Archive

Posts Tagged ‘Geometry’

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

Project Euler Problem 144 – Laser light escaping an ellipse

February 5, 2017 Leave a comment

ellipse

Project Euler – problem 144 – visualization of the solution in Matlab

Some of the easy Putnam 2016 Problems

December 11, 2016 Leave a comment

Here are a few of the problems of the Putnam 2016 contest. I choose to only list problems which I managed to solve. Most of them are pretty straightforward, so maybe the solutions posted here may be very similar to the official ones. You can find a complete list of the problems on other sites, for example here.

A1. Find the smallest integer {j} such that for every polynomial {p} with integer coefficients and every integer {k}, the number

\displaystyle p^{(j)}(k),

that is the {j}-th derivative of {p} evaluated at {k}, is divisible by {2016}.

Hints. Successive derivatives give rise to terms containing products of consecutive numbers. The product of {j} consecutive numbers is divisible by {j!}. Find the smallest number such that {2016 | j!}. Prove that {j-1} does not work by choosing {p = x^{j-1}}. Prove that {j} works by working only on monomials…

Read more…

IMO 2016 – Problem 1

July 26, 2016 Leave a comment

IMO 2016, Problem 1. Triangle {BCF} has a right angle at {B}. Let {A} be the point on line {CF} such that {FA=FB} and {F} lies between {A} and {C}. Point {D} is chosen such that {DA=DC} and {AC} is the bisector of {\angle DAB}. Point {E} is chosen such that {EA=ED} and {AD} is the bisector of {\angle EAC}. Let {M} be the midpoint of {CF}. Let {X} be the point such that {AMXE} is a parallelogram (where {AM || EX} and {AE || MX}). Prove that the lines {BD, FX} and {ME} are concurrent.

Read more…

Balkan Mathematical Olympiad – 2016 Problems

Problem 1. Find all injective functions {f: \mathbb R \rightarrow \mathbb R} such that for every real number {x} and every positive integer {n},

\displaystyle \left|\sum_{i=1}^n i\left(f(x+i+1)-f(f(x+i))\right)\right|<2016

Problem 2. Let {ABCD} be a cyclic quadrilateral with {AB<CD}. The diagonals intersect at the point {F} and lines {AD} and {BC} intersect at the point {E}. Let {K} and {L} be the orthogonal projections of {F} onto lines {AD} and {BC} respectively, and let {M}, {S} and {T} be the midpoints of {EF}, {CF} and {DF} respectively. Prove that the second intersection point of the circumcircles of triangles {MKT} and {MLS} lies on the segment {CD}.

Problem 3. Find all monic polynomials {f} with integer coefficients satisfying the following condition: there exists a positive integer {N} such that {p} divides {2(f(p)!)+1} for every prime {p>N} for which {f(p)} is a positive integer.

Problem 4. The plane is divided into squares by two sets of parallel lines, forming an infinite grid. Each unit square is coloured with one of {1201} colours so that no rectangle with perimeter {100} contains two squares of the same colour. Show that no rectangle of size {1\times1201} or {1201\times1} contains two squares of the same colour.

Optimal triangles with vertices on fixed circles

October 5, 2015 Leave a comment

Let {x,y,z>0} and suppose {ABC} is a triangle such that there exists a point {O} which satisfies {OA=x}, {OB = y}, {OC = z}. What is the relative position of {O} with respect to the triangle {ABC} such that

a) The area is maximized;

b) The perimeter is maximized.

This was inspired by this answer on Math Overflow.

Read more…

Curve of constant curvature on a sphere

September 29, 2015 Leave a comment

In general, curves of constant curvature in {\Bbb{R}^3} are not necessarily simple to describe. Here is a non trivial example. If instead of curves living in {\Bbb{R}^3} we consider only curves lying on the unit sphere, the situation changes.

Problem. If a smooth, three dimensional curve has constant curvature and is contained in the unit sphere, then it is a piece of a circle.

Read more…

%d bloggers like this: