### Archive

Archive for the ‘Algebra’ Category

## Balkan Mathematical Olympiad 2017 – Problems

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 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

## IMC 2016 – Day 2 – Problem 8

July 28, 2016 2 comments

Problem 8. Let ${n}$ be a positive integer and denote by ${\Bbb{Z}_n}$ the ring of integers modulo ${n}$. Suppose that there exists a function ${f:\Bbb{Z}_n \rightarrow \Bbb{Z}_n}$ satisfying the following three properties:

• (i) ${f(x) \neq x}$,
• (ii) ${x = f(f(x))}$,
• (iii) ${f(f(f(x+1)+1)+1) = x}$ for all ${x \in \Bbb{Z}_n}$.

Prove that ${n \equiv 2}$ modulo ${4}$.

## IMC 2016 – Day 1 – Problem 2

Problem 2. Let ${k}$ and ${n}$ be positive integers. A sequence ${(A_1,...,A_k)}$ of ${n\times n}$ matrices is preferred by Ivan the Confessor if ${A_i^2 \neq 0}$ for ${1\leq i \leq k}$, but ${A_iA_j = 0}$ for ${1\leq i,j \leq k}$ with ${i \neq j}$. Show that if ${k \leq n}$ in al preferred sequences and give an example of a preferred sequence with ${k=n}$ for each ${n}$.

## 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. 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.

## SEEMOUS 2016 – Problems

March 5, 2016 3 comments

Problem 1. Let ${f}$ be a continuous and decreasing real valued function defined on ${[0,\pi/2]}$. Prove that

$\displaystyle \int_{\pi/2-1}^{\pi/2} f(x)dx \leq \int_0^{\pi/2} f(x)\cos x dx \leq \int_0^1 f(x) dx.$

When do we have equality?

Problem 2. a) Prove that for every matrix ${X \in \mathcal{M}_2(\Bbb{C})}$ there exists a matrix ${Y \in \mathcal{M}_2(\Bbb{C})}$ such that ${Y^3 = X^2}$.

b) Prove that there exists a matrix ${A \in \mathcal{M}_3(\Bbb{C})}$ such that ${Z^3 \neq A^2}$ for all ${Z \in \mathcal{M}_3(\Bbb{C})}$.

Problem 3. Let ${A_1,A_2,...,A_k}$ be idempotent matrices (${A_i^2 = A_i}$) in ${\mathcal{M}_n(\Bbb{R})}$. Prove that

$\displaystyle \sum_{i=1}^k N(A_i) \geq \text{rank} \left(I-\prod_{i=1}^k A_i\right),$

where ${N(A_i) = n-\text{rank}(A_i)}$ and ${\mathcal{M}_n(\Bbb{R})}$ is the set of ${n \times n}$ matrices with real entries.

Problem 4. Let ${n \geq 1}$ be an integer and set

$\displaystyle I_n = \int_0^\infty \frac{\arctan x}{(1+x^2)^n}dx.$

Prove that

a) ${\displaystyle \sum_{i=1}^\infty \frac{I_n}{n} =\frac{\pi^2}{6}.}$

b) ${\displaystyle \int_0^\infty \arctan x \cdot \ln \left( 1+\frac{1}{x^2}\right) dx = \frac{\pi^2}{6}}$.

Some hints follow.

## Problems of the Miklos Schweitzer Competition 2014

November 10, 2014 5 comments

Problem 1. Let ${n}$ be a positive integer. Let ${\mathcal{F}}$ be a familiy of sets that contains more than half of all subsets of an ${n}$-element set ${X}$. Prove that from ${\mathcal{F}}$ we can select ${\lceil \log_2 n\rceil+1 }$ sets that form a separating family of ${X}$, i.e., for any two distinct elements of ${X}$ there is a selected set containing exactly one of the two elements.

Problem 2. let ${k \geq 1}$ and let ${I_1,...,I_k}$ be non-degenerate subintervals of the interval ${[0,1]}$. Prove that

$\displaystyle \sum \frac{1}{|I_i \cup I_j|} \geq k^2,$

where the summation is over all pairs of indices ${(i,j)}$ such that ${I_i}$ and ${I_j}$ are not disjoint.

Problem 3. We have ${4n+5}$ points in the plane, no three of them collinear. The points are colored with two colors. Prove that from the points we can form ${n}$ empty triangles (they have no colored points in their interiors) with pairwise disjoint interiors, such that all points occuring as vertices of the ${n}$ triangles have the same color.

Problem 4. For a positive integer ${n}$, let ${f(n)}$ be the number of sequences ${a_1,...,a_k}$ of positive integers such that ${a_i \geq 2}$ and ${a_1...a_k = n}$ for ${k \geq 1}$. We make the convention ${f(1)=1}$. Let ${\alpha}$ be the unique real number greater than ${1}$ such that ${\sum_{n=1}^\infty n^{-\alpha}=2}$. Prove that

• (i) ${ \sum_{ k = 1}^n f(k)= O(n^\alpha)}$.
• (ii) There exists no number ${\beta<\alpha}$ such that ${f(n)=O(n^\beta)}$.

