MathDB

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Part of 2014 Romania National Olympiad

Problems(6)

p (2q + 1) + q (2p + 1) = 2 (p^2 + q^2) 2014 Romania NMO VII p1

Source:

8/15/2024
Find all primes pp and qq, with p≤qp \le q, so that p(2q+1)+q(2p+1)=2(p2+q2).p (2q + 1) + q (2p + 1) = 2 (p^2 + q^2).
number theory
strage sum of number of divisors

Source: Romanian National Olympiad 2014, Grade X, Problem 1

3/2/2019
Let be a natural number n. n. Calculate ∑k=1n2#{d∈N∣1≤d≤k≤d2≤n2∧k≡0(modd)}. \sum_{k=1}^{n^2}\#\left\{ d\in\mathbb{N}| 1\le d\le k\le d^2\le n^2\wedge k\equiv 0\pmod d \right\} .
Here, # \# means cardinal.
number theory
Romania National Olympiad 2014,grade 8 ,P1

Source:

4/29/2014
Let a,b,c∈(0,∞)a,b,c\in \left( 0,\infty \right).Prove the inequality a−bca+2(b+c)+b−cab+2(c+a)+c−abc+2(a+b)≥0.\frac{a-\sqrt{bc}}{a+2\left( b+c \right)}+\frac{b-\sqrt{ca}}{b+2\left( c+a \right)}+\frac{c-\sqrt{ab}}{c+2\left( a+b \right)}\ge 0.
inequalitiesinequalities proposed
Kush do e zgjidhe?

Source: Olimpiada Korce Albania

12/27/2017
Find x, y, z ∈Z\in Z\\x2+y2+z2=2n(x+y+z)x^2+y^2+z^2=2^n(x+y+z)\\n∈Nn\in N
number theory
Continuous functions whose sums of functional powers have distinct monotonies

Source: Romania National Olympiad 2014, Grade XI, Problem 1

3/3/2019
Find all continuous functions f:R⟶R f:\mathbb{R}\longrightarrow\mathbb{R} that satisfy: (i)id+f \text{(i)}\text{id}+f is nondecreasing (ii) \text{(ii)} There is a natural number m m such that id+f+f2⋯+fm \text{id}+f+f^2\cdots +f^m is nonincreasing.
Here, id \text{id} represents the identity function, and ^ denotes functional power.
functionalgebra
Injectivity of aA, Aa, for A ring

Source: Romania National Olympiad 2014, Grade XII, Problem 1

3/3/2019
For a ring A, A, and an element a a of it, define sa,da:A⟶A,sa(x)=ax,da=xa. s_a,d_a:A\longrightarrow A, s_a(x)=ax,d_a=xa.
a) Prove that if A A is finite, then sa s_a is injective if and only if da d_a is injective. b) Give example of a ring which has an element b b for which sb s_b is injective and db d_b is not, or, conversely, sb s_b is not injective, but db d_b is.
functionRing Theory