problem stringlengths 10 5.15k | answer stringlengths 0 1.22k | solution stringlengths 0 11.1k | reward float64 0 1 | length float64 172 8.19k | correct_length float64 -1 8.19k | incorrect_length float64 -1 8.19k |
|---|---|---|---|---|---|---|
The cost of 1 piece of gum is 1 cent. What is the cost of 1000 pieces of gum, in dollars? | 10.00 | 0 | 508.4375 | -1 | 508.4375 | |
The isosceles right triangle $ABC$ has right angle at $C$ and area $12.5$. The rays trisecting $\angle ACB$ intersect $AB$ at $D$ and $E$. What is the area of $\triangle CDE$? | \frac{50-25\sqrt{3}}{2} | 1. **Identify the properties of triangle $ABC$:**
Given that $ABC$ is an isosceles right triangle with a right angle at $C$ and an area of $12.5$, we can find the lengths of the legs. The area of a triangle is given by:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Since the tri... | 0 | 7,951 | -1 | 7,951 |
Triangle \(ABC\) has \(\angle A = 90^\circ\), side \(BC = 25\), \(AB > AC\), and area 150. Circle \(\omega\) is inscribed in \(ABC\), with \(M\) as its point of tangency on \(AC\). Line \(BM\) meets \(\omega\) a second time at point \(L\). Find the length of segment \(BL\). | \frac{45\sqrt{17}}{17} | 0 | 6,895.125 | -1 | 6,895.125 | |
When $1 - i \sqrt{3}$ is converted to the exponential form $re^{i \theta}$, what is $\theta$? | \frac{5\pi}{3} | 0.375 | 3,686.125 | 2,722.5 | 4,264.3 | |
Let \( z = \frac{1+\mathrm{i}}{\sqrt{2}} \). Then calculate the value of \( \left(\sum_{k=1}^{12} z^{k^{2}}\right)\left(\sum_{k=1}^{12} \frac{1}{z^{k^{2}}}\right) \). | 36 | 0.3125 | 6,740.9375 | 5,062.4 | 7,503.909091 | |
In triangle $ABC$, $AB=\sqrt{30}$, $AC=\sqrt{6}$, and $BC=\sqrt{15}$. There is a point $D$ for which $\overline{AD}$ bisects $\overline{BC}$, and $\angle ADB$ is a right angle. The ratio $\frac{[ADB]}{[ABC]}$ can be written in the form $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$... | 65 | Because the problem asks for a ratio, we can divide each side length by $\sqrt{3}$ to make things simpler. We now have a triangle with sides $\sqrt{10}$, $\sqrt{5}$, and $\sqrt{2}$.
We use the same graph as above.
Draw perpendicular from $C$ to $AE$. Denote this point as $F$. We know that $DE = EF = x$ and $BD = CF =... | 0 | 7,666.6875 | -1 | 7,666.6875 |
When point P moves on the circle $C: x^2 - 4x + y^2 = 0$, there exist two fixed points $A(1, 0)$ and $B(a, 0)$, such that $|PB| = 2|PA|$, then $a = \ $. | -2 | 0.5625 | 5,856.5625 | 4,040.111111 | 8,192 | |
Aws plays a solitaire game on a fifty-two card deck: whenever two cards of the same color are adjacent, he can remove them. Aws wins the game if he removes all the cards. If Aws starts with the cards in a random order, what is the probability for him to win? | \frac{\left( \binom{26}{13} \right)^2}{\binom{52}{26}} | 0 | 8,192 | -1 | 8,192 | |
Let $p, q, r, s, t, u, v, w$ be distinct elements in the set $\{-8, -6, -4, -1, 1, 3, 5, 14\}$. What is the minimum possible value of $(p+q+r+s)^2 + (t+u+v+w)^2$? | 10 | 0 | 8,192 | -1 | 8,192 | |
If $\frac{5}{33}$ is expressed in decimal form, what digit is in the 92nd place to the right of the decimal point? | 5 | 1 | 1,605.0625 | 1,605.0625 | -1 | |
In a row of 10 chairs, one of which is broken and cannot be used, Mary and James randomly select their seats. What is the probability that they do not sit next to each other? | \frac{7}{9} | 0.5 | 6,853.0625 | 5,514.125 | 8,192 | |
A class has a total of 54 students. Now, using the systematic sampling method based on the students' ID numbers, a sample of 4 students is drawn. It is known that students with ID numbers 3, 29, and 42 are in the sample. What is the ID number of the fourth student in the sample? | 16 | 0.4375 | 6,532.25 | 4,464.714286 | 8,140.333333 | |
(Full score for this problem is 12 points) Given $f(x) = e^x - ax - 1$.
