Olympiad problems

2015 IMC, 3

Let $F(0)=0$, $F(1)=\frac32$, and $F(n)=\frac{5}{2}F(n-1)-F(n-2)$ for $n\ge2$. Determine whether or not $\displaystyle{\sum_{n=0}^{\infty}\, \frac{1}{F(2^n)}}$ is a rational number. (Proposed by Gerhard Woeginger, Eindhoven University of Technology)

2015 IMC, 1

Tags: Matrices , college contests , IMC2015

For any integer $n\ge 2$ and two $n\times n$ matrices with real entries $A,\; B$ that satisfy the equation $$A^{-1}+B^{-1}=(A+B)^{-1}\;$$ prove that $\det (A)=\det(B)$. Does the same conclusion follow for matrices with complex entries? (Proposed by Zbigniew Skoczylas, Wroclaw University of Technology)

2015 IMC, 6

Tags: college contests , series , inequalities , IMC2015

Prove that $$\sum\limits_{n = 1}^{\infty}\frac{1}{\sqrt{n}\left(n+1\right)} < 2.$$ Proposed by Ivan Krijan, University of Zagreb

2015 IMC, 9

Tags: college contests , linear algebra , Matrices , IMC2015

An $n \times n$ complex matrix $A$ is called \emph{t-normal} if $AA^t = A^t A$ where $A^t$ is the transpose of $A$. For each $n$, determine the maximum dimension of a linear space of complex $n \times n$ matrices consisting of t-normal matrices. Proposed by Shachar Carmeli, Weizmann Institute of Science

2015 IMC, 5

Tags: college contests , geometry , vector geometry , IMC2015

Let $n\ge2$, let $A_1,A_2,\ldots,A_{n+1}$ be $n+1$ points in the $n$-dimensional Euclidean space, not lying on the same hyperplane, and let $B$ be a point strictly inside the convex hull of $A_1,A_2,\ldots,A_{n+1}$. Prove that $\angle A_iBA_j>90^\circ$ holds for at least $n$ pairs $(i,j)$ with $\displaystyle{1\le i<j\le n+1}$. Proposed by Géza Kós, Eötvös University, Budapest

2015 IMC, 4

Tags: number theory , trig identities , college contests , IMC2015

Determine whether or not there exist 15 integers $m_1,\ldots,m_{15}$ such that~ $$\displaystyle \sum_{k=1}^{15}\,m_k\cdot\arctan(k) = \arctan(16). \eqno(1)$$ (Proposed by Gerhard Woeginger, Eindhoven University of Technology)

2015 IMC, 7

Tags: college contests , Integral inequality , IMC2015

Compute $$ \lim_{A\to+\infty}\frac1A\int_1^A A^{\frac1x}\, dx . $$ Proposed by Jan Šustek, University of Ostrava

2015 IMC, 10

Tags: college contests , Polynomials , Inequality , IMC2015

Let $n$ be a positive integer, and let $p(x)$ be a polynomial of degree $n$ with integer coefficients. Prove that $$ \max_{0\le x\le1} \big|p(x)\big| > \frac1{e^n}. $$ Proposed by Géza Kós, Eötvös University, Budapest

2015 IMC, 2

Tags: numbers , college contests , inequalities , IMC2015

For a positive integer $n$, let $f(n)$ be the number obtained by writing $n$ in binary and replacing every 0 with 1 and vice versa. For example, $n=23$ is 10111 in binary, so $f(n)$ is 1000 in binary, therefore $f(23) =8$. Prove that \[\sum_{k=1}^n f(k) \leq \frac{n^2}{4}.\] When does equality hold? (Proposed by Stephan Wagner, Stellenbosch University)

2015 IMC, 8

Tags: college contests , combinatorics , IMC2015

Consider all $26^{26}$ words of length 26 in the Latin alphabet. Define the $\emph{weight}$ of a word as $1/(k+1)$, where $k$ is the number of letters not used in this word. Prove that the sum of the weights of all words is $3^{75}$. Proposed by Fedor Petrov, St. Petersburg State University