\(\dfrac{9}{2}\left(\dfrac{2}{12}-\dfrac{6}{12}\right)\le x\le\dfrac{2}{3}\left(\dfrac{4}{12}-\dfrac{6}{12}-\dfrac{9}{12}\right)\)
\(\Leftrightarrow\dfrac{9}{2}.\dfrac{-4}{12}\le x\le\dfrac{2}{3}.\dfrac{-11}{12}\)\(\Leftrightarrow\dfrac{-3}{2}\le x\le\dfrac{-11}{18}\)
\(\Rightarrow x=-1\)
ĐỀ 1:
1. Tính:
a. \(\dfrac{7}{23}\left[\left(\dfrac{-8}{6}\right)-\dfrac{45}{18}\right]\)
b. \(\dfrac{1}{5}:\dfrac{1}{10}-\dfrac{1}{3}\left(\dfrac{6}{5}-\dfrac{9}{4}\right)\)
c. \(\dfrac{3}{5}.\left(\dfrac{-8}{3}\right)-\dfrac{3}{5}:\left(-6\right)\)
d. \(\dfrac{1}{2}\left(\dfrac{4}{3}+\dfrac{2}{5}\right)-\dfrac{3}{4}\left(\dfrac{8}{9}+\dfrac{16}{3}\right)\)
e. \(\dfrac{6}{7}:\left(\dfrac{3}{26}-\dfrac{3}{13}\right)+\dfrac{6}{7}\left(\dfrac{1}{10}-\dfrac{8}{5}\right)\)
2. Tìm x, biết:
a. \(1\dfrac{2}{5}x+\dfrac{3}{7}=\dfrac{4}{5}\)
b. |x - 1,5| = 2
c. \(\dfrac{4}{5}-\left|x-\dfrac{1}{6}\right|=\dfrac{2}{3}\)
d. 3x . 2x = 216
e. \(\dfrac{1}{2}\left(x-\dfrac{1}{3}\right)+\dfrac{-1}{2}=\dfrac{3}{4}\)
* Lm nhanh nha
Ths you <3
ĐỀ 2:
1. Tính:
A. \(\left(-\dfrac{2}{3}\right)^2+\left(-\dfrac{7}{8}\right)+\left(-\dfrac{11}{12}\right)\)
B. \(\left(\dfrac{-1}{3}\right)^2:\dfrac{1}{6}-2.\left(\dfrac{-1}{2}\right)^3\)\
C. \(\dfrac{-1}{5}-\left(\dfrac{1}{2}+\dfrac{3}{4}\right)^2:\dfrac{5}{8}\)
D. \(\left|\dfrac{-3}{2}+1,2\right|+1\dfrac{2}{3}:6\)
2. Tìm x, biết:
a. \(\dfrac{2}{3}x-\dfrac{1}{3}x=\dfrac{5}{12}\)
b. \(\left(x-\dfrac{12}{7}\right):1\dfrac{1}{5}=\dfrac{4}{7}\)
c. \(\dfrac{2}{5}+\left|x+1\right|=\dfrac{3}{4}\)
3. Tìm x, y biết:
a. \(\dfrac{x}{18}=\dfrac{y}{15}\)và x - y = -30
b. 7x = 9x và 10x - 8x = 68
c. \(\left(x-\dfrac{1}{2}\right)^{50}+\left(y+\dfrac{1}{3}\right)^{40}=0\)
* Lm nhanh nha
Tìm số nguyên x biết:
a. \(-4\dfrac{3}{5}.2\dfrac{4}{23}\)<x<\(-2\dfrac{3}{5}:1\dfrac{6}{15}\) b. \(-4\dfrac{1}{3}.\left(\dfrac{1}{2}-\dfrac{1}{6}\right)< x< -\dfrac{2}{3}\left(\dfrac{1}{3}-\dfrac{1}{2}-\dfrac{3}{4}\right)\)
Phương trình : \(\dfrac{1}{\left(x-1\right)\left(x-2\right)}+\dfrac{1}{\left(x-2\right)\left(x-3\right)}=\dfrac{1}{\left(x-3\right)\left(x-4\right)}+\dfrac{1}{\left(x-1\right)\left(x-4\right)}\)
Bài 1: Tính
a) \(\dfrac{1}{5}:\dfrac{1}{10}-\dfrac{1}{3}\left(\dfrac{6}{5}-\dfrac{9}{4}\right)\)
b) \(\dfrac{1}{2}\left(\dfrac{4}{3}+\dfrac{2}{5}\right)-\dfrac{3}{4}\left(\dfrac{8}{9}+\dfrac{16}{3}\right)\)
c) \(\dfrac{6}{7}:\left(\dfrac{3}{26}-\dfrac{3}{13}\right)+\dfrac{6}{7}\left(\dfrac{1}{10}-\dfrac{8}{5}\right)\)
Rút gọn:
\(A=\left[\left(\dfrac{3}{1+x}-\dfrac{x}{x^2+x+1}\right):\dfrac{2x^2+3x}{x+1}+\dfrac{3}{x+1}\right]\cdot\dfrac{x^2+x}{1+3x}\)
