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 manager of a company planned to distribute a $50 bonus to each employee from the company fund, but the fund contained $5 less than what was needed. Instead the manager gave each employee a $45 bonus and kept the remaining $95 in the company fund. The amount of money in the company fund before any bonuses were pai... | 995 | Let $n$ be the number of employees in the company. According to the problem, the manager initially planned to give each employee a $50$ bonus, but the fund was $5$ short. Therefore, the total amount required for the $50$ bonus per employee would be $50n$, and the amount in the fund was $50n - 5$.
However, the manager ... | 1 | 1,219.0625 | 1,219.0625 | -1 |
Eric builds a small pyramid for a school project. His pyramid has a height of twelve inches and a square base that measures ten inches on each side. Eric wants to find the smallest cube-shaped box to put his pyramid in so that he can safely bring it to school right side up. What is the volume of this box, in inches cub... | 1728 | 0.75 | 4,088 | 2,720 | 8,192 | |
"My phone number," said the trip leader to the kids, "is a five-digit number. The first digit is a prime number, and the last two digits are obtained from the previous pair (which represents a prime number) by rearrangement, forming a perfect square. The number formed by reversing this phone number is even." What is th... | 26116 | 0.5 | 5,931.625 | 3,890.75 | 7,972.5 | |
In the polar coordinate system, the equation of circle C is $\rho=4\sqrt{2}\cos(\theta-\frac{\pi}{4})$. A Cartesian coordinate system is established with the pole as the origin and the positive x-axis as the polar axis. The parametric equation of line $l$ is $\begin{cases} x=t+1 \\ y=t-1 \end{cases}$ (where $t$ is the ... | 2\sqrt{3} | 1 | 4,999.0625 | 4,999.0625 | -1 | |
How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots?
(Two rectangles are different if they do not share all four vertices.) | 100 | 0.3125 | 7,369.4375 | 6,549.6 | 7,742.090909 | |
Let $n$ be the smallest positive integer that is a multiple of $75$ and has exactly $75$ positive integral divisors, including $1$ and itself. Find $\frac{n}{75}$.
| 432 | 0.4375 | 7,591.8125 | 6,820.142857 | 8,192 | |
Let \(\lfloor x\rfloor\) denote the greatest integer which is less than or equal to \(x\). For example, \(\lfloor\pi\rfloor = 3\). \(S\) is the integer equal to the sum of the 100 terms shown:
$$
S = \lfloor\pi\rfloor + \left\lfloor\pi + \frac{1}{100}\right\rfloor + \left\lfloor\pi + \frac{2}{100}\right\rfloor + \left... | 314 | 0.6875 | 5,593.3125 | 4,412.090909 | 8,192 | |
The values of $a$, $b$, $c$ and $d$ are 1, 2, 3 and 4, but not necessarily in that order. What is the largest possible value of the sum of the four products $ab$, $bc$, $cd$ and $da$? | 25 | 0.9375 | 3,592.875 | 3,460.266667 | 5,582 | |
If $\sec y + \tan y = 3,$ then find $\sec y - \tan y.$ | \frac{1}{3} | 1 | 2,171.875 | 2,171.875 | -1 | |
The Grammar club has 20 members: 10 boys and 10 girls. A 4-person committee is chosen at random. What is the probability that the committee has at least 1 boy and at least 1 girl? | \dfrac{295}{323} | 0.875 | 5,540.5 | 5,161.714286 | 8,192 | |
Find $a$ such that $ax^2+12x+9$ is the square of a binomial. | 4 | 1 | 1,822.625 | 1,822.625 | -1 | |
There is a cube of size \(10 \times 10 \times 10\) made up of small unit cubes. A grasshopper is sitting at the center \(O\) of one of the corner cubes. It can jump to the center of a cube that shares a face with the one in which the grasshopper is currently located, provided that the distance to point \(O\) increases.... | \frac{27!}{(9!)^3} | 0.4375 | 6,623 | 5,384 | 7,586.666667 | |
Riders on a Ferris wheel travel in a circle in a vertical plane. A particular wheel has radius $20$ feet and revolves at the constant rate of one revolution per minute. How many seconds does it take a rider to travel from the bottom of the wheel to a point $10$ vertical feet above the bottom? | 10 | 1. **Understanding the problem**: We need to find the time it takes for a rider to move from the bottom of a Ferris wheel to a point 10 feet above the bottom. The Ferris wheel has a radius of 20 feet and completes one revolution per minute.
