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 |
|---|---|---|---|---|---|---|
From the 16 vertices of a $3 \times 3$ grid comprised of 9 smaller unit squares, what is the probability that any three chosen vertices form a right triangle? | 9/35 | 0 | 8,184.0625 | -1 | 8,184.0625 | |
Find the point on the line
\[y = -3x + 5\]that is closest to the point $(-4,-2).$ | \left( \frac{17}{10}, -\frac{1}{10} \right) | 0.9375 | 3,334.1875 | 3,010.333333 | 8,192 | |
How many natural numbers between 200 and 400 are divisible by 8? | 26 | 0.4375 | 4,833.9375 | 5,261.571429 | 4,501.333333 | |
Two numbers are independently selected from the set of positive integers less than or equal to 7. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction. | \frac{36}{49} | 0.0625 | 6,683.75 | 2,645 | 6,953 | |
There are more than 20 and fewer than 30 children in Miss Tree's class. They are all standing in a circle. Anna notices that there are six times as many children between her and Zara going round the circle clockwise, as there are going round anti-clockwise. How many children are there in the class? | 23 | 0.5 | 6,091.5625 | 3,991.125 | 8,192 | |
Dr. Fu Manchu has a bank account that has an annual interest rate of 6 percent, but it compounds monthly. If this is equivalent to a bank account that compounds annually at a rate of $r$ percent, then what is $r$? (Give your answer to the nearest hundredth.) | 6.17 | 0.5625 | 6,753.125 | 5,634 | 8,192 | |
A semicircle with diameter $d$ is contained in a square whose sides have length 8. Given the maximum value of $d$ is $m - \sqrt{n},$ find $m+n.$ | 544 | It is easy after getting the image, after drawing labeling the lengths of those segments, assume the radius is $x$, we can see $x=\sqrt{2}(8-x)$ and we get $2x=32-\sqrt{512}$ and we have the answer $\boxed{544}$ ~bluesoul | 0 | 8,192 | -1 | 8,192 |
The area of a rectangle remains unchanged when it is made $2 \frac{1}{2}$ inches longer and $\frac{2}{3}$ inch narrower, or when it is made $2 \frac{1}{2}$ inches shorter and $\frac{4}{3}$ inch wider. Its area, in square inches, is: | 20 | 1. **Set up the equations based on the problem statement:**
We are given that the area of the rectangle remains unchanged under two transformations:
- When the rectangle is made $2 \frac{1}{2}$ inches longer and $\frac{2}{3}$ inch narrower.
- When the rectangle is made $2 \frac{1}{2}$ inches shorter and $\frac... | 0.9375 | 3,849 | 3,559.466667 | 8,192 |
Given that point \( P \) lies in the plane of triangle \( \triangle ABC \) and satisfies the condition \( PA - PB - PC = \overrightarrow{BC} \), determine the ratio of the area of \( \triangle ABP \) to the area of \( \triangle ABC \). | 2:1 | 0 | 7,639.5 | -1 | 7,639.5 | |
Calculate $\fbox{2,3,-1}$. | \frac{26}{3} | 0 | 7,075.8125 | -1 | 7,075.8125 | |
Determine the value of
\[\frac{\frac{2016}{1} + \frac{2015}{2} + \frac{2014}{3} + \dots + \frac{1}{2016}}{\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots + \frac{1}{2017}}.\] | 2017 | 0.8125 | 4,730.5625 | 3,931.769231 | 8,192 | |
My school's math club has 6 boys and 8 girls. I need to select a team to send to the state math competition. We want 6 people on the team. In how many ways can I select the team to have 3 boys and 3 girls? | 1120 | 1 | 2,036.625 | 2,036.625 | -1 | |
Given the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{3}=1\) with the left and right foci \(F_{1}\) and \(F_{2}\) respectively, a line \(l\) passes through the right focus and intersects the ellipse at points \(P\) and \(Q\). Find the maximum area of the inscribed circle of \(\triangle F_{1}PQ\). | \frac{9\pi}{16} | 0 | 8,192 | -1 | 8,192 | |
Find the smallest multiple of 9 that does not contain any odd digits. | 288 | 0.5 | 6,604.625 | 5,414.25 | 7,795 | |
For a point $P = (a,a^2)$ in the coordinate plane, let $l(P)$ denote the line passing through $P$ with slope $2a$. Consider the set of triangles with vertices of the form $P_1 = (a_1, a_1^2), P_2 = (a_2, a_2^2), P_3 = (a_3, a_3^2)$, such that the intersection of the lines $l(P_1), l(P_2), l(P_3)$ form an equilateral tr... | y = -\frac{1}{4} |
Let \( P_1 = (a_1, a_1^2) \), \( P_2 = (a_2, a_2^2) \), and \( P_3 = (a_3, a_3^2) \) be points in the coordinate plane. The lines \( l(P_1) \), \( l(P_2) \), and \( l(P_3) \) have equations with slopes equal to \( 2a_1 \), \( 2a_2 \), and \( 2a_3 \) respectively. The line equation for \( P = (a, a^2) \) with slope \( ... | 0 | 8,189.0625 | -1 | 8,189.0625 |