\(B=\left[\dfrac{a}{2a-6}-\dfrac{a^2}{a^2-9}+\dfrac{a}{2a-9}\cdot\left(\dfrac{3}{a}+\dfrac{1}{3-a}\right)\right]:\dfrac{a^2-5a-6}{18-2a^2}\)
B = 1+ \(\dfrac{1}{2}\left(1+2\right)+\dfrac{1}{3}\left(1+2+3\right)+\dfrac{1}{4}\left(1+2+3+4\right)+.......+\dfrac{1}{x}\left(1+2+3+4+....+x\right)\)
Tìm số nguyên dương x để B = 115
\(\dfrac{3}{8}\cdot27\dfrac{1}{5}-51\dfrac{1}{5}\cdot\dfrac{3}{8}+19\)
\(\dfrac{35\dfrac{1}{6}}{\left(\dfrac{-4}{5}\right)}-\dfrac{46\dfrac{1}{6}}{\left(\dfrac{-4}{5}\right)}\)
\(\dfrac{\left(\dfrac{-3}{4}+\dfrac{2}{5}\right)}{\dfrac{3}{7}+\dfrac{\left(\dfrac{3}{5}+\dfrac{-1}{4}\right)}{\dfrac{3}{7}}}\)
Đáp án đề thi vòng 1:
Bài 1:
a, \(A=\dfrac{50-\dfrac{4}{13}+\dfrac{2}{15}-\dfrac{2}{17}}{100-\dfrac{8}{13}+\dfrac{4}{15}-\dfrac{4}{17}}=\dfrac{50-\dfrac{4}{13}+\dfrac{2}{15}-\dfrac{2}{17}}{2\left(50-\dfrac{4}{13}+\dfrac{2}{15}-\dfrac{2}{17}\right)}=\dfrac{1}{2}\)
Vậy \(A=\dfrac{1}{2}\)
b, \(B=\dfrac{1}{19}+\dfrac{9}{19.29}+\dfrac{9}{29.39}+...+\dfrac{9}{1999.2009}\)
\(=\dfrac{9}{9.19}+\dfrac{9}{19.29}+\dfrac{9}{29.39}+...+\dfrac{9}{1999.2009}\)
\(=\dfrac{9}{10}\left(\dfrac{10}{9.19}+\dfrac{10}{19.29}+\dfrac{10}{29.39}+...+\dfrac{10}{1999.2009}\right)\)
\(=\dfrac{9}{10}\left(\dfrac{1}{9}-\dfrac{1}{19}+\dfrac{1}{19}-\dfrac{1}{29}+\dfrac{1}{29}-\dfrac{1}{39}+...+\dfrac{1}{1999}-\dfrac{1}{2009}\right)\)
\(=\dfrac{9}{10}\left(\dfrac{1}{9}-\dfrac{1}{2009}\right)\)
\(=\dfrac{200}{2009}\)
Vậy \(B=\dfrac{200}{2009}\)
Bài 2:
a, Giải:
Ta có: \(\left(\dfrac{b}{3c}\right)^3=\dfrac{a}{b}.\dfrac{b}{3c}.\dfrac{c}{9a}=\dfrac{1}{27}\Rightarrow\left(\dfrac{b}{3c}\right)^3=\left(\dfrac{1}{3}\right)^3\)
\(\Rightarrow\dfrac{b}{3c}=\dfrac{1}{3}\Rightarrow b=c\left(đpcm\right)\)
b, Ta có: \(\dfrac{1}{1.3}+\dfrac{1}{2.4}+\dfrac{1}{3.5}+\dfrac{1}{4.6}+...+\dfrac{1}{2013.2015}+\dfrac{1}{2014.2016}\)
\(=\dfrac{1}{2}\left(\dfrac{2}{1.3}+\dfrac{2}{2.4}+\dfrac{2}{3.5}+\dfrac{2}{4.6}+...+\dfrac{2}{2013.2015}+\dfrac{2}{2014.2016}\right)\)
\(=\dfrac{1}{2}\left[\left(\dfrac{2}{1.3}+\dfrac{2}{3.5}+...+\dfrac{2}{2013.2015}\right)+\left(\dfrac{2}{2.4}+\dfrac{2}{4.6}+...+\dfrac{2}{2014.2016}\right)\right]\)
\(=\dfrac{1}{2}\left[\left(\dfrac{1}{1}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+...+\dfrac{1}{2013}-\dfrac{1}{2015}\right)+\left(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+...+\dfrac{1}{2014}-\dfrac{1}{2016}\right)\right]\)
\(=\dfrac{1}{2}\left[\left(1-\dfrac{1}{2015}\right)+\left(\dfrac{1}{2}-\dfrac{1}{2016}\right)\right]\)
\(=\dfrac{1}{2}\left(\dfrac{3}{2}-\dfrac{1}{2015}-\dfrac{1}{2016}\right)=\dfrac{3}{4}-\dfrac{1}{2.2015}-\dfrac{1}{2.2016}< \dfrac{3}{4}\)
\(\Rightarrowđpcm\)