(1) Find the intervals where $f(x)$ is monotonically increasing.
(2) If $f(x)$ is monotonically increasing on the domain $\mathbb{R}$, find the range of possible values for $a$.
(3) Does there exist a value of $a$ such that $f(x)$ is monotonically... | a = 1 | 0.8125 | 4,773.75 | 4,202.307692 | 7,250 | |
On an east-west shipping lane are ten ships sailing individually. The first five from the west are sailing eastwards while the other five ships are sailing westwards. They sail at the same constant speed at all times. Whenever two ships meet, each turns around and sails in the opposite direction. When all ships hav... | 25 | 0.875 | 5,522.6875 | 5,141.357143 | 8,192 | |
Given that an isosceles trapezoid is circumscribed around a circle, find the ratio of the area of the trapezoid to the area of the circle if the distance between the points where the circle touches the non-parallel sides of the trapezoid is related to the radius of the circle as $\sqrt{3}: 1$. | \frac{8\sqrt{3}}{3\pi} | 0 | 8,155.8125 | -1 | 8,155.8125 | |
The bases \(AB\) and \(CD\) of the trapezoid \(ABCD\) are 367 and 6 respectively, and its diagonals are mutually perpendicular. Find the scalar product of the vectors \(\overrightarrow{AD}\) and \(\overrightarrow{BC}\). | 2202 | 0.6875 | 5,302.875 | 4,504.454545 | 7,059.4 | |
Solve for $x$: $0.05x + 0.07(30 + x) = 15.4$. | 110.8333333 | 0 | 3,935.1875 | -1 | 3,935.1875 | |
Determine the value of $x$ for which $9^{x+6} = 5^{x+1}$ can be expressed in the form $x = \log_b 9^6$. Find the value of $b$. | \frac{5}{9} | 0 | 8,152 | -1 | 8,152 | |
The function $f$ is defined on positive integers as follows:
\[f(n) = \left\{
\begin{array}{cl}
n + 10 & \text{if $n < 10$}, \\
f(n - 5) & \text{if $n \ge 10$}.
\end{array}
\right.\]Find the maximum value of the function. | 19 | 0.8125 | 5,046.4375 | 4,320.538462 | 8,192 | |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given vectors $\overrightarrow{m} = (2\sin B, -\sqrt{3})$ and $\overrightarrow{n} = (\cos 2B, 2\cos^2 B - 1)$ and $\overrightarrow{m} \parallel \overrightarrow{n}$:
(1) Find the measure of acute angle $B$;
(2) If $b ... | \sqrt{3} | 0.0625 | 7,864.4375 | 5,723 | 8,007.2 | |
In $\triangle ABC$, $\tan A= \frac {3}{4}$ and $\tan (A-B)=- \frac {1}{3}$, find the value of $\tan C$. | \frac {79}{3} | 0.6875 | 5,033.625 | 3,598 | 8,192 | |
How many distinct four-digit even numbers can be formed using the digits 0, 1, 2, 3? | 10 | 0.3125 | 6,828.0625 | 5,153.2 | 7,589.363636 | |
Find the number of solutions to the equation
\[\sin x = \left( \frac{1}{3} \right)^x\]
on the interval \( (0, 150 \pi) \). | 75 | 0 | 7,808.3125 | -1 | 7,808.3125 | |
A certain clothing factory produces jackets and $T$-shirts, with each jacket priced at $100$ yuan and each $T$-shirt priced at $60$ yuan. During a promotional period, the factory offers two discount options to customers:<br/>① Buy one jacket and get one $T$-shirt for free;<br/>② Both the jacket and $T$-shirt are paid a... | 3480 | 0.3125 | 6,601.75 | 5,240.8 | 7,220.363636 | |
A "progressive number" refers to a positive integer in which each digit is greater than the digit to its left (such as 1 458). If the four-digit "progressive numbers" are arranged in ascending order, then the 30th number is \_\_\_\_\_\_\_\_. | 1359 | 0.1875 | 7,859.875 | 7,280 | 7,993.692308 | |
64 people are in a single elimination rock-paper-scissors tournament, which consists of a 6-round knockout bracket. Each person has a different rock-paper-scissors skill level, and in any game, the person with the higher skill level will always win. For how many players $P$ is it possible that $P$ wins the first four r... | 49 | Note that a sub-bracket, that is, a subset of games of the tournament that themselves constitute a bracket, is always won by the person with the highest skill level. Therefore, a person wins her first four rounds if and only if she has the highest skill level among the people in her 16-person sub-bracket. This is possi... | 0 | 7,795.5625 | -1 | 7,795.5625 |
