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 |
|---|---|---|---|---|---|---|
In trapezoid \( KLMN \), diagonal \( KM \) is equal to 1 and is also its height. From points \( K \) and \( M \), perpendiculars \( KP \) and \( MQ \) are drawn to sides \( MN \) and \( KL \), respectively. Find \( LM \) if \( KN = MQ \) and \( LM = MP \). | \sqrt{2} | 0 | 8,192 | -1 | 8,192 | |
Calculate the lengths of arcs of curves given by the parametric equations.
$$
\begin{aligned}
& \left\{\begin{array}{l}
x=2(t-\sin t) \\
y=2(1-\cos t)
\end{array}\right. \\
& 0 \leq t \leq \frac{\pi}{2}
\end{aligned}
$$ | 8 - 4\sqrt{2} | 0.5625 | 6,078.375 | 4,434.444444 | 8,192 | |
Given that the sequence $\{a\_n\}$ is an arithmetic sequence with the sum of its first $n$ terms denoted as $S\_n$. It is known that $a\_1 + a\_5 + a\_9 = 27$. Find $a\_5$ and $S\_9$. | 81 | 1 | 2,217.625 | 2,217.625 | -1 | |
Given vectors $\overrightarrow{O A} \perp \overrightarrow{O B}$, and $|\overrightarrow{O A}|=|\overrightarrow{O B}|=24$. Find the minimum value of $|t \overrightarrow{A B}-\overrightarrow{A O}|+\left|\frac{5}{12} \overrightarrow{B O}-(1-t) \overrightarrow{B A}\right|$ for $t \in[0,1]$. | 26 | 0 | 8,192 | -1 | 8,192 | |
If $\sqrt{8 + x} + \sqrt{15 - x} = 6$, what is the value of $(8 + x)(15 - x)$? | \frac{169}{4} | 1 | 3,847.125 | 3,847.125 | -1 | |
Two cars, Car A and Car B, start from point A and point B, respectively, and move towards each other simultaneously. They meet after 3 hours, after which Car A turns back towards point A and Car B continues moving forward. Once Car A reaches point A, it turns towards point B again and meets Car B half an hour later. Ho... | 432 | 0 | 7,742.25 | -1 | 7,742.25 | |
A circle is tangent to the sides of an angle at points $A$ and $B$. The distance from a point $C$ on the circle to the line $AB$ is 6. Find the sum of the distances from point $C$ to the sides of the angle, given that one of these distances is nine times smaller than the other. | 12 | 0 | 8,192 | -1 | 8,192 | |
Find a positive integral solution to the equation $\frac{1+3+5+\dots+(2n-1)}{2+4+6+\dots+2n}=\frac{115}{116}$. | 115 | 1. **Identify the series in the numerator and denominator:**
The numerator is the sum of the first $n$ odd numbers, which can be expressed as:
\[
1 + 3 + 5 + \dots + (2n-1)
\]
The denominator is the sum of the first $n$ even numbers, which can be expressed as:
\[
2 + 4 + 6 + \dots + 2n
\]
2. **... | 0.9375 | 2,144 | 1,740.8 | 8,192 |
A circle lies inside the parabola defined by the equation \( y = 4x^2 \) and is tangent to the parabola at two points. Determine how much higher the center of the circle is compared to the points of tangency. | \frac{1}{8} | 0.3125 | 7,618.5 | 6,356.8 | 8,192 | |
Euler's Bridge: The following figure is the graph of the city of Konigsburg in 1736 - vertices represent sections of the cities, edges are bridges. An Eulerian path through the graph is a path which moves from vertex to vertex, crossing each edge exactly once. How many ways could World War II bombers have knocked out s... | 13023 | The number of ways to destroy bridges to create an Eulerian path depends on ensuring that exactly 0 or 2 vertices have an odd degree. The specific graph of Konigsburg can be analyzed to find the number of such configurations, resulting in 13023 ways. | 0 | 8,192 | -1 | 8,192 |
Real numbers $x, y$, and $z$ are chosen from the interval $[-1,1]$ independently and uniformly at random. What is the probability that $|x|+|y|+|z|+|x+y+z|=|x+y|+|y+z|+|z+x|$? | \frac{3}{8} | We assume that $x, y, z$ are all nonzero, since the other case contributes zero to the total probability. If $x, y, z$ are all positive or all negative then the equation is obviously true. Otherwise, since flipping the signs of all three variables or permuting them does not change the equality, we assume WLOG that $x, ... | 0 | 8,192 | -1 | 8,192 |
The inclination angle of the line $\sqrt{3}x+y+2024=0$ is $\tan^{-1}\left(-\frac{\sqrt{3}}{1}\right)$. Calculate the angle in radians. | \frac{2\pi}{3} | 1 | 1,658.4375 | 1,658.4375 | -1 | |
Let $\frac {35x - 29}{x^2 - 3x + 2} = \frac {N_1}{x - 1} + \frac {N_2}{x - 2}$ be an identity in $x$. The numerical value of $N_1N_2$ is: | -246 | 1. **Expression Setup**: Given the identity
