problem stringlengths 1 13.6k | solution stringlengths 0 18.5k ⌀ | answer stringlengths 0 575 ⌀ | problem_type stringclasses 8
values | question_type stringclasses 4
values | problem_is_valid stringclasses 1
value | solution_is_valid stringclasses 1
value | source stringclasses 8
values | synthetic bool 1
class | __index_level_0__ int64 0 742k |
|---|---|---|---|---|---|---|---|---|---|
2.043. $\frac{\left(m^{2}-\frac{1}{n^{2}}\right)^{m} \cdot\left(n+\frac{1}{m}\right)^{n-m}}{\left(n^{2}-\frac{1}{m^{2}}\right)^{n} \cdot\left(m-\frac{1}{n}\right)^{m-n}}$. | Solution.
Domain of definition: $\left\{\begin{array}{l}m \neq 0, \\ n \neq 0 .\end{array}\right.$
$$
\frac{\left(m^{2}-\frac{1}{n^{2}}\right)^{m} \cdot\left(n+\frac{1}{m}\right)^{n-m}}{\left(n^{2}-\frac{1}{m^{2}}\right)^{n} \cdot\left(m-\frac{1}{n}\right)^{m-n}}=\frac{\left(\frac{m^{2} n^{2}-1}{n^{2}}\right)^{m} \cd... | (\frac{}{n})^{+n} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,410 |
2.044. $\left(\frac{\sqrt{x-a}}{\sqrt{x+a}+\sqrt{x-a}}+\frac{x-a}{\sqrt{x^{2}-a^{2}}-x+a}\right): \sqrt{\frac{x^{2}}{a^{2}}-1} ; x>a>0$.
2.044. $\left(\frac{\sqrt{x-a}}{\sqrt{x+a}+\sqrt{x-a}}+\frac{x-a}{\sqrt{x^{2}-a^{2}}-x+a}\right): \sqrt{\frac{x^{2}}{a^{2}}-1} ; x>a>0$. | Solution.
$$
\begin{aligned}
& \left(\frac{\sqrt{x-a}}{\sqrt{x+a}+\sqrt{x-a}}+\frac{x-a}{\sqrt{x^{2}-a^{2}}-x+a}\right): \sqrt{\frac{x^{2}}{a^{2}}-1}= \\
& =\left(\frac{\sqrt{x-a}}{\sqrt{x+a}+\sqrt{x-a}}+\frac{(\sqrt{x-a})^{2}}{\sqrt{x-a}(\sqrt{x+a}-\sqrt{x-a})}\right): \sqrt{\frac{x^{2}-a^{2}}{a^{2}}}= \\
& =\left(\f... | 1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,411 |
2.045. $\left(\frac{\sqrt[4]{x^{3}}-\sqrt[4]{x}}{1-\sqrt{x}}+\frac{1+\sqrt{x}}{\sqrt[4]{x}}\right)^{2} \cdot\left(1+\frac{2}{\sqrt{x}}+\frac{1}{x}\right)^{-1 / 2}$. | ## Solution.
Domain of definition: $0<x \neq 1$.
$$
\begin{aligned}
& \left(\frac{\sqrt[4]{x^{3}}-\sqrt[4]{x}}{1-\sqrt{x}}+\frac{1+\sqrt{x}}{\sqrt[4]{x}}\right)^{2} \cdot\left(1+\frac{2}{\sqrt{x}}+\frac{1}{x}\right)^{-1 / 2}=\left(\frac{\sqrt[4]{x}\left(\sqrt[4]{x^{2}}-1\right)}{1-\sqrt{x}}+\frac{1+\sqrt{x}}{\sqrt[4]... | \frac{1-\sqrt{x}}{1-x} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,412 |
2.046. $\frac{\sqrt{1-x^{2}}-1}{x} \cdot\left(\frac{1-x}{\sqrt{1-x^{2}}+x-1}+\frac{\sqrt{1+x}}{\sqrt{1+x}-\sqrt{1-x}}\right)$ | ## Solution.
Domain of definition: $\left\{\begin{array}{l}x \neq 0, \\ -1 \leq x<1 .\end{array}\right.$
$$
\begin{aligned}
& \frac{\sqrt{1-x^{2}}-1}{x} \cdot\left(\frac{1-x}{\sqrt{1-x^{2}}+x-1}+\frac{\sqrt{1+x}}{\sqrt{1+x}-\sqrt{1-x}}\right)=\frac{\sqrt{1-x^{2}}-1}{x} \times \\
& \times\left(\frac{(\sqrt{1-x})^{2}}{... | -1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,413 |
2.047. $\frac{\frac{a-b}{2a-b}-\frac{a^{2}+b^{2}+a}{2a^{2}+ab-b^{2}}}{\left(4b^{4}+4ab^{2}+a^{2}\right):\left(2b^{2}+a\right)} \cdot\left(b^{2}+b+ab+a\right)$. | ## Solution.
Domain of definition: $\left\{\begin{array}{l}a \neq \pm \frac{b}{2}, \\ 2 a^{2}+a b-b^{2} \neq 0, \\ a \neq 0, \\ b \neq 0\end{array},\left\{\begin{array}{l}a \neq \pm \frac{b}{2}, \\ a \neq-b, \\ a \neq 0, \\ b \neq 0 .\end{array}\right.\right.$
$$
\begin{aligned}
& \frac{\frac{a-b}{2 a-b}-\frac{a^{2}+... | \frac{b+1}{b-2a} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,414 |
2.050. $\frac{2\left(x^{4}+4 x^{2}-12\right)+x^{4}+11 x^{2}+30}{x^{2}+6}$. | Solution.
$$
\begin{aligned}
& \frac{2\left(x^{4}+4 x^{2}-12\right)+x^{4}+11 x^{2}+30}{x^{2}+6}=\frac{2\left(x^{2}+6\right)\left(x^{2}-2\right)+\left(x^{2}+6\right)\left(x^{2}+5\right)}{x^{2}+6}= \\
& =\frac{\left(x^{2}+6\right)\left(2\left(x^{2}-2\right)+x^{2}+5\right)}{x^{2}+6}=2 x^{2}-4+x^{2}+5=3 x^{2}+1=1+3 x^{2}
... | 1+3x^{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,415 |
2.051. $\frac{\left(a^{2}-b^{2}\right)\left(a^{2}+\sqrt[3]{b^{2}}+a \sqrt[3]{b}\right)}{a \sqrt[3]{b}+a \sqrt{a}-b \sqrt[3]{b}-\sqrt{a b^{2}}}: \frac{a^{3}-b}{a \sqrt[3]{b}-\sqrt[6]{a^{3} b^{2}}-\sqrt[3]{b^{2}}+a \sqrt{a}} ;$
$$
a=4.91 ; b=0.09
$$ | Solution.
$$
\begin{aligned}
& \frac{\left(a^{2}-b^{2}\right)\left(a^{2}+\sqrt[3]{b^{2}}+a \sqrt[3]{b}\right)}{a \sqrt[3]{b}+a \sqrt{a}-b \sqrt[3]{b}-\sqrt{a b^{2}}}: \frac{a^{3}-b}{a \sqrt[3]{b}-\sqrt[6]{a^{3} b^{2}}-\sqrt[3]{b^{2}}+a \sqrt{a}}= \\
& =\frac{(a-b)(a+b)\left(a^{2}+a \sqrt[3]{b}+\sqrt[3]{b^{2}}\right)}{... | 5 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,416 |
2.053. $\left(\left(1-p^{2}\right)^{-1 / 2}-\left(1+p^{2}\right)^{-1 / 2}\right)^{2}+2\left(1-p^{4}\right)^{-1 / 2}$. | Solution.
Domain of definition: $-1<p<1$.
$$
\begin{aligned}
& \left(\left(1-p^{2}\right)^{-1 / 2}-\left(1+p^{2}\right)^{-1 / 2}\right)^{2}+2\left(1-p^{4}\right)^{-1 / 2}=\left(\frac{1}{\sqrt{1-p^{2}}}-\frac{1}{\sqrt{1+p^{2}}}\right)^{2}+ \\
& +\frac{2}{\sqrt{1-p^{4}}}=\left(\frac{\sqrt{1+p^{2}}-\sqrt{1-p^{2}}}{\sqrt... | \frac{2}{1-p^{4}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,418 |
2.054. $\frac{3 a^{2}+2 a x-x^{2}}{(3 x+a)(a+x)}-2+10 \cdot \frac{a x-3 x^{2}}{a^{2}-9 x^{2}}$. | Solution.
Domain of definition: $\left\{\begin{array}{l}x \neq \pm \frac{a}{3}, \\ x \neq-a .\end{array}\right.$
$$
\begin{aligned}
& \frac{3 a^{2}+2 a x-x^{2}}{(3 x+a)(a+x)}-2+10 \cdot \frac{a x-3 x^{2}}{a^{2}-9 x^{2}}=\frac{-(x+a)(x-3 a)}{(3 x+a)(a+x)}-2+ \\
& +10 \cdot \frac{x(a-3 x)}{(a-3 x)(a+3 x)}=\frac{-x+3 a}... | 1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,419 |
2.055. $\left(\frac{\sqrt[3]{x+y}}{\sqrt[3]{x-y}}+\frac{\sqrt[3]{x-y}}{\sqrt[3]{x+y}}-2\right):\left(\frac{1}{\sqrt[3]{x-y}}-\frac{1}{\sqrt[3]{x+y}}\right)$. | Solution.
Domain of definition: $x \neq \pm y$.
$$
\begin{aligned}
& \left(\frac{\sqrt[3]{x+y}}{\sqrt[3]{x-y}}+\frac{\sqrt[3]{x-y}}{\sqrt[3]{x+y}}-2\right):\left(\frac{1}{\sqrt[3]{x-y}}-\frac{1}{\sqrt[3]{x+y}}\right)= \\
& =\frac{(\sqrt[3]{x+y})^{2}-2 \sqrt[3]{(x+y)(x-y)}+(\sqrt[3]{x-y})^{2}}{\sqrt[3]{x^{2}-y^{2}}}: ... | \sqrt[3]{x+y}-\sqrt[3]{x-y} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,420 |
2.057. $\left(\left(\frac{x}{y-x}\right)^{-2}-\frac{(x+y)^{2}-4 x y}{x^{2}-x y}\right)^{2} \cdot \frac{x^{4}}{x^{2} y^{2}-y^{4}}$.
2.057. $\left(\left(\frac{x}{y-x}\right)^{-2}-\frac{(x+y)^{2}-4xy}{x^{2}-xy}\right)^{2} \cdot \frac{x^{4}}{x^{2}y^{2}-y^{4}}$. | ## Solution.
$$
\begin{aligned}
& \text { D3: }\left\{\begin{array}{l}
x \neq \pm y, \\
x \neq 0, \\
y \neq 0 .
\end{array}\right. \\
& \left(\left(\frac{x}{y-x}\right)^{-2}-\frac{(x+y)^{2}-4 x y}{x^{2}-x y}\right)^{2} \cdot \frac{x^{4}}{x^{2} y^{2}-y^{4}}= \\
& =\left(\frac{(y-x)^{2}}{x^{2}}-\frac{x^{2}+2 x y+y^{2}-4... | \frac{x-y}{x+y} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,422 |
2.058. $\left(\left(\frac{1}{a}+\frac{1}{b+c}\right):\left(\frac{1}{a}-\frac{1}{b+c}\right)\right):\left(1+\frac{b^{2}+c^{2}-a^{2}}{2 b c}\right)$;
$$
a=1 \frac{33}{40} ; b=0.625 ; c=3.2
$$ | Solution.
$$
\begin{aligned}
& \left(\left(\frac{1}{a}+\frac{1}{b+c}\right):\left(\frac{1}{a}-\frac{1}{b+c}\right):\left(1+\frac{b^{2}+c^{2}-a^{2}}{2 b c}\right)=\right. \\
& =\left(\frac{a+b+c}{a(b+c)}: \frac{-a+b+c}{a(b+c)}\right): \frac{2 b c+b^{2}+c^{2}-a^{2}}{2 b c}= \\
& =\left(\frac{a+b+c}{a(b+c)} \cdot \frac{a... | 1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,423 |
2.059. $\left(\left(\frac{x^{2}}{y^{3}}+\frac{1}{x}\right):\left(\frac{x}{y^{2}}-\frac{1}{y}+\frac{1}{x}\right)\right): \frac{(x-y)^{2}+4 x y}{1+y / x}$. | ## Solution.