Lee can make 18 cookies with two cups of flour. How many cookies can he make with three cups of flour? | 27 | 1 | 298.875 | 298.875 | -1 | |
What is the number of square units in the area of trapezoid EFGH with vertices E(0,0), F(0,3), G(5,3), and H(3,0)? | 7.5 | 0 | 3,344.8125 | -1 | 3,344.8125 | |
Given an arithmetic sequence $\{a_{n}\}$ with the sum of the first $n$ terms as $S_{n}$, where the common difference $d\neq 0$, and $S_{3}+S_{5}=50$, $a_{1}$, $a_{4}$, $a_{13}$ form a geometric sequence.<br/>$(1)$ Find the general formula for the sequence $\{a_{n}\}$;<br/>$(2)$ Let $\{\frac{{b}_{n}}{{a}_{n}}\}$ be a ge... | -\frac{1}{27} | 0.375 | 7,739.5625 | 6,985.5 | 8,192 | |
Natural numbers are arranged in a spiral, turning the first bend at 2, the second bend at 3, the third bend at 5, and so on. What is the number at the twentieth bend? | 71 | 0.6875 | 5,450.5625 | 4,659.636364 | 7,190.6 | |
Compute $\dbinom{15}{3}$. | 455 | 0.875 | 3,703.75 | 3,062.571429 | 8,192 | |
In the addition problem shown, $m, n, p$, and $q$ represent positive digits. What is the value of $m+n+p+q$? | 24 | From the ones column, we see that $3 + 2 + q$ must have a ones digit of 2. Since $q$ is between 1 and 9, inclusive, then $3 + 2 + q$ is between 6 and 14. Since its ones digit is 2, then $3 + 2 + q = 12$ and so $q = 7$. This also means that there is a carry of 1 into the tens column. From the tens column, we see that $1... | 0 | 8,192 | -1 | 8,192 |
Compute $\sin(-60^\circ)$. | -\frac{\sqrt{3}}{2} | 0 | 1,936 | -1 | 1,936 | |
Starting with an empty string, we create a string by repeatedly appending one of the letters $H, M, T$ with probabilities $\frac{1}{4}, \frac{1}{2}, \frac{1}{4}$, respectively, until the letter $M$ appears twice consecutively. What is the expected value of the length of the resulting string? | 6 | Let $E$ be the expected value of the resulting string. Starting from the empty string, - We have a $\frac{1}{2}$ chance of not selecting the letter $M$; from here the length of the resulting string is $1+E$. - We have a $\frac{1}{4}$ chance of selecting the letter $M$ followed by a letter other than $M$, which gives a ... | 0.5 | 6,909.75 | 5,856.625 | 7,962.875 |
A teacher offers candy to her class of 50 students, with the mean number of pieces taken by each student being 7. If every student takes at least one candy but no more than 20 candies, what is the greatest number of pieces one student could have taken? | 20 | 0.125 | 7,739.75 | 7,262.5 | 7,807.928571 | |
Find the monic quadratic polynomial, in $x,$ with real coefficients, which has $1 - i$ as a root. | x^2 - 2x + 2 | 1 | 1,742.375 | 1,742.375 | -1 | |
When someone tries to withdraw money from an ATM with a forgotten last digit of the password, calculate the probability that they will press the correct digit in no more than 2 attempts. | \dfrac{1}{5} | 0.0625 | 3,843.625 | 7,276 | 3,614.8 | |
The product of two consecutive page numbers is $18{,}360.$ What is the sum of the two page numbers? | 271 | 1 | 2,866.5625 | 2,866.5625 | -1 | |
Given a circle with a radius of 5, its center lies on the x-axis with an integer horizontal coordinate and is tangent to the line 4x + 3y - 29 = 0.
(1) Find the equation of the circle;
(2) If the line ax - y + 5 = 0 (a ≠ 0) intersects the circle at points A and B, does there exist a real number a such that the line l... | \frac{3}{4} | 0.4375 | 5,924.5625 | 4,917.857143 | 6,707.555556 | |
Find all the solutions to
\[\sqrt[3]{2 - x} + \sqrt{x - 1} = 1.\]Enter all the solutions, separated by commas. | 1,2,10 | 0 | 7,311.875 | -1 | 7,311.875 | |
(1+11+21+31+41)+(9+19+29+39+49)= | 200 | 1. **Identify the terms in each group and their properties**:
The problem gives us two sums:
\[
(1+11+21+31+41) \quad \text{and} \quad (9+19+29+39+49)
\]
Each group contains five terms, and the terms in each group are increasing by 10.
