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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 |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>In an isosceles triangle, the length of each leg is $b$. The distance between the bases of the angle bisectors drawn to the legs is $m$. Find the base of the triangle.<... | medium |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Prove that if the real numbers \( a, b, c \) are each at least 2, then
$$
\log _{b+c} a^{2}+\log _{c+a} b^{2}+\log _{a+b} c^{2} \geqq 3
$$<|Assistant|><think>
The prob... | 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 a strange city called "Impossibility City," people must include the word "impossible" in their speech. At the city gate, there is a question: "In a triangle, what is... | 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 shop sells the same type of pastries in two sizes of gift boxes: a large box costing 85.6 yuan containing 32 pastries, and a small box costing 46.8 yuan containing 15... | 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 large square room, two square carpets were brought in, with the side of one carpet being twice the side of the other. When they were placed in opposite corners 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|>Find all real solutions to
\[\frac{1}{(x - 1)(x - 2)} + \frac{1}{(x - 2)(x - 3)} + \frac{1}{(x - 3)(x - 4)} = \frac{1}{6}.\]Enter all solutions, separated by commas.<|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|>The edge length of cube \( A B C D A_{1} B_{1} C_{1} D_{1} \) is \( a \). Points \( E \) and \( F \) are the midpoints of edges \( B B_{1} \) and \( C C_{1} \) respecti... | 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 the acute angles $\alpha, \beta, \gamma$ that satisfy $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$, prove that $\cot \alpha \cot \beta + \cot \beta \cot \ga... | medium |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Since \( 24 = 3 + 5 + 7 + 9 \), the number 24 can be written as the sum of at least two consecutive odd positive integers.
(a) Can 2005 be written as the sum of at lea... | 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 sum of all values of $x$ such that $2^{x^2-3x-2} = 4^{x - 4}$.<|Assistant|><think>
I need to solve $2^{x² - 3x - 2} = 4^{x - 4}$. I can rewrite the right side ... | 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 the USA, standard letter-size paper is 8.5 inches wide and 11 inches long. What is the largest integer that cannot be written as a sum of a whole number (possibly z... | 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 distance between the roots of the quadratic polynomial \(x^2 + px + q\) is 1. Find the coefficients \(p\) and \(q\), given that they are prime numbers.<|Assistant|>... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Prove that the product of two even or two odd functions is an even function.<|Assistant|><think>
I need to show that the product of two even functions, or two odd funct... | 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 decreasing sequence $a, b, c$ is a geometric progression, and the sequence $19a, \frac{124b}{13}, \frac{c}{13}$ is an arithmetic progression. Find the common ratio ... | medium |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Of the following sets, the one that includes all values of $x$ which will satisfy $2x - 3 > 7 - x$ is:
$\textbf{(A)}\ x > 4 \qquad \textbf{(B)}\ x < \frac {10}{3} \qqu... | 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 \( c \) mentioned in Lagrange's theorem for the function \( y = x^{2} + 6x + 1 \) on the interval \([-1, 3]\).<|Assistant|><think>
Lagrange's Mean Value ... | 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 \( y = f(x) \) is a linear function and \( x_1, x_2, x_3, \ldots \) is an arithmetic progression, prove that the sequence \( f(x_1), f(x_2), f(x_3), \ldots \... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Convex quadrilateral \(ABCD\) is given, and lines are extended from points \(A\), \(B\), \(C\), and \(D\) to points \(B'\), \(C'\), \(D'\), and \(A'\) respectively, suc... | 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 the sum $K A N$ each letter stands for a different digit.
$$
\frac{+G A}{R O O}
$$
What is the answer to the subtraction $R N$ ?
$$
\underline{-K G}
$$
A 10
B 11... | 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 finite set of points \( S \) such that every line passing through two points of \( S \) also passes through a third point of \( S \), show that all the points i... | medium |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Stephen had a 10:00 a.m. appointment 60 km from his home. He averaged 80 km/h for the trip and arrived 20 minutes late for the appointment. At what time did he leave hi... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>How many 10-digit positive integers have all digits either 1 or 2, and have two consecutive 1's?