Domain of definition: $\left\{\begin{array}{l}x \neq 0, \\ y \neq 0, \\ x \neq -y .\end{array}\right.$
$\left(\left(\frac{x^{2}}{y^{3}}+\frac{1}{x}\right):\left(\frac{x}{y^{2}}-\frac{1}{y}+\frac{1}{x}\right)\right): \frac{(x-y)^{2}+4 x y}{1+y / x}=$
$=\left(\frac{x^{3}+y^{3}}{x y^{3}}: \frac{x^{2}-x y+y... | \frac{1}{xy} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,424 |
2.060. $\left(\frac{3}{2 x-y}-\frac{2}{2 x+y}-\frac{1}{2 x-5 y}\right): \frac{y^{2}}{4 x^{2}-y^{2}}$. | ## Solution.
Domain of definition: $\left\{\begin{array}{l}x \neq \pm \frac{y}{2}, \\ x \neq \frac{5 y}{2} .\end{array}\right.$
$$
\begin{aligned}
& \left(\frac{3}{2 x-y}-\frac{2}{2 x+y}-\frac{1}{2 x-5 y}\right): \frac{y^{2}}{4 x^{2}-y^{2}}=\left(\frac{3(2 x+y)-2(2 x-y)}{(2 x-y)(2 x+y)}-\frac{1}{2 x-5 y}\right) \\
& ... | \frac{24}{5y-2x} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,425 |
2.064. $\frac{2 b+a-\frac{4 a^{2}-b^{2}}{a}}{b^{3}+2 a b^{2}-3 a^{2} b} \cdot \frac{a^{3} b-2 a^{2} b^{2}+a b^{3}}{a^{2}-b^{2}}$. | ## Solution.
Domain of definition: $\left\{\begin{array}{l}b \neq 0, \\ b \neq-3 a, \\ b \neq \pm a .\end{array}\right.$
$$
\begin{aligned}
& \frac{2 b+a-\frac{4 a^{2}-b^{2}}{a}}{b^{3}+2 a b^{2}-3 a^{2} b} \cdot \frac{a^{3} b-2 a^{2} b^{2}+a b^{3}}{a^{2}-b^{2}}=\frac{\frac{2 a b-a^{2}-4 a^{2}+b^{2}}{a}}{b\left(b^{2}+... | \frac{-b}{+b} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,427 |
2.066. $\frac{\sqrt{x^{3}}+\sqrt{x y^{2}}-\sqrt{x^{2} y}-\sqrt{y^{3}}}{\sqrt[4]{y^{5}}+\sqrt[4]{x^{4} y}-\sqrt[4]{x y^{4}}-\sqrt[4]{x^{5}}}$.
2.066. $\frac{\sqrt{x^{3}}+\sqrt{x y^{2}}-\sqrt{x^{2} y}-\sqrt{y^{3}}}{\sqrt[4]{y^{5}}+\sqrt[4]{x^{4} y}-\sqrt[4]{x y^{4}}-\sqrt[4]{x^{5}}}$.
The translation is the same as the... | Solution.
Domain of definition: $\left\{\begin{array}{l}x \geq 0, \\ y \geq 0, \\ x \neq y .\end{array}\right.$
$$
\begin{aligned}
& \frac{\sqrt{x^{3}}+\sqrt{x y^{2}}-\sqrt{x^{2} y}-\sqrt{y^{3}}}{\sqrt[4]{y^{5}}+\sqrt[4]{x^{4} y}-\sqrt[4]{x y^{4}}-\sqrt[4]{x^{5}}}=\frac{\left(\sqrt{x^{3}}+\sqrt{x y^{2}}\right)-\left(... | -(\sqrt[4]{x}+\sqrt[4]{y}) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,428 |
2.068. $\frac{\left(\frac{1}{a}+\frac{1}{b}-\frac{2 c}{a b}\right)(a+b+2 c)}{\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{2}{a b}-\frac{4 c^{2}}{a^{2} b^{2}}} ; \quad a=7.4 ; b=\frac{5}{37}$. | Solution.
$$
\begin{aligned}
& \frac{\left(\frac{1}{a}+\frac{1}{b}-\frac{2 c}{a b}\right)(a+b+2 c)}{\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{2}{a b}-\frac{4 c^{2}}{a^{2} b^{2}}}=\frac{\frac{a+b-2 c}{a b} \cdot(a+b+2 c)}{\frac{a^{2}+2 a b+b^{2}-4 c^{2}}{a^{2} b^{2}}}= \\
& =\frac{\frac{(a+b-2 c)(a+b+2 c)}{a b}}{\frac{(a+b... | 1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,430 |
2.069. $\frac{a^{7 / 3}-2 a^{5 / 3} b^{2 / 3}+a b^{4 / 3}}{a^{5 / 3}-a^{4 / 3} b^{1 / 3}-a b^{2 / 3}+a^{2 / 3} b}: a^{1 / 3}$. | Solution.
od3: $\left\{\begin{array}{l}a \neq 0, \\ a^{5 / 3}-a^{4 / 3} b^{1 / 3}-a b^{2 / 3}+a^{2 / 3} b \neq 0 .\end{array}\right.$
$$
\begin{aligned}
& \frac{a^{7 / 3}-2 a^{5 / 3} b^{2 / 3}+a b^{4 / 3}}{a^{5 / 3}-a^{4 / 3} b^{1 / 3}-a b^{2 / 3}+a^{2 / 3} b}: a^{1 / 3}=\frac{a^{3 / 3}\left(a^{4 / 3}-2 a^{2 / 3} b^{... | ^{1/3}+b^{1/3} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,431 |
2.073. $\frac{\sqrt{5-2 \sqrt{6}}}{(\sqrt[4]{3}+\sqrt[4]{2})(\sqrt[4]{3}-\sqrt[4]{2})}$. | Solution.
$$
\begin{aligned}
& \frac{\sqrt{5-2 \sqrt{6}}}{(\sqrt[4]{3}+\sqrt[4]{2})(\sqrt[4]{3}-\sqrt[4]{2})}=\frac{\sqrt{3-2 \sqrt{3 \cdot 2}+2}}{(\sqrt[4]{3})^{2}-(\sqrt[4]{2})^{2}}= \\
& =\frac{\sqrt{(\sqrt{3})^{2}-2 \sqrt{3} \cdot \sqrt{2}+(\sqrt{2})^{2}}}{\sqrt{3}-\sqrt{2}}= \\
& =\frac{\sqrt{(\sqrt{3}-\sqrt{2})^... | 1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,433 |
2.074. $\frac{\left(a^{1 / m}-a^{1 / n}\right)^{2}+4 a^{(m+n) /(m n)}}{\left(a^{2 / m}-a^{2 / n}\right)\left(\sqrt[m]{a^{m+1}}+\sqrt[n]{a^{n+1}}\right)}$. | Solution.
Domain of definition: $\left\{\begin{array}{l}a>0, \text { if } m \text { and } n-\text { are even numbers, } \\ a \neq 0, \\ a \neq 1 .\end{array}\right.$
$$
\begin{aligned}
& \frac{\left(a^{1 / m}-a^{1 / n}\right)^{2}+4 a^{(m+n) /(m n)}}{\left(a^{2 / m}-a^{2 / n}\right)\left(\sqrt[m]{a^{m+1}}+\sqrt[n]{a^{... | \frac{1}{(\sqrt[]{}-\sqrt[n]{})} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,434 |
2.075. $\frac{\left(x^{2 / m}-9 x^{2 / n}\right)\left(\sqrt[m]{x^{1-m}}-3 \sqrt[n]{x^{1-n}}\right)}{\left(x^{1 / m}+3 x^{1 / n}\right)^{2}-12 x^{(m+n) /(m n)}}$. | ## Solution.
Domain of definition: $\left\{\begin{array}{l}\ddot{x} 0, \text { if } m \text { and } n \text { are even numbers, } \\ x \neq 0, \\ x \neq 3^{m n /(m-n)}\end{array}\right.$
$$
\begin{aligned}
& \frac{\left(x^{2 / m}-9 x^{2 / n}\right)\left(\sqrt[m]{x^{1-m}}-3 \sqrt[n]{x^{1-n}}\right)}{\left(x^{1 / m}+3 ... | \frac{x^{1/}+3x^{1/n}}{x} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,435 |
2.077. $\frac{a^{-1}-b^{-1}}{a^{-3}+b^{-3}}: \frac{a^{2} b^{2}}{(a+b)^{2}-3 a b} \cdot\left(\frac{a^{2}-b^{2}}{a b}\right)^{-1} ; \quad a=1-\sqrt{2} ; b=1+\sqrt{2}$. | Solution.
$$
\begin{aligned}
& \frac{a^{-1}-b^{-1}}{a^{-3}+b^{-3}}: \frac{a^{2} b^{2}}{(a+b)^{2}-3 a b} \cdot\left(\frac{a^{2}-b^{2}}{a b}\right)^{-1}=\frac{\frac{1}{a}-\frac{1}{b}}{\frac{1}{a^{3}}+\frac{1}{b^{3}}}: \frac{a^{2} b^{2}}{a^{2}+2 a b+b^{2}-3 a b} \times \\
& \times \frac{a b}{a^{2}-b^{2}}=\frac{\frac{b-a}... | \frac{1}{4} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,436 |
2.078. $\left(\frac{1}{t^{2}+3 t+2}+\frac{2 t}{t^{2}+4 t+3}+\frac{1}{t^{2}+5 t+6}\right)^{2} \cdot \frac{(t-3)^{2}+12 t}{2}$. | Solution.
Domain of definition: $\left\{\begin{array}{l}t \neq-3, \\ t \neq-2, \\ t \neq-1 .\end{array}\right.$
$$
\begin{aligned}
& \left(\frac{1}{t^{2}+3 t+2}+\frac{2 t}{t^{2}+4 t+3}+\frac{1}{t^{2}+5 t+6}\right)^{2} \cdot \frac{(t-3)^{2}+12 t}{2}= \\
& =\left(\frac{1}{(t+2)(t+1)}+\frac{2 t}{(t+3)(t+1)}+\frac{1}{(t+... | 2 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,437 |
2.079. $\left(\sqrt{\sqrt{m}-\sqrt{\frac{m^{2}-9}{m}}}+\sqrt{\sqrt{m}+\sqrt{\frac{m^{2}-9}{m}}}\right)^{2} \cdot \sqrt[4]{\frac{m^{2}}{4}}$. | ## Solution.
Domain of definition: $m \geq 3$.
$$
\begin{aligned}
& \left(\sqrt{\sqrt{m}-\sqrt{\frac{m^{2}-9}{m}}}+\sqrt{\sqrt{m}+\sqrt{\frac{m^{2}-9}{m}}}\right)^{2} \cdot \sqrt[4]{\frac{m^{2}}{4}}= \\
& =\left(\sqrt{\sqrt{m}-\sqrt{\frac{m^{2}-9}{m}}}+2 \sqrt{\left(\sqrt{m}-\sqrt{\frac{m^{2}-9}{m}}\right)}\left(\sqr... | \sqrt{2}(+3) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,438 |
2.080. $\frac{(a-b)^{2}+a b}{(a+b)^{2}-a b}: \frac{a^{5}+b^{5}+a^{2} b^{3}+a^{3} b^{2}}{\left(a^{3}+b^{3}+a^{2} b+a b^{2}\right)\left(a^{3}-b^{3}\right)}$. | ## Solution.
