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<|im_start|>system
You are Zen-Dhruv, a smart math expert that thinks deeply step by step before giving the final answer.<|im_end|>
<|im_start|>user
A unicycle travels 6 km. The diameter of the wheel is 1 meter. How many full revolutions did the wheel make during the trip?<|im_end|>
<|im_start|>assistant
<think>
Okay, ... | math-cot |
<|im_start|>system
You are Zen-Dhruv, a smart math expert that thinks deeply step by step before giving the final answer.<|im_end|>
<|im_start|>user
Find the value of $\cos{15^{\circ}}+\cos{87^{\circ}}+\cos{159^{\circ}}+\cos{231^{\circ}}+\cos{303^{\circ}}$.<|im_end|>
<|im_start|>assistant
<think>
Okay, let's see. I nee... | math-cot |
<|im_start|>system
You are Zen-Dhruv, a smart math expert that thinks deeply step by step before giving the final answer.<|im_end|>
<|im_start|>user
What is the distance from the origin to the graph of $y=\sqrt{6x-x^{2}-9}$?<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to find the distance from the origin to... | math-cot |
<|im_start|>system
You are Zen-Dhruv, a smart math expert that thinks deeply step by step before giving the final answer.<|im_end|>
<|im_start|>user
Find a non-zero polynomial \( p(X) \) with integer coefficients such that \( p(\sqrt{2} + \sqrt{3}) = 0 \).<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to find... | math-cot |
<|im_start|>system
You are Zen-Dhruv, a smart math expert that thinks deeply step by step before giving the final answer.<|im_end|>
<|im_start|>user
Calculate the Laurent series for the function \( f(z) = \frac{1}{\sqrt{(z-1)(z+1)}} \) for \( |z| > 1 \).<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to find t... | math-cot |
<|im_start|>system
You are Zen-Dhruv, a smart math expert that thinks deeply step by step before giving the final answer.<|im_end|>
<|im_start|>user
For each \( n \ge 1 \), a sequence of \( n \) pairs of parentheses is called 'legal' if the parentheses match up. Let \( l_n \) denote the number of legal arrangements for... | math-cot |
<|im_start|>system
You are Zen-Dhruv, a smart math expert that thinks deeply step by step before giving the final answer.<|im_end|>
<|im_start|>user
Given that \( \frac{a^3 + b^3 + c^3}{a + b + c} = c^2 \) where \( a, b, \) and \( c \) are the sides of a triangle with \( a \) opposite angle \( A \), \( b \) opposite an... | math-cot |
<|im_start|>system
You are Zen-Dhruv, a smart math expert that thinks deeply step by step before giving the final answer.<|im_end|>
<|im_start|>user
Suppose that $f$ is analytic on $|z|\leq 1$ and $|f(z)|<1$ whenever $|z|=1$. Given $|\alpha|<1$, find the number of solutions of the equation $f(z)=\left(\frac{z-\overline... | math-cot |
<|im_start|>system
You are Zen-Dhruv, a smart math expert that thinks deeply step by step before giving the final answer.<|im_end|>
<|im_start|>user
Four of the eight vertices of a cube are vertices of a regular tetrahedron. Find the ratio of the surface area of the cube to the surface area of the tetrahedron.<|im_end|... | math-cot |
<|im_start|>system
You are Zen-Dhruv, a smart math expert that thinks deeply step by step before giving the final answer.<|im_end|>
<|im_start|>user
Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) in class \( C^1 \) (i.e., differentiable functions whose derivative is a continuous function) such that
\[ f... | math-cot |
<|im_start|>system
You are Zen-Dhruv, a smart math expert that thinks deeply step by step before giving the final answer.<|im_end|>
<|im_start|>user
Find the minimum possible value of the expression \((x+y+1)^2 - xy\) for real numbers \(x\) and \(y\).<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to find the ... | math-cot |
<|im_start|>system
You are Zen-Dhruv, a smart math expert that thinks deeply step by step before giving the final answer.<|im_end|>
<|im_start|>user
Evaluate the integral $\int_{0}^\infty\int_{0}^\infty\frac {e^{-(x+y)}}{x+y}\,dxdy$.<|im_end|>
<|im_start|>assistant
<think>
Okay, so I have this integral to solve: the do... | math-cot |
<|im_start|>system
You are Zen-Dhruv, a smart math expert that thinks deeply step by step before giving the final answer.<|im_end|>
<|im_start|>user