2. **Rearrange using associative and commutative properties**:
W... | 0 | 425.75 | -1 | 425.75 |
The sequence $\{a_n\}$ satisfies $a_n=13-3n$, $b_n=a_n⋅a_{n+1}⋅a_{n+2}$, $S_n$ is the sum of the first $n$ terms of $\{b_n\}$. Find the maximum value of $S_n$. | 310 | 0.1875 | 7,052.25 | 5,298 | 7,457.076923 | |
Given \( A=\left\{x \mid \log _{3}\left(x^{2}-2 x\right) \leqslant 1\right\}, B=(-\infty, a] \cup(b,+\infty) \), where \( a < b \), if \( A \cup B=\mathbf{R} \), what is the minimum value of \( a - b \) ? | -1 | 0.0625 | 8,095 | 8,192 | 8,088.533333 | |
Amy writes down four integers \(a > b > c > d\) whose sum is 50. The pairwise positive differences of these numbers are \(2, 3, 4, 5, 7,\) and \(10\). What is the sum of the possible values for \(a\)? | 35 | 0.0625 | 7,980.125 | 4,802 | 8,192 | |
In the xy-plane, consider a right triangle $ABC$ with the right angle at $C$. The hypotenuse $AB$ is of length $50$. The medians through vertices $A$ and $B$ are described by the lines $y = x + 5$ and $y = 2x + 2$, respectively. Determine the area of triangle $ABC$. | 500 | 0 | 8,192 | -1 | 8,192 | |
Let $c$ and $d$ be real numbers such that
\[c^3 - 18c^2 + 25c - 75 = 0 \quad \text{and} \quad 9d^3 - 72d^2 - 345d + 3060 = 0.\]Compute $c + d.$ | 10 | 0 | 8,192 | -1 | 8,192 | |
A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. The areas of the three triangles are 3, 7, and 7, as shown. What is the area of the shaded quadrilateral? | 18 | 1. **Identify the triangles and their areas**: We are given that the areas of triangles $EFA$, $FAB$, and $FBD$ are $3$, $7$, and $7$ respectively. We need to find the area of the shaded quadrilateral $CEDF$.
2. **Analyze triangles $EFA$ and $FAB$**: These triangles share an altitude from $F$ to line $AB$. Since their... | 0 | 8,192 | -1 | 8,192 |
What is the base ten equivalent of $54321_6$? | 7465 | 0.9375 | 3,882.25 | 3,594.933333 | 8,192 | |
Gamma and Delta both participated in a two-day science quiz. Each attempted questions totaling 600 points after the second day. On the first day, Gamma scored 210 points out of 350 points attempted, and on the second day scored 150 points out of 250 points attempted. Delta, who also did not attempt 350 points on the fi... | \frac{359}{600} | 0.25 | 7,597.5625 | 7,547.5 | 7,614.25 | |
A sweater costs 160 yuan, it was first marked up by 10% and then marked down by 10%. Calculate the current price compared to the original. | 0.99 | 0 | 564.875 | -1 | 564.875 | |
Jia participates in a shooting practice with 4 bullets, among which one is a blank (a "blank" means a bullet without a projectile).
(1) If Jia shoots only once, calculate the probability of the shot being a blank;
(2) If Jia shoots a total of 3 times, calculate the probability of a blank appearing in these three sh... | 1 - \frac{\sqrt{3}\pi}{150} | 0 | 3,564.9375 | -1 | 3,564.9375 | |
The polynomial $P(x)$ is cubic. What is the largest value of $k$ for which the polynomials $Q_1(x) = x^2 + (k-29)x - k$ and $Q_2(x) = 2x^2+ (2k-43)x + k$ are both factors of $P(x)$? | 30 | We can see that $Q_1$ and $Q_2$ must have a root in common for them to both be factors of the same cubic.
Let this root be $a$.
We then know that $a$ is a root of $Q_{2}(x)-2Q_{1}(x) = 2x^{2}+2kx-43x+k-2x^{2}-2kx+58x+2k = 15x+3k = 0$ , so $x = \frac{-k}{5}$.
