Multilingual-MATH
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MATH datasets translated by Gemini-2.5-pro. • 3 items • Updated • 1
level stringclasses 2
values | question stringclasses 5
values | answer stringclasses 5
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Level 5 | Let \[f(x) = \left\{
\begin{array}{cl} ax+3, &\text{ if }x>2, \\
x-5 &\text{ if } -2 \le x \le 2, \\
2x-b &\text{ if } x <-2.
\end{array}
\right.\]Find $a+b$ if the piecewise function is continuous (which means that its graph can be drawn without lifting your pencil from the paper). | For the piecewise function to be continuous, the cases must "meet" at $2$ and $-2$. For example, $ax+3$ and $x-5$ must be equal when $x=2$. This implies $a(2)+3=2-5$, which we solve to get $2a=-6 \Rightarrow a=-3$. Similarly, $x-5$ and $2x-b$ must be equal when $x=-2$. Substituting, we get $-2-5=2(-2)-b$, which implies... |
Level 5 | A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he increases the number of members in each row by... | Let $x$ be the number of band members in each row for the original formation, when two are left over. Then we can write two equations from the given information: $$rx+2=m$$ $$(r-2)(x+1)=m$$ Setting these equal, we find: $$rx+2=(r-2)(x+1)=rx-2x+r-2$$ $$2=-2x+r-2$$ $$4=r-2x$$ We know that the band has less than 100 memb... |
Level 3 | What is the degree of the polynomial $(4 +5x^3 +100 +2\pi x^4 + \sqrt{10}x^4 +9)$? | This polynomial is not written in standard form. However, we don't need to write it in standard form, nor do we need to pay attention to the coefficients. We just look for the exponents on $x$. We have an $x^4$ term and no other term of higher degree, so $\boxed{4}$ is the degree of the polynomial. |
Level 3 | Evaluate $\left\lceil3\left(6-\frac12\right)\right\rceil$. | Firstly, $3\left(6-\frac12\right)=18-1-\frac12=17-\frac12$. Because $0\le\frac12<1$, we have $\left\lceil17-\frac12\right\rceil=\boxed{17}$. |
Level 3 | Sam is hired for a 20-day period. On days that he works, he earns $\$$60. For each day that he does not work, $\$$30 is subtracted from his earnings. At the end of the 20-day period, he received $\$$660. How many days did he not work? | Call $x$ the number of days Sam works and $y$ the number of days he does not. We can set up the following system of equations to represent the given information: \begin{align*}
x+y &= 20 \\
60x - 30y &= 660 \\
\end{align*} The first equation represents the total number of days Sam works, and the second equation represe... |
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