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## Task A-4.2.
Let $x_{1}, x_{2}$, and $x_{3}$ be distinct roots of the polynomial $P(x)=x^{3}-3 x+1$. Determine
$$
\frac{1}{x_{1}^{2}-3}+\frac{1}{x_{2}^{2}-3}+\frac{1}{x_{3}^{2}-3}
$$
|
## First Solution.
For the root $x_{1}$ of the polynomial $P$, we have
$$
\begin{aligned}
x_{1}^{3}-3 x_{1}+1 & =0 \\
x_{1}^{3}-3 x_{1} & =-1 \\
x_{1}^{2}-3 & =-\frac{1}{x_{1}} \\
\frac{1}{x_{1}^{2}-3} & =-x_{1} .
\end{aligned}
$$
Similarly, this holds for $x_{2}$ and $x_{3}$, so for the desired expression we have
$$
\frac{1}{x_{1}^{2}-3}+\frac{1}{x_{2}^{2}-3}+\frac{1}{x_{3}^{2}-3}=-\left(x_{1}+x_{2}+x_{3}\right)
$$
By Vieta's formulas, $x_{1}+x_{2}+x_{3}=0$, so the expression is equal to zero.
|
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]
] |
[
"0",
""
] |
id_48140
| "Example 7 In $\\triangle A B C$, $\\angle A B C=50^{\\circ}$, $\\angle A C B=30^{\\circ}, R$ is a p(...TRUNCATED) | "Solution: As shown in Figure 7, construct a regular $\\triangle DAB$ on one side of $AB$ inside $\\(...TRUNCATED) |
80
|
[
true,
false
] | [[151646,151644,13314,220,22,758,57960,55114,362,425,356,54876,57960,4044,362,425,356,28,20,15,61,35(...TRUNCATED) | [[4092,311,356,73418,594,11,432,594,220,21,17,13,22,11616,11,892,374,51033,382,14190,11,8365,358,126(...TRUNCATED) |
[
"80",
""
] |
id_28412
| "Let $f(x) = 1 + 2x + 3x^2 + 4x^3 + 5x^4$ and let $\\zeta = e^{2\\pi i/5} = \\cos \\frac{2\\pi}{5} +(...TRUNCATED) | "1. Define the function \\( f(x) = 1 + 2x + 3x^2 + 4x^3 + 5x^4 \\).\n2. Let \\(\\zeta = e^{2\\pi i/5(...TRUNCATED) |
125
|
[
true,
false
] | [[151646,151644,10061,400,69,2075,8,284,220,16,488,220,17,87,488,220,18,87,61,17,488,220,19,87,61,18(...TRUNCATED) | [[7877,8460,7493,369,17767,282,11520,89,1915,61,74,8,1124,3593,40,6099,429,369,264,729,17767,282,207(...TRUNCATED) |
[
"125",
""
] |
id_32595
| "[ Systems of nonlinear algebraic equations ]\n\nGiven the system of equations:\n\n$$\n\\left\\{\\be(...TRUNCATED) | "Note that the product $x_{1} \\ldots x_{k}$ satisfies the equation $x+1 / x^{\\prime}=1 \\Rightarro(...TRUNCATED) |
1
|
[
true,
false
] | [[151646,151644,58,14917,315,73998,46876,292,37906,10444,22043,279,1849,315,37906,1447,14085,198,59,(...TRUNCATED) | [[374,279,1985,315,678,3681,7332,11,892,525,6785,11,17767,1207,4698,1124,8,1969,387,6785,13,2055,11,(...TRUNCATED) |
[
"1",
""
] |
id_26351
| "7.2 On the Island of Truth and Lies, there are knights who always tell the truth, and liars who alw(...TRUNCATED) | "# Answer: 11\n\nSolution. The formation cannot consist of only liars (in such a case, 4-8 would be (...TRUNCATED) |
11
|
[
true,
false
] | [[151646,151644,22,13,17,1913,279,10720,315,29098,323,77086,11,1052,525,70530,879,2677,3291,279,8046(...TRUNCATED) | [[220,16,18,1969,387,264,72624,382,67691,11,2309,220,16,17,25,2379,1053,330,3862,374,264,46609,3403,(...TRUNCATED) |