We then know that $\frac{-k}{5}$ is a root of $Q_{1}$ so w... | 0.8125 | 4,817.6875 | 4,094.615385 | 7,951 |
The polynomial $(x+y)^{10}$ is expanded in decreasing powers of $x$. The second and third terms have equal values when evaluated at $x=p$ and $y=q$, where $p$ and $q$ are positive numbers whose sum of $p$ plus twice $q$ equals one. Determine the value of $p$. | \frac{9}{13} | 1 | 2,541.6875 | 2,541.6875 | -1 | |
Evaluate $\lfloor\sqrt{17}\rfloor^2$. | 16 | 1 | 1,803.875 | 1,803.875 | -1 | |
Compute $\dbinom{16}{15}$. | 16 | 1 | 1,705.8125 | 1,705.8125 | -1 | |
Suppose that on a parabola with vertex $V$ and a focus $F$ there exists a point $A$ such that $AF=20$ and $AV=21$. What is the sum of all possible values of the length $FV?$ | \frac{40}{3} | 1. **Understanding the Parabola and its Properties**:
- The parabola $\mathcal{P}$ is defined such that for any point $T$ on $\mathcal{P}$, the distance from $T$ to the directrix $\ell$ is equal to the distance from $T$ to the focus $F$.
- Let $V$ be the vertex of the parabola, which is equidistant from the focu... | 0.6875 | 5,904.625 | 5,176.090909 | 7,507.4 |
A partition of a number \( n \) is a sequence of positive integers, arranged in descending order, whose sum is \( n \). For example, \( n=4 \) has 5 partitions: \( 1+1+1+1=2+1+1=2+2=3+1=4 \). Given two different partitions of the same number, \( n=a_{1}+a_{2}+\cdots+a_{k}=b_{1}+b_{2}+\cdots+b_{l} \), where \( k \leq l ... | 20 | 0 | 8,192 | -1 | 8,192 | |
Find the range of the function
\[f(x) = \frac{\sin^3 x + 6 \sin^2 x + \sin x + 2 \cos^2 x - 8}{\sin x - 1},\]as $x$ ranges over all real numbers such that $\sin x \neq 1.$ Enter your answer using interval notation. | [2,12) | 0.8125 | 3,890.875 | 3,996.230769 | 3,434.333333 | |
In $\triangle ABC$ with right angle at $C$, altitude $CH$ and median $CM$ trisect the right angle. If the area of $\triangle CHM$ is $K$, then the area of $\triangle ABC$ is | 4K | 1. **Draw the triangle and identify key components**: In $\triangle ABC$, where $\angle C = 90^\circ$, draw altitude $CH$ from $C$ to hypotenuse $AB$ and median $CM$ from $C$ to the midpoint $M$ of $AB$. Since $CM$ trisects the right angle, $\angle MCB = 30^\circ$ and $\angle MCH = 15^\circ$.
2. **Properties of median... | 0.4375 | 6,908.875 | 5,259.142857 | 8,192 |
The stem-and-leaf plot shows the number of minutes and seconds of one ride on each of the 21 top-rated water slides in the world. In the stem-and-leaf plot, $1 \ 45$ represents 1 minute, 45 seconds, which is equivalent to 105 seconds. What is the median of this data set? Express your answer in seconds.
\begin{tabular}... | 135 | 0.0625 | 8,003.9375 | 6,517 | 8,103.066667 | |
Let $f(x)=x^3+3$ and $g(x) = 2x^2 + 2x +1$. What is $g(f(-2))$? | 41 | 1 | 2,191.25 | 2,191.25 | -1 | |
Simplify and write the result as a common fraction: $$\sqrt[4]{\sqrt[3]{\sqrt{\frac{1}{65536}}}}$$ | \frac{1}{\sqrt[3]{4}} | 0 | 7,430.375 | -1 | 7,430.375 | |
Two boards, one five inches wide and the other eight inches wide, are nailed together to form an X. The angle at which they cross is 45 degrees. If this structure is painted and the boards are separated, what is the area of the unpainted region on the five-inch board? (Neglect the holes caused by the nails.) | 40 \sqrt{2} | 0 | 6,833.5 | -1 | 6,833.5 | |
A wire is cut into two pieces, one of length $a$ and the other of length $b$. The piece of length $a$ is bent to form an equilateral triangle, and the piece of length $b$ is bent to form a regular hexagon. The triangle and the hexagon have equal area. What is $\frac{a}{b}$? | \frac{\sqrt6}2 | 0 | 2,851.5625 | -1 | 2,851.5625 | |
Find the only value of \( x \) in the open interval \((- \pi / 2, 0)\) that satisfies the equation
$$
\frac{\sqrt{3}}{\sin x} + \frac{1}{\cos x} = 4.
