Problems of SEEMOUS 2020

March 11, 2020 1 comment

Problem 1. Consider ${A \in \mathcal M_{2020}(\Bbb{C})}$ such that

$\displaystyle A + A^\times = I_{2020}$

$\displaystyle A\cdot A^\times = I_{2020}$

where ${A^\times}$ is the adjugate matrix of ${A}$, i.e., the matrix whose elements are ${a_{ij} = (-1)^{i+j}d_{ji}}$ where ${d_{ji}}$ is the determinant obtained from ${A}$, eliminating the line ${j}$ and the column ${i}$.

Find the maximum number of matrices verifying the two equations above such that any two of them are not similar.

Problem 2. Let ${k>1}$ be a real number. Calculate:

(a) ${L = \displaystyle \lim_{n\rightarrow \infty} \int_0^1 \left( \frac{k}{\sqrt[n]{x}+k-1}\right)^n dx}$.

(b) ${ \displaystyle \lim_{n\rightarrow \infty} n \left[ L- \int_0^1 \left( \frac{k}{\sqrt[n]{x}+k-1}\right)^n dx\right]}$.

Problem 3. Let ${n}$ be a positive integer, ${k \in \Bbb{C}}$ and ${A \in \mathcal M_n(\Bbb{C})}$ such that ${tr(A) \neq 0}$ and

$\displaystyle \text{rank} A + \text{rank} (tr(A)I_n-kA) = n.$

Find the rank of ${A}$.

Problem 4. Consider ${0, ${D=R \setminus \{kT+a | k \in \Bbb{Z}\}}$ and let ${f: D \rightarrow \Bbb{R}}$ be a ${T}$-periodic and differentiable function which satisfies ${f'>1}$ on ${(0,a)}$ and

$\displaystyle f(0)=0,\ \lim_{x \rightarrow a, x

(a) Prove that for every ${n \in \Bbb{N}^*}$ the equation ${f(x)=x}$ has a unique solution in the interval ${(nT,nT+a)}$ denoted ${x_n}$.

(b) Let ${y_n = nT+a-x_n}$ and ${z_n = \int_0^{y_n} f(x)dx}$. Prove that ${\lim_{n\rightarrow \infty} y_n = 0}$ and study the convergence of the series ${\sum_{n=1}^\infty y_n}$ and ${\sum_{n=1}^\infty z_n}$.

Source: https://seemous2020.auth.gr/

Putnam 2019 – Problem B2

B2. For all ${n \geq 1}$, let

$\displaystyle a_n = \sum_{k=1}^{n-1} \frac{\sin\left( \frac{(2k-1)\pi}{2n}\right) }{\cos^2\left(\frac{(k-1)\pi}{2n}\right) \cos^2\left( \frac{k\pi}{2n}\right)}.$

Determine

$\displaystyle \lim_{n \rightarrow \infty} \frac{a_n}{n^3}.$

Putnam 2019 – Problem B5

B4. Let ${F_m}$ be the ${m}$th Fibonacci number, defined by ${F_1=F_2=1}$ and ${F_m = F_{m-1}+F_{m-2}}$ for all ${m \geq 3}$. Let ${p(x)}$ be the polynomial of degree ${1008}$ such that ${p(2n+1) = F_{2n+1}}$ for all ${n = 0,1,...,1008}$. Find integers ${j}$ and ${k}$ such that ${p(2019) = F_j-F_k}$.

Putnam 2019 – Problem B4

B4. Let ${\mathcal F}$ be the set of functions ${f(x,y)}$ that are twice continuously differentiable for ${x \geq 1,\ y \geq 1}$ and that satisfy the following two equations (where scripts denote partial derivatives):

$\displaystyle xf_x+yf_y = xy \ln (xy)$

$\displaystyle x^2f_{xx}+y^2f_{yy} = xy.$

For each ${f\in \mathcal F}$, let

$\displaystyle m(f) = \min_{s \geq 1} \left( f(s+1,s+1)-f(s+1,s)-f(s,s+1)+f(s,s)\right).$

Determine ${m(f)}$ and show that it is independent of the choice of ${f}$.

Putnam 2019 A2

A2. In the triangle ${\Delta ABC}$, let ${G}$ be the centroid, and let ${I}$ be the center of the inscribed circle. Let ${\alpha}$ and ${\beta}$ be the angles at the vertices ${A}$ and ${B}$, respectively. Suppose that the segment ${IG}$ is parallel to ${AB}$ and that ${\beta = 2\tan^{-1}(1/3)}$. Find ${\alpha}$.

Putnam 2019 B3

B3. Let ${Q}$ be a ${n\times n}$ real orthogonal matrix and let ${u \in \Bbb{R}^n}$ be a unit column vector (that is ${u^Tu=1}$). Let ${P=I-2uu^T}$, where ${I}$ is the ${n\times n}$ identity matrix. Show that if ${1}$ is not an eigenvalue of ${Q}$, then ${1}$ is an eigenvalue of ${PQ}$.
$\displaystyle A^3+B^3+C^3-3ABC$
where ${A,B,C}$ are non-negative integers.