Domain of definition: $\left\{\begin{array}{l}a \neq b, \\ a \neq-b .\end{array}\right.$
$$
\begin{aligned}
& \frac{(a-b)^{2}+a b}{(a+b)^{2}-a b}: \frac{a^{5}+b^{5}+a^{2} b^{3}+a^{3} b^{2}}{\left(a^{3}+b^{3}+a^{2} b+a b^{2}\right)\left(a^{3}-b^{3}\right)}=\frac{a^{2}-2 a b+b^{2}+a b}{a^{2}+2 a b+b^{2}-a ... | -b | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,439 |
2.084. $\left(\frac{2-b}{b-1}+2 \cdot \frac{a-1}{a-2}\right):\left(b \cdot \frac{a-1}{b-1}+a \cdot \frac{2-b}{a-2}\right)$;
$a=\sqrt{2}+0.8 ; b=\sqrt{2}-0.2$. | Solution.
$$
\begin{aligned}
& \left(\frac{2-b}{b-1}+2 \cdot \frac{a-1}{a-2}\right):\left(b \cdot \frac{a-1}{b-1}+a \cdot \frac{2-b}{a-2}\right)=\frac{(2-b)(a-2)+2(a-1)(b-1)}{(b-1)(a-2)} \\
& \frac{b(a-1)(a-2)+a(2-b)(b-1)}{(b-1)(a-2)}=\frac{a b-2}{(b-1)(a-2)} \cdot \frac{(b-1)(a-2)}{a^{2} b-a b^{2}-2 a+2 b}= \\
& =\fr... | 1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,443 |
2.087. $\frac{\sqrt{3}\left(a-b^{2}\right)+\sqrt{3} b \sqrt[3]{8 b^{3}}}{\sqrt{2\left(a-b^{2}\right)^{2}+(2 b \sqrt{2 a})^{2}}} \cdot \frac{\sqrt{2 a}-\sqrt{2 c}}{\sqrt{\frac{3}{a}}-\sqrt{\frac{3}{c}}}$.
2.087. $\frac{\sqrt{3}\left(a-b^{2}\right)+\sqrt{3} b \sqrt[3]{8 b^{3}}}{\sqrt{2\left(a-b^{2}\right)^{2}+(2 b \sqrt... | ## Solution.
$$
\text { Domain of definition: }\left\{\begin{array}{l}
a>0 \\
c>0 \\
2\left(a-b^{2}\right)^{2}+(2 b \sqrt{2 a})^{2} \neq 0
\end{array}\right.
$$
$$
\begin{aligned}
& \frac{\sqrt{3}\left(a-b^{2}\right)+\sqrt{3} b \sqrt[3]{8 b^{3}}}{\sqrt{2\left(a-b^{2}\right)^{2}+(2 b \sqrt{2 a})^{2}}} \cdot \frac{\sqr... | -\sqrt{ac} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,444 |
2.093. $\left(\frac{\sqrt{3}+1}{1+\sqrt{3}+\sqrt{t}}+\frac{\sqrt{3}-1}{1-\sqrt{3}+\sqrt{t}}\right) \cdot\left(\sqrt{t}-\frac{2}{\sqrt{t}}+2\right)$. | ## Solution.
Domain of definition: $\left\{\begin{array}{l}t>0, \\ t \neq(\sqrt{3}-1)^{2} .\end{array}\right.$
$$
\begin{aligned}
& \left(\frac{\sqrt{3}+1}{1+\sqrt{3}+\sqrt{t}}+\frac{\sqrt{3}-1}{1-\sqrt{3}+\sqrt{t}}\right) \cdot\left(\sqrt{t}-\frac{2}{\sqrt{t}}+2\right)= \\
& =\frac{(\sqrt{3}+1)(1-\sqrt{3}+\sqrt{t})+... | 2\sqrt{3} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,447 |
2.101. $\left(\frac{1}{a+\sqrt{2}}-\frac{a^{2}+4}{a^{3}+2 \sqrt{2}}\right):\left(\frac{a}{2}-\frac{1}{\sqrt{2}}+\frac{1}{a}\right)^{-1}$. | ## Solution.
Domain of definition: $\left\{\begin{array}{l}a \neq 0, \\ a \neq-\sqrt{2} .\end{array}\right.$
$\left(\frac{1}{a+\sqrt{2}}-\frac{a^{2}+4}{a^{3}+2 \sqrt{2}}\right):\left(\frac{a}{2}-\frac{1}{\sqrt{2}}+\frac{1}{a}\right)^{-1}=\left(\frac{1}{a+\sqrt{2}}-\frac{a^{2}+4}{a^{3}+(\sqrt{2})^{3}}\right) /$
$$
\b... | -\frac{\sqrt{2}}{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,453 |
2.102. $\left(\frac{(a-1)^{-1}}{a^{-3}}-(1-a)^{-1}\right) \cdot \frac{1+a(a-2)}{a^{2}-a+1} \cdot \sqrt{\frac{1}{(a+1)^{2}}}$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
2.102. $\left(\frac{(a-1)^{-1}}{a^{-3}}-(1-a)^{-1}\... | ## Solution.
Domain of definition: $\left\{\begin{array}{l}a \neq 0, \\ a \neq \pm 1 .\end{array}\right.$
$$
\begin{aligned}
& \left(\frac{(a-1)^{-1}}{a^{-3}}-(1-a)^{-1}\right) \cdot \frac{1+a(a-2)}{a^{2}-a+1} \cdot \sqrt{\frac{1}{(a+1)^{2}}}=\left(\frac{\frac{1}{a-1}}{\frac{1}{a^{3}}}-\frac{1}{1-a}\right) \times \\
... | 1-for\in(-\infty,-1);\,-1for\in(-1,0)\cup(0,1)\cup(1,\infty) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,454 |
2.104. $\left(\frac{a}{b} \sqrt[3]{b-\frac{4 a^{6}}{b^{3}}}-a^{2} \sqrt[3]{\frac{b}{a^{6}}-\frac{4}{b^{3}}}+\frac{2}{a b} \sqrt[3]{a^{3} b^{4}-4 a^{9}}\right): \frac{\sqrt[3]{b^{2}-2 a^{3}}}{b^{2}}$. | Solution.
Domain of definition: $\left\{\begin{array}{l}a \neq 0, \\ b \neq 0 .\end{array}\right.$
$$
\begin{aligned}
& \left(\frac{a}{b} \sqrt[3]{b-\frac{4 a^{6}}{b^{3}}}-a^{2} \sqrt[3]{\frac{b}{a^{6}}-\frac{4}{b^{3}}}+\frac{2}{a b} \sqrt[3]{a^{3} b^{4}-4 a^{9}}\right): \frac{\sqrt[3]{b^{2}-2 a^{3}}}{b^{2}}= \\
& =\... | (+b)\sqrt[3]{b^{2}+2^{3}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,455 |
2.105. $\left(\frac{1+\sqrt{1-x}}{1-x+\sqrt{1-x}}+\frac{1-\sqrt{1+x}}{1+x-\sqrt{1+x}}\right)^{2} \cdot \frac{x^{2}-1}{2}-\sqrt{1-x^{2}}$. | Solution.
Domain of definition: $\left\{\begin{array}{l}-1<x<1, \\ x \neq 0 .\end{array}\right.$
$$
\begin{aligned}
& \left(\frac{1+\sqrt{1-x}}{1-x+\sqrt{1-x}}+\frac{1-\sqrt{1+x}}{1+x-\sqrt{1+x}}\right)^{2} \cdot \frac{x^{2}-1}{2}-\sqrt{1-x^{2}}= \\
& \left.\left.=\left(\frac{1+\sqrt{1-x}}{\sqrt{1-x}(\sqrt{1-x}+1}\ri... | -1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,456 |
2.106. $\frac{4 a^{2}-b^{2}}{a^{6}-8 b^{6}} \cdot \sqrt{a^{2}-2 b \sqrt{a^{2}-b^{2}}} \cdot \frac{4^{4}+2 a^{2} b^{2}+4 b^{4}}{4 a^{2}+4 a b+b^{2}} \cdot \sqrt{a^{2}+2 b \sqrt{a^{2}-b^{2}}}$; $a=4 / 3 ; b=0.25$. | Solution.
$$
\begin{aligned}
& \frac{4 a^{2}-b^{2}}{a^{6}-8 b^{6}} \cdot \sqrt{a^{2}-2 b \sqrt{a^{2}-b^{2}}} \cdot \frac{a^{4}+2 a^{2} b^{2}+4 b^{4}}{4 a^{2}+4 a b+b^{2}} \cdot \sqrt{a^{2}+2 b \sqrt{a^{2}-b^{2}}}= \\
& =\frac{(2 a-b)(2 a+b)}{\left(a^{2}\right)^{3}-\left(2 b^{2}\right)^{3}} \cdot \frac{a^{4}+2 a^{2} b^... | \frac{29}{35} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,457 |
2.107. $\frac{1+(a+x)^{-1}}{1-(a+x)^{-1}} \cdot\left(1-\frac{1-\left(a^{2}+x^{2}\right)}{2 a x}\right) ; \quad x=\frac{1}{a-1}$. | Solution.
Domain of definition: $\left\{\begin{array}{l}a \neq 1, \\ a \neq 0, \\ x \neq 0, \\ x \neq-a, \\ x \neq 1-a .\end{array}\right.$
$$
\begin{aligned}
& \frac{1+(a+x)^{-1}}{1-(a+x)^{-1}} \cdot\left(1-\frac{1-\left(a^{2}+x^{2}\right)}{2 a x}\right)=\frac{1+\frac{1}{a+x}}{1-\frac{1}{a+x}} \cdot \frac{2 a x-1+a^... | \frac{^{3}}{2(-1)} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,458 |
2.108. $\left(\frac{a}{b}+\frac{b}{a}+2\right) \cdot\left(\frac{a+b}{2 a}-\frac{b}{a+b}\right):\left(\left(a+2 b+\frac{b^{2}}{a}\right) \cdot\left(\frac{a}{a+b}+\frac{b}{a-b}\right)\right) ;$ $a=0.75 ; b=4 / 3$. | Solution.
$$
\begin{aligned}
& \left(\frac{a}{b}+\frac{b}{a}+2\right) \cdot\left(\frac{a+b}{2 a}-\frac{b}{a+b}\right):\left(\left(a+2 b+\frac{b^{2}}{a}\right) \cdot\left(\frac{a}{a+b}+\frac{b}{a-b}\right)\right)= \\
& =\frac{a^{2}+2 a b+b^{2}}{a b} \cdot \frac{a^{2}+2 a b+b^{2}-2 a b}{2 a(a+b)}:\left(\frac{a^{2}+2 a b... | -\frac{7}{24} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,459 |
2.109. $\left(-4 a^{3} \sqrt{\frac{\sqrt{a x}}{a^{2}}}\right)^{3}+\left(-10 a \sqrt{x} \cdot \sqrt{(a x)^{-1}}\right)^{2}+\left(-2\left(\sqrt[3]{a^{4} \sqrt{\frac{x}{a}}}\right)^{2}\right)^{3} ;$
$$
a=3 \frac{4}{7} ; x=0.28
$$ | ## Solution.
$$
\begin{aligned}
& \left(-4 a^{3} \sqrt{\frac{\sqrt{a x}}{a^{2}}}\right)^{3}+\left(-10 a \sqrt{x} \cdot \sqrt{(a x)^{-1}}\right)^{2}+\left(-2\left(\sqrt[3]{a \sqrt{\frac{x}{a}}}\right)^{2}\right)^{3}= \\
& =\frac{-64 a^{3} \sqrt{a x}}{a^{2}}+\frac{100 a^{2} x}{a x}-\frac{8 a^{2} \sqrt{x}}{\sqrt{a}}=-64 ... | 100 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,460 |
2.110. $\frac{\sqrt{c-d}}{c^{2} \sqrt{2 c}} \cdot\left(\sqrt{\frac{c-d}{c+d}}+\sqrt{\frac{c^{2}+c d}{c^{2}-c d}}\right) ; \quad c=2 ; d=1 / 4$. | Solution.
$$
\begin{aligned}
& \frac{\sqrt{c-d}}{c^{2} \sqrt{2 c}} \cdot\left(\sqrt{\frac{c-d}{c+d}}+\sqrt{\frac{c^{2}+c d}{c^{2}-c d}}\right)=\frac{\sqrt{c-d}}{c^{2} \sqrt{2 c}} \cdot\left(\frac{\sqrt{c-d}}{\sqrt{c+d}}+\sqrt{\frac{c(c+d)}{c(c-d)}}\right)= \\
& =\frac{\sqrt{c-d}}{c^{2} \sqrt{2 c}} \cdot\left(\frac{\sq... | \frac{1}{3} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,461 |
2.111. $\frac{\left(a b^{-1}+a^{-1} b+1\right)\left(a^{-1}-b^{-1}\right)^{2}}{a^{2} b^{-2}+a^{-2} b^{2}-\left(a b^{-1}+a^{-1} b\right)}$. | Solution.