\[
\frac {35x - 29}{x^2 - 3x + 2} = \frac {N_1}{x - 1} + \frac {N_2}{x - 2}
\]
we start by expressing the right-hand side over a common denominator:
\[
\frac {N_1}{x - 1} + \frac {N_2}{x - 2} = \frac{N_1(x-2) + N_2(x-1)}{(x-1)(x-2)}
\]
2. **Simplify the ... | 1 | 2,816.875 | 2,816.875 | -1 |
In triangle $ABC$, $AB=3$, $BC=4$, and $\angle B=60^{\circ}$. Find the length of $AC$. | \sqrt {13} | 0 | 3,614.4375 | -1 | 3,614.4375 | |
How many of the fractions $ \frac{1}{2023}, \frac{2}{2023}, \frac{3}{2023}, \cdots, \frac{2022}{2023} $ simplify to a fraction whose denominator is prime? | 22 | 0.625 | 6,096.0625 | 5,416.4 | 7,228.833333 | |
Jun Jun is looking at an incorrect single-digit multiplication equation \( A \times B = \overline{CD} \), where the digits represented by \( A \), \( B \), \( C \), and \( D \) are all different from each other. Clever Jun Jun finds that if only one digit is changed, there are 3 ways to correct it, and if only the orde... | 17 | 0 | 8,192 | -1 | 8,192 | |
A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. The first cut is $\frac{1}{2}$ foot from the top face. The second cut is $\frac{1}{3}$ foot below the first cut, and the third cut is $\frac{1}{17}$ foot below the second cut. From the top to the bottom the pieces are labe... | 11 | 1. **Understanding the Problem**: We have a cube with a volume of 1 cubic foot, and it is cut into four pieces by three cuts parallel to the top face. The cuts are made at $\frac{1}{2}$ foot, $\frac{1}{3}$ foot, and $\frac{1}{17}$ foot below each preceding cut. The pieces are then rearranged end to end.
2. **Calculati... | 0 | 7,935.3125 | -1 | 7,935.3125 |
In Vila Par, all the truth coins weigh an even quantity of grams and the false coins weigh an odd quantity of grams. The eletronic device only gives the parity of the weight of a set of coins. If there are $2020$ truth coins and $2$ false coins, determine the least $k$, such that, there exists a strategy that allows to... | 21 |
In the given problem, we have 2020 true coins, each weighing an even number of grams, and 2 false coins, each weighing an odd number of grams. The electronic device available can detect the parity (even or odd) of the total weight of a set of coins. We need to determine the minimum number of measurements, \( k \), req... | 0.375 | 7,029 | 5,918.833333 | 7,695.1 |
If $A=20^{\circ}$ and $B=25^{\circ}$, then the value of $(1+\tan A)(1+\tan B)$ is | 2 |
1. **Expanding the Expression**:
\[
(1+\tan A)(1+\tan B) = 1 + \tan A + \tan B + \tan A \tan B
\]
2. **Using the Angle Sum Identity for Tangent**:
Since $A + B = 45^\circ$, we know that $\tan(45^\circ) = 1$. Using the tangent sum formula:
\[
\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}
\... | 1 | 1,839.4375 | 1,839.4375 | -1 |
2008 persons take part in a programming contest. In one round, the 2008 programmers are divided into two groups. Find the minimum number of groups such that every two programmers ever be in the same group. | 11 | 0 | 8,192 | -1 | 8,192 | |
Using the bar graph, what is the positive difference between the number of students at the school with the largest enrollment and the number of students at the school with the smallest enrollment?
[asy]
size(250);
defaultpen(fontsize(9));
fill((0,0)--(40,0)--(40,20)--(0,20)--cycle,lightgray);
draw((0,20)--(0,0)--(40... | 650 | 0.6875 | 2,689.1875 | 2,606.545455 | 2,871 | |
Ioana has three ropes whose lengths are 39 inches, 52 inches and 65 inches. She wants to cut the ropes into equal length pieces for magic tricks. No rope is to be wasted. What is the greatest number of inches possible in the length of each piece? | 13 | 1 | 1,198.875 | 1,198.875 | -1 | |
Given that a certain middle school has 3500 high school students and 1500 junior high school students, if 70 students are drawn from the high school students, calculate the total sample size $n$. | 100 | 0.75 | 3,147.875 | 2,750.666667 | 4,339.5 | |
A square is divided into five congruent rectangles, as shown. If the perimeter of each of these five rectangles is 36 inches, what is the perimeter of the square, in inches?
