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Balkan Mathematica Olympiad 2017 – Problem 3

May 10, 2017 Leave a comment

Problem 3. Let {\mathbb{N}} denote the set of positive integers. Find all functions {f:\mathbb{N}\longrightarrow\mathbb{N}} such that

\displaystyle n+f(m)\mid f(n)+nf(m)

for all {m,n\in \mathbb{N}}

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Balkan Mathematical Olympiad 2017 – Problems

May 10, 2017 Leave a comment

Problem 1. Find all ordered pairs of positive integers { (x, y)} such that:

\displaystyle x^3+y^3=x^2+42xy+y^2.

Problem 2. Consider an acute-angled triangle {ABC} with {AB<AC} and let {\omega} be its circumscribed circle. Let {t_B} and {t_C} be the tangents to the circle {\omega} at points {B} and {C}, respectively, and let {L} be their intersection. The straight line passing through the point {B} and parallel to {AC} intersects {t_C} in point {D}. The straight line passing through the point {C} and parallel to {AB} intersects {t_B} in point {E}. The circumcircle of the triangle {BDC} intersects {AC} in {T}, where {T} is located between {A} and {C}. The circumcircle of the triangle {BEC} intersects the line {AB} (or its extension) in {S}, where {B} is located between {S} and {A}.

Prove that {ST}, {AL}, and {BC} are concurrent.

Problem 3. Let {\mathbb{N}} denote the set of positive integers. Find all functions {f:\mathbb{N}\longrightarrow\mathbb{N}} such that

\displaystyle n+f(m)\mid f(n)+nf(m)

for all {m,n\in \mathbb{N}}

Problem 4. On a circular table sit {\displaystyle {n> 2}} students. First, each student has just one candy. At each step, each student chooses one of the following actions:

  • (A) Gives a candy to the student sitting on his left or to the student sitting on his right.
  • (B) Separates all its candies in two, possibly empty, sets and gives one set to the student sitting on his left and the other to the student sitting on his right.

At each step, students perform the actions they have chosen at the same time. A distribution of candy is called legitimate if it can occur after a finite number of steps. Find the number of legitimate distributions.

(Two distributions are different if there is a student who has a different number of candy in each of these distributions.)

Source: AoPS

Missing digit – short puzzle

April 15, 2017 Leave a comment

The number 2^{29} has 9 digits, all different; which digit is missing?

Mathematical Mind-Benders, Peter Winkler

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Project Euler tips

March 28, 2017 Leave a comment

A few years back I started working on Project Euler problems mainly because it was fun from a mathematical point of view, but also to improve my programming skills. After solving about 120 problems I seem to have hit a wall, because the numbers involved in some of the problems were just too big for my simple brute-force algorithms.

Recently, I decided to try and see if I can do some more of these problems. I cannot say that I’ve acquired some new techniques between 2012-2016 concerning the mathematics involved in these problems. My research topics are usually quite different and my day to day programming routines are more about constructing new stuff which works fast enough than optimizing actual code. Nevertheless, I have some experience coding in Matlab, and I realized that nested loops are to be avoided. Vectorizing the code can speed up things 100 fold.

So the point of Project Euler tasks is making things go well for large numbers. Normally all problems are tested and should run within a minute on a regular machine. This brings us to choosing the right algorithms, the right simplifications and finding the complexity of the algorithms involved.

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Project Euler Problem 285

March 25, 2017 Leave a comment

Another quite nice problem from Project Euler is number 285. The result of the problem depends on the computation of a certain probability, which in turn is related to the computation of a certain area. Below is an illustration of the computation for k equal to 10.


To save you some time, here’s a picture of the case k=1 which I ignored and spent quite a lot of time debugging… Plus, it only affects the last three digits or so after the decimal point…


Here’s a Matlab code which can construct the pictures above and can compute the result for low cases. To solve the problem, you should compute explicitly all these areas.

function problem285(k)

N = 100000;

a = rand(1,N);
b = rand(1,N);

ind = find(abs(sqrt((k*a+1).^2+(k*b+1).^2)-k)<0.5);

axis equal

M = k;
pl = 1;

for k=1:M
if mod(k,100)==0
r1 = (k+0.5)/k;
r2 = (k-0.5)/k;

f1 = @(x) (x<=(-1/k+r1)).*(x>=(-1/k-r1)).*(sqrt(r1^2-(x+1/k).^2)-1/k).*(x>=0).*(x<=1); f1 = @(x) f1(x).*(f1(x)>=0);
f2 = @(x) (x<=(-1/k+r2)).*(x>=(-1/k-r2)).*(sqrt(r2^2-(x+1/k).^2)-1/k).*(x>=0).*(x<=1); f2 = @(x) f2(x).*(f2(x)>=0);

if k == pl
thetas = linspace(0,pi/2,200);
hold on
plot([0 1 1 0 0],[0 0 1 1 0],'k','LineWidth',3);
hold off
axis off

A(k) = integral(@(x) f1(x)-f2(x),0,1);


xs = xlim;
ys = ylim;

w = 0.01;
axis([xs(1)-w xs(2)+w ys(1)-w ys(2)+w]);


Fireworks – Project Euler 317

March 19, 2017 Leave a comment

You can read the text of the problem here. The idea is that we have an explosion at a given height in a uniform gravitational field (no friction/wind). Supposing that all particles go outwards from the explosion point with a constant initial speed, what is the shape of the body formed by these particles? A hint is given in the picture below with a nice Matlab simulation.


Now what happens if we add a little bit of wind? Here’s the perturbation obtained when adding a 10m/s uniform wind speed vs the initial configuration. Something like this would be a lot more challenging 🙂


SEEMOUS 2017 Problems

March 5, 2017 1 comment

Problem 1. Let {A \in \mathcal M_2(\Bbb{R})}. Suppose

\displaystyle A =\begin{pmatrix} a& b\\ c & d \end{pmatrix}


\displaystyle a^2+b^2+c^2+d^2<1/5.

Show that {I+A} is invertible.

Problem 2. Let {A,B \in \mathcal M_n(\Bbb{R})}.

  • a) Prove that there exists {a>0} such that for every {\varepsilon \in (-a,a),\varepsilon\neq 0} the matrix equation

    \displaystyle AX+\varepsilon X = B,\ X \in \mathcal M_n(\Bbb{R})

    has a unique solution {X(\varepsilon) \in \mathcal M_n(\Bbb{R})}.

  • b) Prove that if {B^2 = I_n} and {A} is diagonalizable then

    \displaystyle \lim_{\varepsilon \rightarrow 0} \text{tr}(BX(\varepsilon)) = n-\text{rank}(A).

Problem 3. Let {f: \Bbb{R} \rightarrow \Bbb{R}} be a continuous function. Prove that

\displaystyle \int_0^4 f(x(x-3)^2)dx = 2 \int_1^3 f(x(x-3)^2)dx

Problem 4. a) Let {n \geq 0} be an integer. Compute {\displaystyle \int_0^1 (1-t)^n e^t dt}.

b) Let {k \geq 0} be a fixed integer and let {(x_n)_{n \geq k}} be the sequence defined by

\displaystyle x_n = \sum_{i=k}^n {i \choose k}\left(e-1-\frac{1}{1!}-\frac{1}{2!}-...-\frac{1}{i!}\right).

Prove that the sequence converges and find its limit.

Source: link

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