2. **Setting up the equation**: The vertical position of the rider as a functi... | 0.4375 | 7,035.5 | 6,573.571429 | 7,394.777778 |
Consider a 4x4 grid of points (equally spaced). How many rectangles, of any size, can be formed where each of its four vertices are points on this grid? | 36 | 0.4375 | 7,056.75 | 5,597.142857 | 8,192 | |
To investigate the height of high school students, a stratified sampling method is used to draw a sample of 100 students from three grades. 24 students are sampled from grade 10, 26 from grade 11. If there are 600 students in grade 12, then the total number of students in the school is $\_\_\_\_\_\_$. | 1200 | 0.4375 | 5,090.375 | 3,002 | 6,714.666667 | |
Five couples were at a party. If each person shook hands exactly once with everyone else except his/her spouse, how many handshakes were exchanged? (Note: One obviously doesn't shake hands with oneself.) | 40 | 1 | 2,956.25 | 2,956.25 | -1 | |
Let $ABC$ be an equilateral triangle and a point M inside the triangle such that $MA^2 = MB^2 +MC^2$ . Draw an equilateral triangle $ACD$ where $D \ne B$ . Let the point $N$ inside $\vartriangle ACD$ such that $AMN$ is an equilateral triangle. Determine $\angle BMC$ .
| 150 | 0 | 8,192 | -1 | 8,192 | |
Given the function $f(x)=\frac{1}{3}x^{3}+ax^{2}+bx$ $(a,b\in \mathbb{R})$ attains a maximum value of $9$ at $x=-3$.
$(I)$ Find the values of $a$ and $b$;
$(II)$ Find the maximum and minimum values of the function $f(x)$ in the interval $[-3,3]$. | - \frac{5}{3} | 0.9375 | 3,491.8125 | 3,465.866667 | 3,881 | |
What is the maximum number of L-shaped 3-cell pieces that can be cut out from a rectangular grid of size:
a) $5 \times 10$ cells;
b) $5 \times 9$ cells? | 15 | 0.5625 | 7,705.25 | 7,326.666667 | 8,192 | |
A function $f$ has the property that $f(3x-1)=x^2+x+1$ for all real numbers $x$. What is $f(5)$? | 7 | 1 | 2,452 | 2,452 | -1 | |
If $P = \sqrt{1988 \cdot 1989 \cdot 1990 \cdot 1991 + 1} + \left(-1989^{2}\right)$, calculate the value of $P$. | 1988 | 0.75 | 3,825.0625 | 3,669.25 | 4,292.5 | |
Given eleven books consisting of three Arabic, two English, four Spanish, and two French, calculate the number of ways to arrange the books on the shelf keeping the Arabic books together, the Spanish books together, and the English books together. | 34560 | 0.25 | 5,032.5 | 4,950.75 | 5,059.75 | |
For any positive integer $n$, the value of $n!$ is the product of the first $n$ positive integers. Calculate the greatest common divisor of $8!$ and $10!$. | 40320 | 0.8125 | 3,491.875 | 2,569.230769 | 7,490 | |
Mena listed the numbers from 1 to 30 one by one. Emily copied these numbers and substituted every digit 2 with digit 1. Both calculated the sum of the numbers they wrote. By how much is the sum that Mena calculated greater than the sum that Emily calculated? | 103 | 0.125 | 5,191.25 | 4,962 | 5,224 | |
A group of students during a sports meeting lined up for a team photo. When they lined up in rows of 5, there were two students left over. When they formed rows of 6 students, there were three extra students, and when they lined up in rows of 8, there were four students left over. What is the fewest number of students ... | 59 | 0 | 7,279.8125 | -1 | 7,279.8125 | |
Seven students are standing in a line for a graduation photo. Among them, student A must be in the middle, and students B and C must stand together. How many different arrangements are possible? | 192 | 0.125 | 6,874.375 | 7,940 | 6,722.142857 | |
