Found problems: 85335
Given positive numbers $a_1, \ldots , a_n$, $b_1, \ldots , b_n$, $c_1, \ldots , c_n$. Let $m_k$ be the maximum of the products $a_ib_jc_l$ over the sets $(i, j, l)$ for which $max(i, j, l) = k$. Prove that
$$(a_1 + \ldots + a_n) (b_1 +\ldots + b_n) (c_1 +\ldots + c_n) \leq n^2 (m_1 + \ldots + m_n).$$
Grogg’s favorite positive integer is $n\ge2$, and Grogg has a lucky coin that comes up heads with some fixed probability $p$, where $0<p<1$. Once each day, Grogg flips his coin, and if it comes up heads, he does two things:
[list=1]
[*] He eats a cookie. [/*]
[*] He then flips the coin $n$ more times. If the result of these $n$ flips is $n-1$ heads and $1$ tail (in any order), he eats another cookie. [/*]
[/list]
He never eats a cookie except as a result of his coin flips. Find all possible values of $n$ and $p$ such that the expected value of the number of cookies that Grogg eats each day is exactly $1$.
In a badminton tournament there were 16 participants. Each pair of participants played at most one game and there were no draws. After the tournament it turned out that each participant has won a different number of games.
Prove that each participant has lost a different number of games.
Determine, with proof, the smallest positive integer $c$ such that for any positive integer $n$, the decimal representation of the number $c^n+2014$ has digits all less than $5$.
[i]Proposed by Evan Chen[/i]
Find all polynomials with integer coefficients $P$ such that for all positive integers $n$, the sequence $$0, P(0), P(P(0)), \cdots$$ is eventually constant modulo $n$.
[i]Proposed by Ivan Chan Kai Chin[/i]
Find the number of integers $1\leq k\leq1336$ such that $\binom{1337}{k}$ divides $\binom{1337}{k-1}\binom{1337}{k+1}$.
[i]Proposed by Tristan Shin[/i]
Determine the largest positive integer $n$ such that the following statement holds:
If $a_1,a_2,a_3,a_4,a_5,a_6$ are six distinct positive integers less than or equal to $n$, then there exist $3$ distinct positive integers ,from these six, say $a,b,c$ s.t. $ab>c,bc>a,ca>b$.
Albert and Bob and Charlie are each thinking of a number. Albert's number is one more than twice Bob's. Bob's number is one more than twice Charlie's, and Charlie's number is two more than twice Albert's. What number is Albert thinking of?
$\mathrm{(A) \ } -\frac{11}{7} \qquad \mathrm{(B) \ } -2 \qquad \mathrm {(C) \ } -1 \qquad \mathrm{(D) \ } -\frac{4}{7} \qquad \mathrm{(E) \ } \frac{1}{2}$
Let $D$ be the foot of perpendicular from $A$ to the Euler line (the line passing through the circumcentre and the orthocentre) of an acute scalene triangle $ABC$. A circle $\omega$ with centre $S$ passes through $A$ and $D$, and it intersects sides $AB$ and $AC$ at $X$ and $Y$ respectively. Let $P$ be the foot of altitude from $A$ to $BC$, and let $M$ be the midpoint of $BC$. Prove that the circumcentre of triangle $XSY$ is equidistant from $P$ and $M$.
Suppose that each of $n$ people knows exactly one piece of information and all $n$ pieces are different. Every time person $A$ phones person $B$, $A$ tells $B$ everything he knows, while tells $A$ nothing. What is the minimum of phone calls between pairs of people needed for everyone to know everything?
Let $n > 3$ be an integer. Prove that there exist positive integers $x_1,..., x_n$ in geometric progression and positive integers $y_1,..., y_n$ in arithmetic progression such that $x_1<y_1<x_2<y_2<...<x_n<y_n$
Every term of an infinite geometric progression is also a term of a given infinite arithmetic progression. Prove that the common ratio of the geometric progression is an integer. (4)
Let $a$ and $b$ be two real numbers and let$M(a,b)=\max\left \{ 3a^{2}+2b; 3b^{2}+2a\right \}$. Find the values of $a$ and $b$ for which $M(a,b)$ is minimal.
Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ which are relatively prime to $n$. Over all integers $1\le n \le 100$, find the maximum value of $\phi(n^2+2n)-\phi(n^2)$.
[i]Proposed by Vincent Huang[/i]
Let $ABCD$ be a convex quadrilateral with $AB=CD=10$, $BC=14$, and $AD=2\sqrt{65}$. Assume that the diagonals of $ABCD$ intersect at point $P$, and that the sum of the areas of $\triangle APB$ and $\triangle CPD$ equals the sum of the areas of $\triangle BPC$ and $\triangle APD$. Find the area of quadrilateral $ABCD$.
Let $ABC$ be a triangle with circumcenter $O$ such that $AC = 7$. Suppose that the circumcircle of $AOC$ is tangent to $BC$ at $C$ and intersects the line $AB$ at $A$ and $F$. Let $FO$ intersect $BC$ at $E$. Compute $BE$.
The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$.
[i]Proposed by Mariusz Skalba, Poland[/i]
Let $p$ be an odd positive integer. Find all values of the natural numbers $n\ge 2$ for which holds $$\sum_{i=1}^{n} \prod_{j\ne i} (x_i-x_j)^p\ge 0$$ where $x_1,x_2,..,x_n$ are any real numbers.
Consider the lattice of all algebraically closed subfields of the complex field $ \mathbb{C}$ whose transcendency degree (over $ \mathbb{Q}$) is finite. Prove that this lattice is not modular.
[i]L. Babai[/i]
For a positive integer $n$, we consider all its divisors (including $1$ and itself). Suppose that $p\%$ of these divisors have their unit digit equal to $3$. (For example $n = 117$, has six divisors, namely $1,3,9,13,39,117$. Two of these divisors namely $3$ and $13$, have unit digits equal to $3$. Hence for $n = 117$, $p =33.33...$). Find, when $n$ is any positive integer, the maximum possible value of $p$.
On a cyclic quadrilateral $ABCD$, there is a point $P$ on side $AD$ such that the triangle $CDP$ and the quadrilateral $ABCP$ have equal perimeters and equal areas. Prove that two sides of $ABCD$ have equal lengths.
A train traveling from Aytown to Beetown meets with an accident after $ 1$ hr. It is stopped for $ \frac{1}{2}$ hr., after which it proceeds at four-fifths of its usual rate, arriving at Beetown $ 2$ hr. late. If the train had covered $ 80$ miles more before the accident, it would have been just $ 1$ hr. late. The usual rate of the train is:
$ \textbf{(A)}\ \text{20 mph} \qquad
\textbf{(B)}\ \text{30 mph} \qquad
\textbf{(C)}\ \text{40 mph} \qquad
\textbf{(D)}\ \text{50 mph} \qquad
\textbf{(E)}\ \text{60 mph}$
How many ways are there to fill the $2 \times 2$ grid below with $0$'s and $1$'s such that no row or column has duplicate entries?
[asy]
size(2cm);
draw(unitsquare);
draw( (0.5,0)--(0.5,1) );
draw( (0,0.5)--(1,0.5) );
[/asy]
Find all positive integers $n$ such that the sum of the squares of the five smallest divisors of $n$ is a square.
Circles $o_1$ and $o_2$ with centers in $O_1$ and $O_2$, respectively, intersect in two different points $A$ and $B$, wherein angle $O_1AO_2$ is obtuse. Line $O_1B$ intersects circle $o_2$ in point $C \neq B$. Line $O_2B$ intersects circle $o_1$ in point $D \neq B$. Show that point $B$ is incenter of triangle $ACD$.