21}=0$$
(RM Kuznec)
Let $p = 2017$. Given a positive integer $n$, an $n\times n$ matrix $A$ is formed with each element $a_{ij}$ randomly selected, with equal probability, from $\{0,1,\ldots,p - 1\}$. Let $q_n$ be probability that $\det A\equiv 1\pmod{p}$. Let $q=\displaystyle\lim_{n\rightarrow\infty} q_n$. If $d_1, d_2, d_3, \ldots$ are the digits after the decimal point in the base $p$ expansion of $q$, then compute the remainder when $\displaystyle\sum_{k = 1}^{p^2} d_k$ is divided by $10^9$.
[i]Proposed by Ashwin Sah[/i]