The train schedule in Hummut is hopelessly unreliable. Train A will enter Intersection X from the west at a random time between 9:00 am and 2:30 pm; each moment in that interval is equally likely. Train B will enter the same intersection from the north at a random time between 9:30 am and 12:30 pm, independent of Train... | \frac{13}{48} | Suppose we fix the time at which Train B arrives at Intersection X; then call the interval during which Train A could arrive (given its schedule) and collide with Train B the 'disaster window.' We consider two cases: (i) Train B enters Intersection $X$ between 9:30 and 9:45. If Train B arrives at 9:30, the disaster win... | 0 | 7,806.1875 | -1 | 7,806.1875 |
How many different positive, six-digit integers can be formed using the digits 2, 2, 5, 5, 9 and 9? | 90 | 0.9375 | 2,920.25 | 2,568.8 | 8,192 | |
On a certain math exam, $10\%$ of the students got $70$ points, $25\%$ got $80$ points, $20\%$ got $85$ points, $15\%$ got $90$ points, and the rest got $95$ points. What is the difference between the mean and the median score on this exam? | 1 | 1. **Calculate the percentage of students scoring 95 points**:
Given that $10\%$ scored 70 points, $25\%$ scored 80 points, $20\%$ scored 85 points, and $15\%$ scored 90 points, the remaining percentage of students who scored 95 points is:
\[
100\% - (10\% + 25\% + 20\% + 15\%) = 100\% - 70\% = 30\%
\]
2.... | 0.8125 | 4,116.125 | 3,908.692308 | 5,015 |
There are 17 people at a party, and each has a reputation that is either $1,2,3,4$, or 5. Some of them split into pairs under the condition that within each pair, the two people's reputations differ by at most 1. Compute the largest value of $k$ such that no matter what the reputations of these people are, they are abl... | 7 | First, note that $k=8$ fails when there are $15,0,1,0,1$ people of reputation 1, 2, 3, 4, 5, respectively. This is because the two people with reputation 3 and 5 cannot pair with anyone, and there can only be at maximum $\left\lfloor\frac{15}{2}\right\rfloor=7$ pairs of people with reputation 1. Now, we show that $k=7$... | 0 | 8,068.9375 | -1 | 8,068.9375 |
What is the smallest positive integer $a$ such that $a^{-1}$ is undefined $\pmod{55}$ and $a^{-1}$ is also undefined $\pmod{66}$? | 10 | 0.125 | 3,152.4375 | 5,140.5 | 2,868.428571 | |
Given the parabola C: y^2 = 4x, point B(3,0), and the focus F, point A lies on C. If |AF| = |BF|, calculate the length of |AB|. | 2\sqrt{2} | 0.625 | 5,046.1875 | 5,064.8 | 5,015.166667 | |
How many ways are there to rearrange the letters of CCAMB such that at least one C comes before the A?
*2019 CCA Math Bonanza Individual Round #5* | 40 | 0.1875 | 7,605.3125 | 5,063 | 8,192 | |
A line $l$ is tangent to the circle $x^{2}+y^{2}=1$ and the sum of its intercepts on the two coordinate axes is equal to $\sqrt{3}$. Find the area of the triangle formed by line $l$ and the two coordinate axes. | \frac{3}{2} | 0.5 | 7,424.6875 | 6,657.375 | 8,192 | |
Bryan has some 3 cent stamps and some 4 cent stamps. What is the least number of stamps he can combine so the value of the stamps is 33 cents? | 9 | 1 | 3,550.875 | 3,550.875 | -1 | |
If $9:y^3 = y:81$, what is the value of $y$? | 3\sqrt{3} | 0.5 | 3,610.5 | 3,340.25 | 3,880.75 | |
Given that the coordinates of a point on the terminal side of angle $\alpha$ are $(\frac{\sqrt{3}}{2},-\frac{1}{2})$, determine the smallest positive value of angle $\alpha$. | \frac{11\pi}{6} | 0.5625 | 2,187.125 | 1,906.333333 | 2,548.142857 | |
Given that $a$, $b$, and $c$ represent the sides opposite to angles $A$, $B$, and $C$ of $\triangle ABC$ respectively, and $2\sin \frac{7\pi }{6}\sin (\frac{\pi }{6}+C)+ \cos C=-\frac{1}{2}$.