[asy]
draw((0,0)--(0,5)--(5,5)--(5,0)--cycle);
draw((1,0)--(1,5));
draw((2,0)--(2,5));
draw((3,0)--(3,5));
draw((4,0)--(4,5));
[/asy] | 60 | 1 | 2,029.3125 | 2,029.3125 | -1 | |
How many integers between $500$ and $1000$ contain both the digits $3$ and $4$? | 10 | 0.1875 | 7,629.875 | 5,194 | 8,192 | |
Let the circles $k_1$ and $k_2$ intersect at two points $A$ and $B$ , and let $t$ be a common tangent of $k_1$ and $k_2$ that touches $k_1$ and $k_2$ at $M$ and $N$ respectively. If $t\perp AM$ and $MN=2AM$ , evaluate the angle $NMB$ . | \[
\boxed{\frac{\pi}{4}}
\] | [asy] size(15cm,0); draw((0,0)--(0,2)--(4,2)--(4,-3)--(0,0)); draw((-1,2)--(9,2)); draw((0,0)--(2,2)); draw((2,2)--(1,1)); draw((0,0)--(4,2)); draw((0,2)--(1,1)); draw(circle((0,1),1)); draw(circle((4,-3),5)); dot((0,0)); dot((0,2)); dot((2,2)); dot((4,2)); dot((4,-3)); dot((1,1)); dot((0,1)); label("A",(0,0),NW); labe... | 0 | 8,192 | -1 | 8,192 |
A parallelogram is generated by the vectors $\begin{pmatrix} 3 \\ 1 \\ 1 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ -1 \\ -1 \end{pmatrix}$. Find the cosine of the angle $\theta$ between the diagonals of the parallelogram. | -\frac{\sqrt{3}}{3} | 0 | 2,504.625 | -1 | 2,504.625 | |
A list of five positive integers has mean $12$ and range $18$. The mode and median are both $8$. How many different values are possible for the second largest element of the list? | 6 | 1. **Define Variables:**
Let the integers in the list be $a_1, a_2, a_3, a_4, a_5$ such that $a_1 \leq a_2 \leq a_3 \leq a_4 \leq a_5$. Given that the mode and median are both $8$, we know $a_3 = 8$. Since the mode is $8$, at least one other number must be $8$. Without loss of generality, let $a_2 = 8$.
2. **Use th... | 0.125 | 8,006.6875 | 6,855.5 | 8,171.142857 |
Given that \(a\) and \(b\) are real numbers, and the equation \( x^{4} + a x^{3} + b x^{2} + a x + 1 = 0 \) has at least one real root, find the minimum value of \(a^{2} + b^{2}\). | 4/5 | 0.3125 | 7,942.125 | 7,504 | 8,141.272727 | |
Let $n$ be the answer to this problem. Find the minimum number of colors needed to color the divisors of $(n-24)$! such that no two distinct divisors $s, t$ of the same color satisfy $s \mid t$. | 50 | We first answer the following question. Find the minimum number of colors needed to color the divisors of $m$ such that no two distinct divisors $s, t$ of the same color satisfy $s \mid t$. Prime factorize $m=p_{1}^{e_{1}} \ldots p_{k}^{e_{k}}$. Note that the elements $$\begin{aligned} & 1, p_{1}, p_{1}^{2}, \ldots, p_... | 0 | 8,192 | -1 | 8,192 |
Five friends sat in a movie theater in a row containing $5$ seats, numbered $1$ to $5$ from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved tw... | 2 | To solve this problem, we need to analyze the movements of each friend and determine Ada's original seat based on the given movements and the final seating arrangement.
1. **Initial Setup**: There are 5 seats, and each friend occupies one seat. Ada leaves, creating one empty seat.
2. **Movements**:
- **Bea** moves... | 0.0625 | 7,862.75 | 6,125 | 7,978.6 |
Let the ordered triples $(x,y,z)$ of complex numbers that satisfy
\begin{align*}
x + yz &= 7, \\
y + xz &= 10, \\
z + xy &= 10.
\end{align*}be $(x_1,y_1,z_1),$ $(x_2,y_2,z_2),$ $\dots,$ $(x_n,y_n,z_n).$ Find $x_1 + x_2 + \dots + x_n.$ | 7 | 0.625 | 6,158.25 | 5,591.5 | 7,102.833333 | |
How much greater, in square inches, is the area of a circle of radius 20 inches than a circle of diameter 20 inches? Express your answer in terms of $\pi$. | 300\pi | 1 | 1,298.3125 | 1,298.3125 | -1 | |
A component is made up of 3 identical electronic components in parallel. The component works normally if at least one of the electronic components works normally. It is known that the service life $\xi$ (in years) of this type of electronic component follows a normal distribution, and the probability that the service l... | 0.488 | 0.0625 | 7,111.625 | 4,821 | 7,264.333333 | |
Find the ordered pair $(j,k)$ that satisfies the equations $5j-42k=1$ and $2k-j=3$. | (-4,-\frac{1}{2}) | 1 | 2,145.375 | 2,145.375 | -1 | |
The number halfway between $\frac{1}{8}$ and $\frac{1}{10}$ is | \frac{1}{9} |
#### Step-by-step Calculation:
1. **Identify the numbers between which the midpoint is to be found:**
Given numbers are $\frac{1}{8}$ and $\frac{1}{10}$.
