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 |
|---|---|---|---|---|---|---|
A number is called a visible factor number if it is divisible by each of its non-zero digits. For example, 102 is divisible by 1 and 2, so it is a visible factor number. How many visible factor numbers are there from 100 through 150, inclusive? | 19 | 0 | 7,987.4375 | -1 | 7,987.4375 | |
If two poles $20''$ and $80''$ high are $100''$ apart, then the height of the intersection of the lines joining the top of each pole to the foot of the opposite pole is: | 16'' | 1. **Identify the equations of the lines**:
- The line from the top of the first pole (20'') to the foot of the second pole can be described by considering the slope and y-intercept. The slope is calculated as the change in height over the change in horizontal distance, which is \(\frac{0 - 20}{100 - 0} = -\frac{20... | 0 | 4,751.4375 | -1 | 4,751.4375 |
A graph shows the number of books read in June by the top readers in a school library. The data points given are:
- 4 readers read 3 books each
- 5 readers read 5 books each
- 2 readers read 7 books each
- 1 reader read 10 books
Determine the mean (average) number of books read by these readers. | 5.0833 | 0.125 | 593.1875 | 636.5 | 587 | |
Find the smallest positive real number $x$ such that
\[\lfloor x^2 \rfloor - x \lfloor x \rfloor = 8.\] | \frac{89}{9} | 0 | 8,192 | -1 | 8,192 | |
132009 students are taking a test which comprises ten true or false questions. Find the minimum number of answer scripts required to guarantee two scripts with at least nine identical answers. | 513 | 0.4375 | 7,379.8125 | 6,335.571429 | 8,192 | |
Lead used to make twelve solid lead balls, each with a radius of 2 cm, is reused to form a single larger solid lead sphere. What is the radius of this larger sphere? | \sqrt[3]{96} | 0.125 | 5,757.375 | 6,099.5 | 5,708.5 | |
The following is Xiaoying's process of solving a linear equation. Please read carefully and answer the questions.
解方程:$\frac{{2x+1}}{3}-\frac{{5x-1}}{6}=1$
Solution:
To eliminate the denominators, we get $2\left(2x+1\right)-\left(5x-1\right)=1$ ... Step 1
Expanding the brackets, we get $4x+2-5x+1=1$ ... Step 2 ... | x = -3 | 0.75 | 2,930.75 | 2,577.833333 | 3,989.5 | |
If $P_1P_2P_3P_4P_5P_6$ is a regular hexagon whose apothem (distance from the center to midpoint of a side) is $2$, and $Q_i$ is the midpoint of side $P_iP_{i+1}$ for $i=1,2,3,4$, then the area of quadrilateral $Q_1Q_2Q_3Q_4$ is | 4\sqrt{3} | 1. **Identify the Geometry of the Hexagon and Midpoints:**
Given a regular hexagon $P_1P_2P_3P_4P_5P_6$, each side of the hexagon is equal, and each internal angle is $120^\circ$. The apothem, which is the distance from the center to the midpoint of any side, is given as $2$.
2. **Calculate the Side Length of the H... | 0 | 7,445.3125 | -1 | 7,445.3125 |
Isosceles triangle $ABE$ of area 100 square inches is cut by $\overline{CD}$ into an isosceles trapezoid and a smaller isosceles triangle. The area of the trapezoid is 75 square inches. If the altitude of triangle $ABE$ from $A$ is 20 inches, what is the number of inches in the length of $\overline{CD}$?
[asy]
draw((-... | 5 | 1 | 3,398.75 | 3,398.75 | -1 | |
The maximum area of a right-angled triangle with a hypotenuse of length 8 is | 16 | 1 | 2,628.6875 | 2,628.6875 | -1 | |
Joe wants to find all the four-letter words that begin and end with the same letter. How many combinations of letters satisfy this property? | 17576 | 1 | 2,720.3125 | 2,720.3125 | -1 | |
A set $\mathcal{T}$ of distinct positive integers has the following property: for every integer $y$ in $\mathcal{T},$ the arithmetic mean of the set of values obtained by deleting $y$ from $\mathcal{T}$ is an integer. Given that 1 belongs to $\mathcal{T}$ and that 1764 is the largest element of $\mathcal{T},$ what is t... | 42 | 0.0625 | 8,192 | 8,192 | 8,192 | |
An urn is filled with coins and beads, all of which are either silver or gold. Twenty percent of the objects in the urn are beads. Forty percent of the coins in the urn are silver. What percent of objects in the urn are gold coins? | 48\% | 1. **Identify the percentage of beads and coins:**
Given that 20% of the objects in the urn are beads, the remaining objects must be coins. Therefore, the percentage of coins in the urn is:
\[
100\% - 20\% = 80\%
\]
2. **Determine the composition of the coins:**
It is stated that 40% of the coins are si... | 1 | 2,418.4375 | 2,418.4375 | -1 |
If $10^{\log_{10}9} = 8x + 5$ then $x$ equals: | \frac{1}{2} | 1. **Understanding the Expression**: The expression given is $10^{\log_{10}9}$. By the property of logarithms and exponents, $a^{\log_a b} = b$, we can simplify this expression:
\[
10^{\log_{10}9} = 9
\]
2. **Setting Up the Equation**: We substitute the simplified expression into the equation given in the pro... | 1 | 1,503.75 | 1,503.75 | -1 |
The price of an item is decreased by 20%. To bring it back to its original value and then increase it by an additional 10%, the price after restoration must be increased by what percentage. | 37.5\% | 0.0625 | 4,652.1875 | 871 | 4,904.266667 | |
$A$, $B$, $C$ are three piles of rocks. The mean weight of the rocks in $A$ is $40$ pounds, the mean weight of the rocks in $B$ is $50$ pounds, the mean weight of the rocks in the combined piles $A$ and $B$ is $43$ pounds, and the mean weight of the rocks in the combined piles $A$ and $C$ is $44$ pounds. What is the gr... | 59 | 1. **Define Variables:**
Let $A_n$ be the total number of rocks in pile $A$, and $B_n$ be the total number of rocks in pile $B$. Let the total weight of rocks in pile $C$ be $m$, and the total number of rocks in $C$ be $n$.
