MathDB

4

Part of 2015 District Olympiad

Problems(8)

Romanian District Olympiad 2015, Grade V, Problem 4

Source:

9/25/2018
a) Show that the three last digits of 10382 1038^2 are equal with 4. 4.
b) Show that there are infinitely many perfect squares whose last three digits are equal with 4. 4.
c) Prove that there is no perfect square whose last four digits are equal to 4. 4.
modular arithmeticnumber theoryarithmetic
Romanian District Olympiad 2015, Grade VI, Problem 4

Source:

9/25/2018
Determine all pairs of natural numbers, the components of which have the same number of digits and the double of their product is equal with the number formed by concatenating them.
Wordybase 10 arithmeticarithmetic
Romanian District Olympiad 2015, Grade VII, Problem 4

Source:

9/25/2018
At the exterior of the square ABCD ABCD it is constructed the isosceles triangle ABE ABE with ∠ABE=120∘.M \angle ABE=120^{\circ} . M is the intersection of the bisector line of the angle ∠EAB \angle EAB with its perpendicular that passes through B;N B; N is the intersection of the AB AB with its perpendicular that passe through M;P M; P is the intersection of CN CN with MB. MB. If G G is the center of gravity of the triangle ABE, ABE, prove that PG PG and AE AE are parallel.
geometrycenter of gravity
condition for a parallelepiped to be a cube

Source: Romanian District Olympiad 2015, Grade VIII, Problem 4

9/25/2018
Consider the rectangular parallelepiped ABCDA′B′C′D′ ABCDA'B'C'D' and the point O O to be the intersection of AB′ AB' and A′B. A'B. On the edge BC, BC, pick a point N N such that the plane formed by the triangle B′AN B'AN has to be parallel to the line AC′, AC', and perpendicular to DO′. DO'. Prove, then, that this parallelepiped is a cube.
Pure geometry3D geometrygeometry
find all functions

Source: Romanian District Olympiad 2015, Grade IX, Problem 4

9/25/2018
Find the functions f:N⟶N f:\mathbb{N}\longrightarrow\mathbb{N} that satisfy the following relation: \gcd\left( x,f(y)\right)\cdot\text{lcm}\left(f(x), y\right) = \gcd (x,y)\cdot\text{lcm}\left( f(x), f(y)\right) , \forall x,y\in\mathbb{N} .
functionalgebra
cauchy´s functional equation

Source: Romanian District Olympiad 2015, Grade X, Problem 4

9/25/2018
Let f:(0,∞)⟶(0,∞) f: (0,\infty)\longrightarrow (0,\infty) a non-constant function having the property that f\left( x^y\right) = \left( f(x)\right)^{f(y)}, \forall x,y>0. Show that f(xy)=f(x)f(y) f(xy)=f(x)f(y) and f(x+y)=f(x)+f(y), f(x+y)=f(x)+f(y), for all x,y>0. x,y>0.
functionalgebrafunctional equation
sequence of integer parts of sequence

Source: Romanian District Olympiad 2015, Grade XI, Problem 4

9/26/2018
Let (xn)n≥1 \left( x_n\right)_{n\ge 1} be a sequence of real numbers of the interval [1,∞). [1,\infty) . Suppose that the sequence ([xnk])n≥1 \left( \left[ x_n^k\right]\right)_{n\ge 1} is convergent for all natural numbers k. k. Prove that (xn)n≥1 \left( x_n\right)_{n\ge 1} is convergent.
Here, [β] [\beta ] means the greatest integer smaller than β. \beta .
Sequencesreal analysisepsilon-deltacontestromania
a sufficient condition for nilpotence

Source: Romanian District Olympiad 2015, Grade XII, Problem 4

9/26/2018
Let m m be a non-negative ineger, n≥2 n\ge 2 be a natural number, A A be a ring which has exactly n n elements, and an element a a of A A such that 1−ak 1-a^k is invertible, for all k∈{m+1,m+2,...,m+n−1}. k\in\{ m+1,m+2,...,m+n-1\} . Prove that a a is nilpotent.
Ring Theoryabstract algebranilpotencesuperior algebra