2. **Convert fractions to a common denominator:**
\[
\frac{1}{8} = \frac{5}{40}, \quad \frac{1}{10} = \frac{4}{40}
\]
3. **Calculate the arithmetic me... | 0 | 2,583.9375 | -1 | 2,583.9375 |
George walks $1$ mile to school. He leaves home at the same time each day, walks at a steady speed of $3$ miles per hour, and arrives just as school begins. Today he was distracted by the pleasant weather and walked the first $\frac{1}{2}$ mile at a speed of only $2$ miles per hour. At how many miles per hour must Geor... | 6 | 1. **Calculate the normal time to get to school**: George walks 1 mile to school at a speed of 3 miles per hour. The time taken to walk this distance is calculated by the formula:
\[
\text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{1 \text{ mile}}{3 \text{ mph}} = \frac{1}{3} \text{ hours}
\]
2. **C... | 1 | 1,461.75 | 1,461.75 | -1 |
Find the arithmetic mean of the prime numbers in this list: 21, 23, 25, 27, 29 | 26 | 1 | 590.375 | 590.375 | -1 | |
In triangle $\triangle ABC$, the sides opposite $\angle A$, $\angle B$, and $\angle C$ are $a$, $b$, and $c$ respectively. Given that $\tan A + \tan C = \sqrt{3}(\tan A \tan C - 1)$,
(Ⅰ) find angle $B$
(Ⅱ) If $b=2$, find the maximum area of $\triangle ABC$. | \sqrt{3} | 0.875 | 5,939 | 5,873.928571 | 6,394.5 | |
Given the quadratic function \( y = ax^2 + bx + c \) with its graph intersecting the \( x \)-axis at points \( A \) and \( B \), and its vertex at point \( C \):
(1) If \( \triangle ABC \) is a right-angled triangle, find the value of \( b^2 - 4ac \).
(2) Consider the quadratic function
\[ y = x^2 - (2m + 2)x + m^2 +... | -1 | 0.0625 | 8,087.5 | 6,520 | 8,192 | |
Given an ellipse $C:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with eccentricity $\frac{1}{2}$, a circle $\odot E$ with center at the origin and radius equal to the minor axis of the ellipse is tangent to the line $x-y+\sqrt{6}=0$. <br/>$(1)$ Find the equation of the ellipse $C$; <br/>$(2)$ A line passin... | \frac{6\sqrt{6}}{11} | 0 | 5,909.8125 | -1 | 5,909.8125 | |
A merchant's cumulative sales from January to May reached 38.6 million yuan. It is predicted that the sales in June will be 5 million yuan, the sales in July will increase by x% compared to June, and the sales in August will increase by x% compared to July. The total sales in September and October are equal to the tota... | 20 | 0.3125 | 7,486.375 | 5,934 | 8,192 | |
The sum of the $x$-coordinates of the vertices of a triangle in the Cartesian plane equals $10$. Find the sum of the $x$-coordinates of the midpoints of the sides of the triangle. | 10 | 1 | 2,589.625 | 2,589.625 | -1 | |
On Arbor Day, a class at a certain school divided into 10 small groups to participate in tree planting activities. The number of trees planted by the 10 groups is shown in the table below:
| Number of Trees Planted | 5 | 6 | 7 |
|--------------------------|-----|-----|-----|
| Number of Groups | 3 | 4 ... | 0.6 | 0.875 | 2,978.5625 | 2,706.642857 | 4,882 | |
In a $3 \times 3$ table, numbers are placed such that each number is 4 times smaller than the number in the adjacent cell to the right and 3 times smaller than the number in the adjacent cell above. The sum of all the numbers in the table is 546. Find the number in the central cell. | 24 | 0 | 7,134.125 | -1 | 7,134.125 | |
What is the sum of the mean, median, and mode of the numbers $2,3,0,3,1,4,0,3$? | 7.5 | 1. **Organize the Data**: First, we arrange the given numbers in ascending order:
\[
0, 0, 1, 2, 3, 3, 3, 4
\]
2. **Finding the Mode**: The mode is the number that appears most frequently in the data set. From the ordered list, the number $3$ appears three times, which is more than any other number. Thus, the... | 1 | 2,238.1875 | 2,238.1875 | -1 |
