MathDB

3

Part of 2016 District Olympiad

Problems(6)

find angle (kind of easy)

Source: Romanian District Olympiad 2016, Grade VII, Problem 3

10/4/2018
Let be a triangle ABC ABC with ∠BAC=90∘. \angle BAC = 90^{\circ } . On the perpendicular of BC BC through B, B, consider D D such that AD=BC. AD=BC. Find ∠BAD. \angle BAD.
geometryPure geometry
diophantine with integer parameter: x^3\pm 24x+k=0

Source: Romanian District Olympiad 2016, Grade VIII, Problem 3

10/4/2018
a) Prove that, for any integer k, k, the equation x3−24x+k=0 x^3-24x+k=0 has at most an integer solution.
b) Show that the equation x3+24x−2016=0 x^3+24x-2016=0 has exactly one integer solution.
algebrapolynomialequationdepressed cubiccubic equationcubic function
LHS < 1 implies LHS’ >1

Source: Romanian District Olympiad 2016, Grade IX, Problem 3

10/4/2018
Let be nonnegative real numbers a,b,c, a,b,c, holding the inequality: ∑cycab+c+1≤1. \sum_{\text{cyc}} \frac{a}{b+c+1} \le 1. Prove that ∑cyc1b+c+1≥1. \sum_{\text{cyc}} \frac{1}{b+c+1} \ge 1.
inequalities
Casey variable modulus expression

Source: Romanian District Olympiad, Grade X, Problem 3

10/5/2018
Let α,β \alpha ,\beta be real numbers. Find the greatest value of the expression ∣αx+βy∣+∣αx−βy∣ |\alpha x +\beta y| +|\alpha x-\beta y| in each of the following cases:
a) x,y∈R x,y\in \mathbb{R} and ∣x∣,∣y∣≤1 |x|,|y|\le 1 b) x,y∈C x,y\in \mathbb{C} and ∣x∣,∣y∣≤1 |x|,|y|\le 1
complex numbersalgebra
functional equation: f(x+1/n)<f(x)+1/n

Source: Romanian District Olympiad 2016, Grade XI, Problem 3

10/5/2018
Find the continuous functions f:R⟶R f:\mathbb{R}\longrightarrow\mathbb{R} having the following property: f\left( x+\frac{1}{n}\right) \le f(x) +\frac{1}{n}, \forall n\in\mathbb{Z}^* , \forall x\in\mathbb{R} .
functionalgebrafunctional equationcontinuityreal analysis
Primes and automorphisms

Source: Romanian District Olympiad 2016, Grade XII, Problem 3

10/5/2018
Let be a group G G of order 1+p, 1+p, where p p is and odd prime. Show that if p p divides the number of automorphisms of G, G, then p≡3(mod4). p\equiv 3\pmod 4.
modular arithmeticgroup theorysuperior algebra