(1) Find $C$;
(2) If $c=2\sqrt{3}$, find the maximum area of $\triangle ABC$. | 3\sqrt{3} | 0.875 | 5,830.6875 | 5,673.928571 | 6,928 | |
If we write $\sqrt{5}+\frac{1}{\sqrt{5}} + \sqrt{7} + \frac{1}{\sqrt{7}}$ in the form $\dfrac{a\sqrt{5} + b\sqrt{7}}{c}$ such that $a$, $b$, and $c$ are positive integers and $c$ is as small as possible, then what is $a+b+c$? | 117 | 1 | 5,966.375 | 5,966.375 | -1 | |
Find all functions $ f: (0, \infty) \mapsto (0, \infty)$ (so $ f$ is a function from the positive real numbers) such that
\[ \frac {\left( f(w) \right)^2 \plus{} \left( f(x) \right)^2}{f(y^2) \plus{} f(z^2) } \equal{} \frac {w^2 \plus{} x^2}{y^2 \plus{} z^2}
\]
for all positive real numbers $ w,x,y,z,$ satisfying $ wx ... | f(x) = x \text{ or } f(x) = \frac{1}{x} |
To find all functions \( f: (0, \infty) \to (0, \infty) \) satisfying the given functional equation:
\[
\frac {\left( f(w) \right)^2 + \left( f(x) \right)^2}{f(y^2) + f(z^2) } = \frac {w^2 + x^2}{y^2 + z^2}
\]
for all positive real numbers \( w, x, y, z \) such that \( wx = yz \), we proceed as follows:
### Step 1:... | 0.125 | 8,037.1875 | 6,953.5 | 8,192 |
Let $\mathbf{A}$ be a matrix such that
\[\mathbf{A} \begin{pmatrix} 5 \\ -2 \end{pmatrix} = \begin{pmatrix} -15 \\ 6 \end{pmatrix}.\]Find $\mathbf{A}^5 \begin{pmatrix} 5 \\ -2 \end{pmatrix}.$ | \begin{pmatrix} -1215 \\ 486 \end{pmatrix} | 0.8125 | 4,044.625 | 3,191 | 7,743.666667 | |
In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\left(a-c\right)\left(a+c\right)\sin C=c\left(b-c\right)\sin B$.
$(1)$ Find angle $A$;
$(2)$ If the area of $\triangle ABC$ is $\sqrt{3}$, $\sin B\sin C=\frac{1}{4}$, find the value of $a$. | 2\sqrt{3} | 1 | 4,236.375 | 4,236.375 | -1 | |
If real numbers $\alpha, \beta, \gamma$ form a geometric sequence with a common ratio of 2, and $\sin \alpha, \sin \beta, \sin \gamma$ also form a geometric sequence, then $\cos \alpha = $ ? | -\frac{1}{2} | 0.9375 | 4,949.8125 | 4,733.666667 | 8,192 | |
Given the function $$f(x)= \begin{cases} ( \frac {1}{2})^{x} & ,x≥4 \\ f(x+1) & ,x<4\end{cases}$$, find the value of $f(\log_{2}3)$. | \frac {1}{24} | 0.75 | 4,779.5625 | 3,999 | 7,121.25 | |
Let \( S = \{1, 2, \cdots, 2016\} \). For any non-empty finite sets of real numbers \( A \) and \( B \), find the minimum value of
\[ f = |A \Delta S| + |B \Delta S| + |C \Delta S| \]
where
\[ X \Delta Y = \{a \in X \mid a \notin Y\} \cup \{a \in Y \mid a \notin X\} \]
is the symmetric difference between sets \( X \) a... | 2017 | 0 | 8,192 | -1 | 8,192 | |
A bottle of cola costs 2 yuan, and two empty bottles can be exchanged for one more bottle of cola. With 30 yuan, what is the maximum number of bottles of cola that you can drink? | 29 | 0.25 | 6,149 | 6,762.5 | 5,944.5 | |
Lunasa, Merlin, and Lyrica each have a distinct hat. Every day, two of these three people, selected randomly, switch their hats. What is the probability that, after 2017 days, every person has their own hat back? | 0 | Imagine that the three hats are the vertices of an equilateral triangle. Then each day the exchange is equivalent to reflecting the triangle along one of its three symmetry axes, which changes the orientation of the triangle (from clockwise to counterclockwise or vice versa). Thus, an even number of such exchanges must... | 0.5625 | 5,278 | 3,011.555556 | 8,192 |
In a convex pentagon \( P Q R S T \), the angle \( P R T \) is half of the angle \( Q R S \), and all sides are equal. Find the angle \( P R T \). | 30 | 0 | 8,047.4375 | -1 | 8,047.4375 | |
