Search found 110 matches
- Wed Feb 01, 2017 10:34 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies: 184
- Views: 112409
Re: IMO Marathon
$\text{Solution to Problem } 50$ A solution can be found following this link : https://artofproblemsolving.com/community/c6h348658p1871421 $\text{Problem } 51$ Let $A$ be a set of $N$ residues $\pmod{N^{2}}$. Prove that there exists a set $B$ of of $N$ residues $\pmod{N^{2}}$ such that $A + B = \{a+...
- Wed Feb 01, 2017 7:44 pm
- Forum: Algebra
- Topic: 2009 IMO SL A3
- Replies: 2
- Views: 2960
2009 IMO SL A3
Determine all functions $ f$ from the set of positive integers to the set of positive integers such that, for all positive integers $ a$ and $ b$, there exists a non-degenerate triangle with sides of lengths \[ a, f(b) \text{ and } f(b + f(a) - 1).\] (A triangle is non-degenerate if its vertices are...
- Wed Feb 01, 2017 7:32 pm
- Forum: Number Theory
- Topic: $2008$ ISL N$5$
- Replies: 2
- Views: 2657
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...
- Tue Jan 31, 2017 8:50 pm
- Forum: Number Theory
- Topic: $2008$ ISL N$5$
- Replies: 2
- Views: 2657
$2008$ 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...
- Tue Jan 31, 2017 6:16 pm
- Forum: Geometry
- Topic: Geometry Marathon : Season 3
- Replies: 146
- Views: 187697
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...
- Tue Jan 31, 2017 2:44 am
- Forum: Geometry
- Topic: USA TST 2017
- Replies: 2
- Views: 2925
Re: USA TST 2017
Hints for the solution.
First part:
Second part:
This was a rather bland problem for a TST.
First part:
- Tue Jan 31, 2017 12:04 am
- Forum: Combinatorics
- Topic: $2017$ USA TST P1
- Replies: 1
- Views: 2549
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...
- Mon Jan 30, 2017 11:02 pm
- Forum: Combinatorics
- Topic: $2017$ USA TST P1
- Replies: 1
- Views: 2549
$2017$ 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 ...
- Sat Jan 07, 2017 6:24 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies: 184
- Views: 112409
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 $2010$
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 $2010$
- Tue Nov 15, 2016 2:37 am
- Forum: Number Theory
- Topic: Infinitely many primes divide $1!+2!+\cdots +n!$
- Replies: 1
- Views: 2424
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 $10^{2012}$ .