$$ | -\frac{4\pi}{9} | 0.5625 | 6,533.5625 | 5,243.666667 | 8,192 | |
Let $t(x) = \sqrt{3x+1}$ and $f(x)=5-t(x)$. What is $t(f(5))$? | 2 | 1 | 1,892.75 | 1,892.75 | -1 | |
Medians $\overline{AD}$ and $\overline{BE}$ of $\triangle ABC$ intersect at an angle of $45^\circ$. If $AD = 12$ and $BE = 16$, then calculate the area of $\triangle ABC$. | 64\sqrt{2} | 0.625 | 6,906.9375 | 6,135.9 | 8,192 | |
Find the largest real number $\lambda$ with the following property: for any positive real numbers $p,q,r,s$ there exists a complex number $z=a+bi$($a,b\in \mathbb{R})$ such that $$ |b|\ge \lambda |a| \quad \text{and} \quad (pz^3+2qz^2+2rz+s) \cdot (qz^3+2pz^2+2sz+r) =0.$$ | \sqrt{3} |
To find the largest real number \(\lambda\) such that for any positive real numbers \(p, q, r, s\), there exists a complex number \(z = a + bi\) (\(a, b \in \mathbb{R}\)) satisfying
\[
|b| \ge \lambda |a|
\]
and
\[
(pz^3 + 2qz^2 + 2rz + s) \cdot (qz^3 + 2pz^2 + 2sz + r) = 0,
\]
we proceed as follows:
The answer is \(... | 0 | 8,192 | -1 | 8,192 |
A $2$ by $2$ square is divided into four $1$ by $1$ squares. Each of the small squares is to be painted either green or red. In how many different ways can the painting be accomplished so that no green square shares its top or right side with any red square? There may be as few as zero or as many as four small green... | 6 | To solve this problem, we need to consider all possible configurations of green and red squares in the $2 \times 2$ grid such that no green square shares its top or right side with a red square. We will analyze each case based on the number of green squares.
#### Case 1: No green squares
- All squares are red.
- **Num... | 0 | 8,153.6875 | -1 | 8,153.6875 |
Someone observed that $6! = 8 \cdot 9 \cdot 10$. Find the largest positive integer $n$ for which $n!$ can be expressed as the product of $n - 3$ consecutive positive integers.
| 23 | 0 | 8,192 | -1 | 8,192 | |
Peter's family ordered a 12-slice pizza for dinner. Peter ate one slice and shared another slice equally with his brother Paul. What fraction of the pizza did Peter eat? | \frac{1}{8} | 1. **Calculate the fraction of the pizza Peter ate alone:**
Peter ate one whole slice out of a 12-slice pizza. Therefore, the fraction of the pizza he ate alone is:
\[
\frac{1}{12}
\]
2. **Calculate the fraction of the pizza Peter shared with Paul:**
Peter shared another slice equally with his brother P... | 1 | 1,734.625 | 1,734.625 | -1 |
The function $g(x),$ defined for $0 \le x \le 1,$ has the following properties:
(i) $g(0) = 0.$
(ii) If $0 \le x < y \le 1,$ then $g(x) \le g(y).$
(iii) $g(1 - x) = 1 - g(x)$ for all $0 \le x \le 1.$
(iv) $g\left( \frac{2x}{5} \right) = \frac{g(x)}{3}$ for $0 \le x \le 1.$
Find $g\left( \frac{3}{5} \right).$ | \frac{2}{3} | 0.0625 | 8,038.4375 | 5,735 | 8,192 | |
The number $121_b$, written in the integral base $b$, is the square of an integer, for | $b > 2$ | 1. **Convert the base-b number to base-10**: The number $121_b$ in base $b$ can be expressed in base 10 as:
\[
1 \cdot b^2 + 2 \cdot b^1 + 1 \cdot b^0 = b^2 + 2b + 1
\]
2. **Factorize the expression**: The expression $b^2 + 2b + 1$ can be rewritten by recognizing it as a perfect square:
\[
b^2 + 2b + 1 ... | 0 | 6,075.4375 | -1 | 6,075.4375 |
The diagonal of a regular 2006-gon \(P\) is called good if its ends divide the boundary of \(P\) into two parts, each containing an odd number of sides. The sides of \(P\) are also called good. Let \(P\) be divided into triangles by 2003 diagonals, none of which have common points inside \(P\). What is the maximum numb... | 1003 | 0 | 8,192 | -1 | 8,192 | |
In trapezoid $ABCD$, the sides $AB$ and $CD$ are equal. The perimeter of $ABCD$ is | 34 | 1. **Identify the Shape and Given Information**: We are given a trapezoid $ABCD$ where $AB$ and $CD$ are equal in length. We are also given that the perimeter of the trapezoid is one of the options provided.