The sum of the digits of the time 19 minutes ago is two less than the sum of the digits of the time right now. Find the sum of the digits of the time in 19 minutes. (Here, we use a standard 12-hour clock of the form hh:mm.) | 11 | Let's say the time 19 minutes ago is $h$ hours and $m$ minutes, so the sum of the digits is equivalent to $h+m \bmod 9$. If $m \leq 40$, then the time right now is hours and $m+19$ minutes, so the sum of digits is equivalent \bmod 9 to $h+m+19 \equiv h+m+1(\bmod 9)$, which is impossible. If $m>40$ and $h<12$, then the ... | 0 | 8,192 | -1 | 8,192 |
If $\frac{1}{3}$ of $x$ is equal to 4, what is $\frac{1}{6}$ of $x$? | 2 | Since $\frac{1}{3}$ of $x$ is equal to 4, then $x$ is equal to $3 \times 4$ or 12. Thus, $\frac{1}{6}$ of $x$ is equal to $12 \div 6=2$. Alternatively, since $\frac{1}{6}$ is one-half of $\frac{1}{3}$, then $\frac{1}{6}$ of $x$ is equal to one-half of $\frac{1}{3}$ of $x$, which is $4 \div 2$ or 2. | 1 | 1,209.25 | 1,209.25 | -1 |
When studying the traffic situation in a certain city, road density refers to the number of vehicles passing through a certain section of road divided by time, and vehicle density is the number of vehicles passing through a certain section of road divided by the length of that section. The traffic flow is defined as $v... | \frac{28800}{7} | 0.125 | 6,480.3125 | 7,277.5 | 6,366.428571 | |
If a four-digit natural number $\overline{abcd}$ has digits that are all different and not equal to $0$, and satisfies $\overline{ab}-\overline{bc}=\overline{cd}$, then this four-digit number is called a "decreasing number". For example, the four-digit number $4129$, since $41-12=29$, is a "decreasing number"; another ... | 8165 | 0.125 | 7,615.4375 | 7,545.5 | 7,625.428571 | |
The operation $a \nabla b$ is defined by $a \nabla b=\frac{a+b}{a-b}$ for all integers $a$ and $b$ with $a \neq b$. If $3 \nabla b=-4$, what is the value of $b$? | 5 | Using the definition, $3 \nabla b=\frac{3+b}{3-b}$. Assuming $b \neq 3$, the following equations are equivalent: $3 \nabla b =-4$, $\frac{3+b}{3-b} =-4$, $3+b =-4(3-b)$, $3+b =-12+4 b$, $15 =3 b$ and so $b=5$. | 1 | 1,870.375 | 1,870.375 | -1 |
Each of two boxes contains four chips numbered $1$, $2$, $3$, and $4$. Calculate the probability that the product of the numbers on the two chips is a multiple of $4$. | \frac{1}{2} | 0.25 | 7,466.3125 | 5,289.25 | 8,192 | |
A two-digit integer $AB$ equals $\frac{1}{9}$ of the three-digit integer $CCB$, where $C$ and $B$ represent distinct digits from 1 to 9. What is the smallest possible value of the three-digit integer $CCB$? | 225 | 0.1875 | 7,821.4375 | 6,810 | 8,054.846154 | |
Define $A\star B$ as $A\star B = \frac{(A+B)}{3}$. What is the value of $(2\star 10) \star 5$? | 3 | 1 | 1,255.5 | 1,255.5 | -1 | |
What is the largest 4-digit integer congruent to $7 \pmod{19}$? | 9982 | 0.875 | 5,099.625 | 4,657.857143 | 8,192 | |
Given that \( x \) is a four-digit number and the sum of its digits is \( y \). When the value of \( \frac{x}{y} \) is minimized, \( x = \) _______ | 1099 | 0.125 | 7,837 | 7,123 | 7,939 | |
If the set $\{1, a, \frac{b}{a}\}$ equals the set $\{0, a^2, a+b\}$, then find the value of $a^{2017} + b^{2017}$. | -1 | 0.125 | 7,904.1875 | 8,192 | 7,863.071429 | |
Add \(53_8 + 27_8\). Express your answer in base \(8\). | 102_8 | 0.8125 | 4,262.3125 | 3,355.461538 | 8,192 | |
What is the maximum number of kings that can be placed on a chessboard so that no two of them attack each other? | 16 | 0.4375 | 7,328.4375 | 6,218.142857 | 8,192 | |
Let $ABC$ be a triangle. There exists a positive real number $k$, such that if the altitudes of triangle $ABC$ are extended past $A$, $B$, and $C$, to $A'$, $B'$, and $C'$, as shown, such that $AA' = kBC$, $BB' = kAC$, and $CC' = kAB$, then triangle $A'B'C'$ is equilateral.