Bài 3:
a, \(VP=\left(x+y\right)\left(x-y\right)=x^2-xy+xy-y^2=x^2-y^2=VT\)
\(\Rightarrowđpcm\)
b, Giải:
a, b, c là độ dài các cạnh của một tam giác nên \(a+b>c,a+c>b,b+c>a\) ( bất đẳng thức tam giác )
\(\Rightarrow a+b-c>0,a-b+c>0,-a+b+c>0\) (*)
Ta có: \(\left\{{}\begin{matrix}a^2-\left(b-c\right)^2\le a^2\\b^2-\left(c-a\right)^2\le b^2\\c^2-\left(a-b\right)^2\le c^2\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}\left(a+b-c\right)\left(a-b+c\right)\le a^2\\\left(b+c-a\right)\left(b-c+a\right)\le b^2\\\left(c+a-b\right)\left(c-a+b\right)\le c^2\end{matrix}\right.\)
Kết hợp (*) ta có: \(\left[\left(a+b-c\right)\left(a-b+c\right)\left(-a+b+c\right)\right]^2\le\left(abc\right)^2\)
\(\Rightarrow\left(a+b-c\right)\left(a-b+c\right)\left(-a+b+c\right)\le abc\left(đpcm\right)\)
Vậy \(\left(a+b-c\right)\left(a-b+c\right)\left(-a+b+c\right)\le abc\)
Bài 4:
Giải:
Vẽ \(CD\perp BI\) tại D, CD cắt AB tại E
\(\Delta BCE\) cân tại B do BD vừa là đường cao, vừa là đường phân giác
\(\Rightarrow BD\) cũng là đường trung tuyến của \(\Delta BCE\)
\(\Rightarrow BE=BC,CE=2CD\)
Mặt khác: \(\widehat{BIC}=180^o-\left(\widehat{IBC}+\widehat{ICB}\right)\)
\(=180^o-\left(\dfrac{\widehat{ABC}}{2}+\dfrac{\widehat{ACB}}{2}\right)=135^o\)
\(\Rightarrow\widehat{DIC}=45^o\Rightarrow\Delta DIC\) vuông cân tại D
Do đó \(CI^2=DI^2+CD^2=2CD^2\)
Ta có: \(AE=BE-AB=BC-AB\)
\(\Delta ACE\) vuông tại A \(\Rightarrow CE^2=AE^2+AC^2\)
\(\Rightarrow4CD^2=\left(BC-AB\right)^2+AC^2\)
\(\Rightarrow2CI^2=\left(BC-AB\right)^2+AC^2\)
\(\Rightarrow CI^2=\dfrac{\left(BC-AB\right)^2+AC^2}{2}\left(đpcm\right)\)
Vậy \(CI^2=\dfrac{\left(BC-AB\right)^2+AC^2}{2}\)
Bài 5:
a, Áp dụng bất đẳng thức \(\left|a\right|+\left|b\right|\ge\left|a+b\right|\) ta có:
\(\left|x-2013\right|+\left|x-2016\right|=\left|x-2013\right|+\left|2016-x\right|\ge x-2013+2016-x=3\)
Kết hợp với giả thiết, ta có:
\(\left|x-2014\right|+\left|y-2015\right|\le0\)
Điều này chỉ xảy ra khi:
\(\left\{{}\begin{matrix}\left|x-2014\right|=0\\\left|y-2015\right|=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x=2014\\y=2015\end{matrix}\right.\)
Thay vào \(\left|x-2013\right|+\left|x-2014\right|+\left|y-2015\right|+\left|x-2016\right|=3\), ta thấy thỏa mãn
Vậy \(x=2014,y=2015\)
b, Giải:
Giả sử không có hai số nào trong 2013 số tự nhiên \(a_1,a_2,...,a_{2013}\) bằng nhau
Do đó, ta có: \(\dfrac{1}{a_1}+\dfrac{1}{a_2}+...+\dfrac{1}{a_{2013}}\le1+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2013}< 1+\dfrac{1}{2}+\dfrac{1}{2}+...+\dfrac{1}{2}=1+1006=1007\)
Mâu thuẫn với giả thiết
Vậy ít nhất hai trong 2013 số tự nhiên đã cho bằng nhau.