2. **Assumption of Rectangle Presence**: The solution assumes the presence of a rectangle within the trapezoid,... | 0 | 7,842.9375 | -1 | 7,842.9375 |
A school organized a trip to the Expo Park for all third-grade students and rented some large buses. Initially, the plan was to have 28 people on each bus. After all the students boarded, it was found that 13 students could not get on the buses. So, they decided to have 32 people on each bus, and this resulted in 3 emp... | 125 | 0 | 936.25 | -1 | 936.25 | |
The Seattle weather forecast suggests a 60 percent chance of rain each day of a five-day holiday. If it does not rain, then the weather will be sunny. Stella wants exactly two days to be sunny during the holidays for a gardening project. What is the probability that Stella gets the weather she desires? Give your answer... | \frac{4320}{15625} | 0 | 2,553.9375 | -1 | 2,553.9375 | |
Given 6 parking spaces in a row and 3 cars that need to be parked such that no two cars are next to each other, calculate the number of different parking methods. | 24 | 0.125 | 6,802.1875 | 6,607.5 | 6,830 | |
In the geometric sequence $\{a_{n}\}$, $a_{5}a_{8}=6$, $a_{3}+a_{10}=5$, then $\frac{a_{20}}{a_{13}}=$ \_\_\_\_\_\_. | \frac{2}{3} | 0.0625 | 7,831.4375 | 6,197 | 7,940.4 | |
Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $2500$ are not factorial tails? | 499 | 0 | 8,192 | -1 | 8,192 | |
The highest temperatures from April 1st to April 6th in a certain region were 28℃, 21℃, 22℃, 26℃, 28℃, and 25℃, respectively. Calculate the variance of the highest temperature data for these six days. | \frac{22}{3} | 0.6875 | 3,244.125 | 2,815.272727 | 4,187.6 | |
Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$, then the average value (arithmetic mean) of the integers remaining is $32$. If the least integer in $S$ is also removed, then the average value of the integers remaining is $35$. If the greatest integer is then re... | 36.8 | Let $S = \{a_1, a_2, a_3, \hdots, a_n\}$ where $a_1 < a_2 < a_3 < \hdots < a_n$. We are given the following conditions:
1. Removing the greatest integer $a_n$ from $S$ results in an average of $32$ for the remaining integers.
2. Removing both the greatest integer $a_n$ and the least integer $a_1$ from $S$ results in an... | 0.4375 | 7,070.0625 | 5,627.571429 | 8,192 |
Given that the sequence {a<sub>n</sub>} is an arithmetic sequence, a<sub>1</sub> < 0, a<sub>8</sub> + a<sub>9</sub> > 0, a<sub>8</sub> • a<sub>9</sub> < 0. Find the smallest value of n for which S<sub>n</sub> > 0. | 16 | 0.125 | 7,898.375 | 6,093 | 8,156.285714 | |
Mr. Green measures his rectangular garden by walking two of the sides and finds that it is $15$ steps by $20$ steps. Each of Mr. Green's steps is $2$ feet long. Mr. Green expects a half a pound of potatoes per square foot from his garden. How many pounds of potatoes does Mr. Green expect from his garden? | 600 | 1. **Convert steps to feet**:
Mr. Green's garden measures $15$ steps by $20$ steps. Given that each step is $2$ feet long, we convert the dimensions from steps to feet:
\[
15 \text{ steps} \times 2 \text{ feet/step} = 30 \text{ feet}
\]
\[
20 \text{ steps} \times 2 \text{ feet/step} = 40 \text{ feet}... | 0.9375 | 1,775.6875 | 1,347.933333 | 8,192 |
An experimenter selects 4 out of 8 different chemical substances to place in 4 distinct bottles. If substances A and B should not be placed in bottle 1, the number of different ways of arranging them is ____. | 1260 | 0.1875 | 7,787.6875 | 6,884.333333 | 7,996.153846 | |
In the famous book "Algorithm for Direct Calculation" by the Chinese mathematician Cheng Dawei of the Ming Dynasty, there is a well-known math problem:
"One hundred mantou for one hundred monks, three big monks have no dispute, three small monks share one, how many big and small monks are there?" | 75 | 0.125 | 7,145.5625 | 1,537 | 7,946.785714 | |
