Eunice-Labs/aria-7b-merged
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You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Twenty-five coins are being separated into piles in the following manner. First, they are randomly divided into two groups. Then, any one of the existing groups is agai... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>The first term of a sequence is 934. Each following term is equal to the sum of the digits of the previous term, multiplied by 13.
Find the 2013th term of the sequence... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Given the function \( y = \frac{\sqrt{4 - x^2}}{\lg (|x| + x)} \), determine its domain:
(A) \( [-2, 0) \cup (0, 2] \).
(B) \( \left(0, \frac{1}{2}\right) \cup \left(\f... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Find all extrema of the function \( y = \frac{2}{3} \cos \left(3x - \frac{\pi}{6}\right) \) on the interval \( (0, \pi / 2) \).<|Assistant|><think>
To find the extrema ... | medium |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Given \( x, y, z \in \mathbf{R}_{+} \) and \( n \in \mathbf{N}^{*} \), prove that \(\frac{x}{n x+y+z}+\frac{y}{x+n y+z}+\frac{z}{x+y+n z} \leqslant \frac{3}{n+2} \).<|A... | medium |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Xiaohu uses 6 equilateral triangles with side lengths of 1 to form shapes on a table without overlapping. Each triangle must share at least one side fully with another ... | medium |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>A sequence of two distinct numbers is extended in two ways: one to form a geometric progression and the other to form an arithmetic progression. The third term of the g... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Two people, A and B, are collaborating to type a document. Initially, A types 100 characters per minute, and B types 200 characters per minute. When they reach half of ... | medium |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>The sum \( A = \left\lfloor \frac{8}{9} \right\rfloor + \left\lfloor \frac{8^2}{9} \right\rfloor + \cdots + \left\lfloor \frac{8^{2014}}{9} \right\rfloor \) when divide... | medium |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Say that a 4-digit positive integer is *mixed* if it has 4 distinct digits, its leftmost digit is neither the biggest nor the smallest of the 4 digits, and its rightmos... | medium |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Evaluate $\sqrt{12 +\!\sqrt{12 + \!\sqrt{12 + \!\sqrt{12 + \cdots}}}}$.<|Assistant|><think>
I need to evaluate the infinite nested radical $x = \sqrt{12 + \sqrt{12 + \s... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>In a sequence, each term after the first two terms is the mean of all the terms which come before that term. The first term is 8 and the tenth term is 26. What is the s... | medium |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Let \( a \) and \( b \) be integers such that
\[ |a + b| > |1 + ab|. \]
Prove that \( ab = 0 \).<|Assistant|><think>
We are given that \( a \) and \( b \) are integer... | medium |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Let \( a, b, c, \) and \( d \) be four distinct positive real numbers that satisfy the equations:
\[
\left(a^{2012}-c^{2012}\right)\left(a^{2012}-d^{2012}\right)=2011
... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>The product \( 29 \cdot 28 \) when \( P \) is divided by \( D \), the quotient is \( Q \), and the remainder is \( R \). When \( Q \) is divided by \( D' \), the quotie... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Given a regular $n$-gon $A_{1} A_{2} \ldots A_{n}$ on a plane:
a) Prove that if $n$ is an even number, then it is possible to choose the signs plus and minus in the ex... | medium |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>In trapezoid \(A B C D\), the bases \(A D\) and \(B C\) are 8 and 18, respectively. It is known that the circumscribed circle of triangle \(A B D\) is tangent to lines ... | medium |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>The time on a digital clock is 5:55. How many minutes will pass before the clock next shows a time with all digits identical?
(A) 71
(B) 72
(C) 255
(D) 316
(E) 436<|Ass... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Two parabolas are the graphs of the equations $y=2x^2-10x-10$ and $y=x^2-4x+6$. Find all points where they intersect. List the points in order of increasing $x$-coordin... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Let \( A A_1 \) be the median of triangle \( ABC \). Prove that angle \( A \) is acute if and only if \( A A_1 > \frac{1}{2} BC \).<|Assistant|><think>
I need to prove ... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Find all monotonic and invertible functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy
\[ f(x) + f^{-1}(x) \equiv 2x, \quad x \in \mathbb{R}. \]<|Assistan... | medium |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Suppose that $a$ varies inversely with $b^2$. If $a=9$ when $b=2$, find the value of $a$ when $b=3$.<|Assistant|><think>
The problem states that $a$ varies inversely wi... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>A train passes a bystander in $t_{1}$ seconds, and at the same speed it crosses a bridge of length $a$ meters in $t_{2}$ seconds. Find the length and speed of the train... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>A finite sequence of positive integers $ m_i$ for $ i\equal{}1,2,...,2006$ are defined so that $ m_1\equal{}1$ and $ m_i\equal{}10m_{i\minus{}1} \plus{}1$ for ... | medium |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>The first 9 positive odd integers are placed in the magic square so that the sum of the numbers in each row, column and diagonal are equal. Find the value of $A + E$ .... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Find the point \( M^{\prime} \) that is symmetric to the point \( M \) with respect to the plane.
