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9 tháng 1 2020

Because: \(\hept{\begin{cases}a-b=1\\b-c=2\end{cases}}\Rightarrow\left(a-b\right)+\left(b-c\right)=1+2\)

That means: \(a-c=1+2=3\)

We have: \(a^2+b^2+c^2-ab-bc-ca=\frac{1}{2}\left[\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2\right]\)

\(=\frac{1}{2}\left[1+2^2+3^2\right]=7\)

So...

P/s: Is that true?

10 tháng 1 2020

thanks ban nhiu

31 tháng 10 2017

Trần Phun ơi giúp em mình với

NV
3 tháng 8 2021

\(\dfrac{\sqrt{ab+2c^2}}{\sqrt{1+ab-c^2}}=\dfrac{\sqrt{ab+2c^2}}{\sqrt{a^2+b^2+ab}}=\dfrac{ab+2c^2}{\sqrt{\left(a^2+b^2+ab\right)\left(ab+2c^2\right)}}\ge\dfrac{2\left(ab+2c^2\right)}{a^2+b^2+2ab+2c^2}\)

\(\ge\dfrac{2\left(ab+2c^2\right)}{a^2+b^2+a^2+b^2+2c^2}=\dfrac{ab+2c^2}{a^2+b^2+c^2}=ab+2c^2\)

Tương tự và cộng lại:

\(VT\ge ab+bc+ca+2\left(a^2+b^2+c^2\right)=2+ab+bc+ca\)

10 tháng 10 2021

1, Áp dụng BĐT cosi cho a,b,c>0

\(ab+bc\ge2\sqrt{ab^2c}=2b\sqrt{ac}\\ bc+ca\ge2\sqrt{abc^2}=2c\sqrt{ab}\\ ca+ab\ge2\sqrt{a^2bc}=2a\sqrt{bc}\)

Cộng VTV 3 BĐT trên:

\(\Leftrightarrow2\left(ab+bc+ac\right)\ge2\left(b\sqrt{ac}+a\sqrt{bc}+c\sqrt{ab}\right)\\ \Leftrightarrow ab+bc+ca\ge a\sqrt{bc}+b\sqrt{ac}+c\sqrt{ab}\)

10 tháng 10 2021

\(2,\)

Ta có

 \(\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2\ge0\\ \Leftrightarrow2a^2+2b^2+2c^2-2ab-2ac-2bc\ge0\\ \Leftrightarrow a^2+b^2+c^2-ab-ac-bc\ge0\\ \Leftrightarrow a^2+b^2+c^2\ge ab+bc+ca\)

Áp dụng BĐT cm ở câu 1

Suy ra đpcm

 

19 tháng 5 2018

\(\sum\dfrac{a}{b^2+bc+c^2}\ge\dfrac{\left(a+b+c\right)^2}{ab^2+abc+ac^2+bc^2+abc+ba^2+ca^2+abc+cb^2}=\dfrac{\left(a+b+c\right)^2}{\left(a+b+c\right)\left(ab+bc+ac\right)}=\dfrac{a+b+c}{ab+bc+ac}\)

25 tháng 5 2018

Đúng rầu đấy

câu 1: A rectangle has a length of 60cm and a width of 30cm. It is cut into 2 indentical squares, 2 identical rectangles and a shaded small square. Find the area of the shaded square. Find the area of the shaded square. câu 2.The number of ordered pairs (x; y) where x, y ∈ N* such that x2y2 - 2(x + y) is perfect square is .......... câu 3.Let ABCD be the square with the side length 56cm. If E and F lie on CD, C respectively such that CF = 14cm and EAF = 45o then CE = ........cm. câu...
Đọc tiếp

câu 1: A rectangle has a length of 60cm and a width of 30cm. It is cut into 2 indentical squares, 2 identical rectangles and a shaded small square. Find the area of the shaded square.
Find the area of the shaded square.
Đề thi violympic toán tiếng anh lớp 8 vòng 10
câu 2.The number of ordered pairs (x; y) where x, y ∈ N* such that x2y2 - 2(x + y) is perfect square is ..........

câu 3.Let ABCD be the square with the side length 56cm. If E and F lie on CD, C respectively such that CF = 14cm and EAF = 45o then CE = ........cm.

câu 4.