Each edge length of a rectangular solid is a prime number. If the volume of the rectangular solid is 385 cubic units, what is the total surface area, in square units, of the rectangular solid? | 334 | 1 | 1,422.4375 | 1,422.4375 | -1 | |
Find the ratio of the area of $\triangle BCX$ to the area of $\triangle ACX$ in the diagram if $CX$ bisects $\angle ACB$. Express your answer as a common fraction. [asy]
import markers;
real t=27/(27+30);
pair A=(-15.57,0);
pair B=(8.43,0);
pair C=(0,25.65);
pair X=t*A+(1-t)*B;
draw(C--A--B--C--X);
label("$A$",A,SW)... | \frac{9}{10} | 0.5 | 6,747.125 | 6,633.25 | 6,861 | |
Find $\frac{1}{3}+\frac{2}{7}$. | \frac{13}{21} | 1 | 1,392.8125 | 1,392.8125 | -1 | |
The café has enough chairs to seat $310_5$ people. If $3$ people are supposed to sit at one table, how many tables does the café have? | 26 | 0.0625 | 2,785.3125 | 5,392 | 2,611.533333 | |
Given an ellipse with a chord passing through the focus and perpendicular to the major axis of length $\sqrt{2}$, and a distance from the focus to the corresponding directrix of $1$, determine the eccentricity of the ellipse. | \frac{\sqrt{2}}{2} | 0 | 4,952.5625 | -1 | 4,952.5625 | |
A semicircular sheet of iron with a radius of 6 is rolled into the lateral surface of a cone. The volume of this cone is \_\_\_\_\_\_. | 9\sqrt{3}\pi | 1 | 2,834.8125 | 2,834.8125 | -1 | |
In a rectangular array of points, with 5 rows and $N$ columns, the points are numbered consecutively from left to right beginning with the top row. Thus the top row is numbered 1 through $N,$ the second row is numbered $N + 1$ through $2N,$ and so forth. Five points, $P_1, P_2, P_3, P_4,$ and $P_5,$ are selected so tha... | 149 | 0 | 8,192 | -1 | 8,192 | |
Find the maximum value of the area of triangle $\triangle ABC$ that satisfies $AB=4$ and $AC=2BC$. | \frac{16}{3} | 0.875 | 5,929.25 | 5,606 | 8,192 | |
Given a sequence \( a_{1}, a_{2}, \cdots, a_{n}, \cdots \) such that \( a_{1}=a_{2}=1 \), \( a_{3}=2 \), and for any natural number \( n \), \( a_{n} a_{n+1} a_{n+2} \neq 1 \). Additionally, it holds that \( a_{n} a_{n+1} a_{n+2} a_{n+3} = a_{1} + a_{n+1} + a_{n+2} + a_{n+3} \). Determine the value of \( a_{1} + a_{2} ... | 200 | 0.0625 | 7,872.3125 | 3,077 | 8,192 | |
The positive integers \(a\) and \(b\) are such that the numbers \(15a + 165\) and \(16a - 155\) are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares? | 481 | 0 | 8,192 | -1 | 8,192 | |
Independent trials are conducted, in each of which event \( A \) can occur with a probability of 0.001. What is the probability that in 2000 trials, event \( A \) will occur at least two and at most four times? | 0.541 | 0.0625 | 7,888.6875 | 8,192 | 7,868.466667 | |
The function \( f(x) = \frac{x^{2}}{8} + x \cos x + \cos (2x) \) (for \( x \in \mathbf{R} \)) has a minimum value of ___ | -1 | 0 | 8,192 | -1 | 8,192 | |
A cylindrical barrel with a radius of 5 feet and a height of 15 feet is full of water. A solid cube with side length 7 feet is set into the barrel so that one edge of the cube is vertical. Calculate the square of the volume of water displaced, $v^2$, when the cube is fully submerged. | 117649 | 0.75 | 5,523.5625 | 5,095.166667 | 6,808.75 | |
Let $N = \sum_{k = 1}^{1000} k ( \lceil \log_{\sqrt{2}} k \rceil - \lfloor \log_{\sqrt{2}} k \rfloor )$
Find the remainder when $N$ is divided by 1000. ($\lfloor{k}\rfloor$ is the greatest integer less than or equal to $k$, and $\lceil{k}\rceil$ is the least integer greater than or equal to $k$.) | 477 | The ceiling of a number minus the floor of a number is either equal to zero (if the number is an integer); otherwise, it is equal to 1. Thus, we need to find when or not $\log_{\sqrt{2}} k$ is an integer.
The change of base formula shows that $\frac{\log k}{\log \sqrt{2}} = \frac{2 \log k}{\log 2}$. For the $\log 2$ t... | 0.8125 | 4,000 | 3,805.384615 | 4,843.333333 |
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