[
"11",
""
] |
id_9612
| "5. Given an ellipse $\\frac{x^{2}}{2}+k y^{2}=1$ with its foci on the $x$-axis, points $A$ and $B$ (...TRUNCATED) | "5. $\\frac{2 \\sqrt{3}}{3}$.\n\nLet $|O A|=r_{1},|O C|=r_{2}$. Then, since $\\triangle A B C$ is an(...TRUNCATED) |
\frac{2\sqrt{3}}{3}
|
[
true,
false
] | [[151646,151644,20,13,16246,458,57397,57960,37018,45340,47822,17,3417,90,17,92,10,74,379,47822,17,51(...TRUNCATED) | [[14,17,488,595,296,29456,7252,382,54815,11,272,3212,284,11995,241,296,17946,58,18,608,320,16,14,17,(...TRUNCATED) |
[
"\\frac{2\\sqrt{3}}{3}",
""
] |
id_28899
| "55. A person worked for 24 consecutive days and earned 1900 yuan. He works full days from Monday to(...TRUNCATED) |
Reference answer: 18
|
18
|
[
true,
false
] | [[151646,151644,20,20,13,362,1697,6439,369,220,17,19,23921,2849,323,15303,220,16,24,15,15,72283,13,1(...TRUNCATED) | [[15,72283,1817,1899,13,1913,7728,11,566,4278,4279,95214,11,27644,220,20,15,72283,1817,13,7270,374,1(...TRUNCATED) |
[
"18",
""
] |
id_16781
| "4. In $\\triangle A B C$, $A B$ $=B C=12, \\angle B=90^{\\circ}$, as shown in the figure. Fold alon(...TRUNCATED) | "$$\n\\begin{array}{l}\n\\text { 4. } 8 \\frac{2}{3} . \\\\\n\\because B D \\cdot D C=2: 1, \\\\\n\\(...TRUNCATED) |
8\frac{2}{3}
|
[
true,
false
] | [[151646,151644,19,13,758,57960,55114,362,425,356,54876,400,32,425,3,400,28,33,356,28,16,17,11,1124,(...TRUNCATED) | [[537,5961,382,92014,11,2474,44742,3156,43990,14043,362,311,422,11,1221,42108,1969,6144,16036,11,714(...TRUNCATED) |
[
"\\frac{26}{3}",
""
] |
id_44932
| "Let $e > 0$ be a given real number. Find the least value of $f(e)$ (in terms of $e$ only) such that(...TRUNCATED) | "1. **Apply the AM-GM Inequality:**\n The Arithmetic Mean-Geometric Mean (AM-GM) Inequality states(...TRUNCATED) |
\frac{1}{4e^2}
|
[
true,
false
] | [[151646,151644,10061,400,68,861,220,15,3,387,264,2661,1931,1372,13,7379,279,3245,897,315,400,69,202(...TRUNCATED) | [[19,21103,230,220,15,13,15,16,20,21,17,20,1124,701,892,374,2686,1091,220,15,13,15,21,18,19,22,21,20(...TRUNCATED) |
[
"\\frac{1}{4e^2}",
""
] |
id_15079
| "B1. The set $S$ contains 6 distinct positive integers. The average of the two smallest numbers is 5(...TRUNCATED) | "Solution: Let $S=\\left\\{s_{1}, \\ldots, s_{6}\\right\\}$, where $s_{1}<s_{2}<\\cdots<s_{6}$. Then(...TRUNCATED) |
15.5
|
[
true,
false
] | [[151646,151644,33,16,13,576,738,400,50,3,5610,220,21,12460,6785,25780,13,576,5461,315,279,1378,2463(...TRUNCATED) | [[773,282,374,220,17,18,382,4416,11,421,358,738,384,284,220,17,16,11,282,284,220,17,18,11,1221,272,3(...TRUNCATED) |
[
"\\frac{31}{2}",
""
] |
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