### Archive

Posts Tagged ‘Algebra’

## 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 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 ${n}$.

## IMC 2016 – Day 1 – Problem 3

Problem 3. Let ${n}$ be a positive integer. Also let ${a_1,a_2,...,a_n}$ and ${b_1,b_2,...,b_n}$ be reap numbers such that ${a_i+b_i >0}$ for ${i = 1,2,...,n}$. Prove that

$\displaystyle \sum_{i=1}^n \frac{a_ib_i -b_i^2}{a_i+b_i} \leq \frac{\sum_{i=1}^n a_i \cdot \sum_{i=1}^n b_i - \left(\sum_{i=1}^n b_i \right)^2 }{\sum_{i=1}^n (a_i+b_i)}.$

## SEEMOUS 2016 – Problems

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.

## IMO 2015 Day 2

Problem 4. Triangle ${ABC}$ has circumcircle ${\Omega}$ and circumcenter ${O}$. A circle ${\Gamma}$ with center ${A}$ intersects the segment ${BC}$ at points ${D}$ and ${E}$, such that ${B}$, ${D}$, ${E}$, and ${C}$ are all different and lie on line ${BC}$ in this order. Let ${F}$ and ${G}$ be the points of intersection of ${\Gamma}$ and ${\Omega}$, such that ${A}$, ${F}$, ${B}$, ${C}$, and ${G}$ lie on ${\Omega}$ in this order. Let ${K}$ be the second point of intersection of the circumcircle of triangle ${BDF}$ and the segment ${AB}$. Let ${L}$ be the second point of intersection of the circumcircle of triangle ${CGE}$ and the segment ${CA}$.

Suppose that the lines ${FK}$ and ${GL}$ are different and intersect at the point ${X}$. Prove that ${X}$ lies on the line ${AO}$.

Problem 5. Let ${\mathbb R}$ be the set of real numbers. Determine all functions ${f:\mathbb R\rightarrow\mathbb R}$ that satisfy the equation

$\displaystyle f(x+f(x+y))+f(xy)=x+f(x+y)+yf(x)$

for all real numbers ${x}$ and ${y}$.

Problem 6. The sequence ${a_1,a_2,\dots}$ of integers satisfies the conditions:

(i) ${1\le a_j\le2015}$ for all ${j\ge1}$, (ii) ${k+a_k\neq \ell+a_\ell}$ for all ${1\le k<\ell}$. Prove that there exist two positive integers ${b}$ and ${N}$ for which

$\displaystyle \left\vert\sum_{j=m+1}^n(a_j-b)\right\vert\le1007^2$

for all integers ${m}$ and ${n}$ such that ${n>m\ge N}$.

## Vojtech Jarnik Competition 2015 – Problems Category 2

Problem 1. Let ${A}$ and ${B}$ be two ${3 \times 3}$ matrices with real entries. Prove that

$\displaystyle A - (A^{-1}+(B^{-1}-A)^{-1})^{-1} = ABA,$

provided all the inverses appearing on the left-hand side of the equality exist.

Problem 2. Determine all pairs ${(n,m)}$ of positive integers satisfying the equation

$\displaystyle 5^n = 6m^2+1.$

Problem 3. Determine the set of real values ${x}$ for which the following series converges, and find its sum:

$\displaystyle \sum_{n=1}^\infty \left( \sum_{k_i \geq 0, k_1+2k_2+...+nk_n = n} \frac{(k_1+...+k_n)!}{k_1!...k_n!} x^{k_1+...+k_n}\right).$

Problem 4. Find all continuously differentiable functions ${f : \Bbb{R} \rightarrow \Bbb{R}}$, such that for every ${a \geq 0}$ the following relation holds:

$\displaystyle \int_{D(a)} xf\left( \frac{ay}{\sqrt{x^2+y^2}}\right) dxdydz = \frac{\pi a^3}{8}(f(a)+\sin a -1),$

where ${D(a) = \left\{ (x,y,z) : x^2+y^2+z^2 \leq a^2,\ |y| \leq \frac{x}{\sqrt{3}}\right\}}$

## Seemous 2015 Problems

Problem 1. Prove that for every ${x \in (0,1)}$ the following inequality holds:

$\displaystyle \int_0^1 \sqrt{1+\cos^2 y}dy > \sqrt{x^2+\sin^2 x}.$

Problem 2. For any positive integer ${n}$, let the functions ${f_n : \Bbb{R} \rightarrow \Bbb{R}}$ be defined by ${f_{n+1}(x)=f_1(f_n(x))}$, where ${f_1(x)=3x-4x^3}$. Solve the equation ${f_n(x)=0}$.

Problem 3. For an integer ${n>2}$, let ${A,B,C,D \in \mathcal{M}_n(\Bbb{R})}$ be matrices satisfying:

$\displaystyle AC-BD = I_n,$

$\displaystyle AD+BC = O_n,$

where ${I_n}$ is the identity matrix and ${O_n}$ is the zero matrix in ${\mathcal{M}_n(\Bbb{R})}$.

Prove that:

• (a) ${CA-DB = I_n}$ and ${DA+CB = O_n}$.
• (b) ${\det(AC) \geq 0}$ and ${(-1)^n \det(BD) \geq 0}$.

Problem 4. Let ${I\subset \Bbb{R}}$ be an open interval which contains ${0}$ and ${f: I \rightarrow \Bbb{R}}$ be a function of class ${C^{2016}(I)}$ such that ${f(0)=0, f'(0)=1, f''(0) = ... = f^{(2015)}(0)=0,\ f^{(2016)}(0)<0}$.

• (i) Prove that there exists ${\delta>0}$ such that ${0 for ${x \in (0,\delta)}$.
• (ii) With ${\delta}$ determined at (i) define the sequence ${(a_n)}$ by

$\displaystyle a_1 = \frac{\delta}{2},\ a_{n+2}=f(a_n),\ n \geq 1.$

Study the convergence of the series ${\displaystyle \sum_{n=1}^\infty a_n^r}$ for ${r \in \Bbb{R}}$.

Hints:

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