messages listlengths 3 3 |
|---|
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let $\\mathcal{C}$ be the family of circles in $\\mathbb{R}^2$ defined by the following properties:\n- For each $n \\in \\mathbb{Z}$, the circle $C_n$ with ce... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the minimum value of \\( n \\) such that \\(\\frac{2}{3} < \\frac{m}{n} < \\frac{7}{10}\\) for \\( (m, n) \\in \\mathbb{N} \\)."
},
{
"role": "as... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For an injective function \\( f: \\mathbb{R} \\to \\mathbb{R} \\) satisfying \\( f(x + f(y)) = 2012 + f(x + y) \\), find \\( f(2013) \\)."
},
{
"role"... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( f : \\mathbb{N} \\to \\mathbb{N} \\) be a function defined by \\( f(1) = 1 \\), \\( f(2n) = f(n) \\), and \\( f(2n + 1) = f(2n) + 1 \\). Find the numb... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given the quadratic equation \\(2x^2 - 2x + 1 = 0\\) with roots \\(x_1\\) and \\(x_2\\), find the value of \\({x_1}^{x_2} - {x_2}^{x_1}\\)."
},
{
"rol... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Determine the Galois group \\( Gal(\\mathbb{Q}(\\sqrt[4]{2})/ \\mathbb{Q}) \\)."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A polynomial \\( P(x) \\) of degree 7 satisfies \\( P(n) = 2^n \\) for \\( n = 0 \\) to 7. Find \\( P(10) \\)."
},
{
"role": "assistant",
"content... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the position of points \\( P \\), \\( Q \\), and \\( R \\) on sides \\( [AB] \\), \\( [AC] \\), and \\( [BC] \\) of an acute-angled triangle \\( ABC \\) ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In a 50x50 grid of intersections, each street between intersections can be painted with one of 900 distinct colors. The government requires that for any selec... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) be a continuous function such that for all \\( x \\in \\mathbb{R} \\), \\( f(x) + f(x+1009) \\in \\mathbb{... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let $z = z(x, y)$ be the implicit function defined by the equation $3e^z + z^3 = x - y + f(x^2y^2, e^{1+2xy})$, where $f$ is a differentiable function. Determ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all solutions of ordered pairs \\((x, y)\\) of positive integers for the equation \\(x^2 + 2 = y^3\\)."
},
{
"role": "assistant",
"content": ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(ABCD\\) be a square with side length 1. How many points \\(P\\) inside the square (not on its sides) have the property that the square can be cut into ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "There are $N$ permutations $(a_{1}, a_{2},..., a_{30})$ of $1, 2, \\ldots, 30$ such that for $m \\in \\left\\{{2, 3, 5}\\right\\}$, $m$ divides $a_{n+m} - ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the sum of the first 49 even squares."
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. I need to find the sum of the firs... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "What is the maximum $k$ such that the set $\\{1, 2, \\ldots, 2021\\}$ can be partitioned into $k$ sets such that the sum of the elements of one set is not div... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Simplify the expression \\(\\binom{n}{1} + \\binom{n+1}{2} + \\binom{n+2}{3} + \\cdots + \\binom{n+m+1}{m}\\)."
},
{
"role": "assistant",
"content... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For how many choices of starting numbers below $2023$ will Felix never write a number of more than four digits on the board, given the following rules: if the... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Calculate the determinant of the matrix defined by:\n\\[ \\begin{vmatrix} a & b & b & \\cdots & b \\\\ -b & a & b & \\cdots & b \\\\ -b & -b & a & \\cdots & b... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve the equation \\( \\sqrt{x}+\\sqrt[3]{x+7}=\\sqrt[4]{x+80} \\)."
},
{
"role": "assistant",
"content": "<think>\nOkay, let's try to solve this... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the limit\n\n\\[\n\\lim_{x \\to -177} \\frac{x^{100} + 177x^{99} + x + 1}{x^2 + 178x + 177}.\n\\]"
},
{
"role": "assistant",
"content": "<thi... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve the system of equations:\n\\[\n\\left\\{\n\\begin{matrix}\nx^2 + y^2 - 5x + 3 = 0 \\\\\nxy + 2\\sqrt{xy} - 5y - 3 = 0\n\\end{matrix}\n\\right.\n\\]"
}... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Andrew runs on skis faster than Victor but slower than Eugene. They simultaneously started from point A, in the same direction along a circular path and stopp... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all injective functions \\( f: \\mathbb{R}^+_* \\to \\mathbb{R}^+_* \\) such that\n\\[ f(f(x)y)(f(x) + y) = xyf(x + y) \\]\nfor all \\( x, y \\in \\mathb... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In a grid of $P$s and $O$s arranged as follows:\n$$P$$\n$$P \\text{ } O \\text{ } P$$\n$$P \\text{ } O \\text{ } P \\text{... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Suzanne, the medium, spends her 10-hour work day helping ghosts and ghouls one at a time without taking a break. When she is helping a ghost, there is a 30% c... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the volume of the solid generated by revolving the region bounded by the lines \\( x = 0 \\), \\( y = 0 \\), \\( x = 1 \\), and the curve \\( y = e^{x^2}... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the number of injective mappings \\( f: \\mathbb{A} \\rightarrow \\mathbb{B} \\) such that \\( f(i) \\neq i \\) for all \\( i \\in \\mathbb{A} \\), where... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Evaluate the integral $\\int_{4}^{6} \\frac{x^{3} - 2x^{2} -4x +13}{x^{2} - x -6} \\ dx$."