Given P(x) = (x2 - 1/2 x - 1/2)1008
If P(x) = a2016x2016 + a2015x2015 + ..... + a1x + a0
then the value of the sum a0 + a2 + a4 + .... + a2014 is ...........

Write your answer by decimal in simplest form câu 5.Let ABC be an isoceles triangle (AB = AC) and its area is 501cm2. BD is the internal bisector of the angle ABC (D ∈ AC), E is a point on the opposite ray of CA such that CE = CB. I is a point on BC such that CI = 1/2 BI. The line EI meets AB at K, BD meets KC at H. Find the area of the triangle AHC. câu 6. all roots of the polynomial P(x) = x2 + 5x - 1 are also roots of the polynomial Q(x) = x3 + ax2 + bx + c then the value of a + b + 6c is ............ câu 7.Suppose that the polynomial f(x) = x5 - x4 - 4x3 + 2x2 + 4x + 1 has 5 solutions x1; x2; x3; x4; x5. The other polynomial k(x) = x2 - 4.
Find the value of P = k(x1) x k(x2) x k(x3) x k(x4) x k(x5) câu 8.The smallest value of Đề thi violympic toán tiếng anh lớp 8 vòng 10 is ............. câu 9.Let ABCD be a trapezoid with bases AB, CD and O be the intersection of AC and BD. If the areas of triangle OAB, triangle OCD are 16cm2, 40cm2respectively and M is the midpoint of BD, then the area of the triangle AMD is .........cm2. câu 10.Bottle A contains 15% syrup. Bottle B contains 40% syrup. When these 2 bottles of syrup are mixed, the syrup content is 30% and the total volume is 600ml. How much syrup is in the bottle A at first? câu 11.Let ABCD.A'B'C'D' be a cube with AC' = √3cm. Find the total surface area of this cube. câu 12.Let ABC be a triangle with AB = 3cm, AC = 7cm. The internal bisector of the angle BAC intersects BC at D. The line passing through D and parallel to AC cuts AB at E. Find the measure of DE. câu 12.Given two numbers x, y such that (4y2 - 12y + 25)(4x2 + 6x + 4) = 28
The ratio of y to x is ........ câu 13.Mr.Joseph drives car from A to B at a constant speed. If the speed of the car is increased by 20%, it takes him one hour less than the usual time. If he drives at the constant speed for the first 100km before increasing the speed by 30%, it also takes him one hour less than the usual. The distance of AB is ..........km. câu 14.A triangle ABC has  = 120o and the bisector AD (D ∈ BC). If AB = 40cm, AD = 30cm, then AC = ..... cm. câu 15.As shown in the figure, the length of BE is .............
Đề thi violympic toán tiếng anh lớp 8 vòng 10 câu 16.

Let f(x) the polynomial given by f(x) = (1 + 2x + 3x2 + 4x3 + 5x4 + 84x5)
Suppose that f(x) = ao + a1x + a2x2 + ..... + ..... + a50x50.
The value of T = a1 + a2 + .... + a50 is .........

  • a. 9910 - 1
  • b. 9910
  • c. 10010
  • d. 10010 - 1
  • câu 17.Find the area of the trapezoid ABCD, BC // AD, AB = CD = 5cm, BC = 10cm, AD = 16cm.
    The area of the trapezoid ABCD is .........cm2.
  • câu 18.Find the least possible value of A = 4x2 - 3x + 1/4x + 2015, where x varies in the set of positive real numbers. The least possible value of A is
  • câu 19.
0
NV
12 tháng 5 2021

Đề bài có nhầm lẫn gì ko nhỉ?

\(T=\dfrac{ab}{a^2+b^2+ab}+\dfrac{bc}{b^2+c^2+2bc}+\dfrac{ca}{c^2+a^2+ca}\le\dfrac{ab}{2ab+ab}+\dfrac{bc}{2bc+bc}+\dfrac{ca}{2ca+ca}=1\)

13 tháng 5 2021

kh bt nx