Domain of definition: $\left\{\begin{array}{l}a \neq 0, \\ b \neq 0 .\end{array}\right.$
$$
\begin{aligned}
& \frac{\left(a b^{-1}+a^{-1} b+1\right)\left(a^{-1}-b^{-1}\right)^{2}}{a^{2} b^{-2}+a^{-2} b^{2}-\left(a b^{-1}+a^{-1} b\right)}=\frac{\left(\frac{a}{b}+\frac{b}{a}+1\right)\left(\frac{1}{a}-\frac{1}... | \frac{1}{} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,462 |
2.112. $\left(\sqrt[3]{\left(\frac{1}{2}\right)^{-3}-t^{3}}+\sqrt[3]{\frac{t^{5}+2 t^{4}+4 t^{3}}{4-4 t+t^{2}}}\right):\left(\frac{1}{\sqrt{2}-\sqrt{t}}+\frac{1}{\sqrt{2}+\sqrt{t}}\right)$. | Solution.
Domain of definition: $\left\{\begin{array}{l}t \geq 0, \\ t \neq 2 .\end{array}\right.$
$$
\begin{aligned}
& \left(\sqrt[3]{\left(\frac{1}{2}\right)^{-3}-t^{3}}+\sqrt[3]{\frac{t^{5}+2 t^{4}+4 t^{3}}{4-4 t+t^{2}}}\right):\left(\frac{1}{\sqrt{2}-\sqrt{t}}+\frac{1}{\sqrt{2}+\sqrt{t}}\right)= \\
& =\left(\sqrt... | \frac{\sqrt[3]{8-^{3}}}{\sqrt{2}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,463 |
2.114. $\left(\frac{9-4 a^{-2}}{3 a^{-1 / 2}+2 a^{-3 / 2}}-\frac{1+a^{-1}-6 a^{-2}}{a^{-1 / 2}+3 a^{-3 / 2}}\right)^{4}$. | ## Solution.
Domain of definition: $\left\{\begin{array}{l}a \neq 0, \\ a \neq-3, \\ a \neq-\frac{2}{3} .\end{array}\right.$
$$
\begin{aligned}
& \left(\frac{9-4 a^{-2}}{3 a^{-1 / 2}+2 a^{-3 / 2}}-\frac{1+a^{-1}-6 a^{-2}}{a^{-1 / 2}+3 a^{-3 / 2}}\right)^{4}=\left(\frac{9-\frac{4}{a^{2}}}{\frac{3}{a^{1 / 2}}+\frac{2}{... | 16^{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,465 |
2.118. $\left(\frac{2}{\sqrt{3}-1}+\frac{3}{\sqrt{3}-2}+\frac{15}{3-\sqrt{3}}\right) \cdot(\sqrt{3}+5)^{-1}$. | Solution.
$\left(\frac{2}{\sqrt{3}-1}+\frac{3}{\sqrt{3}-2}+\frac{15}{3-\sqrt{3}}\right) \cdot(\sqrt{3}+5)^{-1}=$
$$
\begin{aligned}
& =\left(\frac{2(\sqrt{3}+1)}{(\sqrt{3}-1)(\sqrt{3}+1)}+\frac{3(\sqrt{3}+2)}{(\sqrt{3}-2)(\sqrt{3}+2)}+\frac{15(3+\sqrt{3})}{(3-\sqrt{3})(3+\sqrt{3})}\right) \cdot \frac{1}{\sqrt{3}+5}= ... | \frac{1}{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,468 |
2.119. $\frac{\sqrt[4]{7 \sqrt[3]{54}+15 \sqrt[3]{128}}}{\sqrt[3]{4 \sqrt[4]{32}}+\sqrt[3]{9 \sqrt[4]{162}}}$.
2.119. $\frac{\sqrt[4]{7 \sqrt[3]{54}+15 \sqrt[3]{128}}}{\sqrt[3]{4 \sqrt[4]{32}}+\sqrt[3]{9 \sqrt[4]{162}}}$.
The translation is the same as the original text because it is a mathematical expression, which ... | Solution.
$$
\begin{aligned}
& \frac{\sqrt[4]{7 \sqrt[3]{54}+15 \sqrt[3]{128}}}{\sqrt[3]{4 \sqrt[4]{32}}+\sqrt[3]{9 \sqrt[4]{162}}}=\frac{\sqrt[4]{\sqrt[3]{27 \cdot 2}}+15 \sqrt[3]{64 \cdot 2}}{\sqrt[3]{4 \sqrt[4]{16 \cdot 2}}+\sqrt[3]{9 \sqrt[4]{81 \cdot 2}}}=\frac{\sqrt[4]{7 \cdot 3 \sqrt[3]{2}+15 \cdot 4 \sqrt[3]{2... | \frac{3}{5} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,469 |
2.120. $\frac{5 \sqrt[3]{4 \sqrt[3]{192}}+7 \sqrt[3]{18 \sqrt[3]{81}}}{\sqrt[3]{12 \sqrt[3]{24}+6 \sqrt[3]{375}}}$.
2.120. $\frac{5 \sqrt[3]{4 \sqrt[3]{192}}+7 \sqrt[3]{18 \sqrt[3]{81}}}{\sqrt[3]{12 \sqrt[3]{24}+6 \sqrt[3]{375}}}$.
The expression is already in a mathematical format and does not require translation. H... | Solution.
$$
\begin{aligned}
& \frac{5 \sqrt[3]{4 \sqrt[3]{192}}+7 \sqrt[3]{18 \sqrt[3]{81}}}{\sqrt[3]{12 \sqrt[3]{24}+6 \sqrt[3]{375}}}=\frac{5 \sqrt[3]{4 \sqrt[3]{64 \cdot 3}}+7 \sqrt[3]{18 \sqrt[3]{27 \cdot 3}}}{\sqrt[3]{12 \sqrt[3]{8 \cdot 3}+6 \sqrt[3]{125 \cdot 3}}}= \\
& =\frac{5 \sqrt[3]{4 \cdot 4 \sqrt[3]{3}}... | \frac{31}{3} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,470 |
2.121. $\sqrt[4]{32 \sqrt[3]{4}}+\sqrt[4]{64 \sqrt[3]{\frac{1}{2}}}-3 \sqrt[3]{2 \sqrt[4]{2}}$. | Solution.
$\sqrt[4]{32 \sqrt[3]{4}}+\sqrt[4]{64 \sqrt{\frac{1}{2}}}-3 \sqrt[3]{2 \sqrt[4]{2}}=\sqrt[4]{2^{5} \cdot 2^{2 / 3}}+\sqrt[4]{2^{6} \cdot 2^{-1 / 3}}-3 \sqrt[3]{2 \cdot 2^{1 / 4}}=$
$=2^{17 / 12}+2^{17 / 12}-3 \cdot 2^{5 / 12}=2 \cdot 2^{17 / 12}-3 \cdot 2^{5 / 12}=2^{5 / 12}(4-3)=2^{5 / 12}=$
$=\sqrt[12]{2... | \sqrt[12]{32} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,471 |
2.122. $5 \sqrt{48 \sqrt{\frac{2}{3}}}+\sqrt{32 \sqrt[3]{\frac{9}{4}}}-11 \sqrt[3]{12 \sqrt{8}}$. | Solution.
$5 \sqrt{48 \sqrt{\frac{2}{3}}}+\sqrt{32 \sqrt[3]{\frac{9}{4}}}-1 \sqrt[3]{12 \sqrt{8}}=5 \sqrt{16 \cdot 3 \sqrt[3]{\frac{2}{3}}}+\sqrt{16 \cdot 2 \sqrt[3]{\frac{9}{4}}}-$
$-11 \sqrt[3]{12 \cdot 2 \sqrt{2}}=5 \cdot 4 \sqrt{\sqrt[33]{\frac{2}{3}}}+4 \sqrt{2 \sqrt{\frac{9}{4}}}-11 \sqrt[3]{8 \cdot 3 \sqrt{2}}... | 2\sqrt[6]{18} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,472 |
2.123. $2 \sqrt{40 \sqrt{12}}+3 \sqrt{5 \sqrt{48}}-2 \sqrt[4]{75}-4 \sqrt{15 \sqrt{27}}$. | Solution.
$$
\begin{aligned}
& 2 \sqrt{40 \sqrt{12}}+3 \sqrt{5 \sqrt{48}}-2 \sqrt[4]{75}-4 \sqrt{15 \sqrt{27}}= \\
& =2 \sqrt{40 \sqrt{4 \cdot 3}}+3 \sqrt{5 \sqrt{16 \cdot 3}}-2 \sqrt[4]{25 \cdot 3}-4 \sqrt{15 \sqrt{9 \cdot 3}}= \\
& =2 \sqrt{40 \cdot 2 \sqrt{3}}+3 \sqrt{5 \cdot 4 \sqrt{3}}-2 \sqrt{\sqrt{25 \cdot 3}}-... | 0 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,473 |
2.126. $(4+\sqrt{15})(\sqrt{10}-\sqrt{6}) \cdot \sqrt{4-\sqrt{15}}=2$.
| ## Решение.
Возведем обе части равенства в квадрат. Тогда
$$
\begin{aligned}
& (4+\sqrt{15})^{2}(\sqrt{10}-\sqrt{6})^{2}(4-\sqrt{15})=4 \\
& (4+\sqrt{15})(4-\sqrt{15})(4+\sqrt{15})(10-2 \sqrt{60}+6)=4 \\
& \left(4^{2}-(\sqrt{15})^{2}\right)(4+\sqrt{15})(16-2 \sqrt{60})=4 \\
& (16-15)(4+\sqrt{15}) \cdot 2 \cdot(8-\sqr... | 2 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,475 |
2.127. $\sqrt{3-\sqrt{5}} \cdot(3+\sqrt{5}) \cdot(\sqrt{10}-\sqrt{2})=8$. | ## Solution.
