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[
{
"role": "system",
"content": "You are a helpful, expert AI tutor. Solve the problem by breaking it down step-by-step."
},
{
"role": "user",
"content": "Let $a, b$ and $c$ be numbers with $0 < a < b < c$. Which of the following is\nimpossible?\n$\\mathrm{(A)} \\ a + c < b \\qquad \\mathrm{(B)}... |
[
{
"role": "system",
"content": "You are a helpful, expert AI tutor. Solve the problem by breaking it down step-by-step."
},
{
"role": "user",
"content": "8. Through the vertex $A$ of a regular tetrahedron $ABCD$, make a section in the shape of an isosceles triangle, and let the angle between the... |
[
{
"role": "system",
"content": "You are a helpful, expert AI tutor. Solve the problem by breaking it down step-by-step."
},
{
"role": "user",
"content": "Three. (Full marks 25 points) On the blackboard, all natural numbers from 1 to 1997 are written. Students $A$ and $B$ take turns to perform th... |
[
{
"role": "system",
"content": "You are a helpful, expert AI tutor. Solve the problem by breaking it down step-by-step."
},
{
"role": "user",
"content": "In triangle $ABC$, $AB = 4$, $BC = 6$, and $CA = 8$. The point $D$ lies on line segment $\\overline{AC}$, and the circle centered at $B$ passi... |
[
{
"role": "system",
"content": "You are a helpful, expert AI tutor. Solve the problem by breaking it down step-by-step."
},
{
"role": "user",
"content": "Find all roots of the polynomial $x^4+2x^3-7x^2-12x+9$. Enter your answer as a list of numbers separated by commas. If a root occurs more than... |
[
{
"role": "system",
"content": "You are a helpful, expert AI tutor. Solve the problem by breaking it down step-by-step."
},
{
"role": "user",
"content": "11. Electoral Districts. In Anchuria, presidential elections are being prepared, in which President Miraflores wants to win. Exactly half of t... |
[
{
"role": "system",
"content": "You are a helpful, expert AI tutor. Solve the problem by breaking it down step-by-step."
},
{
"role": "user",
"content": "4. The number of integer solutions $(m, n)$ for the equation $m^{2}-n^{2}+3 m-7 n-2=0$ is $($.\n(A) 1\n(B) 2\n(C) 3\n(D) 4"
},
{
"role... |
[
{
"role": "system",
"content": "You are a helpful, expert AI tutor. Solve the problem by breaking it down step-by-step."
},
{
"role": "user",
"content": "3. Let the garden have the shape of an equilateral triangle with a side of 1. Prove that Georgy Konstantinovich will have to buy hedgehogs of ... |
[
{
"role": "system",
"content": "You are a helpful, expert AI tutor. Solve the problem by breaking it down step-by-step."
},
{
"role": "user",
"content": "Let $f(x)$ be a fourth-degree polynomial with real coefficients satisfying \n$$|f(-2)|=|f(-1)|=|f(1)|=|f(2)|=|f(3)|=|f(4)|=18.$$\nThe function... |
[
{
"role": "system",
"content": "You are a helpful, expert AI tutor. Solve the problem by breaking it down step-by-step."
},
{
"role": "user",
"content": "Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Determine, with certainty, the largest possible value of the expression $$ \\frac{a}... |
[
{
"role": "system",
"content": "You are a helpful, expert AI tutor. Solve the problem by breaking it down step-by-step."
},
{
"role": "user",
"content": "4. In $\\triangle A B C$, it is known that $M$ is the midpoint of $A B$, $N$ is a point on side $B C$, and $C N=2 B N$. Connecting $A N$ and $... |
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