Let the polynomial $x^{10} = a_0 + a_1(x+1) + \ldots + a_9(x+1)^9 + a_{10}(x+1)^{10}$, find the sum $a_1 + a_3 + a_5 + a_7 + a_9$. | -512 | 0.6875 | 5,359.4375 | 4,174.454545 | 7,966.4 | |
The school organized a picnic with several participants. The school prepared many empty plates. Each attendee counts the empty plates and takes one empty plate to get food (each person can only take one plate, no more). The first attendee counts all the empty plates, the second attendee counts one less plate than the f... | 1006 | 0 | 7,821.4375 | -1 | 7,821.4375 | |
In a right triangle ABC with sides 9, 12, and 15, a small circle with center Q and radius 2 rolls around the inside of the triangle, always remaining tangent to at least one side of the triangle. When Q first returns to its original position, through what distance has Q traveled? | 24 | 0 | 8,164.1875 | -1 | 8,164.1875 | |
In convex quadrilateral $ABCD$, $AB=BC=13$, $CD=DA=24$, and $\angle D=60^\circ$. Points $X$ and $Y$ are the midpoints of $\overline{BC}$ and $\overline{DA}$ respectively. Compute $XY^2$ (the square of the length of $XY$). | \frac{1033}{4}+30\sqrt{3} | 0 | 8,192 | -1 | 8,192 | |
Alex picks his favorite point $(x, y)$ in the first quadrant on the unit circle $x^{2}+y^{2}=1$, such that a ray from the origin through $(x, y)$ is $\theta$ radians counterclockwise from the positive $x$-axis. He then computes $\cos ^{-1}\left(\frac{4 x+3 y}{5}\right)$ and is surprised to get $\theta$. What is $\tan (... | \frac{1}{3} | $x=\cos (\theta), y=\sin (\theta)$. By the trig identity you never thought you'd need, $\frac{4 x+3 y}{5}=\cos (\theta-\phi)$, where $\phi$ has sine $3 / 5$ and cosine $4 / 5$. Now $\theta-\phi=\theta$ is impossible, since $\phi \neq 0$, so we must have $\theta-\phi=-\theta$, hence $\theta=\phi / 2$. Now use the trusty... | 0.9375 | 2,797.8125 | 2,438.2 | 8,192 |
Given an ellipse $C: \frac{x^{2}}{4} + y^{2} = 1$, with $O$ being the origin of coordinates, and a line $l$ intersects the ellipse $C$ at points $A$ and $B$, and $\angle AOB = 90^{\circ}$.
(Ⅰ) If the line $l$ is parallel to the x-axis, find the area of $\triangle AOB$;
(Ⅱ) If the line $l$ is always tangent to the ci... | \frac{2\sqrt{5}}{5} | 0 | 7,079.125 | -1 | 7,079.125 | |
A ball inside a rectangular container of width 7 and height 12 is launched from the lower-left vertex of the container. It first strikes the right side of the container after traveling a distance of $\sqrt{53}$ (and strikes no other sides between its launch and its impact with the right side). Find the height at which ... | 2 | Let $h$ be this height. Then, using the Pythagorean theorem, we see that $h^{2} + 7^{2} = 53$, so $h = 2$. | 1 | 4,185.6875 | 4,185.6875 | -1 |
Denis has cards with numbers from 1 to 50. How many ways are there to choose two cards such that the difference of the numbers on the cards is 11, and their product is divisible by 5?
The order of the selected cards does not matter: for example, selecting cards with numbers 5 and 16, as well as selecting cards with nu... | 15 | 0.375 | 7,517.5625 | 6,393.5 | 8,192 | |
Find the area of a triangle with angles $\frac{1}{7} \pi$ , $\frac{2}{7} \pi$ , and $\frac{4}{7} \pi $ , and radius of its circumscribed circle $R=1$ . | \frac{\sqrt{7}}{4} | 0 | 5,399.0625 | -1 | 5,399.0625 | |
Compute the following expression:
\[
\frac{(1 + 15) \left( 1 + \dfrac{15}{2} \right) \left( 1 + \dfrac{15}{3} \right) \dotsm \left( 1 + \dfrac{15}{17} \right)}{(1 + 17) \left( 1 + \dfrac{17}{2} \right) \left( 1 + \dfrac{17}{3} \right) \dotsm \left( 1 + \dfrac{17}{15} \right)}.