},
{
"role": "assistant",
"content": "<think>\nOkay, le... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Two distinct numbers $a$ and $b$ are randomly and uniformly chosen from the set $\\{3, 8, 16, 18, 24\\}$. What is the probability that there exist integers $c... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let $ABCD$ be a tetrahedron with $AB=41$, $AC=7$, $AD=18$, $BC=36$, $BD=27$, and $CD=13$, as shown in the figure. Let $d$ be the distance between the midpoin... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A rectangle with a perimeter of 176 is divided into five congruent rectangles. What is the perimeter of one of the five congruent rectangles?"
},
{
"r... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Tom Sawyer can paint a fence by himself in 3 hours, and Huck Finn can paint the same fence by himself in 4 hours. They start painting together at 12:00 noon b... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many distinct amounts of money can be created from the following denominations: 1 $2 bill, 3 $5 bills, 2 $10 bills, and 1 $20 bill (excluding $0)?"
},
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the smallest positive integer $n$ such that $n = a^2 + b^2 - c^2$ has no solution in positive integers $a$, $b$, and $c$ with $a < b < c$. If all nonnega... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve the Cauchy problem for the differential equation \\( txx'' - t(x')^2 - xx' = 0 \\) with initial conditions \\( x(1) = 1 \\) and \\( x'(1) = 1 \\)."
},... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all integer solutions to the Diophantine equation $x^y - y^x = x + y$."
},
{
"role": "assistant",
"content": "<think>\nAlright, so I need to ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "If a radio call sign consists of 4 letters, with the first letter being either 'A' or 'B' and no restrictions on the other 3 letters, what is the probability ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Mr. Crisper's class is studying the number $2014$ in different bases from $5$ to $2014$. Each student converts $2014$ from their assigned base to base $10$. W... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(a, b, c\\) be positive numbers such that \\(a + b + c = 1\\). Find the minimum value of \n\\[ P = \\frac{a}{1 + b^2 + c^2} + \\frac{b}{1 + c^2 + a^2} +... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "What are the roots of the equation \\(x^n + x^{n-1} + x^{n-2} + \\cdots + x + 1 = 0\\)?"
},
{
"role": "assistant",
"content": "<think>\nOkay, so I... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In a trapezoid with bases of lengths \\(a\\) and \\(b\\), a line segment \\(EF\\) parallel to the bases divides the trapezoid into two equal areas. Determine ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In $\\triangle ABC$, let $DEF$ be the intouch triangle and let $I_A$ be the $A$-excenter. Let $P$ be the foot from $D$ to $\\overline{EF}$, let $N=\\overline{... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In triangle \\( \\Delta ABC \\), let \\( P \\) and \\( Q \\) be isogonal conjugates. Define \\( D \\) as the intersection of \\( AQ \\) and the circumcircle o... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A box contains 5 red balls, 10 blue balls, and 5 yellow balls. Four balls are selected at random without replacement. If a red or yellow ball is selected, 3 n... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let $A=\\{1,2,3,4\\}$, and $f$ and $g$ be randomly chosen (not necessarily distinct) functions from $A$ to $A$. The probability that the range of $f$ and the ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( x, y, z \\geq 0 \\). Calculate the maximum value of the expression\n\n\\[\n\\frac{4\\sum\\limits_{cyc}{(x^3y-xy^3)}}{\\sum\\limits_{cyc}(x^2-y^2)^2}.\... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all pairs \\((a, b)\\) of positive integers such that \\(2a - 1\\) and \\(2b + 1\\) are coprime and \\(a + b\\) divides \\(4ab + 1\\)."