Select 5 different letters from the word "equation" to arrange in a row, including the condition that the letters "qu" are together and in the same order. | 480 | 0.5625 | 6,563.3125 | 6,493.333333 | 6,653.285714 | |
Choose one digit from 0, 2, 4, and two digits from 1, 3, 5 to form a three-digit number without repeating digits. The total number of different three-digit numbers that can be formed is ( )
A 36 B 48 C 52 D 54 | 48 | 0 | 5,140.875 | -1 | 5,140.875 | |
Given that the volume of the parallelepiped formed by vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ is 4, find the volume of the parallelepiped formed by the vectors $\mathbf{2a} + \mathbf{b}$, $\mathbf{b} + 4\mathbf{c}$, and $\mathbf{c} - 5\mathbf{a}$. | 232 | 0 | 7,229.375 | -1 | 7,229.375 | |
A biased coin lands heads with a probability of $\frac{2}{3}$ and tails with $\frac{1}{3}$. A player can choose between Game C and Game D. In Game C, the player tosses the coin five times and wins if either the first three or the last three outcomes are all the same. In Game D, she tosses the coin five times and wins i... | \frac{29}{81} | 0 | 8,120.9375 | -1 | 8,120.9375 | |
Square $ABCD$ has area $36,$ and $\overline{AB}$ is parallel to the x-axis. Vertices $A,$ $B$, and $C$ are on the graphs of $y = \log_{a}x,$ $y = 2\log_{a}x,$ and $y = 3\log_{a}x,$ respectively. What is $a?$ | \sqrt[6]{3} | 1. **Identify the properties of the square and the logarithmic functions**:
- The square $ABCD$ has an area of $36$, so each side of the square is $\sqrt{36} = 6$ units.
- The vertices $A$, $B$, and $C$ lie on the graphs of $y = \log_a x$, $y = 2\log_a x$, and $y = 3\log_a x$ respectively. Since $\overline{AB}$ ... | 0.1875 | 7,588.5625 | 4,973.666667 | 8,192 |
Tanya wrote a certain two-digit number on a piece of paper; to Sveta, who was sitting opposite her, the written number appeared different and was 75 less. What number did Tanya write? | 91 | 0.125 | 5,404.8125 | 540 | 6,099.785714 | |
Let $p$ and $q$ be positive integers such that\[\frac{3}{5} < \frac{p}{q} < \frac{2}{3}\]and $q$ is as small as possible. What is $q - p$? | 11 | 0 | 3,753.1875 | -1 | 3,753.1875 | |
The volume of the parallelepiped determined by the three-dimensional vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ is 4. Find the volume of the parallelepiped determined by the vectors $\mathbf{a} + \mathbf{b},$ $\mathbf{b} + 3 \mathbf{c},$ and $\mathbf{c} - 7 \mathbf{a}.$ | 80 | 0.4375 | 6,105.5625 | 3,423 | 8,192 | |
Find all values of $k$ so that
\[x^2 - (k - 3) x - k + 6 > 0\]for all $x.$ | (-3,5) | 1 | 2,532.375 | 2,532.375 | -1 | |
A spherical balloon collapses into a wet horizontal surface and settles into a shape of a hemisphere while keeping the same volume. The minor radius of the original balloon, when viewed as an ellipsoid due to unequal pressure distribution, was $4\sqrt[3]{3}$ cm. Find the major radius of the original balloon, assuming t... | 8\sqrt[3]{3} | 0.4375 | 7,256.4375 | 6,053.571429 | 8,192 | |
What is the greatest common factor of the numbers 2835 and 8960? | 35 | 1 | 2,366.875 | 2,366.875 | -1 | |
Cindy wishes to arrange her coins into $X$ piles, each consisting of the same number of coins, $Y$. Each pile will have more than one coin and no pile will have all the coins. If there are 13 possible values for $Y$ given all of the restrictions, what is the smallest number of coins she could have? | 144 | 0.9375 | 3,451.1875 | 3,374.4 | 4,603 | |
Given that $F_{1}$ and $F_{2}$ are the two foci of the ellipse $\frac{x^{2}}{20} + \frac{y^{2}}{4} = 1$, a line passing through $F_{2}$ intersects the ellipse at points $A$ and $B$. If $|F_{1}A| + |F_{1}B| = 5\sqrt{5}$, then $|AB| = $ ______. | 3\sqrt{5} | 0.25 | 7,614.125 | 5,880.5 | 8,192 | |
Given an isosceles trapezoid \(ABCD\), where \(AD \parallel BC\), \(BC = 2AD = 4\), \(\angle ABC = 60^\circ\), and \(\overrightarrow{CE} = \frac{1}{3} \overrightarrow{CD}\), calculate the value of \(\overrightarrow{CA} \cdot \overrightarrow{BE}\). | -10 | 0.875 | 5,168.6875 | 4,827.571429 | 7,556.5 | |
The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to
[asy] unitsize(3mm); defaultpen(linewidth(0.8pt)); path p1=(0,0)--(3,0)--(3,3)--(0,3)--(0,0); path p2=(0,1)--(1,1)--(1,0); path p3=(2,0)--(2,1)--(3,1); path p4=(3,2)--(... | 56 | 1. **Understanding the Tiling Pattern**: The problem states that the plane is tiled by congruent squares and congruent pentagons. We need to determine the percentage of the plane covered by the pentagons.