\] | 496 | 0 | 6,647.5 | -1 | 6,647.5 | |
Consider a pentagonal prism with seven faces, fifteen edges, and ten vertices. One of its faces will be used as the base for a new pyramid. Calculate the maximum value of the sum of the number of exterior faces, vertices, and edges of the combined solid (prism and pyramid). | 42 | 0.0625 | 6,299.25 | 3,098 | 6,512.666667 | |
(In the preliminaries of optimal method and experimental design) When using the 0.618 method to find the optimal amount to add in an experiment, if the current range of excellence is $[628, 774]$ and the good point is 718, then the value of the addition point for the current experiment is ________. | 684 | 0.25 | 7,390.25 | 6,027.25 | 7,844.583333 | |
The strengths of the two players are equal, meaning they have equal chances of winning each game. They agreed that the prize would go to the first player to win 6 games. They had to stop the game after the first player won 5 games and the second won 3. In what proportion should the prize be fairly divided? | 7:1 | 0.25 | 6,681.375 | 6,248.75 | 6,825.583333 | |
What is the intersection point of the line $y = 2x + 5$ and the line perpendicular to it that passes through the point $(5, 5)$? | (1, 7) | 1 | 1,782.8125 | 1,782.8125 | -1 | |
Given that $ a,b,c,d$ are rational numbers with $ a>0$ , find the minimal value of $ a$ such that the number $ an^{3} + bn^{2} + cn + d$ is an integer for all integers $ n \ge 0$ . | \frac{1}{6} | 0.0625 | 8,080.5625 | 6,409 | 8,192 | |
The expression $64x^6-729y^6$ can be factored as $(ax+by)(cx^2+dxy+ey^2)(fx+gy)(hx^2+jxy+ky^2)$. If $a$, $b$, $c$, $d$, $e$, $f$, $g$, $h$, $j$, and $k$ are all integers, find their sum. | 30 | 0.375 | 6,015.6875 | 4,747.666667 | 6,776.5 | |
Find the number of natural numbers not exceeding 2022 and not belonging to either the arithmetic progression \(1, 3, 5, \ldots\) or the arithmetic progression \(1, 4, 7, \ldots\). | 674 | 0.875 | 5,516.75 | 5,134.571429 | 8,192 | |
What is the $y$-intercept of the line $x - 2y = 5$? | -\frac{5}{2} | 1 | 1,826.6875 | 1,826.6875 | -1 | |
The height of a right-angled triangle, dropped to the hypotenuse, divides this triangle into two triangles. The distance between the centers of the inscribed circles of these triangles is 1. Find the radius of the inscribed circle of the original triangle. | \frac{\sqrt{2}}{2} | 0 | 8,192 | -1 | 8,192 | |
Call a number prime-looking if it is composite but not divisible by $2, 3,$ or $5.$ The three smallest prime-looking numbers are $49, 77$, and $91$. There are $168$ prime numbers less than $1000$. How many prime-looking numbers are there less than $1000$? | 100 | 1. **Identify the total number of integers less than 1000**:
There are 999 integers from 1 to 999.
2. **Calculate the number of prime numbers less than 1000**:
It is given that there are 168 prime numbers less than 1000.
3. **Exclude the primes 2, 3, and 5 from the count**:
Since 2, 3, and 5 are primes, we ... | 0.3125 | 7,206.125 | 6,348 | 7,596.181818 |
Victoria wants to order at least 550 donuts from Dunkin' Donuts for the HMMT 2014 November contest. However, donuts only come in multiples of twelve. Assuming every twelve donuts cost \$7.49, what is the minimum amount Victoria needs to pay, in dollars? | 344.54 | The smallest multiple of 12 larger than 550 is $552=12 \cdot 46$. So the answer is $46 \cdot \$7.49$. To make the multiplication easier, we can write this as $46 \cdot(\$7.5-\$0.01)=\$345-\$0.46=\$344.54$. | 1 | 1,897.125 | 1,897.125 | -1 |
Find the ratio of the area of $\triangle BCX$ to the area of $\triangle ACX$ in the diagram if $CX$ bisects $\angle ACB$. In the diagram, $AB = 34$, $BC = 35$, and $AC = 39$. Let the coordinates be:
- $A = (-17,0)$,
- $B = (17,0)$,
- $C = (0,30)$.
Express your answer as a common fraction. | \frac{35}{39} | 0.5625 | 5,940.5 | 4,980.666667 | 7,174.571429 | |
Let $p$ and $q$ be the roots of the equation $x^2 - 7x + 12 = 0$. Compute the value of:
\[ p^3 + p^4 q^2 + p^2 q^4 + q^3. \] | 3691 | 1 | 3,420 | 3,420 | -1 | |
Find the number of counter examples to the statement: | 2 | We are tasked with finding the number of counterexamples to the statement that a number $N$ with the sum of its digits equal to $4$ and none of its digits being $0$ is prime. We will analyze each possible set of digits that sum to $4$ and check if the resulting numbers are prime.
1. **Set $\{1,1,1,1\}$**:
- The num... | 0 | 4,472.1875 | -1 | 4,472.1875 |
Consider the integer \[N = 9 + 99 + 999 + 9999 + \cdots + \underbrace{99\ldots 99}_\text{321 digits}.\]Find the sum of the digits of $N$. | 342 | Observe how adding results in the last term but with a $1$ concatenated in front and also a $1$ subtracted ($09$, $108$, $1107$, $11106$). Then for any index of terms, $n$, the sum is $11...10-n$, where the first term is of length $n+1$. Here, that is $\boxed{342}$.
