MathDB

3

Part of 2014 Romania Team Selection Test

Problems(5)

Coprime numbers!

Source: Romania TST 2014 Day 2 Problem 3

1/21/2015
Determine all positive integers nn such that all positive integers less than nn and coprime to nn are powers of primes.
number theory unsolvednumber theory
Locus with equilateral triangles

Source: Romania TST 2014 Day 1 Problem 3

12/25/2014
Let A0A1A2A_0A_1A_2 be a scalene triangle. Find the locus of the centres of the equilateral triangles X0X1X2X_0X_1X_2 , such that AkA_k lies on the line Xk+1Xk+2X_{k+1}X_{k+2} for each k=0,1,2k=0,1,2 (with indices taken modulo 33).
IMO Shortlistgeometry unsolvedgeometry
Best constant!

Source: Romania TST Day 3 Problem 3

1/21/2015
Determine the smallest real constant cc such that ∑k=1n(1k∑j=1kxj)2≤c∑k=1nxk2\sum_{k=1}^{n}\left ( \frac{1}{k}\sum_{j=1}^{k}x_j \right )^2\leq c\sum_{k=1}^{n}x_k^2 for all positive integers nn and all positive real numbers x1,⋯ ,xnx_1,\cdots ,x_n.
inequalitiesinequalities unsolved
Permutation sum

Source: Romania, 4th TST 2014, Problem 3

12/6/2014
Let n∈Nn \in \mathbb{N} and SnS_{n} the set of all permutations of {1,2,3,...,n}\{1,2,3,...,n\}. For every permutation σ∈Sn\sigma \in S_{n} denote I(σ):={i:σ(i)≤i}I(\sigma) := \{ i: \sigma (i) \le i \}. Compute the sum ∑σ∈Sn1∣I(σ)∣∑i∈I(σ)(i+σ(i))\sum_ {\sigma \in S_{n}} \frac{1}{|I(\sigma )|} \sum_ {i \in I(\sigma)} (i+ \sigma(i)).
algebra unsolvedalgebracombinatorics
Maximizing a sum !

Source: Romania TST Day 5 Problem 3

1/21/2015
Let nn a positive integer and let f ⁣:[0,1]→Rf\colon [0,1] \to \mathbb{R} an increasing function. Find the value of : max⁡0≤x1≤⋯≤xn≤1∑k=1nf(∣xk−2k−12n∣) \max_{0\leq x_1\leq\cdots\leq x_n\leq 1}\sum_{k=1}^{n}f\left ( \left | x_k-\frac{2k-1}{2n} \right | \right )
inequalities unsolvedinequalitiesalgebra