\(\dfrac{1}{4^2}+\dfrac{1}{5^2}+\dfrac{1}{6^2}+...+\dfrac{1}{100^2}< \dfrac{1}{3.4}+\dfrac{1}{4.5}+\dfrac{1}{5.6}+...+\dfrac{1}{99.100}\)

\(\dfrac{1}{3.4}+\dfrac{1}{4.5}+\dfrac{1}{5.6}+...+\dfrac{1}{99.100}=\)

\(=\dfrac{4-3}{3.4}+\dfrac{5-4}{4.5}+\dfrac{6-5}{5.6}+...++\dfrac{100-99}{99.100}=\)

\(=\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{6}+...+\dfrac{1}{99}-\dfrac{1}{100}=\dfrac{1}{3}-\dfrac{1}{100}< \dfrac{1}{3}\)

\(\Rightarrow\dfrac{1}{4^2}+\dfrac{1}{5^2}+\dfrac{1}{6^2}+...+\dfrac{1}{100^2}< \dfrac{1}{3}\)

Đặt số chia là a và thương là b ta có

\(\dfrac{200-13}{a}=b\Rightarrow b=\dfrac{187}{a}\Rightarrow187⋮a\)

\(\Rightarrow a=\left\{1;11;17\right\}\Rightarrow b=\left\{187;17;11\right\}\)

A = \([20;31;42;53;64;75;86;97]\)

Hiệu vận tốc của 2 xe là:

     12-11=1 (km/giờ)

Hai xe gặp nhau sau thời gian là:

     7:1=7 (giờ)

Hai xe gặp nhau lúc:

     7 giờ + 7 giờ = 14 giờ

Chỗ gặp nhau cách A  là:

     12.7=84 (km)

          Đáp số: 2 xe gặp nhau lúc 14 giờ và chỗ gặp nhau cách A 84 km

\(\dfrac{5}{13}\cdot\dfrac{-3}{10}\cdot\dfrac{-3}{5}\)

\(=\dfrac{5}{5}\cdot\dfrac{-3}{13}\cdot\dfrac{-3}{10}\)

\(=\dfrac{3}{13}\cdot\dfrac{3}{10}=\dfrac{9}{130}\)

5/3 . (-3/10).(-3/5) 

=5/3 . 3/10 . 3/5

=45/150

=3/10

\(\dfrac{3}{4}\cdot\dfrac{1}{13}-\dfrac{3}{4}\cdot\dfrac{14}{13}\)

\(=\dfrac{3}{4}\left(\dfrac{1}{13}-\dfrac{14}{13}\right)\)

\(=\dfrac{3}{4}\cdot\dfrac{-13}{13}=-\dfrac{3}{4}\)

3/4×1/13-3/4×14/13

=3/4×(1/13-14/13)

=3/4×(-13/13)

=3/4×(-1)

=(-3/4)

\(\dfrac{10}{-13}:\dfrac{-4}{13}\cdot\dfrac{11}{-10}\)

\(=\dfrac{10}{13}\cdot\dfrac{13}{4}\cdot\dfrac{11}{-10}\)

\(=\dfrac{10}{4}\cdot\dfrac{11}{-10}=-\dfrac{11}{4}\)

\(\dfrac{10}{-13}:\dfrac{-4}{13}.\dfrac{11}{-10}\)

\(=\dfrac{10}{-13}.\dfrac{13}{4}.\dfrac{11}{-10}\)

\(=\dfrac{1430}{520}\)

\(=\dfrac{11}{4}\)

\(\dfrac{x}{15}=\dfrac{-6}{5}\)

\(\Rightarrow x=\dfrac{-6}{5}\cdot15\)

\(\Rightarrow x=-6\cdot3\)

\(\Rightarrow x=-18\)

\(\dfrac{x}{15}=\dfrac{-6}{5}\\ \dfrac{x}{15}=\dfrac{-18}{15}\)

⇒\(x=-18\)

Vậy \(x=-18\)

a) \(\dfrac{-2}{5}< \dfrac{x}{15}< \dfrac{1}{6}\)

\(\Rightarrow\dfrac{-2\cdot6}{5\cdot6}< \dfrac{2\cdot x}{15\cdot2}< \dfrac{5\cdot1}{6\cdot5}\)

\(\Rightarrow-12< 2x< 5\)

\(\Rightarrow-6< x< \dfrac{5}{2}\)

Mà: x nguyên

\(\Rightarrow x\in\left\{-5;-4;-3;-2;-1;0;1;2\right\}\)

b) \(\dfrac{1}{2}-\dfrac{1}{3}-\dfrac{5}{12}< 2x< \dfrac{-12}{31}+\dfrac{-136}{-31}\)

\(\Rightarrow\dfrac{1}{6}-\dfrac{5}{12}< 2x< \dfrac{-12}{31}+\dfrac{136}{31}\)

\(\Rightarrow\dfrac{2}{12}-\dfrac{5}{12}< 2x< \dfrac{124}{31}\)

\(\Rightarrow-\dfrac{3}{12}< 2x< 4\)

\(\Rightarrow-\dfrac{1}{4}< x< 2\)

Mà x nguyên

\(\Rightarrow x\in\left\{0;1\right\}\)

a)\(\dfrac{-2}{5}< \dfrac{x}{15}< \dfrac{1}{6}\)

⇒\(\dfrac{-12}{30}< \dfrac{2.x}{30}< \dfrac{5}{30}\)

⇒\(-12< 2.x< 5\)

⇒\(2.x\)ϵ{\(-10;-8;-6;-4;-2;0;2;4\)}

⇒\(x\left\{-5;-4;-3;-2;-1;0;1;2\right\}\)