2. **Analyzing a Single Tile**: Consider a single tile in the tiling pattern. The tile is a square divided into sm... | 0.0625 | 7,706.625 | 7,072 | 7,748.933333 |
The average of the numbers $1, 2, 3, \dots, 44, 45, x$ is $50x$. What is $x$? | \frac{1035}{2299} | 0.625 | 6,602.6875 | 5,649.1 | 8,192 | |
The year 2013 has arrived, and Xiao Ming's older brother sighed and said, "This is the first year in my life that has no repeated digits." It is known that Xiao Ming's older brother was born in a year that is a multiple of 19. How old is the older brother in 2013? | 18 | 0.0625 | 8,017.9375 | 7,825 | 8,030.8 | |
For rational numbers $a$ and $b$, define the operation "$\otimes$" as $a \otimes b = ab - a - b - 2$.
(1) Calculate the value of $(-2) \otimes 3$;
(2) Compare the size of $4 \otimes (-2)$ and $(-2) \otimes 4$. | -12 | 0.4375 | 2,828.875 | 2,962.857143 | 2,724.666667 | |
Evaluate the expression \[ \frac{a^2 + 2a}{a^2 + a} \cdot \frac{b^2 - 4}{b^2 - 6b + 8} \cdot \frac{c^2 + 16c + 64}{c^2 + 12c + 36} \]
given that \(c = b - 20\), \(b = a + 4\), \(a = 2\), and ensuring none of the denominators are zero. | \frac{3}{4} | 0 | 2,892.375 | -1 | 2,892.375 | |
Let $S$ be a region in the plane with area 10. When we apply the matrix
\[\begin{pmatrix} 2 & 1 \\ 7 & -3 \end{pmatrix}\]to $S,$ we obtain the region $S'.$ Find the area of $S'.$ | 130 | 1 | 1,185.0625 | 1,185.0625 | -1 | |
Find the ordered pair $(a,b)$ of positive integers, with $a < b,$ for which
\[\sqrt{1 + \sqrt{21 + 12 \sqrt{3}}} = \sqrt{a} + \sqrt{b}.\] | (1,3) | 0.875 | 4,159.6875 | 3,583.642857 | 8,192 | |
Two circles of radius $r$ are externally tangent to each other and internally tangent to the ellipse $x^2 + 4y^2 = 8$. Find the value of $r$. | \frac{\sqrt{6}}{2} | 0 | 6,459.8125 | -1 | 6,459.8125 | |
In trapezoid $PQRS$, the parallel sides $PQ$ and $RS$ have lengths of 10 and 30 units, respectively, and the altitude is 18 units. Points $T$ and $U$ are the midpoints of sides $PR$ and $QS$, respectively. What is the area of quadrilateral $TURS$? | 225 | 0.0625 | 8,118.8125 | 7,021 | 8,192 | |
Thirty-nine students from seven classes invented 60 problems, with the students from each class inventing the same number of problems (which is not zero), and the students from different classes inventing different numbers of problems. How many students invented one problem each? | 33 | 0 | 8,146 | -1 | 8,146 | |
A particle begins at a point P on the parabola y = x^2 - 2x - 8 where the y-coordinate is 8. It rolls along the parabola to the nearest point Q where the y-coordinate is -8. Calculate the horizontal distance traveled by the particle, defined as the absolute difference between the x-coordinates of P and Q. | \sqrt{17} - 1 | 0.8125 | 5,247.5 | 5,208.076923 | 5,418.333333 | |
Given a positive integer \(N\) (written in base 10), define its integer substrings to be integers that are equal to strings of one or more consecutive digits from \(N\), including \(N\) itself. For example, the integer substrings of 3208 are \(3, 2, 0, 8, 32, 20, 320, 208\), and 3208. (The substring 08 is omitted from... | 88,888,888 | 0 | 8,192 | -1 | 8,192 | |
Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures $3$ by $4$ by $5$ units. Given that the volume of this set is $\frac{m + n\pi}{p},$ where $m, n,$ and $p$ are positive integers, and $n$ and $p$ are relatively prime, find $m + n + p.$ | 505 | [asy] size(220); import three; currentprojection = perspective(5,4,3); defaultpen(linetype("8 8")+linewidth(0.6)); draw(box((0,-.1,0),(0.4,0.6,0.3))); draw(box((-.1,0,0),(0.5,0.5,0.3))); draw(box((0,0,-.1),(0.4,0.5,0.4))); draw(box((0,0,0),(0.4,0.5,0.3)),linewidth(1.2)+linetype("1")); [/asy]