<|Assistant|><think>
I need to find the number of 10-digit numbers comp... | 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 \(a > 0\), let \(f: (0, +\infty) \rightarrow \mathbf{R}\) be a function such that \(f(a) = 1\). If for any positive real numbers \(x\) and \(y\), the following co... | medium |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>If \( p \) and \( q \) are two prime numbers, and \( q = p + 2 \), prove that \( p^q + q^p \) is divisible by \( p + q \).<|Assistant|><think>
I need to prove that 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|>If 7 is subtracted from a certain three-digit number, the resulting difference is divisible by 7. If 8 is subtracted, the resulting difference is divisible by 8. If 9 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|>A triangle \(ABC\) is given. Find a point \(M\) on the line \(AB\) such that the sum of the radii of the circumscribed circles of triangles \(ACM\) and \(BCM\) is minim... | medium |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Triangle $ABC$ with vertices $A(-2, 0)$, $B(1, 4)$ and $C(-3, 2)$ is reflected over the $y$-axis to form triangle $A'B'C'$. What is the length of a segment drawn from $... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Sasha and Maxim (out of boredom) wrote non-zero digits in the cells of a $100 \times 100$ grid. After this, Sasha said that among the 100 numbers formed by the digits 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|>The graph of \[y^4 - 4x^4 = 2y^2 - 1\]is the union of the graphs of two different conic sections. Which two types of conic sections are they?
(Write your answer as a l... | 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 sum of two positive integers is 60 and their least common multiple is 273. What are the two integers?<|Assistant|><think>
Let the two positive integers be $a$ and $... | 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 \( p > 3 \) be a prime number. Show that \( p^2 - 1 \) is a multiple of 12.<|Assistant|><think>
I need to show that for a prime $p > 3$, $p^2 - 1$ is a multiple of ... | 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 \( AM^2 = CM \cdot MN \) or \( \frac{AM}{MN} = \frac{CM}{AM} \), and since \( AM = MK \), we have \( \frac{AM}{MN} = \frac{CM}{MK} \).
Consider triangles \(... | medium |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Prove that if the real numbers \(a\), \(b\), and \(c\) satisfy \(a^{2}+b^{2}+c^{2}=3\), then
\[
\frac{a^{2}}{2+b+c^{2}}+\frac{b^{2}}{2+c+a^{2}}+\frac{c^{2}}{2+a+b^{2}}... | 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 have a standard deck of 52 cards, with 4 cards in each of 13 ranks. We call a 5-card poker hand a full house if the hand has 3 cards of one rank and 2 cards of anot... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Prove that if \( g = a_{1} x + a_{2} x^{2} + \ldots \) and \( r \) is any rational number, then
\[
(1+g)^{r} = 1 + r g + \frac{r(r-1)}{2!} g^{2} + \ldots + \frac{r(r... | medium |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Solve the system of equations:
\[
\begin{cases}
y + z = x y z \\
z + x = x y z \\
x + y = x y z
\end{cases}
\]<|Assistant|><think>
The system of equations is:
\[
\begi... | 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 you roll four standard, fair six-sided dice, the top faces of the dice can show just one value (for example, $3333$ ), two values (for example, $2666$ ), three val... | medium |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Usually, I go up the escalator in the subway. I have calculated that when I walk up the moving escalator, I ascend 20 steps, and it takes me exactly 60 seconds. My wife... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>How many six-letter words formed from the letters of AMC do not contain the substring AMC? (For example, AMAMMC has this property, but AAMCCC does not.)<|Assistant|><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|>Prove that two consecutive Fibonacci numbers \( F_{n-1} \) and \( F_{n} \) (for \( n \geq 1 \)) are coprime.<|Assistant|><think>
I need to show that \( \gcd(F_n, F_{n-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|>Let \( f(x) \) be an odd function defined on \( \mathbf{R} \), with \( f(1)=2 \). When \( x>0 \), \( f(x) \) is an increasing function. Additionally, for any \( x, y \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|>There are 9 people lost in the mountains, and the food they have is enough to last for 5 days. After one day, these 9 people meet another group of lost people who have ... | 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 \(a, b, c, x, y, z\) are all positive numbers, and \(a^{2}+b^{2}+c^{2}=25\), \(x^{2}+y^{2}+z^{2}=36\), and \(a x+b y+c z= 30\), determine \(\frac{a+b+c}{x+y+... | 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 buses operate on a circular route, traveling at the same speed with an interval of 21 minutes. What will the interval be if 3 buses operate on the same route at the... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>Which parallelograms can be inscribed in a circle? Which parallelograms can be circumscribed around a circle? Which parallelograms can be both inscribed in and circumsc... | easy |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>For \( n \in \mathbf{N}^{*} \), let \( T_{n}=\frac{1}{1^{4}}+\frac{1}{2^{4}}+\frac{1}{3^{4}}+\cdots + \frac{1}{n^{4}} \). Prove that \( T_{n} \leqslant \frac{11}{10} \)... | medium |
You are ARIA, a math reasoning assistant. Think through problems carefully but concisely inside <think> tags, then provide a clean final answer.<|User|>At a Russian-French meeting, there were no representatives from other countries. The total amount of money possessed by the French was greater than the total amount of ... | easy |
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