Found problems: 85335
$a, a_1,a_2,\dots ,a_n$ are natural numbers. We know that for any natural number $k$ which $ak+1$ is square, at least one of $a_1k+1,\dots ,a_n k+1$ is also square.
Prove $a$ is one of $a_1,\dots ,a_n$
[i]Proposed by Mohsen Jamali[/i]
A line through $A$ intersects a circle at points $B,C$ with $B$ between $A,C$. The two tangents from $A$ intersect the circle at $S,T$. $ST$ and $AC$ intersect at $P$. Show that $\frac{AP}{PC}=2\frac{AB}{BC}$.
Let $I$ be the incenter of triangle $ABC$, and let $A_1$, $B_1$, and $C_1$ be arbitrary points lying on segments $AI$,$BI$, and $CI$, respectively. The perpendicular bisectors of segments $AA_1$, $BB_1$, and $CC_1$ form triangles $A_2B_2C_2$. Prove that the circumcenter of triangle $A_2B_2C_2$ coincides with the circumcenter of triangle $ABC$ if and only if $I$ is the orthocenter of triangle $A_1B_1C_1$.
There are $n\ge 2$ line segments in the plane such that every two segments cross and no three segments meet at a point. Geoff has to choose an endpoint of each segment and place a frog on it facing the other endpoint. Then he will clap his hands $n-1$ times. Every time he claps,each frog will immediately jump forward to the next intersection point on its segment. Frogs never change the direction of their jumps. Geoff wishes to place the frogs in such a way that no two of them will ever occupy the same intersection point at the same time.
(a) Prove that Geoff can always fulfill his wish if $n$ is odd.
(b) Prove that Geoff can never fulfill his wish if $n$ is even.
Let $p> 3$ be a prime number. Determine the number of all ordered sixes $(a, b, c, d, e, f)$ of positive integers whose sum is $3p$ and all fractions $\frac{a + b}{c + d},\frac{b + c}{d + e},\frac{c + d}{e + f},\frac{d + e}{f + a},\frac{e + f}{a + b}$ have integer values.
Compute the positive real number $x$ satisfying $$x^{(2x^6)}=3.$$
If $a$ and $b$ are each randomly and independently chosen in the interval $[-1, 1]$, what is the probability that $|a|+|b|<1$?
Let $a_1$, $a_2$, $\ldots$ be an infinite sequence of positive integers. Suppose that there is an integer $N > 1$ such that, for each $n \geq N$, the number
$$\frac{a_1}{a_2} + \frac{a_2}{a_3} + \cdots + \frac{a_{n-1}}{a_n} + \frac{a_n}{a_1}$$
is an integer. Prove that there is a positive integer $M$ such that $a_m = a_{m+1}$ for all $m \geq M$.
[i]Proposed by Bayarmagnai Gombodorj, Mongolia[/i]
Two sets $S_1$ and $S_2$, which are not necessarily distinct, are each selected randomly and independently from each other among the $512$ subsets of $S = \{1, 2, \ldots ,9\}$. Let $\sigma(X)$ denote the sum of the elements of set $X$. Note that $\sigma(\emptyset) = 0$ where $\emptyset$ denotes the empty set. If $S_1 \cup S_2$ stands for the union of $S_1$ and $S_2$, the probability that $\sigma(S_1 \cup S_2)$ is divisible by $3$ can be expressed as a common fraction of the form $\tfrac{m}{2^n}$ where $m$ is odd and $n$ is a positive integer. Find $m + n$.
For each prime $p$, let $\mathbb S_p = \{1, 2, \dots, p-1\}$. Find all primes $p$ for which there exists a function $f\colon \mathbb S_p \to \mathbb S_p$ such that
\[ n \cdot f(n) \cdot f(f(n)) - 1 \; \text{is a multiple of} \; p \]
for all $n \in \mathbb S_p$.
[i]Andrew Wen[/i]
Prove that for the sides $a, b, c$, the angles $A, B, C$ and the area $S$ of the triangle holds $$\cot A+ \cot B + \cot C = \frac{a^2+b^2+c^2}{4S}.$$
Let $n$ be a positive integer. Let $f(n)$ be the probability that, if divisors $a, b, c$ of $n$ are selected uniformly at random with replacement, then $\gcd(a, \text{lcm}(b, c)) = \text{lcm}(a, \gcd(b, c))$. Let $s(n)$ be the sum of the distinct prime divisors of $n$. If $f(n) < \frac{1}{2018}$, compute the smallest possible value of $s(n)$.
We call a natural number $n$ good if each of the numbers $n$, $ n+1$, $n+2$ and $n+3$ are divided by the sum of their digits. (For example, $n = 60398$ is good.) Does the penultimate digit of a good number ending in eight have to be nine?