The set can be broken into... | 0.5625 | 6,273.3125 | 5,308.555556 | 7,513.714286 |
In triangle $ABC$, $AB = 18$ and $BC = 12$. Find the largest possible value of $\tan A$. | \frac{2\sqrt{5}}{5} | 0 | 6,832.3125 | -1 | 6,832.3125 | |
Determine the maximum number of bishops that we can place in a $8 \times 8$ chessboard such that there are not two bishops in the same cell, and each bishop is threatened by at most one bishop.
Note: A bishop threatens another one, if both are placed in different cells, in the same diagonal. A board has as diagonals ... | 20 | To solve this problem, we need to determine the maximum number of bishops that can be placed on an \(8 \times 8\) chessboard such that each bishop is threatened by at most one other bishop. The key here is to understand the attacking capability of bishops on a chessboard.
Bishops move diagonally, which means they cont... | 0 | 7,877.625 | -1 | 7,877.625 |
In triangle $ABC$, $AB = 5$, $AC = 5$, and $BC = 6$. The medians $AD$, $BE$, and $CF$ intersect at the centroid $G$. Let the projections of $G$ onto $BC$, $AC$, and $AB$ be $P$, $Q$, and $R$, respectively. Find $GP + GQ + GR$. | \frac{68}{15} | 0.875 | 5,875.5 | 5,544.571429 | 8,192 | |
It takes 60 grams of paint to paint a cube on all sides. How much paint is needed to paint a "snake" composed of 2016 such cubes? The beginning and end of the snake are shown in the diagram, and the rest of the cubes are indicated by ellipses. | 80660 | 0.625 | 5,105.5 | 4,337.5 | 6,385.5 | |
Given the broadcast time of the "Midday News" program is from 12:00 to 12:30 and the news report lasts 5 minutes, calculate the probability that Xiao Zhang can watch the entire news report if he turns on the TV at 12:20. | \frac{1}{6} | 0.0625 | 1,334.625 | 514 | 1,389.333333 | |
In a new school $40$ percent of the students are freshmen, $30$ percent are sophomores, $20$ percent are juniors, and $10$ percent are seniors. All freshmen are required to take Latin, and $80$ percent of the sophomores, $50$ percent of the juniors, and $20$ percent of the seniors elect to take Latin. The probability t... | 25 | We see that $40\% \cdot 100\% + 30\% \cdot 80\% + 20\% \cdot 50\% + 10\% \cdot 20\% = 76\%$ of students are learning Latin. In addition, $30\% \cdot 80\% = 24\%$ of students are sophomores learning Latin. Thus, our desired probability is $\dfrac{24}{76}=\dfrac{6}{19}$ and our answer is $6+19=\boxed{025}$. | 1 | 2,078.5625 | 2,078.5625 | -1 |
Crestview's school has expanded its official colors to include blue along with the original purple and gold. The students need to design a flag using three solid-colored horizontal stripes. Using one, two, or all three of the school colors, how many different flags are possible if adjacent stripes may be of the same co... | 27 | 0.8125 | 4,698.3125 | 4,258.923077 | 6,602.333333 | |
On the sides \( BC \) and \( AC \) of triangle \( ABC \), points \( M \) and \( N \) are taken respectively such that \( CM:MB = 1:3 \) and \( AN:NC = 3:2 \). Segments \( AM \) and \( BN \) intersect at point \( K \). Find the area of quadrilateral \( CMKN \), given that the area of triangle \( ABC \) is 1. | 3/20 | 0.1875 | 8,153.1875 | 8,142.666667 | 8,155.615385 | |
A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe? | \frac {16!}{2^8} |
To solve this problem, we need to determine the number of ways the spider can put on its socks and shoes such that each sock is put on before its corresponding shoe on each leg.