Let's square both sides of the equation. Then
$$
(\sqrt{3-\sqrt{5}})^{2}(3+\sqrt{5})^{2}(\sqrt{2}(\sqrt{5}-1))^{2}=64
$$
$(3-\sqrt{5})(3+\sqrt{5})^{2} \cdot 2(\sqrt{5}-1)^{2}=64$,
$(3-\sqrt{5})(3+\sqrt{5})(3+\sqrt{5})(5-2 \sqrt{5}+1)=32$,
$\left(3^{2}-(\sqrt{5})^{2}\right)(3+\sqrt{5})(6-2 \sqrt{5})=32... | 32 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,476 |
2.128. $\frac{\sqrt[3]{\sqrt{3}+\sqrt{6}} \cdot \sqrt[6]{9-6 \sqrt{2}}-\sqrt[6]{18}}{\sqrt[6]{2}-1}=-\sqrt[3]{3}$. | ## Solution.
Let's transform the left side of the equation:
$$
\begin{aligned}
& \frac{\sqrt[3]{\sqrt{3}+\sqrt{6}} \cdot \sqrt[6]{9-6 \sqrt{2}}-\sqrt[6]{18}}{\sqrt[6]{2}-1}=\frac{\sqrt[3]{(\sqrt{3}+\sqrt{6})^{2}} \cdot \sqrt[6]{9-6 \sqrt{2}}-\sqrt[6]{18}}{\sqrt[6]{2}-1}= \\
&=\frac{\sqrt[6]{(3+2 \sqrt{18}+6)(9-6 \sqr... | -\sqrt[3]{3} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,477 |
2.129. $\frac{25 \cdot \sqrt[4]{2}+2 \sqrt{5}}{\sqrt{250}+5 \sqrt[4]{8}}-\sqrt{\frac{\sqrt{2}}{5}+\frac{5}{\sqrt{2}}+2}=-1$. | ## Solution.
Let's set
$$
X=\frac{25 \cdot \sqrt[4]{2}+2 \sqrt{5}}{\sqrt{250}+5 \sqrt[4]{8}}=\frac{\sqrt[4]{5^{8} \cdot 2}+\sqrt[4]{5^{2} \cdot 2^{4}}}{\sqrt[4]{5^{6} \cdot 2^{2}}+\sqrt[4]{5^{4} \cdot 2^{3}}}=\frac{\sqrt[4]{5^{2} \cdot 2}\left(\sqrt[4]{5^{6}}+\sqrt[4]{2^{3}}\right)}{\sqrt[4]{5^{2} \cdot 2} \cdot \sqr... | -1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,478 |
2.130. $\frac{\sqrt{\sqrt[4]{27}+\sqrt{\sqrt{3}-1}}-\sqrt{\sqrt[4]{27}-\sqrt{\sqrt{3}-1}}}{\sqrt{\sqrt[4]{27}-\sqrt{2 \sqrt{3}}+1}}=\sqrt{2}$.
| ## Решение.
Возведем обе части равенства в квадрат. Тогда
$$
\begin{aligned}
& \frac{2 \sqrt[4]{27}-2 \sqrt{(\sqrt[4]{27})^{2}-(\sqrt{\sqrt{3}-1})^{2}}}{\sqrt[4]{27}-\sqrt{2 \sqrt{3}+1}}=... | 1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,479 |
2.131. $\left(\frac{4}{3-\sqrt{5}}\right)^{2}-\left(\frac{6-5 \sqrt{6}}{5-\sqrt{6}}\right)^{2}=2 \sqrt{61+24 \sqrt{5}}$. | ## Solution.
\[
\begin{aligned}
& \frac{16}{9-6 \sqrt{5}+5}-\left(-\frac{\sqrt{6}(\sqrt{6}-5)}{\sqrt{6}-5}\right)^{2}=2 \sqrt{61+24 \sqrt{5}} \\
& \frac{16}{14-6 \sqrt{5}}-6=2 \sqrt{61+24 \sqrt{5}}, \quad \frac{8}{7-3 \sqrt{5}}-6=2 \sqrt{61+24 \sqrt{5}} \\
& \frac{4}{7-3 \sqrt{5}}-3=\sqrt{61+24 \sqrt{5}}, \quad \frac{... | 3\sqrt{5}+4=\sqrt{61+24\sqrt{5}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,480 |
2.132. $\frac{1}{\sqrt{7}-\sqrt{6}}=\frac{3}{\sqrt{6}-\sqrt{3}}+\frac{4}{\sqrt{7}+\sqrt{3}}$. | ## Solution.
By multiplying the numerator and denominator of each fraction by the expression conjugate to its denominator, we have
$$
\frac{\sqrt{7}+\sqrt{6}}{(\sqrt{7}-\sqrt{6})(\sqrt{7}+\sqrt{6})}=\frac{3(\sqrt{6}+\sqrt{3})}{(\sqrt{6}-\sqrt{3})(\sqrt{6}+\sqrt{3})}+\frac{4(\sqrt{7}-\sqrt{3})}{(\sqrt{7}+\sqrt{3})(\sq... | \sqrt{6}=\sqrt{6} | Algebra | proof | Yes | Yes | olympiads | false | 50,481 |
2.134. $\frac{\sqrt{2}-1}{\sqrt{2}+1}=\sqrt[3]{\frac{10-7 \sqrt{2}}{10+7 \sqrt{2}}}$.
2.134. $\frac{\sqrt{2}-1}{\sqrt{2}+1}=\sqrt[3]{\frac{10-7 \sqrt{2}}{10+7 \sqrt{2}}}$. | ## Solution.
By multiplying the numerator and denominator of each fraction by the expression conjugate to its denominator, we have
$$
\begin{aligned}
& \frac{(\sqrt{2}-1)(\sqrt{2}-1)}{(\sqrt{2}+1)(\sqrt{2}-1)}=\sqrt[3]{\frac{(10-7 \sqrt{2})(10-7 \sqrt{2})}{(10+7 \sqrt{2})(10-7 \sqrt{2})}}, \quad \frac{(\sqrt{2}-1)^{2... | proof | Algebra | proof | Yes | Yes | olympiads | false | 50,482 |
2.135. $\frac{x^{3}-a^{-2 / 3} \cdot b^{-1}\left(a^{2}+b^{2}\right) x+b^{1 / 2}}{b^{3 / 2} \cdot x^{2}} ; x=a^{2 / 3} b^{-1 / 2}$. | ## Solution.
Domain of definition: $\left\{\begin{array}{l}a \neq 0, \\ b \neq 0 .\end{array}\right.$
$$
\begin{aligned}
& \frac{\left(a^{2 / 3} b^{-1 / 2}\right)^{3}-a^{2 / 3} \cdot b^{-1}\left(a^{2}+b^{2}\right)^{2 / 3} b^{-1 / 2}+b^{1 / 2}}{b^{3 / 2} \cdot\left(a^{2 / 3} b^{-1 / 2}\right)^{2}}= \\
& =\frac{a^{2} b... | 0 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,483 |
2.136. $\frac{1-b}{\sqrt{b}} \cdot x^{2}-2 x+\sqrt{b} ; \quad x=\frac{\sqrt{b}}{1-\sqrt{b}}$.
2.136. $\frac{1-b}{\sqrt{b}} \cdot x^{2}-2 x+\sqrt{b} ; \quad x=\frac{\sqrt{b}}{1-\sqrt{b}}$. | ## Solution.
Domain of definition: $0<b \neq 1$.
$$
\begin{aligned}
& \frac{1-b}{\sqrt{b}} \cdot\left(\frac{\sqrt{b}}{1-\sqrt{b}}\right)^{2}-2 \cdot \frac{\sqrt{b}}{1-\sqrt{b}}+\sqrt{b}=\frac{(1-\sqrt{b})(1+\sqrt{b})}{\sqrt{b}} \cdot \frac{b}{(1-\sqrt{b})^{2}}-\frac{2 \sqrt{b}}{1-\sqrt{b}}+ \\
& +\sqrt{b}=\frac{(1+\s... | 0 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,484 |
2.137. $\left(\frac{x+2 b}{x-2 b}+\frac{x+2 a}{x-2 a}\right): \frac{x}{2} ; \quad x=\frac{4 a b}{a+b}$. | ## Solution.
Domain of definition: $a \neq-b \neq 0$.
$$
\begin{aligned}
& \left(\frac{\frac{4 a b}{a+b}+2 b}{\frac{4 a b}{a+b}-2 b}+\frac{\frac{4 a b}{a+b}+2 a}{\frac{4 a b}{a+b}-2 a}\right): \frac{4 a b}{2(a+b)}= \\
& =\left(\frac{4 a b+2 a b+b^{2}}{a+b}: \frac{4 a b-2 a b-2 b^{2}}{a+b}+\frac{4 a b+2 a^{2}+2 a b}{a... | \frac{+b}{} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,485 |
2.138. $(x+1)(x+2)(x+3)(x+4) \quad x=\frac{\sqrt{7}-5}{2}$. | Solution.
$$
\begin{aligned}
& \left(\frac{\sqrt{7}-5}{2}+1\right) \cdot\left(\frac{\sqrt{7}-5}{2}+2\right) \cdot\left(\frac{\sqrt{7}-5}{2}+3\right) \cdot\left(\frac{\sqrt{7}-5}{2}+4\right)= \\
& =\left(\frac{\sqrt{7}-5}{2}+1\right) \cdot\left(\frac{\sqrt{7}-5}{2}+4\right) \cdot\left(\frac{\sqrt{7}-5}{2}+2\right) \cdo... | -\frac{3}{4} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,486 |
2.139. $\frac{(z-1)(z+2)(z-3)(z+4)}{23} ; \quad z=\frac{\sqrt{3}-1}{2}$. | Solution.
$$
\begin{aligned}
& \frac{\left(\frac{\sqrt{3}-1}{2}-1\right) \cdot\left(\frac{\sqrt{3}-1}{2}+2\right) \cdot\left(\frac{\sqrt{3}-1}{2}-3\right) \cdot\left(\frac{\sqrt{3}-1}{2}+4\right)}{23}= \\
& =\frac{\left(\left(\frac{\sqrt{3}-1}{2}\right)^{2}+\frac{\sqrt{3}-1}{2}-2\right) \cdot\left(\left(\frac{\sqrt{3}... | \frac{3}{4} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,487 |
2.140. $\frac{x(x+1)(x+2)(x+3)}{(x-1)(x+4)} ; x=\frac{\sqrt{5}-3}{2}$. | ## Solution.
$$
\begin{aligned}
& \frac{\frac{\sqrt{5}-3}{2} \cdot\left(\frac{\sqrt{5}-3}{2}+1\right) \cdot\left(\frac{\sqrt{5}-3}{2}+2\right) \cdot\left(\frac{\sqrt{5}-3}{2}+3\right)}{\left(\frac{\sqrt{5}-3}{2}-1\right) \cdot\left(\frac{\sqrt{5}-3}{2}+4\right)}= \\
& =\frac{\frac{\sqrt{5}-3}{2} \cdot\left(\frac{\sqrt... | \frac{1}{5} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,488 |
2.141. $\frac{(1-y)(y+2)}{y^{2}(y+1)^{2}} ; \quad y=\frac{\sqrt{3}-1}{2}$. | Solution.
$$
\begin{aligned}
& \frac{\left(1-\frac{\sqrt{3}-1}{2}\right) \cdot\left(\frac{\sqrt{3}-1}{2}+2\right)}{\left(\frac{\sqrt{3}-1}{2}\right)^{2} \cdot\left(\frac{\sqrt{3}-1}{2}+1\right)^{2}}=\frac{-\left(\frac{\sqrt{3}-1}{2}-1\right) \cdot\left(\frac{\sqrt{3}-1}{2}+2\right)}{\left(\frac{\sqrt{3}-1}{2} \cdot\le... | 6 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,489 |
### 2.142.
$$
\frac{\frac{1}{\sqrt{3+x} \cdot \sqrt{x+2}}+\frac{1}{\sqrt{3-x} \cdot \sqrt{x-2}}}{\frac{1}{\sqrt{3+x} \cdot \sqrt{x+2}}-\frac{1}{\sqrt{3-x} \cdot \sqrt{x-2}}} ; \quad x=\sqrt{6} \text {. }
$$ | Solution.
$$
\begin{aligned}
& \frac{\frac{1}{\sqrt{3+\sqrt{6}} \cdot \sqrt{\sqrt{6}+2}}+\frac{1}{\sqrt{3-\sqrt{6}} \cdot \sqrt{\sqrt{6}-2}}}{1}= \\
& \sqrt{3+\sqrt{6}} \cdot \sqrt{\sqrt{6}+2}-\sqrt{\sqrt{3-\sqrt{6}} \cdot \sqrt{\sqrt{6}-2}} \\
& \sqrt{3-\sqrt{6}} \cdot \sqrt{\sqrt{6}-2}+\sqrt{3+\sqrt{6}} \cdot \sqrt{... | -\frac{\sqrt{6}}{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,490 |
2.144. $\frac{2 a \sqrt{1+x^{2}}}{x+\sqrt{1+x^{2}}} ; \quad x=\frac{1}{2} \cdot\left(\sqrt{\frac{a}{b}}-\sqrt{\frac{b}{a}}\right) \quad a>0, b>0$.