At the midpoint of line segment $ AB$ which is $ p$ units long, a perpendicular $ MR$ is erected with length $ q$ units. An arc is described from $ R$ with a radius equal to $ \frac{1}{2}AB$, meeting $ AB$ at $ T$. Then $ AT$ and $ TB$ are the roots of:
$ \textbf{(A)}\ x^2\plus{}px\plus{}q^2\equal{}0 \\
\textbf{(B)}\ x^2\minus{}px\plus{}q^2\equal{}0 \\
\textbf{(C)}\ x^2\plus{}px\minus{}q^2\equal{}0 \\
\textbf{(D)}\ x^2\minus{}px\minus{}q^2\equal{}0 \\
\textbf{(E)}\ x^2\minus{}px\plus{}q\equal{}0$
A positive integer $c$ is given. The sequence $\{p_{k}\}$ is constructed by the following rule: $p_{1}$ is arbitrary prime and for $k\geq 1$ the number $p_{k+1}$ is any prime divisor of $p_{k}+c$ not present among the numbers $p_{1}$, $p_{2}$, $\dots$, $p_{k}$. Prove that the sequence $\{p_{k}\}$ cannot be infinite.
[i]Proposed by A. Golovanov[/i]
The polynomial $P$ has integer coefficients and $P(x)=5$ for five different integers $x$. Show that there is no integer $x$ such that $-6\le P(x)\le 4$ or $6\le P(x)\le 16$.
Let $p>13$ be a prime of the form $2q+1$, where $q$ is prime. Find the number of ordered pairs of integers $(m,n)$ such that $0\le m<n<p-1$ and
\[3^m+(-12)^m\equiv 3^n+(-12)^n\pmod{p}.\]
[i]Alex Zhu.[/i]
[hide="Note"]The original version asked for the number of solutions to $2^m+3^m\equiv 2^n+3^n\pmod{p}$ (still $0\le m<n<p-1$), where $p$ is a Fermat prime.[/hide]
Let $G$ be the centroid of triangle $ABC$. Find the biggest $\alpha$ such that there exists a triangle for which there are at least three angles among $\angle GAB, \angle GAC, \angle GBA, \angle GBC, \angle GCA, \angle GCB$ which are $\geq \alpha$.
Eve and Odette play a game on a $3\times 3$ checkerboard, with black checkers and white checkers. The rules are as follows:
$\text{I.}$ They play alternately.
$\text{II.}$ A turn consists of placing one checker on an unoccupied square of the board.
$\text{III.}$ In her turn, a player may select either a white checker or a black checker and need not always use the same colour.
$\text{IV.}$ When the board is full, Eve obtains one point for every row, column or diagonal that has an even number of black checkers, and Odette obtains one point for very row, column or diagonal that has an odd number of black checkers.
$\text{V.}$ The player obtaining at least five of the eight points WINS.
$\text{(a)}$ Is a $4-4$ tie possible? Explain.
$\text{(b)}$ Describe a winning strategy for the girl who is first to play.
Let $f(x)$ be a polynomial with real coefficients such that $f(0) = 1,$ $f(2)+f(3)=125,$ and for all $x$, $f(x)f(2x^{2})=f(2x^{3}+x).$ Find $f(5).$
Find all real values of $a$ for which the equation $x(x+1)^3=(2x+a)(x+a+1)$ has four distinct real roots.
There is a smallest positive real number $a$ such that there exists a positive real number $b$ such that all the roots of the polynomial $x^3-ax^2+bx-a$ are real. In fact, for this value of $a$ the value of $b$ is unique. What is this value of $b$?
$\textbf{(A)}\ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 12$
Let $S$ be a great circle with pole $P$. On any great circle through $P$, two points $A$ and $B$ are chosen equidistant from $P$. For any [i] spherical triangle [/i] $ABC$ (the sides are great circles ares), where $C$ is on $S$, prove that the great circle are $CP$ is the angle bisector of angle $C$.
[b] Note. [/b] A great circle on a sphere is one whose center is the center of the sphere. A pole of the great circle $S$ is a point $P$ on the sphere such that the diameter through $P$ is perpendicular to the plane of $S$.
Find the sum of all positive two-digit integers that are divisible by each of their digits.
A regular octagon $P$ is given whose incircle $k$ has diameter $1$. About $k$ is circumscribed a regular $16$-gon, which is also inscribed in $P$, cutting from $P$ eight isosceles triangles. To the octagon $P$, three of these triangles are added so that exactly two of them are adjacent and no two of them are opposite to each other. Every $11$-gon so obtained is said to be $P'$. Prove the following statement: Given a finite set $M$ of points lying in $P$ such that every two points of this set have a distance not exceeding $1$, one of the $11$-gons $P'$ contains all of $M$.