#### Step-by-step Analysis:
1. **Total Items to Wear:** The spider has 8 legs, and for each leg, it has one sock and one shoe, making a tot... | 0.4375 | 4,435.0625 | 3,558 | 5,117.222222 |
Given that \( n! \), in decimal notation, has exactly 57 ending zeros, find the sum of all possible values of \( n \). | 1185 | 0.8125 | 5,120.6875 | 4,835.923077 | 6,354.666667 | |
How many two-digit numbers have digits whose sum is a perfect square less than or equal to 25? | 17 | 0.9375 | 4,630.625 | 4,393.2 | 8,192 | |
Given a geometric sequence $\{a_n\}$ with a sum of the first $n$ terms as $S_n$, and $S_{10}:S_5 = 1:2$, find the value of $\frac{S_5 + S_{10} + S_{15}}{S_{10} - S_5}$. | -\frac{9}{2} | 0.375 | 6,980.4375 | 5,381.666667 | 7,939.7 | |
In a certain year, a specific date was never a Sunday in any month. Determine this date. | 31 | 0.0625 | 7,970 | 6,670 | 8,056.666667 | |
Given $$\frac{1}{C_5^m} - \frac{1}{C_6^m} = \frac{7}{10C_7^m}$$, find $C_{21}^m$. | 210 | 0.9375 | 4,510.9375 | 4,265.533333 | 8,192 | |
Find $4^{-1} \pmod{35}$, as a residue modulo 35. (Give an answer between 0 and 34, inclusive.) | 9 | 1 | 2,215.3125 | 2,215.3125 | -1 | |
A student accidentally added five to both the numerator and denominator of a fraction, changing the fraction's value to $\frac12$. If the original numerator was a 2, what was the original denominator? | 9 | 1 | 1,684.6875 | 1,684.6875 | -1 | |
Let $r$ be a complex number such that $r^5 = 1$ and $r \neq 1.$ Compute
\[(r - 1)(r^2 - 1)(r^3 - 1)(r^4 - 1).\] | 5 | 0.6875 | 6,262.625 | 5,385.636364 | 8,192 | |
Two people, A and B, participate in a general knowledge competition, with a total of 4 different questions, including 2 multiple-choice questions and 2 true/false questions. A and B each draw one question (without repetition).
$(1)$ What is the probability that A draws a multiple-choice question and B draws a true/fa... | \frac{5}{6} | 0.8125 | 3,979.875 | 3,007.846154 | 8,192 | |
Assume every 7-digit whole number is a possible telephone number except those that begin with $0$ or $1$. What fraction of telephone numbers begin with $9$ and end with $0$? | \frac{1}{80} | 1. **Determine the total number of valid 7-digit phone numbers ($b$):**
- The first digit can be any digit from 2 to 9 (since 0 and 1 are not allowed), giving us 8 choices.
- Each of the remaining six digits can be any digit from 0 to 9, giving us 10 choices for each digit.
- Therefore, the total number of val... | 1 | 1,604.375 | 1,604.375 | -1 |
Let $S(n)$ equal the sum of the digits of positive integer $n$. For example, $S(1507) = 13$. For a particular positive integer $n$, $S(n) = 1274$. Which of the following could be the value of $S(n+1)$?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 1239\qquad\textbf{(E)}\ 1265$
| 1239 | 0 | 4,374.0625 | -1 | 4,374.0625 | |
Let $f(x)$ and $g(x)$ be two monic cubic polynomials, and let $s$ be a real number. Two of the roots of $f(x)$ are $s + 2$ and $s + 8$. Two of the roots of $g(x)$ are $s + 5$ and $s + 11$, and
\[f(x) - g(x) = 2s\] for all real numbers $x$. Find $s$. | \frac{81}{4} | 0.25 | 7,554.9375 | 5,643.75 | 8,192 | |
In an isosceles triangle \(ABC\) with base \(AC\) equal to 37, the exterior angle at vertex \(B\) is \(60^\circ\). Find the distance from vertex \(C\) to line \(AB\). | 18.5 | 0 | 7,414.25 | -1 | 7,414.25 | |
If each digit of a four-digit natural number $M$ is not $0$, and the five times of the digit in the thousandth place is equal to the sum of the digits in the hundredth, tenth, and unit places, then this four-digit number is called a "modest number". For example, for the four-digit number $2163$, since $5\times 2=1+6+3$... | 3816 | 0.0625 | 7,955.3125 | 5,955 | 8,088.666667 | |
A fly trapped inside a cubical box with side length $1$ meter decides to relieve its boredom by visiting each corner of the box. It will begin and end in the same corner and visit each of the other corners exactly once. To get from a corner to any other corner, it will either fly or crawl in a straight line. What is th... | $4\sqrt{2}+4\sqrt{3}$ | 1. **Understanding the Problem**: A fly is inside a cubical box with side length $1$ meter. It starts at one corner, visits each of the other corners exactly once, and returns to the starting corner. The fly moves in straight lines between corners. We need to find the maximum possible length of its path.
