Search found 108 matches

by rah4927
Wed Feb 01, 2017 7:32 pm
Forum: Number Theory
Topic: $2008$ ISL N$5$
Replies: 2
Views: 1466

Re: $2008$ ISL N$5$

A way to do lemma $4$ (perhaps the same as above). $x|a$ and $x|b$ imply $x|(a,b)$. Now use condition $2$ by fixing $xy=n$ and letting $x$ be a prime divisor of $xy=n$. Use the gcd lemma mentioned before to see that you are left with an ugly $(p_1-1,p_2-1,\cdots)$. WLOG let $p_1$ be the smallest pri...
by rah4927
Tue Jan 31, 2017 8:50 pm
Forum: Number Theory
Topic: $2008$ ISL N$5$
Replies: 2
Views: 1466

$2008$ ISL N$5$

For every $ n\in\mathbb{N}$ let $ d(n)$ denote the number of (positive) divisors of $ n$. Find all functions $ f: \mathbb{N}\to\mathbb{N}$ with the following properties: $ d\left(f(x)\right) = x$ for all $ x\in\mathbb{N}$. $ f(xy)$ divides $ (x - 1)y^{xy - 1}f(x)$ for all $ x$, $ y\in\mathbb{N}$. So...
by rah4927
Tue Jan 31, 2017 6:16 pm
Forum: Geometry
Topic: Geometry Marathon : Season 3
Replies: 146
Views: 62884

Re: Geometry Marathon : Season 3

Solution to Problem 17 We will apply barycentric coordinates on $\triangle ABC$. The calculations are quite routine and requires one major insight. $A=(1:0:0),A_1=(0:b:c),A_2=(0:s-c:s-b)$ Let the circumcircle of $\triangle AA_1A_2$ have the equation $$-a^2yz-b^2zx-c^2cy+(ux+vy+wz)(x+y+z)=0$$ This c...
by rah4927
Tue Jan 31, 2017 2:44 am
Forum: Geometry
Topic: USA TST 2017
Replies: 2
Views: 1684

Re: USA TST 2017

Hints for the solution.
First part:
Try to prove that $AA1$ is a symmedian.
Second part:
Find the inverse of $A2$ with respect to the inversion about the circumcircle of $\triangle ABC$. What property does it satisfy?
This was a rather bland problem for a TST.
by rah4927
Tue Jan 31, 2017 12:04 am
Forum: Combinatorics
Topic: $2017$ USA TST P1
Replies: 1
Views: 1489

Re: $2017$ USA TST P1

Let me first rephrase the problem so that it becomes a bit easier to deal with. Imagine a bipartite graph, the left side (call it $S$) containing teams(we only need to look at $S<n$), and the right side (call it $C$) containing $n$ colours. We join the two vertices in $S$ and $C$ if the team corresp...
by rah4927
Mon Jan 30, 2017 11:02 pm
Forum: Combinatorics
Topic: $2017$ USA TST P1
Replies: 1
Views: 1489

$2017$ USA TST P1

In a sports league, each team uses a set of at most $t$ signature colors. A set $S$ of teams is color-identifiable if one can assign each team in $S$ one of their signature colors, such that no team in $S$ is assigned any signature color of a different team in $S$. For all positive integers $n$ and ...
by rah4927
Sat Jan 07, 2017 6:24 pm
Forum: International Mathematical Olympiad (IMO)
Topic: IMO Marathon
Replies: 184
Views: 63501

Re: IMO Marathon

$\text{Problem }50$

In every $1\times1$ square of an $m\times n$ table we have drawn one of two diagonals. Prove that there is a path including these diagonals either from left side to the right side, or from the upper side to the lower side.

Source: Iran TST $2010$
by rah4927
Tue Nov 15, 2016 2:37 am
Forum: Number Theory
Topic: Infinitely many primes divide $1!+2!+\cdots +n!$
Replies: 1
Views: 1455

Infinitely many primes divide $1!+2!+\cdots +n!$

For a positive integer $n$ define $S_n=1!+2!+\cdots +n!$. Prove that there exists an integer $n$ such that $S_n$ has a prime divisor greater than $10^{2012}$ .
by rah4927
Mon Nov 14, 2016 7:40 am
Forum: Geometry
Topic: Prove parallel
Replies: 2
Views: 1859

Re: Prove parallel

Hint : Forget about the trapezoid. Think about how the second circle looks like in $\triangle DBC$.Apply POP .
by rah4927
Thu Nov 10, 2016 1:09 pm
Forum: Algebra
Topic: Euclidean algorithm with reals ?
Replies: 0
Views: 1059

Euclidean algorithm with reals ?

$4.$ Three nonnegative real numbers $ r_1$, $ r_2$, $ r_3$ are written on a blackboard. These numbers have the property that there exist integers $ a_1$, $ a_2$, $ a_3$, not all zero, satisfying $ a_1r_1 + a_2r_2 + a_3r_3 = 0$. We are permitted to perform the following operation: find two numbers $ ...