Given the square ABCD. Draw M, N, P, Q on the sides AB, BC, CD, DA respectively. Let E, F, G be the midpoints of MQ, MP, NP respectively. What figure is quadrilateral MNPQ when A, E, F, G, B are collinear ?

Find the smallest positive integer n such that n(n + 1)(n + 2)\(⋮247\)

Given an isosceles trapezoid ABCD (BC // AD). Let M, N be the midpoints of BC, AD respectively. Draw point P on the opposite ray of AB. PN cuts BD at Q. Prove that MN is the bisector ray of \(\widehat{PMQ}\)

Given the rhombus ABCD with \(\widehat{B}=40^0\). Let E be the midpoint of BC. Draw \(AF\perp DE\). Find \(\widehat{DFC}\)

Calculate :

\(\dfrac{1}{3+1}+\dfrac{2}{3^2+1}+\dfrac{4}{3^4+1}+\dfrac{8}{3^8+1}+...+\dfrac{2^{2006}}{3^{2^{2006}}+1}\)

a/ Factorise : x⁵ + x - 1

b/ Write 10000000099 as a product of 2 integer which are greater than 1

a) Prove that :

\(\left\{{}\begin{matrix}ax+by=m\\cx+dy=n\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x=\dfrac{bn-dm}{bc-ad}\\y=\dfrac{cm-an}{bc-ad}\end{matrix}\right.\)

b) Prove that :

\(\left\{{}\begin{matrix}ax+by+cz=m\\dx+ey+fz=n\\gx+hy+iz=p\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}y=\dfrac{\left(fa-dc\right)\left(pa-gm\right)-\left(ia-gc\right)\left(na-dm\right)}{\left(fa-dc\right)\left(ha-gb\right)-\left(ea-db\right)\left(ia-gc\right)}\\z=\dfrac{\left(ha-gb\right)\left(na-dm\right)-\left(ea-db\right)\left(pa-gm\right)}{\left(fa-dc\right)\left(ha-gb\right)-\left(ea-db\right)\left(ia-gc\right)}\end{matrix}\right.\)

Prove that :

\(ax^2+bx+c=a\left(x+\dfrac{b-\sqrt{b^2-4ac}}{2a}\right)\left(x+\dfrac{b+\sqrt{b^2-4ac}}{2a}\right)\)

\(\left(a\ne0;b^2-4ac\ge0\right)\)

Apply the formula above to factorise the following polynomials :

\(6x^2-5x-6;3x^3+5x^2-12x;42-17x^2-15x^4\)

Given a regular hexagon ABCDEF. Let G, I be the midpoint of AF, a point on CF such that \(IC=\dfrac{1}{3}IF\). Which kind of triangle is \(\Delta IGE\) ? Why ?

Given \(\Delta ABC\). Let D, E, F be the midpoints of BC, CA and AB respectively. A line passing through A cuts DE, DF at M, N respectively. BN cuts CM at P. Find \(\widehat{FPE}\) 

I wrote the natural numbers from 1 to n. However, I forgot a number. So, the average of the numbers is \(\dfrac{599}{17}\). Find n and the number I forgot.

Find the sum of the roots of x³ + 9x - 5 and x³ - 15x² + 84x - 165, given that each polynomial has only 1 root.

Find \(n\in Z^+\) : \(\dfrac{7n-12}{2^n}+\dfrac{2n-14}{3^n}+\dfrac{24n}{6^n}=1\)

Given \(\left|a-b\right|=\left|b-c\right|=\left|c-d\right|=\left|d-e\right|=\left|e-a\right|\)

Prove that a = b = c = d = e

\(\Delta ABC\) has \(\widehat{A}=70^0\). The perpendicular bisector of BC cuts the bisector of the exterior angle at vertex A at T. Find \(\widehat{TBC}\)

Given 10 numbers such that each number equal the sum of the squares of 9 remaining numbers. Find 10 numbers

\(\Delta ABC\) isosceles at A has AB = AC = 8 cm ; the median BD = 6 cm. Find BC.

Given \(\left|x\right|-3=\left|y\right|+4=10-\left|z\right|\left(x,y,z\in R\right)\)

Find the maximum value of y(x + z)

There are only three species in Magic Wood: 12 snakes, 23 mice and 31 cats. Whenever a snake eats a cat, it becomes a rat, but when the cat eats the mouse it becomes a snake. Moreover, when the snake eats the mouse, it becomes a cat. Find the maximum number of animals in Magic Woods when no one eats another.

In a box, there is 1 ball numbered 1 ; 2 balls are numbered 2, ..., 100
balls numbered 100. I take the ball out of the box without looking. How many balls do I take at least to ensure that there are 10 balls numbered the same number ?