2. **Identify... | 0 | 8,192 | -1 | 8,192 |
Call a positive integer $n$ weird if $n$ does not divide $(n-2)$!. Determine the number of weird numbers between 2 and 100 inclusive. | 26 | We claim that all the weird numbers are all the prime numbers and 4. Since no numbers between 1 and $p-2$ divide prime $p,(p-2)$! will not be divisible by $p$. We also have $2!=2$ not being a multiple of 4. Now we show that all other numbers are not weird. If $n=p q$ where $p \neq q$ and $p, q \geq 2$, then since $p$ a... | 0.0625 | 8,168.4375 | 7,815 | 8,192 |
From 5 differently colored balls, select 4 balls to place into 3 distinct boxes, with the requirement that no box is left empty. The total number of different ways to do this is ______. (Answer with a number) | 180 | 0.9375 | 5,303.8125 | 5,194.2 | 6,948 | |
Evaluate the expression given by $$2+\cfrac{3}{4+\cfrac{5}{6+\cfrac{7}{8}}}.$$ | \frac{137}{52} | 0.9375 | 4,252.25 | 3,989.6 | 8,192 | |
A carton contains milk that is $2\%$ fat, an amount that is $40\%$ less fat than the amount contained in a carton of whole milk. What is the percentage of fat in whole milk? | \frac{10}{3} | 1. **Understanding the Problem:**
The problem states that a carton of milk contains 2% fat, which is 40% less than the fat content in a carton of whole milk. We need to find the percentage of fat in the whole milk.
2. **Setting Up the Equation:**
Let $x$ be the percentage of fat in whole milk. According to the p... | 0.5 | 2,449 | 3,143.375 | 1,754.625 |
Throw a fair die, and let event $A$ be that the number facing up is even, and event $B$ be that the number facing up is greater than $2$ and less than or equal to $5$. Then, the probability of the complement of event $B$ is ____, and the probability of event $A \cup B$ is $P(A \cup B) = $ ____. | \dfrac{5}{6} | 1 | 2,264.3125 | 2,264.3125 | -1 | |
Simplify $\dfrac{18}{17}\cdot\dfrac{13}{24}\cdot\dfrac{68}{39}$. | 1 | 0.625 | 5,613.5 | 4,066.4 | 8,192 | |
How many of the 729 smallest positive integers written in base 9 use 7 or 8 (or both) as a digit? | 386 | 0.0625 | 7,701.5 | 5,270 | 7,863.6 | |
Compute \( \frac{2^{3}-1}{2^{3}+1} \cdot \frac{3^{3}-1}{3^{3}+1} \cdot \frac{4^{3}-1}{4^{3}+1} \cdot \frac{5^{3}-1}{5^{3}+1} \cdot \frac{6^{3}-1}{6^{3}+1} \). | 43/63 | Use the factorizations \( n^{3}-1=(n-1)\left(n^{2}+n+1\right) \) and \( n^{3}+1=(n+1)\left(n^{2}-n+1\right) \) to write \( \frac{1 \cdot 7}{3 \cdot 3} \cdot \frac{2 \cdot 13}{4 \cdot 7} \cdot \frac{3 \cdot 21}{5 \cdot 13} \cdot \frac{4 \cdot 31}{6 \cdot 21} \cdot \frac{5 \cdot 43}{7 \cdot 31}=\frac{1 \cdot 2 \cdot 43}{... | 0.5 | 6,672.75 | 5,775.875 | 7,569.625 |
Given that \( a \) and \( b \) are positive integers, and \( b - a = 2013 \). If the equation \( x^{2} - a x + b = 0 \) has a positive integer solution, what is the smallest value of \( a \)? | 93 | 0.9375 | 4,209.6875 | 3,944.2 | 8,192 | |
There are $15$ (not necessarily distinct) integers chosen uniformly at random from the range from $0$ to $999$ , inclusive. Yang then computes the sum of their units digits, while Michael computes the last three digits of their sum. The probability of them getting the same result is $\frac mn$ for relatively pri... | 200 | 0.125 | 8,192 | 8,192 | 8,192 | |
Remove all perfect squares from the sequence of positive integers $1, 2, 3, \ldots$ to obtain a new sequence. What is the 2003rd term of this new sequence? | 2048 | 0.875 | 5,391.375 | 4,991.285714 | 8,192 | |
An ideal gas is used as the working substance of a heat engine operating cyclically. The cycle consists of three stages: isochoric pressure reduction from $3 P_{0}$ to $P_{0}$, isobaric density increase from $\rho_{0}$ to $3 \rho_{0}$, and a return to the initial state, represented as a quarter circle in the $P / P_{0}... | 1/9 | 0 | 8,192 | -1 | 8,192 | |
The pie charts below indicate the percent of students who prefer golf, bowling, or tennis at East Junior High School and West Middle School. The total number of students at East is 2000 and at West, 2500. In the two schools combined, the percent of students who prefer tennis is | 32\% | 1. **Calculate the number of students who prefer tennis at East Junior High School:**
- The total number of students at East Junior High School is 2000.
