MathDB

1

Part of 2013 Romania Team Selection Test

Problems(5)

Binomial expansion of cube root 2-1

Source: Romania TST 1 P1, 2013

4/5/2013
Given an integer n≥2,n\geq 2, let an,bn,cna_{n},b_{n},c_{n} be integer numbers such that (23−1)n=an+bn23+cn43. \left( \sqrt[3]{2}-1\right) ^{n}=a_{n}+b_{n}\sqrt[3]{2}+c_{n}\sqrt[3]{4}. Prove that cn≡1(mod3)c_{n}\equiv 1\pmod{3} if and only if n≡2(mod3).n\equiv 2\pmod{3}.
geometry3D geometrymodular arithmeticsymmetryalgebrapolynomialnumber theory unsolved
[na] divides [nb]

Source: Romania TST 2013 Test 2 Problem 1

4/26/2013
Suppose that aa and bb are two distinct positive real numbers such that ⌊na⌋\lfloor na\rfloor divides ⌊nb⌋\lfloor nb\rfloor for any positive integer nn. Prove that aa and bb are positive integers.
floor functionratiolimitnumber theorygreatest common divisorinequalitiesalgebra proposed
Bounding difference of fractional parts.

Source: Romania TST 2013 Day 3 Problem 1

1/21/2015
Let aa and bb be two square-free, distinct natural numbers. Show that there exist c>0c>0 such that ∣{na}−{nb}∣>cn3 \left | \{n\sqrt{a}\}-\{n\sqrt{b}\} \right |>\frac{c}{n^3} for every positive integer nn.
inequalitiesalgebrapolynomialabsolute valuetriangle inequalitynumber theory unsolvednumber theory
Unit discs!

Source: Romania TST 2013 Day 4 Problem 1

1/21/2015
Fix a point OO in the plane and an integer n≥3n\geq 3. Consider a finite family D\mathcal{D} of closed unit discs in the plane such that: (a) No disc in D\mathcal{D} contains the point OO; and (b) For each positive integer k<nk < n, the closed disc of radius k+1k + 1 centred at OO contains the centres of at least kk discs in D\mathcal{D}. Show that some line through OO stabs at least 2πlog⁡n+12\frac{2}{\pi} \log \frac{n+1}{2} discs in D\mathcal{D}.
logarithmscombinatorics unsolvedcombinatorics
Unusual inequality!

Source: Romania TST 2013 Day 5 Problem 1

1/21/2015
Let nn be a positive integer and let x1x_1, …\ldots, xnx_n be positive real numbers. Show that: min⁡(x1,1x1+x2,⋯ ,1xn−1+xn,1xn)≤2cos⁡πn+2≤max⁡(x1,1x1+x2,⋯ ,1xn−1+xn,1xn). \min\left ( x_1,\frac{1}{x_1}+x_2, \cdots,\frac{1}{x_{n-1}}+x_n,\frac{1}{x_n} \right )\leq 2\cos \frac{\pi}{n+2} \leq\max\left ( x_1,\frac{1}{x_1}+x_2, \cdots,\frac{1}{x_{n-1}}+x_n,\frac{1}{x_n} \right ).
inequalitiestrigonometryinequalities unsolved