[asy]
unitsize(0.6 cm);
pair[] A, B, C;
pa... | \frac{1}{\sqrt{3}} | 0 | 8,192 | -1 | 8,192 | |
Given the hexadecimal system, determine the product of $A$ and $B$. | 6E | 0.5625 | 1,714.0625 | 614 | 3,128.428571 | |
Tim wants to invest some money in a bank which compounds quarterly with an annual interest rate of $7\%$. To the nearest dollar, how much money should he invest if he wants a total of $\$60,\!000$ at the end of $5$ years? | \$42409 | 0 | 8,192 | -1 | 8,192 | |
Many states use a sequence of three letters followed by a sequence of three digits as their standard license-plate pattern. Given that each three-letter three-digit arrangement is equally likely, the probability that such a license plate will contain at least one palindrome (a three-letter arrangement or a three-digit ... | 59 | Note that we can pick the first and second letters/numbers freely with one choice left for the last letter/number for there to be a palindrome. Thus, the probability of no palindrome is \[\frac{25}{26}\cdot \frac{9}{10}=\frac{45}{52}\] thus we have $1-\frac{45}{52}=\frac{7}{52}$ so our answer is $7+52 = \boxed{059}.$
... | 1 | 3,710.5 | 3,710.5 | -1 |
Given a function $f: \{1, 2, 3\} \rightarrow \{1, 2, 3\}$ that satisfies $f(f(x)) = f(x)$, determine the total number of such functions. | 10 | 0.625 | 6,698.125 | 5,801.8 | 8,192 | |
Solve the equations.
4x + x = 19.5
26.4 - 3x = 14.4
2x - 0.5 × 2 = 0.8. | 0.9 | 0.8125 | 684.5625 | 694.769231 | 640.333333 | |
Positive integers $a$, $b$, $c$, and $d$ satisfy $a > b > c > d$, $a + b + c + d = 2010$, and $a^2 - b^2 + c^2 - d^2 = 2010$. Find the number of possible values of $a.$ | 501 | 0.25 | 7,400.4375 | 5,374.75 | 8,075.666667 | |
Given that \( t = \frac{1}{1 - \sqrt[4]{2}} \), simplify the expression for \( t \). | -(1 + \sqrt[4]{2})(1 + \sqrt{2}) | 0 | 6,309.0625 | -1 | 6,309.0625 | |
The base-ten representation for $19!$ is $121,6T5,100,40M,832,H00$, where $T$, $M$, and $H$ denote digits that are not given. What is $T+M+H$? | 12 | 1. **Determine the value of $H$:**
Since $19!$ includes the factors $2^{15}$ and $5^3$ (from the prime factorization of numbers from $1$ to $19$), it has at least three factors of $10$, and thus ends in at least three zeros. Therefore, $H = 0$.
2. **Use the divisibility rule for $9$:**
For a number to be div... | 0.4375 | 7,206.8125 | 5,940.142857 | 8,192 |
How many integer solutions does the inequality
$$
|x| + |y| < 1000
$$
have, where \( x \) and \( y \) are integers? | 1998001 | 0.1875 | 8,006.25 | 7,201.333333 | 8,192 | |
Let $P$ and $Q$ be the midpoints of sides $AB$ and $BC,$ respectively, of $\triangle ABC.$ Suppose $\angle A = 30^{\circ}$ and $\angle PQC = 110^{\circ}.$ Find $\angle B$ in degrees. | 80 | 0 | 8,127.875 | -1 | 8,127.875 | |
Divide a square into 25 smaller squares, where 24 of these smaller squares are unit squares, and the remaining piece can also be divided into squares with a side length of 1. Find the area of the original square. | 25 | 0.125 | 7,105.8125 | 5,872.5 | 7,282 | |
A plane flies between two cities. With the tailwind, it takes 5 hours and 30 minutes, and against the wind, it takes 6 hours. Given that the wind speed is 24 kilometers per hour, and assuming the plane's flying speed is $x$ kilometers per hour, then the speed of the plane with the tailwind is kilometers per hour, a... | 528 | 1 | 2,239.125 | 2,239.125 | -1 | |
David, Ellie, Natasha, and Lucy are tutors in their school science lab. Their working schedule is as follows: David works every fourth school day, Ellie works every fifth school day, Natasha works every sixth school day, and Lucy works every eighth school day. Today, they all happened to be working together. In how man... | 120 | 1 | 1,620.6875 | 1,620.6875 | -1 | |
Calculate: \((56 \times 0.57 \times 0.85) \div(2.8 \times 19 \times 1.7) =\) | 0.3 | 0.25 | 3,255.4375 | 5,858 | 2,387.916667 | |