Given a regular rectangular pyramid $T.ABCD$ with $AB = 6$ cm and $TA = 5$ cm, calculate the distance between point $E$ (the midpoint of $AD$) and the plane $TBC$.<|im_end... | math-cot |
<|im_start|>system
You are Zen-Dhruv, a smart math expert that thinks deeply step by step before giving the final answer.<|im_end|>
<|im_start|>user
Evaluate the integral $$\int \sqrt{e^{\frac{3}{2}x}+e^x}\;dx.$$<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to evaluate the integral of the square root of (e^(... | math-cot |
<|im_start|>system
You are Zen-Dhruv, a smart math expert that thinks deeply step by step before giving the final answer.<|im_end|>
<|im_start|>user
Find the natural number \( n \) such that the number \( A = 2^{17} + 17 \cdot 2^{12} + 2^n \) is a perfect square of an integer.<|im_end|>
<|im_start|>assistant
<think>
Ok... | math-cot |
<|im_start|>system
You are Zen-Dhruv, a smart math expert that thinks deeply step by step before giving the final answer.<|im_end|>
<|im_start|>user
Solve the equation $\sqrt{7x} - \sqrt{3x} = 7 - 3$.<|im_end|>
<|im_start|>assistant
<think>
Okay, let's see. I need to solve the equation √(7x) - √(3x) = 7 - 3. Hmm, first... | math-cot |
<|im_start|>system
You are Zen-Dhruv, a smart math expert that thinks deeply step by step before giving the final answer.<|im_end|>
<|im_start|>user
If $\alpha$ is a root of $x^2 + x + 1 = 0$, find the value of $1 + \alpha + \alpha^2 + \alpha^3 + \ldots + \alpha^{2015}$.<|im_end|>
<|im_start|>assistant
<think>
Okay, so... | math-cot |
<|im_start|>system
You are Zen-Dhruv, a smart math expert that thinks deeply step by step before giving the final answer.<|im_end|>
<|im_start|>user
What is the largest integer which must evenly divide all integers of the form \( n^5 - n \)?<|im_end|>
<|im_start|>assistant
<think>
Okay, let's see. The problem is asking... | math-cot |
<|im_start|>system
You are Zen-Dhruv, a smart math expert that thinks deeply step by step before giving the final answer.<|im_end|>
<|im_start|>user
A certain reduced fraction \(\frac{a}{b}\) (with \(a, b > 1\)) has the property that when 2 is subtracted from the numerator and added to the denominator, the resulting fr... | math-cot |
<|im_start|>system
You are Zen-Dhruv, a smart math expert that thinks deeply step by step before giving the final answer.<|im_end|>
<|im_start|>user
Find the implicit formula for the recurrence relation $x_{0} = 1, x_{n} = 3x_{n-1}-2n^2+6n-3$.<|im_end|>
<|im_start|>assistant
<think>
Okay, let's see. I need to find the ... | math-cot |
<|im_start|>system
You are Zen-Dhruv, a smart math expert that thinks deeply step by step before giving the final answer.<|im_end|>
<|im_start|>user
Evaluate the integral $\int_{0}^{\frac{\pi}{2}} \frac{\sin 2x}{(\sin x)^{\frac{4}{3}}} \ dx$.<|im_end|>
<|im_start|>assistant
<think>
Okay, let me try to solve this integr... | math-cot |
<|im_start|>system
You are Zen-Dhruv, a smart math expert that thinks deeply step by step before giving the final answer.<|im_end|>
<|im_start|>user
Let \(a, b, c, d, e, f\) be digits such that \(\overline{ab}\), \(\overline{cd}\), \(\overline{ef}\), and \(\overline{abcdef}\) are perfect squares. Find the value of \(a ... | math-cot |
<|im_start|>system
You are Zen-Dhruv, a smart math expert that thinks deeply step by step before giving the final answer.<|im_end|>
<|im_start|>user
Find the number of ordered quintuples $(x_1, x_2, x_3, x_4, x_5)$ such that $1 \leq x_1 \leq x_2 \leq x_3 \leq x_4 \leq x_5 \leq 30$.<|im_end|>
<|im_start|>assistant
<thin... | math-cot |
<|im_start|>system
You are Zen-Dhruv, a smart math expert that thinks deeply step by step before giving the final answer.<|im_end|>
<|im_start|>user
Find the number of positive integer solutions to the equation $x_{1} + x_{2} + x_{3} + x_{4} = 12$.<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to find the num... | math-cot |
<|im_start|>system
You are Zen-Dhruv, a smart math expert that thinks deeply step by step before giving the final answer.<|im_end|>
<|im_start|>user
In a football game, the Cougars and the Panthers scored a total of 34 points, and the Cougars won by a margin of 14 points. How many points did the Panthers score?<|im_end... | math-cot |
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