- The percentage of students who prefer tennis at East Junior High School is 22%.
- Therefore, the number of students who prefer tennis at East Junior High Sc... | 0 | 8,044 | -1 | 8,044 |
Given the function $f(x)=\sin ^{2}x+ \frac{ \sqrt{3}}{2}\sin 2x$.
(1) Find the interval(s) where the function $f(x)$ is monotonically decreasing.
(2) In $\triangle ABC$, if $f(\frac{A}{2})=1$ and the area of the triangle is $3\sqrt{3}$, find the minimum value of side $a$. | 2\sqrt{3} | 0.9375 | 5,503.125 | 5,323.866667 | 8,192 | |
In the trapezoid \(ABCD\), the lengths of the bases \(AD = 24\) cm and \(BC = 8\) cm, and the diagonals \(AC = 13\) cm, \(BD = 5\sqrt{17}\) cm are known. Calculate the area of the trapezoid. | 80 | 0.875 | 3,332.9375 | 2,638.785714 | 8,192 | |
\(a_{1}, a_{2}, a_{3}, \ldots\) is an increasing sequence of natural numbers. It is known that \(a_{a_{k}} = 3k\) for any \(k\).
Find
a) \(a_{100}\)
b) \(a_{1983}\). | 3762 | 0 | 8,192 | -1 | 8,192 | |
Given a sequence of real numbers \( a_{1}, a_{2}, a_{3}, \ldots \) that satisfy
1) \( a_{1}=\frac{1}{2} \), and
2) \( a_{1} + a_{2} + \ldots + a_{k} = k^{2} a_{k} \), for \( k \geq 2 \).
Determine the value of \( a_{100} \). | \frac{1}{10100} | 1 | 4,213.3125 | 4,213.3125 | -1 | |
If: (1) \(a, b, c, d\) are all elements of the set \(\{1,2,3,4\}\); (2) \(a \neq b\), \(b \neq c\), \(c \neq d\), \(d \neq a\); (3) \(a\) is the smallest among \(a, b, c, d\). Then, how many different four-digit numbers \(\overline{abcd}\) can be formed? | 24 | 0 | 7,936.125 | -1 | 7,936.125 | |
Jeff wants to calculate the product $0.52 \times 7.35$ using a calculator. However, he mistakenly inputs the numbers as $52 \times 735$ without the decimal points. The calculator then shows a product of $38220$. What would be the correct product if Jeff had correctly entered the decimal points?