},
{
"rol... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For a given chord \\(MN\\) of a circle, consider all triangles \\(ABC\\) such that \\(AB\\) is a diameter of the given circle not intersecting \\(MN\\) and th... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Determine the sum of all positive integers \\( k \\ge 2 \\) such that there exists a positive integer \\( m \\) for which \\( k \\mid m^n + n^m \\) for every ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For non-negative real numbers \\(x, y, z\\), let \\(f(x, y, z) = -4(x+y+z)^2(x^2y^3+y^2z^3+z^2x^3) + 9(x^3+y^3+z^3)(xy^3+yz^3+x^3z) + 9(x+y+z)x^2y^2z^2\\). Fi... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given that \\( f \\) is a real-valued differentiable function such that \\( f(4x) = 2f(x) \\) for all real \\( x \\) and \\( \\int_{0}^{1} f(x) \\, dx = 2 \\)... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A list of five two-digit positive integers is written in increasing order on a blackboard. Each of the five integers is a multiple of 3, and each digit 0, 1, ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the number of mutually non-congruent, non-degenerate trapeziums whose sides are four distinct integers from the set $\\{5,6,7,8,9,10\\}$."
},
{
"... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all positive integers $n$ such that all prime factors of $2^n-1$ are less than or equal to $7$."
},
{
"role": "assistant",
"content": "<think... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Factor the expression \\(6k^4 - 24k^2 + 17k^3 - 68k - 14k^2 + 56\\)."
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. I need to fa... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all functions \\( f \\) from the complex numbers to the complex numbers such that \\( f(f(x)) = x \\)."
},
{
"role": "assistant",
"content": ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Circles $\\mathcal{P}$ and $\\mathcal{Q}$ have radii $p$ and $q,$ respectively, and circle $\\mathcal{R}$ is externally tangent to each of $\\mathcal{P}$ and ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Two cards are dealt at random from a standard deck of 52 cards (13 hearts, 13 clubs, 13 spades, and 13 diamonds). What is the probability that the first card... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Determine the minimum number of swaps required to change a given permutation of \\( n \\) elements into another given permutation."
},
{
"role": "assi... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For real numbers $x$ and positive integers $k$, determine when the limit $\\displaystyle\\lim_{n\\to\\infty} \\left(1+\\dfrac{x}{n^k}\\right)^n$ exists, and f... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Triangle $ABC$ is inscribed in circle $\\omega$ with $AB=5$, $BC=7$, and $AC=3$. The bisector of angle $A$ meets side $\\overline{BC}$ at $D$ and circle $\\om... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Denote \\( S_n := 1 + \\frac{1}{2} + \\ldots + \\frac{1}{n} \\). Also denote \\( F_{n, m} := S_n + S_m - S_{nm} \\). Find \\( \\inf\\{F_{n, m} \\mid n, m \\in... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the sum to infinity of the series \\(1 - \\frac{1}{3 \\cdot 3} + \\frac{1}{5 \\cdot 3^{2}} - \\frac{1}{7 \\cdot 3^{3}} + \\cdots\\)."
},
{
"role"... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the range of \\( u \\) where \\( u = x + y \\) and \\( x^2 + y^2 = 1 \\) with \\( x \\geq 0 \\) and \\( y \\geq 0 \\)."
},
{
"role": "assistant",... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given positive real numbers \\(a\\), \\(b\\), and \\(c\\) such that \\(abc = 1\\), calculate the minimum value of the expression\n\\[\n\\frac{1}{\\sqrt{b + \\... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve the equation $\\sin x \\sin 2x \\sin 3x + \\cos x \\cos 2x \\cos 3x = 1$."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all continuous functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) such that the identity \\( f(x-y) \\cdot f(y-z) \\cdot f(z-x) + 8 = 0 \\) holds for all ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(ABCD\\) be a quadrilateral. Choose points \\(P\\) on \\(AB\\) and \\(Q\\) on \\(CD\\) such that \\(\\frac{AP}{AB} = \\frac{DQ}{DC} = x\\). Compute the ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "How many positive integers \\( n > 2007 \\) satisfy the inequality \\( n - S(n) \\le 2007 \\), where \\( S(n) \\) is the sum of the squares of the digits of \... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Define an \"impossible number\" \\( n \\) if and only if \\( n \\) satisfies all of the following conditions:\n1. \\( \\frac{2n}{3} \\) is an integer.\n2. \\(... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Consider the points \\( O(0, 0, 0) \\), \\( A(\\sqrt{2}, 0, 0) \\), and \\( B(0, \\sqrt{2}, 0) \\). A sphere with radius \\( \\sqrt{6} \\) is tangent to \\( O... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given \\(a + b + c + d = 36\\), find the maximum value of \\(2a + 3b + 6c + 24d\\)."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I nee... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "A rectangle is inscribed in a circle with radius 10, with one of its sides along the diameter of the circle. What must be the ratio of the rectangle's length ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the largest positive integer \\( n \\) such that there exist rectangles \\( A_1, \\ldots, A_n \\) and \\( B_1, \\ldots, B_n \\) on the plane, each with s... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( d(n) \\) be the digit sum of a given integer \\( n \\). Find \\( x \\), where \\( x \\) is the result of applying \\( d(x) \\) an infinite number of t... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the values of real nonzero \\( a \\) such that the roots of the equations \\( x^2 + \\frac{8}{a}x - 2a = 0 \\) and \\( x^2 + \\frac{6}{a}x - a = 0 \\) ar... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "On the parabola \\( y = x^2 \\), points \\( A, B, C \\) are marked such that \\( A \\) is to the left of \\( B \\) and \\( C \\). The bisector of the angle \\... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( n \\) and \\( k \\) be two positive integers such that \\( n \\geq 2^{k-1} \\). In how many subsets of the set \\( S = \\{1, 2, 3, \\ldots, n\\} \\) i... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "In triangle \\(ABC\\) with \\(AB < AC\\), an altitude \\(AD\\) is drawn. Points \\(E\\) and \\(A\\) lie on opposite sides of \\(BC\\), with \\(E\\) on the cir... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find all triples of integers $(m,p,q)$ satisfying the equation\n\n$$2^mp^2+1=q^7$$\n\nwhere $m>0$, and $p$ and $q$ are prime numbers, or show that they do not... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Rationalize the denominator of the expression \\(\\frac{10}{\\sqrt[5]{16}}\\)."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "On rectangle $ABCD$ with $AB > BC$, points $X$ and $Y$ are on $AB$ and $CD$ such that $AXCY$ is a rhombus. The diagonal $BD$ intersects $CX$ and $AY$ at $M$ a... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the equation of the line with slope $\\frac{1}{2}$ and passing through the point (-3,2)."
},
{
"role": "assistant",
"content": "<think>\nOkay... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the smallest value of the expression \\( P = \\frac{1}{\\cos^6 a} + \\frac{1}{\\cos^6 b} + \\frac{1}{\\cos^6 c} \\) where \\( a, b, c \\) form an arithme... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Solve for \\( x \\) in the equation \\( 2^{x+3} = 6^{2x-1} \\)."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to solve the equat... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "For how many values of the coefficient \\( a \\) do the equations \n\\[ x^2 + ax + 1 = 0 \\]\nand \n\\[ x^2 - x - a = 0 \\]\nhave a common real solution?"
}... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( f:[0, \\alpha] \\rightarrow \\mathbb{R} \\) be such that \\( f(0)=0 \\) and \n\\[ f'(t) = f(t)^2 + t \\quad \\forall t \\in [0,\\alpha]. \\]\nDetermin... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given a real constant \\( A > 0 \\), consider the class of all triangles \\( T \\) with area \\( A \\). Find the maximum value of the area of the equilateral ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "10 lines and 10 circles divide the plane into at most \\( n \\) disjoint regions. Compute \\( n \\)."
},
{
"role": "assistant",
"content": "<think... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\(a, b, c > 0\\). Calculate the minimum value of the expression\n\\[ \\left(\\frac{a^2}{b^2}+\\frac{b^2}{c^2}\\right)\\left(\\frac{b^2}{c^2}+\\frac{c^2}{... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the radius of the sphere that touches each edge of a triangular pyramid (tetrahedron) where the length of each edge is $a$."
},
{
"role": "assist... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the value of \\( m \\) such that the curve \\( C: f(x) = \\frac{1}{3}x^3 + \\frac{1}{2}mx^2 + mx - 1 \\) touches the line \\( L: y = x + 2 \\)."
},
{... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Given triangle \\(ABC\\) with \\(\\angle CAB = 30^\\circ\\) and \\(\\angle ABC = 27.5^\\circ\\). Point \\(D\\) is inside triangle \\(ABC\\) such that \\(\\ang... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Determine the orthogonal projection of the straight line \\( p \\) given by \\(\\frac{x-1}{2} = \\frac{y+2}{-1} = \\frac{z+2}{1}\\) onto the plane \\(\\pi\\) ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( T(n) \\) be the sum of all greatest odd divisors of the numbers from \\( 1 \\) to \\( n \\). Find the value of \\( T(2^{2016} + 2^{2015}) \\)."
},
... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find \\(\\lim_{x \\to 0} \\frac{(1+9x)^{10} - (1+10x)^9}{x^2}\\)."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I have to find the limi... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Find the smallest positive integer \\( x \\) such that \\( x^2 \\) ends with 9009."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Define the function \\( f(n) \\) as follows:\n\\[ f(n) = \\begin{cases} \n\\frac{n}{2} & \\text{if } n \\text{ is even} \\\\\n\\frac{n + 1023}{2} & \\text{if ... |
[
{
"role": "system",
"content": "Please reason step by step, and put your final answer within \\boxed{}."
},
{
"role": "user",
"content": "Let \\( A \\in M_n(\\mathbb{R}) \\) and \\( E = \\{ M \\in M_n(\\mathbb{R}) \\mid AMA = 0 \\} \\). Find the dimension of \\( E \\)."
},
{
"role": "ass... |
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