\(a, b, c\) are distinct positive integers such that \(\{a+b, b+c, c+a\} = \left\{n^2, (n+1)^2, (n+2)^2\right\}\), where \(n\) is a positive integer. What is the minimum value of \(a^2 + b^2 + c^2\)? | 1297 | 0.25 | 8,012.3125 | 7,473.25 | 8,192 | |
Let $ S $ be the set of all sides and diagonals of a regular hexagon. A pair of elements of $ S $ are selected at random without replacement. What is the probability that the two chosen segments have the same length? | \frac{33}{105} | 0 | 4,951.625 | -1 | 4,951.625 | |
Given the sequence ${a_n}$ satisfying $a_1=1$, $a_2=2$, $a_3=3$, $a_{n+3}=a_n$ ($n\in\mathbb{N}^*$). If $a_n=A\sin(\omega n+\varphi)+c$ $(ω>0,|\varphi|<\frac{\pi}{2})$, find the real number $A$. | -\frac{2\sqrt{3}}{3} | 0 | 7,127.125 | -1 | 7,127.125 | |
The product of two consecutive negative integers is 2550. What is the sum of the two integers? | -101 | 1 | 2,730.125 | 2,730.125 | -1 | |
In the convex quadrilateral $ABCD$, the sum of $AB+BD+DC$ is at most 2, and the area of the quadrilateral is $1/2$. What can be the length of diagonal $AC$? | \sqrt{2} | 0 | 8,151.375 | -1 | 8,151.375 | |
Let \( x, y, z \) be positive numbers that satisfy the system of equations:
\[
\begin{cases}
x^{2}+xy+y^{2}=27 \\
y^{2}+yz+z^{2}=16 \\
z^{2}+xz+x^{2}=43
\end{cases}
\]
Find the value of the expression \( xy+yz+xz \). | 24 | 0.5625 | 7,137.9375 | 6,318.111111 | 8,192 | |
Which digit is represented by $\Theta$ if $252/\Theta=\underline{3\Theta}+\Theta$, where $\underline{3\Theta}$ represents a two-digit number with $3$ in the tens digit and $\Theta$ in the ones digit? | 6 | 1 | 3,799.5 | 3,799.5 | -1 | |
A bundle of wire was used in the following sequence:
- The first time, more than half of the total length was used, plus an additional 3 meters.
- The second time, half of the remaining length was used, minus 10 meters.
- The third time, 15 meters were used.
- Finally, 7 meters were left.
How many meters of wire were ... | 54 | 0.1875 | 881.625 | 1,024 | 848.769231 | |
For a point $P = (a, a^2)$ in the coordinate plane, let $\ell(P)$ 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 intersections of the lines $\ell(P_1)$ , $\ell(P_2)$ , $\ell... | \[
\boxed{y = -\frac{1}{4}}
\] | Solution 1
Note that the lines $l(P_1), l(P_2), l(P_3)$ are \[y=2a_1x-a_1^2, y=2a_2x-a_2^2, y=2a_3x-a_3^2,\] respectively. It is easy to deduce that the three points of intersection are \[\left(\frac{a_1+a_2}{2},a_1a_2\right),\left(\frac{a_2+a_3}{2},a_2a_3\right), \left(\frac{a_3+a_1}{2},a_3a_1\right).\] The slopes of ... | 0 | 8,192 | -1 | 8,192 |
Given a hyperbola with asymptotes $2x \pm y=0$, that passes through the intersection of the lines $x+y-3=0$ and $2x-y+3t=0$, where $-2 \leq t \leq 5$. Find the maximum possible length of the real axis of the hyperbola. | 4\sqrt{3} | 0.25 | 6,820.75 | 6,315.25 | 6,989.25 | |
Line $m$ in the coordinate plane has the equation $2x - 3y + 30 = 0$. This line is rotated $30^{\circ}$ counterclockwise about the point $(10, 10)$ to form line $n$. Find the $x$-coordinate of the $x$-intercept of line $n$. | \frac{20\sqrt{3} + 20}{2\sqrt{3} + 3} | 0 | 8,136.25 | -1 | 8,136.25 | |
In the xy-plane, what is the length of the shortest path from $(0,0)$ to $(12,16)$ that does not go inside the circle $(x-6)^{2}+(y-8)^{2}= 25$? | 10\sqrt{3}+\frac{5\pi}{3} | 1. **Identify the Points and Circle**:
- Let $A(0,0)$ and $D(12,16)$ be the start and end points, respectively.
- The circle is centered at $O(6,8)$ with radius $5$, given by the equation $(x-6)^2 + (y-8)^2 = 25$.
2. **Calculate Distance $OA$**:
- Using the distance formula, $OA = \sqrt{(6-0)^2 + (8-0)^2} = ... | 0 | 7,760 | -1 | 7,760 |
In the complex plane, the line segment with end-points $-11 + 3i$ and $3 - 7i$ is plotted in the complex plane. Find the complex number corresponding to the mid-point of this line segment. | -4 - 2i | 1 | 1,597.0625 | 1,597.0625 | -1 | |
What is the smallest square of an integer that ends with the longest sequence of the same digits?
