IMO Marathon

Discussion on International Mathematical Olympiad (IMO)
Posts: 181
Joined: Mon Mar 28, 2016 6:21 pm

Re: IMO Marathon

We can't save any units on the outside of the grid. So, we better fill them with triangles. The problem reduces to a 98*98 grid for which we could ignore the outside edges. We notice that out of any 2*1 grid, we must have one unit with a triangle. So there are at least 98*98/2=4802 units filled with triangles. We have 4802 free units left! Let us save them . We now show a construction that saves 4802 units. CHESSBOARD the 98*98.
Frankly, my dear, I don't give a damn.

Posts: 181
Joined: Mon Mar 28, 2016 6:21 pm

Re: IMO Marathon

$\boxed{\text{Problem 48}}$ Prove that, for any positive integer set $\{a_1,a_2,...,a_n\}$ there exists a positive integer $b$ such that the set $\{ba_1,ba_2,...,ba_n\}$ consists of perfect powers.
Source -Number Theory Structures examples and problems.
Frankly, my dear, I don't give a damn.

Nirjhor
Posts: 136
Joined: Thu Aug 29, 2013 11:21 pm
Location: Varies.

Re: IMO Marathon

Solution $\boxed{48}$
Problem $\boxed{49}$

Is it possible to write $\dbinom n 2$ consecutive natural numbers on the edges of $K_n$ such that for every path or cycle of length $3$, if the numbers written on the $3$ edges are $a,b,c$ then $\gcd(a,c)\mid b$ is satisfied? In case of a path, the edge $b$ lies between the edges $a$ and $c$.

Source: some Iran TST.
- What is the value of the contour integral around Western Europe?

- Zero.

- Why?

- Because all the poles are in Eastern Europe.

Revive the IMO marathon.

asif e elahi
Posts: 183
Joined: Mon Aug 05, 2013 12:36 pm

Re: IMO Marathon

Nirjhor wrote: Problem $\boxed{49}$

Is it possible to write $\dbinom n 2$ consecutive natural numbers on the edges of $K_n$ such that for every path or cycle of length $3$, if the numbers written on the $3$ edges are $a,b,c$ then $\gcd(a,c)\mid b$ is satisfied? In case of a path, the edge $b$ lies between the edges $a$ and $c$.

Source: some Iran TST.
I split the problem into $2$ cases.
Case $1$: $\dfrac{n(n-1)}{2}$ is even
Consider the two edges which have $\dfrac{n(n-1)}{2}$ and $\dfrac{n(n-1)}{4}$ written on them. Take another edge which connects both of them. Easy to see that such an edge always exist. Now the number written on that edge must be divisible by $gcd(\dfrac{n(n-1)}{2},\dfrac{n(n-1)}{4})=\dfrac{n(n-1)}{4}$. But $\dfrac{n(n-1)}{4}$ and $\dfrac{n(n-1)}{2}$ are the only numbers which are less or equal to $\dfrac{n(n-1)}{2}$ and divisible by $\dfrac{n(n-1)}{4}$ unless $\dfrac{n(n-1)}{4}=1$. This gives us no solution.

Case $2$: $\dfrac{n(n-1)}{2}$ is odd
Take the edges with numbers $\dfrac{n(n-1)}{2}-1$ and $\dfrac{\dfrac{n(n-1)}{2}-1}{2}$. Also take an edge which connects them. Now the number written on it must be multiple of $\dfrac{\dfrac{n(n-1)}{2}-1}{2}$. Using a similar argument as the previous case, we can show that $\dfrac{\dfrac{n(n-1)}{2}-1}{2}$ must be equal to $1$.
However this gives us a solution $n=3$.

Nirjhor
Posts: 136
Joined: Thu Aug 29, 2013 11:21 pm
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Re: IMO Marathon

The $\dbinom n 2$ consecutive natural numbers don't necessarily have to be the first $\dbinom n 2$ natural numbers.
- What is the value of the contour integral around Western Europe?

- Zero.

- Why?

- Because all the poles are in Eastern Europe.

Revive the IMO marathon.

Nirjhor
Posts: 136
Joined: Thu Aug 29, 2013 11:21 pm
Location: Varies.

Re: IMO Marathon

Solution $\boxed{49}$
Someone post the next problem.
- What is the value of the contour integral around Western Europe?

- Zero.

- Why?

- Because all the poles are in Eastern Europe.

Revive the IMO marathon.

rah4927
Posts: 108
Joined: Sat Feb 07, 2015 9:47 pm

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 $2010$

rah4927
Posts: 108
Joined: Sat Feb 07, 2015 9:47 pm

Re: IMO Marathon

$\text{Solution to Problem } 50$

A solution can be found following this link : https://artofproblemsolving.com/communi ... 58p1871421

$\text{Problem } 51$

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 $A + B = \{a+b|a \in A, b \in B\}$ contains at least half of all the residues $\pmod{N^{2}}$

Nirjhor
Posts: 136
Joined: Thu Aug 29, 2013 11:21 pm
Location: Varies.

Re: IMO Marathon

Solution $\boxed{51}$
Problem $\boxed{52}$

Find all pairs $(f,g)$ of functions from $\mathbb R$ to $\mathbb R$ that satisfy
$g(f(x+y))= f(x) +(2x+y)g(y)$ for all $(x,y)\in \mathbb R^2$.
- What is the value of the contour integral around Western Europe?

- Zero.

- Why?

- Because all the poles are in Eastern Europe.

Revive the IMO marathon.

Nirjhor
Posts: 136
Joined: Thu Aug 29, 2013 11:21 pm
Location: Varies.

Re: IMO Marathon

Solution $\boxed{52}$
Someone post the next problem.
- What is the value of the contour integral around Western Europe?

- Zero.

- Why?

- Because all the poles are in Eastern Europe.

Revive the IMO marathon.