\( M(-1, 0, -1) \)
\( 2x + 6y - 2z + 11 = 0 \)<|Assistant|><think>
I... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Determine the number of solutions to
\[2\sin^3 x - 5 \sin^2 x + 2 \sin x = 0\]in the range $0 \le x \le 2 \pi.$<|Assistant|><think>
I need to solve \(2\sin^3 x - 5 \sin... | medium |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>We consider \( n \) numbers each being 1 or -1 on a board. We take two of them and replace them with 1 if they are equal and -1 otherwise. We repeat this operation unti... | medium |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>A kilogram of one type of candy costs 80 kopecks more than another. Andrey and Yura bought 150 grams of candies each, including both types. Andrey bought twice as much ... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>The first three terms of an arithmetic progression are $x - 1, x + 1, 2x + 3$, in the order shown. The value of $x$ is:
$\textbf{(A)}\ -2\qquad\textbf{(B)}\ 0\qquad\te... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>A rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n+76$ equal squares. Find $n$ .<|Assistant|><think>
Let the dimensi... | medium |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>The magician Magimático calls three people from the audience: Ana, Beto, and Caio, to help in his mathematic magic. He tells each one to think of a number from 1 to 50,... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Given that \( x, y, z \) are distinct real numbers, and \( x + \frac{1}{y} = y + \frac{1}{z} = z + \frac{1}{x} \), prove that \( x^2 y^2 z^2 = 1 \).<|Assistant|><think>... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Given \( 2014 + y = 2015 + x = 2016 + z \) and \( y \times x \times z = 504 \), find \( y \times x + x \times z \).<|Assistant|><think>
We are given \(2014 + y = 2015 +... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Methanol can be gently oxidized with hot copper metal. What is(are) the product(s) of this oxidation? $ \textbf{(A) }\text{Acetic acid}\qquad\textbf{(B) }\text{Carbon d... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Define points on the plane with both coordinates as integers as lattice points. How many lattice points do the curve $y=\frac{1}{5}(x^{2}-x+1)$ pass through?
(A) 0.
(B)... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>If \((x+y)^{2}-(x-y)^{2}=20\), then \(xy\) is equal to:
(a) 0
(b) 1
(c) 2
(d) 5
(e) 10<|Assistant|><think>
I am given the equation \((x + y)^2 - (x - y)^2 = 20\) and ne... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>The numbers from 1 to 6 are to be placed at the intersections of three circles, one number in each of the six squares. The number 6 is already placed. Which number must... | medium |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>David, when submitting a problem for CMIMC, wrote his answer as $100\tfrac xy$ , where $x$ and $y$ are two positive integers with $x<y$ . Andrew interpreted the e... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>The circle graph shown illustrates the results of a survey taken by the Fermat H.S. Student Council to determine the favourite cafeteria food. How many of the 200 stude... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>
Are the vectors \( c_{1} \) and \( c_{2} \), constructed from the vectors \( a \) and \( b \), collinear?
Given:
\[ a = \{1; 0; 1\} \]
\[ b = \{-2; 3; 5\} \]
\[ c_{1}... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>In \(\triangle ABC\), \(AB = 9\), \(BC = 8\), and \(AC = 7\). The bisector of \(\angle A\) meets \(BC\) at \(D\). The circle passing through \(A\) and touching \(BC\) a... | medium |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>For which \( k \in \mathbb{N} \) do there exist non-negative integers \( a \) and \( b \) such that
\[
2^a 3^b = k(k + 1) ?
\]<|Assistant|><think>
I need to find all... | medium |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>There are 2022 points marked on a line such that each two neighboring points are at the same distance from each other. Half of the points are painted red, and the other... | medium |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Find the solutions of the system of equations
$$
\begin{aligned}
a^{3}+3 a b^{2}+3 a c^{2}-6 a b c & =1 \\
b^{3}+3 b a^{2}+3 b c^{2}-6 a b c & =1 \\
c^{3}+3 c a^{2}+3 ... | medium |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Given $a_{1}, a_{2}, \ldots, a_{n}$ as strictly positive real numbers, prove the geometric-harmonic inequality:
$$
\sqrt[n]{a_{1} a_{2} \ldots a_{n}} \geq \frac{n}{\fr... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Find a one-to-one correspondence between the open interval \((0, 1)\) and the closed interval \([0, 1]\).<|Assistant|><think>
To establish a one-to-one correspondence b... | medium |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Seven friends have picked a total of one hundred mushrooms, each having picked a different number. Show that there are three among them who have picked at least 50 mush... | medium |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>176. \(C\) is a sufficient condition for \(B\), and \(B\) is a necessary condition for \(A\). Will \(C\) be a necessary (sufficient) condition for \(A\)?<|Assistant|><t... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Ostap Bender, in an interview for a chess magazine about a simultaneous exhibition in Vasyuki, reported that in one game he had 3 times fewer pieces than his opponent a... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Let \( a_{n}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\ldots+(-1)^{n} \frac{1}{2 n+1} \).
a) Prove that \(\lim _{n \rightarrow \infty} a_{n}=\frac{\pi}{4}\).
b) Prove th... | medium |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>
Given that \( k \) and \( m \) are real numbers, the inequality
$$
\left|x^{2}-k x-m\right| \leqslant 1
$$
holds for all \( x \in [a, b] \). Prove that:
$$
b-a \leqsl... | medium |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Scientists gathered at a congress, some of whom are friends. It turned out that any two of them with the same number of friends at the congress do not have any mutual f... | medium |
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