problem
stringlengths 74
370
| solution
stringlengths 11
51
| answer
stringlengths 1
43
| choices
sequencelengths 4
4
| correct_choice_idx
int64 1
4
| level
stringclasses 3
values | source
stringlengths 24
30
|
|---|---|---|---|---|---|---|
When two dice are rolled together, what is the probability that the number appearing on one of the dice is prime and their sum is at least 6? | $\boxed{\frac{7}{18}}$ | $\frac{7}{18}$ | [
"$\\frac{2}{9}$",
"$\\frac{5}{9}$",
"$\\frac{7}{18}$",
"$\\frac{13}{18}$"
] | 3 | easy | 1403_ordibehesht-Riazi-23 |
When two dice are rolled, what is the probability that the numbers shown are neither consecutive nor equal? | $\boxed{\frac{5}{9}}$ | $\frac{5}{9}$ | [
"$\\frac{5}{12}$",
"$\\frac{5}{9}$",
"$\\frac{2}{3}$",
"$\\frac{1}{6}$"
] | 2 | easy | 1403_ordibehesht-Tajrobi-131 |
Given the sets $E = C \cup D$, $D = (B' - A) \cup (B' - A')$, and $C = (A' - B) \cup (A' - B')$, determine the complement of set $E$, denoted as $E'$. | $\boxed{A \cap B}$ | $A \cap B$ | [
"$A' \\cup B'$",
"$A' \\cap B'$",
"$A \\cup B$",
"$A \\cap B$"
] | 4 | easy | 1403_ordibehesht-Ensani-2 |
If $f$ is a constant function and $f = \{(m, 3m-1), (-1, k^2 - k), (k^2 - k, 2)\}$, what is the value of the product of the members of the domain of $f$? | $\boxed{-2}$ | $-2$ | [
"$2$",
"$-2$",
"$8$",
"$-8$"
] | 2 | easy | 1403_ordibehesht-Ensani-3 |
If $f(x) = \begin{cases} |x| \cdot \text{sign}(-x) & \lfloor x \rfloor \geq 0 \\ 2 - \text{sign}(-x) & \lfloor x \rfloor < 0 \end{cases}$ then what is the value of $f\left(\frac{1}{2}\right) + f\left(-\frac{1}{3}\right)$? | $\boxed{\frac{1}{2}}$ | $\frac{1}{2}$ | [
"$\\frac{1}{2}$",
"$-\\frac{1}{4}$",
"-$\\frac{5}{6}$",
"$\\frac{10}{3}$"
] | 1 | easy | 1403_ordibehesht-Ensani-4 |
If $g(x) = ax + h$ and $f(x) = -mx - h$ pass through the point $(-2, 3)$ and $f\left(-\frac{5}{4}\right) = g(-5)$, what is the value of $\frac{m}{a}$? | $\boxed{4}$ | $4$ | [
"$2$",
"$3$",
"$4$",
"$5$"
] | 3 | easy | 1403_ordibehesht-Ensani-5 |
If $\sqrt{3}$ is the geometric mean of the roots of the equation $mx^2 - 4x + m^2 - 4 = 0$, what is the sum of the roots of this equation? | $\boxed{-4}$ | $-4$ | [
"$1$",
"$-1$",
"$4$",
"$-4$"
] | 4 | easy | 1403_ordibehesht-Ensani-7 |
A cake was initially divided equally among people at a party. Four people left the party, and the remaining cake was divided among the remaining people such that each person's share was $\frac{1}{3}$ more than their previous share. If half of the cake was divided among the initial people, what share of a cake does each person receive? | $\boxed{\frac{1}{12}}$ | $\frac{1}{12}$ | [
"$\\frac{1}{6}$",
"$\\frac{1}{16}$",
"$\\frac{1}{12}$",
"$\\frac{1}{24}$"
] | 3 | easy | 1403_ordibehesht-Ensani-8 |
The housing rent index in $1995$ is $6$ units higher than the rent index in $1994$ and the inflation rate of the rent index in $1994$ is $44\%$. If the inflation rate of this index is the same each year, what is the inflation rate of housing rent in $1995$ compared to $1994$? | $\boxed{20}$ | $20$ | [
"$22$",
"$20$",
"$18$",
"$16$"
] | 2 | easy | 1403_ordibehesht-Ensani-13 |
If the profit function $P(x)$ from selling $x$ units of a product for a company is given by $P(x) = 200(-x^2 - 540x + 112000)$, for how many units sold does the company neither gain profit nor incur a loss? | $\boxed{160}$ | $160$ | [
"$800$",
"$700$",
"$160$",
"$140$"
] | 3 | easy | 1403_ordibehesht-Ensani-14 |
Using the digits $0, 1, 2, 4, 5, 7$, how many three-digit odd numbers can be formed without repeating any digits that are not multiples of $5$? | $\boxed{32}$ | $32$ | [
"$48$",
"$40$",
"$36$",
"$32$"
] | 4 | easy | 1403_ordibehesht-Ensani-15 |
Maryam wants to place $6$ different books in two equal rows (with equal number of books in each row) on a shelf randomly. What is the probability that she will place two books with titles "Mathematics" and "Literature" in the same row? | $\boxed{ \frac{4}{15} }$ | $ \frac{4}{15} $ | [
"$ \\frac{4}{15} $",
"$\\frac{1}{5}$",
"$\\frac{5}{6}$",
"$\\frac{9}{10}$"
] | 1 | easy | 1403_ordibehesht-Ensani-16 |
If the sequences $a_n = \frac{1}{n^2 + 1}$ and $b_n = \frac{2n + 1}{n + 1}$, what is the result of $b_4 - a_3$? | $\boxed{ \frac{1}{7} }$ | $ \frac{1}{7} $ | [
"$ \\frac{1}{2} $",
"$ -\\frac{1}{2} $",
"$ \\frac{1}{7} $",
"$ -\\frac{1}{7} $"
] | 3 | easy | 1403_ordibehesht-Ensani-17 |
In an arithmetic sequence, the sum of the third and the twenty-eighth terms is $61$ units more than the fifth term. What is the twenty-sixth term of this sequence? | $\boxed{61}$ | $61$ | [
"$76$",
"$61$",
"$55$",
"$43$"
] | 2 | easy | 1403_ordibehesht-Ensani-18 |
The first term and the common ratio of a geometric sequence are $1458$ and $\frac{1}{3}$ respectively. If the $n$th term of this sequence is equal to $2$, what is the value of $n$? | $\boxed{7}$ | $7$ | [
"$9$",
"$8$",
"$6$",
"$7$"
] | 4 | easy | 1403_ordibehesht-Ensani-19 |
In the equation $\frac{(2x)^5 \times 21^{3}}{15^3 \times 5^2} = 7^3$, what is the value of $x$? | $\boxed{2.5}$ | $2.5$ | [
"$2.5$",
"$3$",
"$4.5$",
"$5$"
] | 1 | easy | 1403_ordibehesht-Ensani-20 |