2.144. $\frac{2 a \sqrt{1+x^{2}}}{x+\sqrt{1+x^{2}}} ; \quad x=\frac{1}{2} \cdot\left(\sqrt{\frac{a}{b}}-\sqrt{\frac{b}{a}}\right) \quad a>0, b>0$. | Solution.
$$
\begin{aligned}
& \frac{2 a \sqrt{1+\left(\frac{1}{2} \cdot\left(\sqrt{\frac{a}{b}}-\sqrt{\frac{b}{a}}\right)\right)^{2}}}{\frac{1}{2} \cdot\left(\sqrt{\frac{a}{b}}-\sqrt{\frac{b}{a}}\right)+\sqrt{1+\left(\frac{1}{2} \cdot\left(\sqrt{\frac{a}{b}}-\sqrt{\frac{b}{a}}\right)\right)^{2}}}=\frac{2 a \sqrt{1+\f... | +b | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,491 |
2.145. $\frac{1-a x}{1+a x} \cdot \sqrt{\frac{1+b x}{1-b x}} ; \quad x=\frac{1}{a} \cdot \sqrt{\frac{2 a-b}{b}} ; \quad 0<\frac{b}{2}<a<b$. | Solution.
$\frac{1-a \cdot \frac{1}{a} \cdot \sqrt{\frac{2 a-b}{b}}}{1+a \cdot \frac{1}{a} \cdot \sqrt{\frac{2 a-b}{b}}} \cdot \sqrt{\frac{1+b \cdot \frac{1}{a} \cdot \sqrt{\frac{2 a-b}{b}}}{1-b \cdot \frac{1}{a} \cdot \sqrt{\frac{2 a-b}{b}}}}=\frac{1-\sqrt{\frac{2 a-b}{b}}}{1+\sqrt{\frac{2 a-b}{b}}} \times$
$$
\begi... | 1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,492 |
2.146. $\frac{14}{\sqrt[4]{3}+\sqrt[8]{2}}$.
2.146. $\frac{14}{\sqrt[4]{3}+\sqrt[8]{2}}$. | Solution.
$$
\begin{aligned}
& \frac{14}{\sqrt[4]{3}+\sqrt[8]{2}}=\frac{14}{\sqrt[8]{9}+\sqrt[8]{2}}= \\
& =\frac{14\left(\sqrt[8]{9^{7}}-\sqrt[8]{9^{6} \cdot 2}+\sqrt[8]{9^{5} \cdot 2^{2}}-\sqrt[8]{9^{4} \cdot 2^{3}}+\sqrt[8]{9^{3} \cdot 2^{4}}-\sqrt[8]{9^{2} \cdot 2^{5}}+\sqrt[8]{9 \cdot 2^{6}}-\sqrt[8]{2^{7}}\right... | 2(\sqrt[4]{3}-\sqrt[8]{2})(\sqrt{3}+\sqrt[4]{2})(3+\sqrt{2}) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,493 |
2.147. $\frac{4}{\sqrt[4]{13}-\sqrt[4]{9}}$.
Express the above text in English, keeping the original text's line breaks and format, and output the translation result directly.
2.147. $\frac{4}{\sqrt[4]{13}-\sqrt[4]{9}}$. | Solution.
$$
\begin{aligned}
& \frac{4}{\sqrt[4]{13}-\sqrt[4]{9}}=\frac{4\left(\sqrt[4]{13^{3}}+\sqrt[4]{13^{2} \cdot 9}+\sqrt[4]{13 \cdot 9^{2}}+\sqrt[4]{9^{3}}\right)}{(\sqrt[4]{13}-\sqrt[4]{9})\left(\sqrt[4]{13^{2} \cdot 9}+\sqrt[4]{13 \cdot 9^{2}}+\sqrt[4]{9^{3}}\right)}= \\
& =\frac{4(\sqrt[4]{13}+\sqrt[4]{9})\le... | (\sqrt[4]{13}+\sqrt[4]{9})(\sqrt{13}+3) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,494 |
2.148. $\frac{3+\sqrt{2}+\sqrt{3}}{3-\sqrt{2}-\sqrt{3}}$. | ## Solution.
$$
\begin{aligned}
& \frac{(3+(\sqrt{2}+\sqrt{3}))(3+(\sqrt{2}+\sqrt{3}))}{(3-(\sqrt{2}+\sqrt{3}))(3+(\sqrt{2}+\sqrt{3}))}=\frac{(3+(\sqrt{2}+\sqrt{3}))^{2}}{3^{2}-(\sqrt{2}+\sqrt{3})^{2}}= \\
& =\frac{9+6(\sqrt{2}+\sqrt{3})+(\sqrt{2}+\sqrt{3})^{2}}{9-(2+2 \sqrt{6}+3)}=\frac{9+6(\sqrt{2}+\sqrt{3})+2+2 \sq... | -\frac{(4+3\sqrt{2})(5+3\sqrt{3})}{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,495 |
2.149. $\frac{6}{\sqrt{2}+\sqrt{3}+\sqrt{5}}$.
Express the above text in English, keeping the original text's line breaks and format, and output the translation result directly.
2.149. $\frac{6}{\sqrt{2}+\sqrt{3}+\sqrt{5}}$. | ## Solution.
$$
\begin{aligned}
& \frac{6(\sqrt{2}+\sqrt{3}-\sqrt{5})}{(\sqrt{2}+\sqrt{3}+\sqrt{5})(\sqrt{2}+\sqrt{3}-\sqrt{5})}=\frac{6(\sqrt{2}+\sqrt{3}-\sqrt{5})}{2+3-5+2 \sqrt{2 \cdot 3}}=\frac{6(\sqrt{2}+\sqrt{3}-\sqrt{5})}{2 \sqrt{6}}= \\
& =\frac{3(\sqrt{2}+\sqrt{3}-\sqrt{5})}{\sqrt{6}}=\frac{3(\sqrt{2}+\sqrt{3... | \frac{2\sqrt{3}+3\sqrt{2}-\sqrt{30}}{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,496 |
2.150. $\frac{2-\sqrt{2}-\sqrt{3}}{2+\sqrt{2}-\sqrt{3}}$. | ## Solution.
Let's represent the given fraction as $\frac{\sqrt{4}-\sqrt{2}-\sqrt{3}}{\sqrt{4}+\sqrt{2}-\sqrt{3}} \cdot$ Multiply this fraction by $(\sqrt{4}+\sqrt{2}+\sqrt{3})(4+2-3-2 \sqrt{4 \cdot 2})$ and, applying the equality $(\sqrt{a}+\sqrt{b}-\sqrt{c})(\sqrt{a}+\sqrt{b}-\sqrt{c})(a+b-c-2 \sqrt{a b})=(a+b+c)^{2... | \frac{(2\sqrt{6}+1)(3-4\sqrt{2})}{23} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,497 |
2.152. Show that if $z=\sqrt[3]{a+\sqrt{a^{2}+b^{3}}}-\sqrt[3]{\sqrt{a^{2}+b^{3}}-a}$, then $z^{3}+3 b z-2 a=0$ | ## Solution.
$$
\begin{aligned}
& z^{3}=\left(\sqrt[3]{a+\sqrt{a^{2}+b^{3}}}-\sqrt[3]{\sqrt{a^{2}+b^{3}}-a}\right)^{3}= \\
& =\left(\sqrt[3]{a+\sqrt{a^{2}+b^{3}}}+\sqrt[3]{a-\sqrt{a^{2}+b^{3}}}\right)^{3}= \\
& =\left(\sqrt[3]{a+\sqrt{a^{2}+b^{3}}}\right)^{3}+3 \sqrt[3]{\left(a+\sqrt{a^{2}+b^{3}}\right)^{2}\left(a-\sq... | proof | Algebra | proof | Yes | Yes | olympiads | false | 50,498 |
2.154. What is the value of $\sqrt{25-x^{2}}+\sqrt{15-x^{2}}$, given that the difference $\sqrt{25-x^{2}}-\sqrt{15-x^{2}}=2$ (the value of $x$ does not need to be found)? | Solution.
Domain of definition: $\left\{\begin{array}{l}25-x^{2} \geq 0, \\ 15-x^{2} \geq 0\end{array} \Leftrightarrow-\sqrt{15} \leq x \leq \sqrt{15}\right.$.
Multiplying both sides of the equation by $\sqrt{25-x^{2}}+\sqrt{15-x^{2}}$, we have
$$
\begin{aligned}
& \left(\sqrt{25-x^{2}}-\sqrt{15-x^{2}}\right)\left(\... | 5 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,499 |
2.155. Transform $\left(a^{2}+b^{2}\right)\left(c^{2}+d^{2}\right)$ so that it becomes $(a c+b d)^{2}+(a d-b c)^{2}$. | ## Solution.
