MathDB

2013 IMC

Part of IMC

Subcontests

(5)
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IMC 2013/1/5

Does there exist a sequence (an)\displaystyle{\left( {{a_n}} \right)} of complex numbers such that for every positive integer p\displaystyle{p} we have that ∑n=1+∞anp\displaystyle{\sum\limits_{n = 1}^{ + \infty } {a_n^p} } converges if and only if p\displaystyle{p} is not a prime?
Proposed by Tomáš Bárta, Charles University, Prague.

IMC 2013/2/5

Consider a circular necklace with 2013\displaystyle{2013} beads. Each bead can be paintes either green or white. A painting of the necklace is called good if among any 21\displaystyle{21} successive beads there is at least one green bead. Prove that the number of good paintings of the necklace is odd.
Note. Two paintings that differ on some beads, but can be obtained from each other by rotating or flipping the necklace, are counted as different paintings.
Proposed by Vsevolod Bykov and Oleksandr Rybak, Kiev.
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IMC 2013/1/4

Let n⩾3\displaystyle{n \geqslant 3} and let x1,x2,...,xn\displaystyle{{x_1},{x_2},...,{x_n}} be nonnegative real numbers. Define A=∑i=1nxi,B=∑i=1nxi2,C=∑i=1nxi3\displaystyle{A = \sum\limits_{i = 1}^n {{x_i}} ,B = \sum\limits_{i = 1}^n {x_i^2} ,C = \sum\limits_{i = 1}^n {x_i^3} }. Prove that: (n+1)A2B+(n−2)B2⩾A4+(2n−2)AC.\displaystyle{\left( {n + 1} \right){A^2}B + \left( {n - 2} \right){B^2} \geqslant {A^4} + \left( {2n - 2} \right)AC}.
Proposed by Géza Kós, Eötvös University, Budapest.

IMC 2013/2/4

Does there exist an infinite set M\displaystyle{M} consisting of positive integers such that for any a,b∈M\displaystyle{a,b \in M} with a<b\displaystyle{a < b} the sum a+b\displaystyle{a + b} is square-free? Note. A positive integer is called square-free if no perfect square greater than 1\displaystyle{1} divides it.
Proposed by Fedor Petrov, St. Petersburg State University.
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IMC 2013/1/3

There are 2n\displaystyle{2n} students in a school (n∈N,n⩾2)\displaystyle{\left( {n \in {\Bbb N},n \geqslant 2} \right)}. Each week n\displaystyle{n} students go on a trip. After several trips the following condition was fulfiled: every two students were together on at least one trip. What is the minimum number of trips needed for this to happen?
Proposed by Oleksandr Rybak, Kiev, Ukraine.

IMC 2013/2/3

Suppose that v1,v2,...,vd\displaystyle{{v_1},{v_2},...,{v_d}} are unit vectors in Rd\displaystyle{{{\Bbb R}^d}}. Prove that there exists a unitary vector u\displaystyle{u} such that ∣u⋅vi∣≤1d\displaystyle{\left| {u \cdot {v_i}} \right| \leq \frac{1}{{\sqrt d }}} for i=1,2,...,d\displaystyle{i = 1,2,...,d}.
Note. Here ⋅\displaystyle{ \cdot } denotes the usual scalar product on Rd\displaystyle{{{\Bbb R}^d}}.
Proposed by Tomasz Tkocz, University of Warwick.
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Imc 2013/1/2

Let f:R→R\displaystyle{f:{\cal R} \to {\cal R}} be a twice differentiable function. Suppose f(0)=0\displaystyle{f\left( 0 \right) = 0}. Prove there exists ξ∈(−π2,π2)\displaystyle{\xi \in \left( { - \frac{\pi }{2},\frac{\pi }{2}} \right)} such that f′′(ξ)=f(ξ)(1+2tan⁡2ξ).\displaystyle{f''\left( \xi \right) = f\left( \xi \right)\left( {1 + 2{{\tan }^2}\xi } \right)}.
Proposed by Karen Keryan, Yerevan State University, Yerevan, Armenia.

IMC 2013/2/2

Let p,q\displaystyle{p,q} be relatively prime positive integers. Prove that ∑k=0pq−1(−1)⌊kp⌋+⌊kq⌋={0 if pq is even1if pq odd\displaystyle{ \sum_{k=0}^{pq-1} (-1)^{\left\lfloor \frac{k}{p}\right\rfloor + \left\lfloor \frac{k}{q}\right\rfloor} = \begin{cases} 0 & \textnormal{ if } pq \textnormal{ is even}\\ 1 & \textnormal{if } pq \textnormal{ odd}\end{cases}}
Proposed by Alexander Bolbot, State University, Novosibirsk.
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