~BJHHar | 0.0625 | 7,640.4375 | 4,029 | 7,881.2 |
A rectangle has its length increased by $30\%$ and its width increased by $15\%$. Determine the percentage increase in the area of the rectangle. | 49.5\% | 1 | 2,451.25 | 2,451.25 | -1 | |
Suppose we have an (infinite) cone $\mathcal{C}$ with apex $A$ and a plane $\pi$. The intersection of $\pi$ and $\mathcal{C}$ is an ellipse $\mathcal{E}$ with major axis $BC$, such that $B$ is closer to $A$ than $C$, and $BC=4, AC=5, AB=3$. Suppose we inscribe a sphere in each part of $\mathcal{C}$ cut up by $\mathcal{... | \frac{1}{3} | It can be seen that the points of tangency of the spheres with $E$ must lie on its major axis due to symmetry. Hence, we consider the two-dimensional cross-section with plane $ABC$. Then the two spheres become the incentre and the excentre of the triangle $ABC$, and we are looking for the ratio of the inradius to the e... | 0 | 8,192 | -1 | 8,192 |
The red parabola shown is the graph of the equation $x = ay^2 + by + c$. Find $c$. (The grid lines are spaced one unit apart.)
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownle... | -2 | 0.875 | 4,296.375 | 4,374.642857 | 3,748.5 | |
An ordinary clock in a factory is running slow so that the minute hand passes the hour hand at the usual dial position (12 o'clock, etc.) but only every 69 minutes. At time and one-half for overtime, the extra pay to which a $4.00 per hour worker should be entitled after working a normal 8 hour day by that slow running... | $2.60 | 0 | 7,465 | -1 | 7,465 | |
The Red Sox play the Yankees in a best-of-seven series that ends as soon as one team wins four games. Suppose that the probability that the Red Sox win Game $n$ is $\frac{n-1}{6}$. What is the probability that the Red Sox will win the series? | 1/2 | Note that if we imagine that the series always continues to seven games even after one team has won four, this will never change the winner of the series. Notice also that the probability that the Red Sox will win Game $n$ is precisely the probability that the Yankees will win Game $8-n$. Therefore, the probability tha... | 0 | 8,192 | -1 | 8,192 |
Among all triangles $ABC$, find the maximum value of $\cos A + \cos B \cos C$. | \frac{3}{2} | 0 | 7,988.3125 | -1 | 7,988.3125 | |
Using the 3 vertices of a triangle and 7 points inside it (a total of 10 points), how many smaller triangles can the original triangle be divided into?
(1985 Shanghai Junior High School Math Competition, China;
1988 Jiangsu Province Junior High School Math Competition, China) | 15 | 0.4375 | 7,116.25 | 6,097.142857 | 7,908.888889 | |
Given that the sequence $\{a_n\}$ is a geometric sequence, with $a_1=2$, common ratio $q>0$, and $a_2$, $6$, $a_3$ forming an arithmetic sequence.
(I) Find the general term formula for the sequence $\{a_n\}$;
(II) Let $b_n=\log_2{a_n}$, and $$T_{n}= \frac {1}{b_{1}b_{2}}+ \frac {1}{b_{2}b_{3}}+ \frac {1}{b_{3}b_{4}}+…+... | 98 | 1 | 2,633.0625 | 2,633.0625 | -1 | |
Let $a,$ $b,$ and $c$ be the roots of the equation $x^3 - 15x^2 + 25x - 10 = 0$. Calculate the value of $(1+a)(1+b)(1+c)$. | 51 | 1 | 2,340.875 | 2,340.875 | -1 | |
In a right triangular prism $\mathrm{ABC}-\mathrm{A}_{1} \mathrm{~B}_{1} \mathrm{C}_{1}$, the lengths of the base edges and the lateral edges are all 2. If $\mathrm{E}$ is the midpoint of $\mathrm{CC}_{1}$, what is the distance from $\mathrm{C}_{1}$ to the plane $\mathrm{AB} \mathrm{B}_{1} \mathrm{E}$? | \frac{\sqrt{2}}{2} | 0 | 6,072.75 | -1 | 6,072.75 | |
What is the least positive integer with exactly $12$ positive factors? | 72 | 0 | 4,149.4375 | -1 | 4,149.4375 | |
Let $\triangle ABC$ have sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively. The areas of the equilateral triangles with side lengths $a$, $b$, $c$ are $S_{1}$, $S_{2}$, $S_{3}$ respectively. Given $S_{1}-S_{2}+S_{3}=\frac{{\sqrt{3}}}{2}$ and $\sin B=\frac{1}{3}$.<br/>$(1)$ Find the area of $\triangle AB... | \frac{1}{2} | 0.5 | 7,434.3125 | 6,676.625 | 8,192 | |
How many nonnegative solutions are there to the equation $x^2 = -4x$? | 1 | 1 | 2,005.25 | 2,005.25 | -1 | |