Expanding the brackets, we get $a^{2} c^{2}+a^{2} d^{2}+b^{2} c^{2}+b^{2} d^{2}$. Add and subtract the expression $2 a b c d$. Then
$$
\begin{aligned}
& a^{2} c^{2}+2 a b c d+b^{2} d^{2}+a^{2} d^{2}-2 a b c d+b^{2} c^{2}=(a c+b d)^{2}+(a d-b c)^{2} \Rightarrow \\
& \Rightarrow\left(a^{2}+b^{2}\right)\lef... | (^{2}+b^{2})(^{2}+^{2})=(+)^{2}+(-)^{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,500 |
2.156. Calculate the sum of the cubes of two numbers if their sum and product are 11 and 21, respectively. | Solution.
Let $a+b=11$ and $a b=21$. Then
$$
\begin{aligned}
& a^{3}+b^{3}=(a+b)\left(a^{2}-a b+b^{2}\right)=(a+b)\left((a+b)^{2}-3 a b\right)=11\left(11^{2}-3 \cdot 21\right)= \\
& =11(121-63)=638
\end{aligned}
$$
Answer: 638. | 638 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,501 |
2.157. Calculate the value of the expression:
a) $\frac{z^{3}}{3}-z, \quad z=\sqrt[3]{\sqrt{3}+\sqrt{2}}+\sqrt[3]{\sqrt{3}-\sqrt{2}}$;
b) $x^{3}+3 x, \quad x=\sqrt[3]{\sqrt{5}+2}-\sqrt[3]{\sqrt{5}-2}$. | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,502 | ||
3.003. $\frac{\cos (3 \pi-2 \alpha)}{2 \sin ^{2}\left(\frac{5 \pi}{4}+\alpha\right)}=\tan\left(\alpha-\frac{5 \pi}{4}\right)$. | ## Solution.
$$
\begin{aligned}
& \frac{\cos (3 \pi-2 \alpha)}{2 \sin ^{2}\left(\frac{5 \pi}{4}+\alpha\right)}=\frac{-\cos 2 \alpha}{1-\cos \left(\frac{5 \pi}{2}+2 \alpha\right)}=\frac{-\cos 2 \alpha}{1-\cos \left(\frac{4 \pi+\pi}{2}+2 \alpha\right)}= \\
& =\frac{-\cos 2 \alpha}{1-\cos \left(2 \pi+\left(\frac{\pi}{2}+... | proof | Algebra | proof | Yes | Yes | olympiads | false | 50,504 |
3.004. $\frac{\tan 2 \alpha+\cot 3 \beta}{\cot 2 \alpha+\tan 3 \beta}=\frac{\tan 2 \alpha}{\tan 3 \beta}$. | ## Solution.
$$
\begin{aligned}
& \frac{\tan 2 \alpha + \cot 3 \beta}{\cot 2 \alpha + \tan 3 \beta} = \frac{\frac{\sin 2 \alpha}{\cos 2 \alpha} + \frac{\cos 3 \beta}{\sin 3 \beta}}{\frac{\cos 2 \alpha}{\sin 2 \alpha} + \frac{\sin 3 \beta}{\cos 3 \beta}} = \frac{\frac{\sin 2 \alpha \sin 3 \beta + \cos 2 \alpha \cos 3 \... | \frac{\tan2\alpha}{\tan3\beta} | Algebra | proof | Yes | Yes | olympiads | false | 50,505 |
3.005. $\cos \alpha+\cos 2 \alpha+\cos 6 \alpha+\cos 7 \alpha=4 \cos \frac{\alpha}{2} \cos \frac{5 \alpha}{2} \cos 4 \alpha$. | ## Solution.
$\cos \alpha+\cos 2 \alpha+\cos 6 \alpha+\cos 7 \alpha=(\cos \alpha+\cos 7 \alpha)+(\cos 2 \alpha+\cos 6 \alpha)=$ $=2 \cos 4 \alpha \cos 3 \alpha+2 \cos 4 \alpha \cos 2 \alpha=2 \cos 4 \alpha(\cos 3 \alpha+\cos 2 \alpha)=$ $=2 \cos 4 \alpha \cdot 2 \cos \frac{5 \alpha}{2} \cos \frac{\alpha}{2}=4 \cos \fr... | proof | Algebra | proof | Yes | Yes | olympiads | false | 50,506 |
3.006. $\sin 9 \alpha+\sin 10 \alpha+\sin 11 \alpha+\sin 12 \alpha=4 \cos \frac{\alpha}{2} \cos \alpha \sin \frac{21 \alpha}{2}$. | ## Solution.
$\sin 9 \alpha+\sin 10 \alpha+\sin 11 \alpha+\sin 12 \alpha=$
$=(\sin 9 \alpha+\sin 12 \alpha)+(\sin 10 \alpha+\sin 11 \alpha)=$
$=2 \sin \frac{21 \alpha}{2} \cos \frac{3 \alpha}{2}+2 \sin \frac{21 \alpha}{2} \cos \frac{\alpha}{2}=2 \sin \frac{21 \alpha}{2}\left(\cos \frac{3 \alpha}{2}+\cos \frac{\alpha... | proof | Algebra | proof | Yes | Yes | olympiads | false | 50,507 |
3.007. $\cos 2 \alpha-\cos 3 \alpha-\cos 4 \alpha+\cos 5 \alpha=-4 \sin \frac{\alpha}{2} \sin \alpha \cos \frac{7 \alpha}{2}$. | Solution.
$$
\begin{aligned}
& (\cos 2 \alpha+\cos 5 \alpha)-(\cos 3 \alpha+\cos 4 \alpha)= \\
& =2 \cos \frac{7 \alpha}{2} \cos \frac{3 \alpha}{2}-2 \cos \frac{7 \alpha}{2} \cos \frac{\alpha}{2}=2 \cos \frac{7 \alpha}{2}\left(\cos \frac{3 \alpha}{2}-\cos \frac{\alpha}{2}\right)= \\
& =2 \cos \frac{7 \alpha}{2} \cdot\... | proof | Algebra | proof | Yes | Yes | olympiads | false | 50,508 |
3.008. $\sin 4 \alpha-\sin 5 \alpha-\sin 6 \alpha+\sin 7 \alpha=-4 \sin \frac{\alpha}{2} \sin \alpha \sin \frac{11 \alpha}{2}$. | ## Solution.
$$
\begin{aligned}
& \sin 4 \alpha+\sin 7 \alpha-(\sin 5 \alpha+\sin 6 \alpha)= \\
& =2 \sin \frac{11 \alpha}{2} \cos \frac{3 \alpha}{2}-2 \sin \frac{11 \alpha}{2} \cos \frac{\alpha}{2}=2 \sin \frac{11 \alpha}{2}\left(\cos \frac{3 \alpha}{2}-\cos \frac{\alpha}{2}\right)= \\
& =2 \sin \frac{11 \alpha}{2} \... | proof | Algebra | proof | Yes | Yes | olympiads | false | 50,509 |
3.009. $\cos \alpha+\sin \alpha+\cos 3 \alpha+\sin 3 \alpha=2 \sqrt{2} \cos \alpha \sin \left(\frac{\pi}{4}+2 \alpha\right)$. | ## Solution.
$$
\begin{aligned}
& \cos \alpha+\cos \left(\frac{\pi}{2}-\alpha\right)+\cos 3 \alpha+\cos \left(\frac{\pi}{2}-3 \alpha\right)= \\
& =2 \cos \frac{\pi}{4} \cos \left(\frac{\pi}{4}-\alpha\right)+2 \cos \frac{\pi}{4} \cos \left(\frac{\pi}{4}-3 \alpha\right)= \\
& =2 \cos \frac{\pi}{4}\left(\cos \left(\frac{... | proof | Algebra | proof | Yes | Yes | olympiads | false | 50,510 |
3.010. $\operatorname{tg} \alpha+\operatorname{ctg} \alpha+\operatorname{tg} 3 \alpha+\operatorname{ctg} 3 \alpha=\frac{8 \cos ^{2} 2 \alpha}{\sin 6 \alpha}$.
3.010. $\tan \alpha+\cot \alpha+\tan 3 \alpha+\cot 3 \alpha=\frac{8 \cos ^{2} 2 \alpha}{\sin 6 \alpha}$. | Solution.
$$
\begin{aligned}
& \frac{\sin \alpha}{\cos \alpha}+\frac{\cos \alpha}{\sin \alpha}+\frac{\sin 3 \alpha}{\cos 3 \alpha}+\frac{\cos 3 \alpha}{\sin 3 \alpha}=\frac{\sin ^{2} \alpha+\cos ^{2} \alpha}{\sin \alpha \cos \alpha}+\frac{\sin ^{2} 3 \alpha+\cos ^{2} 3 \alpha}{\sin 3 \alpha \cos 3 \alpha} \\
& =\frac{... | proof | Algebra | proof | Yes | Yes | olympiads | false | 50,511 |
3.013.
$$
\frac{\sin 2 \alpha - \sin 3 \alpha + \sin 4 \alpha}{\cos 2 \alpha - \cos 3 \alpha + \cos 4 \alpha} = \tan 3 \alpha
$$ | Solution.
$$
\begin{aligned}
& \frac{(\sin 2 \alpha+\sin 4 \alpha)-\sin 3 \alpha}{(\cos 2 \alpha+\cos 4 \alpha)-\cos 3 \alpha}=\frac{2 \sin 3 \alpha \cos \alpha-\sin 3 \alpha}{2 \cos 3 \alpha \cos \alpha-\cos 3 \alpha}=\frac{\sin 3 \alpha(2 \cos \alpha-1)}{\cos 3 \alpha(2 \cos \alpha-1)}= \\
& =\frac{\sin 3 \alpha}{\c... | proof | Algebra | proof | Yes | Yes | olympiads | false | 50,512 |
3.014. $2 \sin ^{2}(3 \pi-2 \alpha) \cos ^{2}(5 \pi+2 \alpha)=\frac{1}{4}-\frac{1}{4} \sin \left(\frac{5}{2} \pi-8 \alpha\right)$. | ## Solution.
Applying the power reduction formulas $\sin ^{2} \frac{x}{2}=\frac{1-\cos x}{2}$ and $\cos ^{2} \frac{x}{2}=\frac{1+\cos x}{2}$, we represent the left-hand side as follows:
$$
\begin{aligned}
& 2 \cdot \frac{(1-\cos (6 \pi-4 \alpha))(1+\cos (10 \pi+4 \alpha))}{4}=\frac{1}{2}(1-\cos (6 \pi-4 \alpha))(1+\c... | proof | Algebra | proof | Yes | Yes | olympiads | false | 50,513 |
3.015. $\sin 2 \alpha(1+\operatorname{tg} 2 \alpha \operatorname{tg} \alpha)+\frac{1+\sin \alpha}{1-\sin \alpha}=\operatorname{tg} 2 \alpha+\operatorname{tg}^{2}\left(\frac{\pi}{4}+\frac{\alpha}{2}\right)$. | ## Solution.
Let
$$
\begin{aligned}
& X=\sin 2 \alpha(1+\operatorname{tg} 2 \alpha \operatorname{tg} \alpha)=\sin 2 \alpha\left(1+\frac{\sin 2 \alpha}{\cos 2 \alpha} \cdot \frac{\sin \alpha}{\cos \alpha}\right)= \\
& =\frac{\sin 2 \alpha(\cos 2 \alpha \cos \alpha+\sin 2 \alpha \sin \alpha)}{\cos 2 \alpha \cos \alpha}... | proof | Algebra | proof | Yes | Yes | olympiads | false | 50,514 |
3.016. $1-\sin 4 \alpha+\operatorname{ctg}\left(\frac{3}{4} \pi-2 \alpha\right) \cos 4 \alpha=0$. | ## Solution.
$$
\begin{aligned}
& 1-\sin 4 \alpha+\operatorname{ctg}\left(\frac{3}{4} \pi-2 \alpha\right) \cos 4 \alpha= \\
& =\cos ^{2} 2 \alpha+\sin ^{2} 2 \alpha-\sin (2 \cdot 2 \alpha)+\frac{\cos \left(\frac{3 \pi}{4}-2 \alpha\right)}{\sin \left(\frac{3 \pi}{4}-2 \alpha\right)} \cdot \cos (2 \cdot 2 \alpha)= \\
& ... | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,515 |
3.017. $\sin ^{6} \frac{\alpha}{2}-\cos ^{6} \frac{\alpha}{2}=\frac{\sin ^{2} \alpha-4}{4} \cos \alpha$. | Solution.
Let $X=\left(\sin ^{2} \frac{\alpha}{2}\right)^{3}-\left(\cos ^{2} \frac{\alpha}{2}\right)^{3}$.