Problem 5. Let ${\alpha}$ be a non-real algebraic integer of degree two, and let ${P}$ be the set of irreducible elements of the ring ${\Bbb{Z}[\alpha]}$. Prove that

$\displaystyle \sum_{ p \in P} \frac{1}{|p|^2} = \infty.$

Problem 6. Let ${\rho : G \rightarrow GL(V)}$ be a representation of a finite ${p}$-group ${G}$ over a field of characteristic ${p}$. Prove that if the restriction of the linear map ${\sum_{ g \in G} \rho(g)}$ to a finite dimensional subspace ${W}$ of ${V}$ is injective, then the subspace spanned by the subspaces ${\rho(g)W}$ (${g \in G}$) is the direct sum of these subspaces.

Problem 7. Lef ${f: \Bbb{R} \rightarrow \Bbb{R}}$ be a continuous function and let ${g: \Bbb{R} \rightarrow \Bbb{R}}$ be arbitrary. Suppose that the Minkowski sum of the graph of ${f}$ and the graph of ${g}$ (i.e. the set ${\{(x+y,f(x)+g(y) : x,y \in \Bbb{R}\}}$ has Lebesgue measure zero. Does it follow then that the function ${f}$ is of the form ${f(x)=ax+b}$, with suitable constants ${a,b \in \Bbb{R}}$?

Problem 8. Let ${n \geq 1}$ be a fixed integer. Calculate the distance

$\displaystyle \inf_{p,f} \max_{0 \leq x \leq 1} |f(x)-p(x)|,$

where ${p}$ runs over polynomials of degree less than ${n}$ with real coefficients and ${f}$ runs over functions of the form

$\displaystyle f(x) = \sum_{ k = n}^\infty c_kx^k$

defined on the closed interval ${[0,1]}$, where ${c_k\geq 0}$ and ${\sum_{k=n}^\infty c_k =1}$.

Problem 9. Let ${\rho : \Bbb{R}^n \rightarrow \Bbb{R},\ \rho(x)=e^{-\|x\|^2}}$, and let ${K \subset \Bbb{R}^n}$ be a convex body, i.e. a compact convex set with nonempty interior. Define the barycenter ${s_K}$ of the body ${K}$ with respect to the weight function ${\rho}$ by the usual formula

$\displaystyle s_K = \frac{\int_K \rho(x) x dx}{\int_K \rho(x)dx}.$

Prove that the translates of the body ${K}$ have pairwise distinct barycenters with respect to ${\rho}$.

Problem 10. To each vertex of a given triangulation of the two dimensional sphere, we assign a convex subset of the plane. Assume that the three convex sets corresponding to the three vertices of any two dimensional face of the triangulation have at least one point in common. Show that there exist four vertices such that the corresponding convex sets have at least one point in common.

Problem 11. Let ${U}$ be a random variable that is uniformly distributed on the interval ${[0,1]}$, and let

$\displaystyle S_n = 2\sum_{k=1}^n \sin(2kU\pi).$

Show that, as ${n \rightarrow \infty}$, the limit distribution of ${S_n}$ is the Cauchy distribution with density function ${f(x) =\frac{1}{\pi(1+x^2)}}$.

## Monotonic bijection from naturals to pairs of natural numbers

This is a cute problem I found this evening.

Suppose ${\phi : \Bbb{N}^* \rightarrow \Bbb{N}^*\times \Bbb{N}^*}$ is a bijection such that if ${\phi(k) = (i,j),\ \phi(k')=(i',j')}$ and ${k \leq k'}$, then ${ij \leq i'j'}$.

Prove that if ${k = \phi(i,j)}$ then ${k \geq ij}$.

Proof: The trick is to divide the pairs of positive integers into families with the same product.

$\displaystyle \begin{matrix} (1,1) & (1,2) & (1,3) & (1,4) & (1,5) & (1,6) & \cdots \\ & (2,1) & (3,1) & (2,2) & (5,1) & (2,3) & \cdots \\ & & &( 4,1) & & (3,2) & \cdots \\ & & & & & (6,1) & \cdots \end{matrix}$

Note that the ${M}$-th column contains as many elements as the number of divisors of ${M}$. Now we just just use a simple observation. Let ${\phi(k)=(i,j)}$ be on the ${M}$-th column (i.e. ${ij = M}$). If ${n \geq 1}$ then ${\phi(k+n)=(i',j')}$ cannot be on one of the first ${M-1}$ columns. Indeed, the monotonicity property implies ${M = ij \leq i'j'}$. The fact that ${\phi}$ is a bijection assures us that ${\phi(1),...,\phi(k)}$ cover the first ${M-1}$ columns. Moreover, one element from the ${M}$-th column is surely covered, namely ${(i,j) = \phi(k)}$. This means that

$\displaystyle k \geq d(1)+...+d(M-1)+1 \geq M = ij,$

where we have denoted by ${d(n)}$ the number of positive divisors of ${n}$.