MathDB

<AQC =< PQB, <DRQ = 45^o, right isosceles (2004 Romania District VII P4)

Source:

5/24/2020

Consider the isosceles right triangle

ABC

(

AB = AC

) and the points

M, P \in [AB]

so that

AM = BP

. Let

D

be the midpoint of the side

BC

and

R, Q

the intersections of the perpendicular from

A

on

CM

with

CM

and

BC

respectively. Prove that
a)

\angle AQC = \angle PQB

b)

\angle DRQ = 45^o

geometryanglesequal anglesisoscelesright triangle

cosine of angles between planes , distance of point to plane, right trapezoid

Source: 2004 Romania District VIII P4

5/24/2020

In the right trapezoid

ABCD

with

AB \parallel CD, \angle B = 90^o

and

AB = 2DC

. At points

A

and

D

there is therefore a part of the plane

(ABC)

perpendicular to the plane of the trapezoid, on which the points

N

and

P

are taken, (

AP

and

PD

are perpendicular to the plane) such that

DN = a

and

AP = \frac{a}{2}

. Knowing that

M

is the midpoint of the side

BC

and the triangle

MNP

is equilateral, determine:
a) the cosine of the angle between the planes

MNP

and

ABC

. b) the distance from

D

to the plane

MNP

trigonometry3D geometrygeometrypalnesanglestrapezoidright angle

14 specific points on a ’chocolate’ tablet

Source: Romanian District Olympiad, Grade IX, Problem 4

10/7/2018

Divide a

2\times 4

rectangle into

8

unit squares to obtain a set of

15

vertices denoted by

\mathcal{M} .

Find the points

A\in\mathcal{M}

that have the property that the set

\mathcal{M}\setminus \{ A\}

can form

7

pairs

\left( A_1,B_1\right) ,\left( A_2,B_2\right) ,\ldots ,\left( A_7,B_7\right)\in\mathcal{M}\times\mathcal{M}

such that

\overrightarrow{A_1B_1} +\overrightarrow{A_2B_2} +\cdots +\overrightarrow{A_7B_7} =\overrightarrow{O} .

geometryrectanglevectorial geometryanalytic geometry

complex numbers, algebra

Source:

1/10/2020

If

x,y \in (0, \frac{\pi}{2})

such as

(cosx+isiny)^n=cos(nx)+isin(ny)

for two consecutive positive integers, then the relation is true for all positive integers.

complex numbersalgebra

Romania District Olympiad 2004 - Grade XI

Source:

4/10/2011

Let

A=(a_{ij})\in \mathcal{M}_p(\mathbb{C})

such that

a_{12}=a_{23}=\ldots=a_{p-1,p}=1

and

a_{ij}=0

for any other entry.
a)Prove that

A^{p-1}\neq O_p

and

A^p=O_p

.
b)If

X\in \mathcal{M}_{p}(\mathbb{C})

and

AX=XA

, prove that there exist

a_1,a_2,\ldots,a_p\in \mathbb{C}

such that:

X=\left( \begin{array}{ccccc} a_1 & a_2 & a_3 & \ldots & a_p \\ 0 & a_1 & a_2 & \ldots & a_{p-1} \\ 0 & 0 & a_1 & \ldots & a_{p-2} \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & a_1 \end{array} \right)

c)If there exist

B,C\in \mathcal{M}_p(\mathbb{C})

such that

(I_p+A)^n=B^n+C^n,\ (\forall)n\in \mathbb{N}^*

, prove that

B=O_p

or

C=O_p

.

linear algebramatrixcalculusderivativealgebrapolynomiallinear algebra unsolved

integral of f is equal to two integrals of f

Source: Romanian District Olympiad 2004, Grade XII, Problem 4

10/7/2018

Let

a,b\in (0,1)

and a continuous function

f:[0,1]\longrightarrow\mathbb{R}

with the property that \int_0^x f(t)dt=\int_0^{ax} f(t)dt +\int_0^{bx} f(t)dt, \forall x\in [0,1] .
a) Show that if

a+b<1,

then

f=0.

b) Show that if

a+b=1,

then

f

is constant.

Integralreal analysiscalculusintegration

4

Problems(6)