A) $0.3822$
B) $38.22$
C... | 3.822 | 0.3125 | 494.9375 | 459.4 | 511.090909 | |
Evaluate $\left|\frac12 - \frac38i\right|$. | \frac58 | 1 | 1,634.0625 | 1,634.0625 | -1 | |
Emily's broken clock runs backwards at five times the speed of a regular clock. How many times will it display the correct time in the next 24 hours? Note that it is an analog clock that only displays the numerical time, not AM or PM. The clock updates continuously. | 12 | 0.1875 | 7,730.8125 | 5,732.333333 | 8,192 | |
Given $a$, $b$, $c$, $d$, $e$ are 5 elements taken from the set $\{1, 2, 3, 4, 5\}$ without repetition, calculate the probability that $abc+de$ is an odd number. | \frac{2}{5} | 0.25 | 7,727.8125 | 6,809.5 | 8,033.916667 | |
For each positive integer $n$, let $f(n) = n^4 - 360n^2 + 400$. What is the sum of all values of $f(n)$ that are prime numbers? | 802 | 0.375 | 7,547.25 | 6,472.666667 | 8,192 | |
Currently, 7 students are to be assigned to participate in 5 sports events, with the conditions that students A and B cannot participate in the same event, each event must have at least one participant, and each student can only participate in one event. How many different ways can these conditions be satisfied? (Answe... | 15000 | 0.0625 | 8,157.1875 | 8,119 | 8,159.733333 | |
While standing in line to buy concert tickets, Kit moved 60 feet closer to the ticket window over a period of 30 minutes. At this rate, how many minutes will it take her to move the remaining 70 yards to the ticket window? | 105 | 1 | 1,352.9375 | 1,352.9375 | -1 | |
In the non-decreasing sequence of odd integers $\{a_1,a_2,a_3,\ldots \}=\{1,3,3,3,5,5,5,5,5,\ldots \}$ each odd positive integer $k$ appears $k$ times. It is a fact that there are integers $b, c$, and $d$ such that for all positive integers $n$, $a_n=b\lfloor \sqrt{n+c} \rfloor +d$, where $\lfloor x \rfloor$ denotes th... | 2 | 1. **Understanding the sequence**: The sequence $\{a_1, a_2, a_3, \ldots\}$ is defined such that each odd integer $k$ appears exactly $k$ times. For example, $1$ appears once, $3$ appears three times, $5$ appears five times, and so on.
2. **Form of the sequence**: We are given that $a_n = b\lfloor \sqrt{n+c} \rfloor +... | 0 | 7,852.0625 | -1 | 7,852.0625 |
What is the maximum value of \( N \) such that \( N! \) has exactly 2013 trailing zeros? | 8069 | 0.1875 | 7,862.4375 | 6,762.666667 | 8,116.230769 | |
The number $2017$ is prime. Let $S = \sum \limits_{k=0}^{62} \dbinom{2014}{k}$. What is the remainder when $S$ is divided by $2017?$
$\textbf{(A) }32\qquad \textbf{(B) }684\qquad \textbf{(C) }1024\qquad \textbf{(D) }1576\qquad \textbf{(E) }2016\qquad$
| 1024 | 0 | 7,743.1875 | -1 | 7,743.1875 | |
Let $L O V E R$ be a convex pentagon such that $L O V E$ is a rectangle. Given that $O V=20$ and $L O=V E=R E=R L=23$, compute the radius of the circle passing through $R, O$, and $V$. | 23 | Let $X$ be the point such that $R X O L$ is a rhombus. Note that line $R X$ defines a line of symmetry on the pentagon $L O V E R$. Then by symmetry $R X V E$ is also a rhombus, so $R X=O X=V X=23$. This makes $X$ the center of the circle, and the radius is 23. | 0.5625 | 7,315.125 | 6,633.111111 | 8,192 |
What is $(-1)^1+(-1)^2+\cdots+(-1)^{2006}$ ? | 0 | 1 | 2,106.5625 | 2,106.5625 | -1 | |
How many ordered pairs of integers $(a,b)$ satisfy all of the following inequalities? \[ \begin{aligned} a^2 + b^2 &< 16 \\ a^2 + b^2 &< 8a \\ a^2 + b^2 &< 8b \end{aligned}\] | 6 | 0.4375 | 7,585.3125 | 6,805.285714 | 8,192 | |
Six natural numbers (with possible repetitions) are written on the faces of a cube, such that the numbers on adjacent faces differ by more than 1. What is the smallest possible sum of these six numbers? | 18 | 0 | 8,192 | -1 | 8,192 | |
With all angles measured in degrees, the product $\prod_{k=1}^{45} \csc^2(2k-1)^\circ=m^n$, where $m$ and $n$ are integers greater than 1. Find $m+n$. | 91 | Let $p=\sin1\sin3\sin5...\sin89$
\[p=\sqrt{\sin1\sin3\sin5...\sin177\sin179}\]
\[=\sqrt{\frac{\sin1\sin2\sin3\sin4...\sin177\sin178\sin179}{\sin2\sin4\sin6\sin8...\sin176\sin178}}\]
\[=\sqrt{\frac{\sin1\sin2\sin3\sin4...\sin177\sin178\sin179}{(2\sin1\cos1)\cdot(2\sin2\cos2)\cdot(2\sin3\cos3)\cdot....\cdot(2\sin89\cos89... | 0.1875 | 7,905.375 | 6,730.333333 | 8,176.538462 |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $b^2+c^2-a^2+bc=0$,
(1) Find the measure of angle $A$;
(2) If $a= \sqrt {3}$, find the maximum value of the area $S_{\triangle ABC}$ of triangle $ABC$. | \frac { \sqrt {3}}{4} | 0 | 5,508.5 | -1 | 5,508.5 |
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