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<a<T}, {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} f(x) = +\infty \text{ and } \lim_{x \rightarrow a,x<a} \frac{f'(x)}{f^2(x)} = 1.

(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

March 11, 2020 Leave a comment

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}.

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Putnam 2019 – Problem B5

March 9, 2020 Leave a comment

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}.

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Putnam 2019 – Problem B4

March 9, 2020 Leave a comment

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}.

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Putnam 2019 A2

March 3, 2020 5 comments

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}.

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Putnam 2019 B3

January 13, 2020 Leave a comment

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}.

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Putnam 2019 A1

January 13, 2020 Leave a comment

A1. Determine all possible values of the expression

\displaystyle A^3+B^3+C^3-3ABC

where {A,B,C} are non-negative integers.

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