MathDB

A,P,Q are collinear iff AC=AB\sqrt2 (2013 Romania District VII P3)

Source:

5/20/2020

On the sides

(AB)

and

(AC)

of the triangle

ABC

are considered the points

M

and

N

respectively so that

\angle ABC =\angle ANM

. Point

D

is symmetric of point

A

with respect to

B

, and

P

and

Q

are the midpoints of the segments

[MN]

and

[CD]

, respectively. Prove that the points

A, P

and

Q

are collinear if and only if

AC = AB \sqrt {2}

geometrycollinearequal anglesmidpoint

Inequality

Source: Romania District Olympiad 2013,grade IX(problem 3)

3/10/2013

Let

n\in {{\mathbb{N}}^{*}}

and

{{a}_{1}},{{a}_{2}},...,{{a}_{n}}\in \mathbb{R}

so

{{a}_{1}}+{{a}_{2}}+...+{{a}_{k}}\le k,\left( \forall \right)k\in \left\{ 1,2,...,n \right\}.

Prove that

\frac{{{a}_{1}}}{1}+\frac{{{a}_{2}}}{2}+...+\frac{{{a}_{n}}}{n}\le \frac{1}{1}+\frac{1}{2}+...+\frac{1}{n}

inequalitiesinequalities proposed

distance of perpendicular lines, regular hexagonal prism

Source: 2013 Romania District VIII P3

5/19/2020

Let be the regular hexagonal prism

ABCDEFA'B C'D'E'F'

with the base edge of

12

and the height of

12 \sqrt{3}

. We denote by

N

the middle of the edge

CC'

.
a) Prove that the lines

BF'

and

ND

are perpendicular
b) Calculate the distance between the lines

BF'

and

ND

.

prism3D geometrygeometryperpendicular

injective is surjective

Source: Romania District Olympiad 2013,grade X(problem 3)

3/14/2013

Take the function

f:\mathbb{R}\to \mathbb{R}

,

f\left( x \right)=ax,x\in \mathbb{Q},f\left( x \right)=bx,x\in \mathbb{R}\backslash \mathbb{Q}

, where

a

and

b

are two real numbers different from 0. Prove that

f

is injective if and only if

f

is surjective.

functionalgebra proposedalgebra

matrix

Source: Romania District Olympiad 2013,grade XI(problem 3)

3/14/2013

Let

A

be an non-invertible of order

n

,

n>1

, with the elements in the set of complex numbers, with all the elements having the module equal with 1
a)Prove that, for

n=3

, two rows or two columns of the

A

matrix are proportional b)Does the conclusion from the previous exercise remains true for

n=4

?

linear algebramatrixabstract algebracomplex numbersalgebra proposedalgebra

Integral

Source: Romania District Olympiad 2013,grade XII (Problem 3)

3/11/2013

Problem 3. Let

f:\left[ 0,\frac{\pi }{2} \right]\to \left[ 0,\infty \right)

an increasing function .Prove that: (a)

\int_{0}^{\frac{\pi }{2}}{\left( f\left( x \right)-f\left( \frac{\pi }{4} \right) \right)}\left( \sin x-\cos x \right)dx\ge 0.

(b) Exist

a\in \left[ \frac{\pi }{4},\frac{\pi }{2} \right]

such that

\int_{0}^{a}{f\left( x \right)\sin x\ dx=}\int_{0}^{a}{f\left( x \right)\cos x\ dx}.

calculusintegrationfunctiontrigonometrycalculus computations

3

Problems(6)