For example, if the longest sequence of the same digits were five, then a suitable number would be 24677777 (of course, if it were the smallest square, but it is not). Zero is not considered an acceptable digit. | 1444 | 0 | 8,192 | -1 | 8,192 | |
Let \(P_1\) be a regular \(r\)-sided polygon and \(P_2\) be a regular \(s\)-sided polygon with \(r \geq s \geq 3\), such that each interior angle of \(P_1\) is \(\frac{61}{60}\) as large as each interior angle of \(P_2\). What is the largest possible value of \(s\)? | 121 | 0.125 | 7,919.5 | 6,012 | 8,192 | |
What is the probability that in a random sequence of 8 ones and 2 zeros, there are exactly three ones between the two zeros? | 2/15 | 0.9375 | 4,895.5 | 4,675.733333 | 8,192 | |
Rectangle $ABCD$ is 8 cm by 4 cm. $M$ is the midpoint of $\overline{BC}$ , and $N$ is the midpoint of $\overline{CD}$. What is the number of square centimeters in the area of region $AMCN$?
[asy]
draw((0,0)--(32,0)--(32,16)--(0,16)--cycle);
draw((0,16)--(16,0)--(32,8)--cycle);
label("$A$",(0,16),N);
label("$B$",(32,16... | 16 | 0.75 | 6,766.375 | 6,291.166667 | 8,192 | |
A point $P$ is chosen at random from the interior of a square $ABCD$. What is the probability that the triangle $ABP$ has a greater area than each of the triangles $BCP$, $CDP$, and $DAP$? | \frac{1}{4} | 0.0625 | 8,147.5625 | 7,481 | 8,192 | |
A square is inscribed in a circle of radius 1. Find the perimeter of the square. | 4 \sqrt{2} | The square has diagonal length 2, so side length $\sqrt{2}$ and perimeter $4 \sqrt{2}$. | 1 | 1,652.375 | 1,652.375 | -1 |
How many integers are greater than $\sqrt{15}$ and less than $\sqrt{50}$? | 4 | Using a calculator, $\sqrt{15} \approx 3.87$ and $\sqrt{50} \approx 7.07$. The integers between these real numbers are $4,5,6,7$, of which there are 4 . Alternatively, we could note that integers between $\sqrt{15}$ and $\sqrt{50}$ correspond to values of $\sqrt{n}$ where $n$ is a perfect square and $n$ is between 15 a... | 1 | 4,159 | 4,159 | -1 |
The number of distinct points in the $xy$-plane common to the graphs of $(x+y-5)(2x-3y+5)=0$ and $(x-y+1)(3x+2y-12)=0$ is | 1 | To find the number of distinct points common to the graphs of $(x+y-5)(2x-3y+5)=0$ and $(x-y+1)(3x+2y-12)=0$, we need to solve the systems of equations formed by each factor being zero.
#### System 1: $x+y=5$ and $x-y=-1$
1. Adding these two equations:
\[
(x+y) + (x-y) = 5 + (-1) \implies 2x = 4 \implies x = 2
... | 1 | 5,155.3125 | 5,155.3125 | -1 |
The diameters of two pulleys with parallel axes are 80 mm and 200 mm, respectively, and they are connected by a belt that is 1500 mm long. What is the distance between the axes of the pulleys if the belt is tight (with millimeter precision)? | 527 | 0 | 7,934.1875 | -1 | 7,934.1875 | |
Turbo the snail sits on a point on a circle with circumference $1$. Given an infinite sequence of positive real numbers $c_1, c_2, c_3, \dots$, Turbo successively crawls distances $c_1, c_2, c_3, \dots$ around the circle, each time choosing to crawl either clockwise or counterclockwise.
Determine the largest constant $... | 0.5 |
To find the largest constant \( C > 0 \) with the given property, we first need to understand the problem setup. Turbo starts at a point on a circle with a circumference of 1 and moves according to the sequence of positive real numbers \( c_1, c_2, c_3, \ldots \). At each step, Turbo chooses to move either clockwise o... | 0.0625 | 8,147.9375 | 7,487 | 8,192 |
A positive real number $x$ is such that \[
\sqrt[3]{1-x^3} + \sqrt[3]{1+x^3} = 1.
\]Find $x^6.$ | \frac{28}{27} | 0.5625 | 5,941 | 4,190.222222 | 8,192 |
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