Using the power reduction formulas $\sin ^{2} \frac{x}{2}=\frac{1-\cos x}{2}$ and $\cos ^{2} \frac{x}{2}=\frac{1+\cos x}{2}$, we get
$$
\begin{aligned}
& X=\left(\frac{1-\cos \alpha}{2}\right)^{3}-\left(\frac{1... | \frac{\sin^{2}\alpha-4}{4}\cos\alpha | Algebra | proof | Yes | Yes | olympiads | false | 50,516 |
3.018. $\cos \left(\frac{3}{2} \pi+4 \alpha\right)+\sin (3 \pi-8 \alpha)-\sin (4 \pi-12 \alpha)=$ $=4 \cos 2 \alpha \cos 4 \alpha \sin 6 \alpha$. | ## Solution.
$$
\begin{aligned}
& \cos \left(\frac{3}{2} \pi+4 \alpha\right)+\sin (3 \pi-8 \alpha)-\sin (4 \pi-12 \alpha)= \\
& =\sin 4 \alpha+\sin 8 \alpha+\sin 12 \alpha=2 \sin 6 \alpha \cos 2 \alpha+2 \sin 6 \alpha \cos 6 \alpha= \\
& =2 \sin 6 \alpha(\cos 2 \alpha+\cos 6 \alpha)=2 \sin 6 \alpha \cdot 2 \cos 4 \alp... | proof | Algebra | proof | Yes | Yes | olympiads | false | 50,517 |
3.019.
$$
\frac{\cos \left(\frac{5}{2} \pi-6 \alpha\right)+\sin (\pi+4 \alpha)+\sin (3 \pi-\alpha)}{\sin \left(\frac{5}{2} \pi+6 \alpha\right)+\cos (4 \alpha-2 \pi)+\cos (\alpha+2 \pi)}=\tan \alpha
$$ | ## Solution.
$$
\begin{aligned}
& \frac{\cos \left(\frac{5}{2} \pi-6 \alpha\right)+\sin (\pi+4 \alpha)+\sin (3 \pi-\alpha)}{\sin \left(\frac{5}{2} \pi+6 \alpha\right)+\cos (4 \alpha-2 \pi)+\cos (\alpha+2 \pi)}=\frac{\sin 6 \alpha-\sin 4 \alpha+\sin \alpha}{\cos 6 \alpha+\cos 4 \alpha+\cos \alpha}= \\
& =\frac{2 \cos 5... | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,518 |
3.021. $\sin \alpha+\sin \left(\dot{\alpha}+\frac{14}{3} \pi\right)+\sin \left(\alpha-\frac{8}{3} \pi\right)=0$. | ## Solution.
$$
\begin{aligned}
& \sin \alpha + \sin \left(\frac{14}{3} \pi + \alpha\right) - \sin \left(\frac{8}{3} \pi - \alpha\right) = \sin \alpha + \sin \left(\frac{15 \pi - \pi}{3} + \alpha\right) - \\
& - \sin \left(\frac{9 \pi - \pi}{3} - \alpha\right) = \sin \alpha + \sin \left(5 \pi + \left(\alpha - \frac{\p... | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,519 |
3.022. $\operatorname{ctg}^{2} \alpha-\operatorname{ctg}^{2} \beta=\frac{\cos ^{2} \alpha-\cos ^{2} \beta}{\sin ^{2} \alpha \sin ^{2} \beta}$. | ## Solution.
$$
\begin{aligned}
& \frac{\cos ^{2} \alpha}{\sin ^{2} \alpha}-\frac{\cos ^{2} \beta}{\sin ^{2} \beta}=\frac{\sin ^{2} \beta \cos ^{2} \alpha+\cos ^{2} \beta \sin ^{2} \alpha}{\sin ^{2} \alpha \sin ^{2} \beta}= \\
& =\frac{(\sin \beta \cos \alpha-\cos \beta \sin \alpha)(\sin \beta \cos \alpha+\cos \beta \... | proof | Algebra | proof | Yes | Yes | olympiads | false | 50,520 |
3.023. $(\cos \alpha-\cos \beta)^{2}+(\sin \alpha-\sin \beta)^{2}=4 \sin ^{2} \frac{\alpha-\beta}{2}$. | ## Solution.
$$
\begin{aligned}
& \cos ^{2} \alpha-2 \cos \alpha \cos \beta+\cos ^{2} \beta+\sin ^{2} \alpha-2 \sin \alpha \sin \beta+\sin ^{2} \beta= \\
& =\left(\cos ^{2} \alpha+\sin ^{2} \alpha\right)+\left(\cos ^{2} \beta+\sin ^{2} \beta\right)-2(\cos \alpha \cos \beta+\sin \alpha \sin \beta)= \\
& =2-2 \cos (\alp... | 4\sin^{2}\frac{\alpha-\beta}{2} | Algebra | proof | Yes | Yes | olympiads | false | 50,521 |
3.024. $\frac{\left(\tan \alpha+\cos ^{-1} \alpha\right)(\cos \alpha-\cot \alpha)}{(\cos \alpha+\cot \alpha)\left(\tan \alpha-\cos ^{-1} \alpha\right)}=1$. | ## Solution.
$$
\frac{\left(\operatorname{tg} \alpha+\cos ^{-1} \alpha\right)(\cos \alpha-\operatorname{ctg} \alpha)}{(\cos \alpha+\operatorname{ctg} \alpha)\left(\operatorname{tg} \alpha-\cos ^{-1} \alpha\right)}=\frac{\left(\frac{\sin \alpha}{\cos \alpha}+\frac{1}{\cos \alpha}\right)\left(\cos \alpha-\frac{\cos \alp... | proof | Algebra | proof | Yes | Yes | olympiads | false | 50,522 |
3.025. $\frac{\sin 4 \alpha}{1+\cos 4 \alpha} \cdot \frac{\cos 2 \alpha}{1+\cos 2 \alpha}=\operatorname{ctg}\left(\frac{3}{2} \pi-\alpha\right)$.
3.025. $\frac{\sin 4 \alpha}{1+\cos 4 \alpha} \cdot \frac{\cos 2 \alpha}{1+\cos 2 \alpha}=\cot\left(\frac{3}{2} \pi-\alpha\right)$. | Solution.
$$
\begin{aligned}
& \frac{2 \sin 2 \alpha \cos 2 \alpha}{1+2 \cos ^{2} 2 \alpha-1} \cdot \frac{\cos 2 \alpha}{1+\cos 2 \alpha}=\frac{2 \sin 2 \alpha \cos 2 \alpha}{2 \cos ^{2} 2 \alpha} \cdot \frac{\cos 2 \alpha}{1+\cos 2 \alpha}= \\
& =\frac{\sin 2 \alpha \cos 2 \alpha}{\cos 2 \alpha(1+\cos 2 \alpha)}=\fra... | proof | Algebra | proof | Yes | Yes | olympiads | false | 50,523 |
3.026. $\cos ^{2}\left(\alpha-90^{\circ}\right)+\operatorname{ctg}^{2}\left(\alpha-270^{\circ}\right)=\frac{1}{\sin ^{2}\left(\alpha+90^{\circ}\right)}-\cos ^{2}\left(\alpha+180^{\circ}\right)$. | Solution.
$$
\begin{aligned}
& \cos ^{2}\left(\alpha-90^{\circ}\right)+\operatorname{ctg}^{2}\left(\alpha-270^{\circ}\right)=\sin ^{2} \alpha+\operatorname{tg}^{2} \alpha=\sin ^{2} \alpha+\frac{\sin ^{2} \alpha}{\cos ^{2} \alpha}= \\
& =\frac{\sin ^{2} \alpha \cos ^{2} \alpha+\sin ^{2} \alpha}{\cos ^{2} \alpha}=\frac{... | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,524 |
3.027. $\frac{1-\operatorname{tg}\left(90^{\circ}+\alpha\right)}{1+\operatorname{ctg}\left(360^{\circ}-\alpha\right)}=\frac{\operatorname{tg}\left(180^{\circ}+\alpha\right)+1}{\operatorname{ctg}\left(270^{\circ}-\alpha\right)-1}$.
3.027. $\frac{1-\tan\left(90^{\circ}+\alpha\right)}{1+\cot\left(360^{\circ}-\alpha\right... | Solution.
$$
\frac{1-\operatorname{tg}\left(90^{\circ}+\alpha\right)}{1+\operatorname{ctg}\left(360^{\circ}-\alpha\right)}=\frac{1+\operatorname{ctg} \alpha}{1-\operatorname{ctg} \alpha}=\frac{1+\frac{1}{\operatorname{tg} \alpha}}{1-\frac{1}{\operatorname{tg} \alpha}}=\frac{\operatorname{tg} \alpha+1}{\operatorname{tg... | proof | Algebra | proof | Yes | Yes | olympiads | false | 50,525 |
3.030. $\sin ^{2}\left(\frac{15}{8} \pi-2 \alpha\right)-\cos ^{2}\left(\frac{17}{8}-2 \alpha\right)=-\frac{\cos 4 \alpha}{\sqrt{2}}$. | ## Solution.
Using the power reduction formulas $\sin ^{2} \frac{x}{2}=\frac{1-\cos x}{2}$ and $\cos ^{2} \frac{x}{2}=\frac{1+\cos x}{2}$, we represent the left-hand side of the equation as
$\frac{1-\cos \left(\frac{15 \pi}{4}-4 \alpha\right)}{2}-\frac{1+\cos \left(\frac{17 \pi}{4}-4 \alpha\right)}{2}=\frac{1-\cos \l... | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,527 |
3.031. $(\cos \alpha-\cos \beta)^{2}-(\sin \alpha-\sin \beta)^{2}=-4 \sin ^{2} \frac{\alpha-\beta}{2} \cos (\alpha+\beta)$. | ## Solution.
$$
\begin{aligned}
& (\cos \alpha - \cos \beta)^2 - (\sin \alpha - \sin \beta)^2 = \\
& = \cos^2 \alpha - 2 \cos \alpha \cos \beta + \cos^2 \beta - \sin^2 \alpha + 2 \sin \alpha \sin \beta - \sin^2 \beta = \\
& = (\cos^2 \alpha - \sin^2 \alpha) + (\cos^2 \beta - \sin^2 \beta) - 2 (\cos \alpha \cos \beta -... | proof | Algebra | proof | Yes | Yes | olympiads | false | 50,528 |
3.033. $\cos 4 \alpha-\sin 4 \alpha \operatorname{ctg} 2 \alpha=\cos 2 \alpha-2 \cos ^{2} \alpha$.
Translate the above text into English, keeping the original text's line breaks and format, and output the translation result directly.
3.033. $\cos 4 \alpha-\sin 4 \alpha \operatorname{ctg} 2 \alpha=\cos 2 \alpha-2 \cos... | Solution.
$\cos 4 \alpha-\sin 4 \alpha \operatorname{ctg} 2 \alpha=\cos 4 \alpha-\sin 4 \alpha \cdot \frac{\cos 2 \alpha}{\sin 2 \alpha}=$
$$
\begin{aligned}
& =\frac{\sin 2 \alpha \cos 4 \alpha-\cos 2 \alpha \sin 4 \alpha}{\sin 2 \alpha}=\frac{\sin (-2 \alpha)}{\sin 2 \alpha}=\frac{-\sin 2 \alpha}{\sin 2 \alpha}= \\... | proof | Algebra | proof | Yes | Yes | olympiads | false | 50,529 |
3.034. $\sin ^{2}\left(\frac{9 \pi}{8}+\frac{\alpha}{4}\right)-\sin ^{2}\left(\frac{7 \pi}{8}+\frac{\alpha}{4}\right)=\frac{\sin \frac{\alpha}{2}}{\sqrt{2}}$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
3.034. $\sin ^{2}\... | ## Solution.
$$
\begin{aligned}
& \sin ^{2}\left(\frac{9 \pi}{8}+\frac{\alpha}{4}\right)-\sin ^{2}\left(\frac{7 \pi}{8}+\frac{\alpha}{4}\right)=\frac{1-\cos \left(\frac{9 \pi}{4}+\frac{\alpha}{2}\right)}{2}-\frac{1-\cos \left(\frac{7 \pi}{4}+\frac{\alpha}{2}\right)}{2}= \\
& =\frac{1}{2}-\frac{\cos \left(\frac{8 \pi+\... | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 50,530 |
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