Let \(X_{0}\) be the interior of a triangle with side lengths 3, 4, and 5. For all positive integers \(n\), define \(X_{n}\) to be the set of points within 1 unit of some point in \(X_{n-1}\). The area of the region outside \(X_{20}\) but inside \(X_{21}\) can be written as \(a\pi + b\), for integers \(a\) and \(b\). C... | 4112 | 0.25 | 7,538.375 | 6,399.75 | 7,917.916667 | |
Fill the numbers 1, 2, 3 into a 3×3 grid such that each row and each column contains no repeated numbers. How many different ways can this be done? | 12 | 0.75 | 5,407.8125 | 4,479.75 | 8,192 | |
Choose any $2$ numbers from $-5$, $-3$, $-1$, $2$, and $4$. Let the maximum product obtained be denoted as $a$, and the minimum quotient obtained be denoted as $b$. Then the value of $\frac{a}{b}$ is ______. | -\frac{15}{4} | 0.25 | 5,515.9375 | 6,396.25 | 5,222.5 | |
Find the remainder when $3^{1999}$ is divided by $13$. | 3 | 1 | 3,084.25 | 3,084.25 | -1 | |
There are four cards, each with a number on both sides. The first card has 0 and 1, the other three cards have 2 and 3, 4 and 5, and 7 and 8 respectively. If any three cards are selected and arranged in a row, how many different three-digit numbers can be formed? | 168 | 0.0625 | 7,859 | 7,692 | 7,870.133333 | |
Compute
\[\frac{\lfloor \sqrt{1} \rfloor \cdot \lfloor \sqrt{2} \rfloor \cdot \lfloor \sqrt{3} \rfloor \cdot \lfloor \sqrt{5} \rfloor \dotsm \lfloor \sqrt{15} \rfloor}{\lfloor \sqrt{2} \rfloor \cdot \lfloor \sqrt{4} \rfloor \cdot \lfloor \sqrt{6} \rfloor \dotsm \lfloor \sqrt{16} \rfloor}.\] | \frac{3}{8} | 0.1875 | 4,954.8125 | 2,858.333333 | 5,438.615385 | |
In a triangle, the lengths of the three sides are $4$, $x$, and $12-x$.<br/>$(1)$ The range of values for $x$ is ______;<br/>$(2)$ If this is an isosceles triangle, then the perimeter of the isosceles triangle is ______. | 16 | 1 | 2,163.8125 | 2,163.8125 | -1 | |
Consider a two-digit number in base 12 represented by $AB_{12}$, where $A$ and $B$ are duodecimal digits (0 to 11), and $A \neq B$. When this number is reversed to $BA_{12}$, under what condition is a particular prime number a necessary factor of the difference $AB_{12} - BA_{12}$? | 11 | 0.9375 | 4,273.25 | 4,012 | 8,192 | |
Find the sum of the digits of \(11 \cdot 101 \cdot 111 \cdot 110011\). | 48 | There is no regrouping, so the answer is \(2 \cdot 2 \cdot 3 \cdot 4=48\). The actual product is 13566666531. | 0.125 | 7,316.75 | 6,401 | 7,447.571429 |
An athlete's target heart rate, in beats per minute, is $80\%$ of the theoretical maximum heart rate. The maximum heart rate is found by subtracting the athlete's age, in years, from $220$. To the nearest whole number, what is the target heart rate of an athlete who is $26$ years old? | 134 | 1. **Calculate the Maximum Heart Rate**: The maximum heart rate for an athlete is calculated by subtracting the athlete's age from 220. For an athlete who is 26 years old, the calculation is:
\[
\text{Maximum Heart Rate} = 220 - 26 = 194 \text{ beats per minute}
\]
2. **Calculate the Target Heart Rate**: The ... | 0 | 1,553.625 | -1 | 1,553.625 |
Find the smallest solution to the equation \[\lfloor x^2 \rfloor - \lfloor x \rfloor^2 = 19.\] | \sqrt{119} | 0.875 | 5,698.125 | 5,341.857143 | 8,192 | |
Determine the maximum integer value of the expression
\[\frac{3x^2 + 9x + 28}{3x^2 + 9x + 7}.\] | 85 | 0.875 | 3,839.8125 | 3,452.714286 | 6,549.5 | |
How many seconds are in 7.8 minutes? | 468 | 1 | 207.5 | 207.5 | -1 | |
If \( x = 3 \) and \( y = 7 \), then what is the value of \( \frac{x^5 + 3y^3}{9} \)? | 141 | 0 | 4,876.3125 | -1 | 4,876.3125 | |
In the diagram, what is the perimeter of polygon $PQRST$? [asy]
import olympiad;
size(6cm); // ADJUST
pair p = (0, 6);
pair q = (3, 6);
pair r = (3, 3);
pair t = (0, 0);
pair s = (7, 0);
draw(p--q--r--s--t--cycle);
label("$P$", p, NW);
label("$Q$", q, NE);
label("$R$", r, E + NE);
label("$S$", s, SE);
label("$T$",... | 24 | 1 | 2,521.8125 | 2,521.8125 | -1 | |
Each of the natural numbers $1, 2, 3, \ldots, 377$ is painted either red or blue (both colors are present). It is known that the number of red numbers is equal to the smallest red number, and the number of blue numbers is equal to the largest blue number. What is the smallest red number? | 189 | 0.6875 | 5,878.25 | 4,826.545455 | 8,192 |
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