2. **Use Given Averages:**
- For piles $A$ and $B$, the mean weight is given by:
\[
... | 0.4375 | 7,350.8125 | 6,284.142857 | 8,180.444444 |
Simply the expression
\[\frac{(\sqrt{2} - 1)^{1 - \sqrt{3}}}{(\sqrt{2} + 1)^{1 + \sqrt{3}}},\]writing your answer as $a - b \sqrt{c},$ where $a,$ $b,$ and $c$ are positive integers, and $c$ is not divisible by the square of a prime. | 3 - 2 \sqrt{2} | 0.75 | 3,779.6875 | 2,308.916667 | 8,192 | |
In an arithmetic sequence $\{a_n\}$, the sum $a_1 + a_2 + \ldots + a_5 = 30$, and the sum $a_6 + a_7 + \ldots + a_{10} = 80$. Calculate the sum $a_{11} + a_{12} + \ldots + a_{15}$. | 130 | 1 | 4,193.8125 | 4,193.8125 | -1 | |
Let $\alpha \in \left(0, \frac{\pi}{3}\right)$, satisfying $\sqrt{3}\sin\alpha + \cos\alpha = \frac{\sqrt{6}}{2}$.
$(1)$ Find the value of $\cos\left(\alpha + \frac{\pi}{6}\right)$;
$(2)$ Find the value of $\cos\left(2\alpha + \frac{7\pi}{12}\right)$. | \frac{\sqrt{2} - \sqrt{30}}{8} | 0 | 7,213.6875 | -1 | 7,213.6875 | |
Let $f(x) = x^2 + 6x + c$ for all real numbers $x$, where $c$ is some real number. For what values of $c$ does $f(f(x))$ have exactly $3$ distinct real roots? | \frac{11 - \sqrt{13}}{2} | 0 | 8,128.8125 | -1 | 8,128.8125 | |
Calculate the value of $3^{12} \cdot 3^3$ and express it as some integer raised to the third power. | 243 | 0.5 | 1,555.5625 | 1,785.625 | 1,325.5 | |
Given unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}+\overrightarrow{b}|+2\overrightarrow{a}\cdot\overrightarrow{b}=0$, determine the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{2\pi}{3} | 0.0625 | 2,324.9375 | 1,759 | 2,362.666667 | |
A given odd function $f(x)$, defined on $\mathbb{R}$, is symmetric about the line $x=1$, and $f(-1) = 1$. Find the value of $f(1) + f(2) + f(3) + \ldots + f(2009)$. | -1 | 0.9375 | 5,657.3125 | 5,488.333333 | 8,192 | |
In the Cartesian coordinate system, point O is the origin, and the coordinates of three vertices of the parallelogram ABCD are A(2,3), B(-1,-2), and C(-2,-1).
(1) Find the lengths of the diagonals AC and BD;
(2) If the real number t satisfies $ (\vec{AB}+t\vec{OC})\cdot\vec{OC}=0 $, find the value of t. | -\frac{11}{5} | 1 | 2,789.4375 | 2,789.4375 | -1 | |
In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is given by
$$
\begin{cases}
x = 1 + t\cos\alpha \\
y = 2 + t\sin\alpha
\end{cases}
$$
where $t$ is the parameter. In the polar coordinate system, which uses the same unit length as $xOy$ and has the origin $O$ as the pole and the positive $x... | 2\sqrt{7} | 0.9375 | 5,642.5 | 5,472.533333 | 8,192 | |
Determine the value of $b$ that satisfies the equation $295_{b} + 467_{b} = 762_{b}$. | 10 | 0.9375 | 3,070.1875 | 2,728.733333 | 8,192 | |
For real numbers $a$ and $b$, define $a\textdollar b = (a - b)^2$. What is $(x - y)^2\textdollar(y - x)^2$? | 0 | Given the operation $a \textdollar b = (a - b)^2$, we need to evaluate $(x - y)^2 \textdollar (y - x)^2$.
1. **Substitute into the operation definition**:
\[
(x - y)^2 \textdollar (y - x)^2 = \left((x - y)^2 - (y - x)^2\right)^2
\]
2. **Simplify the expression inside the square**:
- Note that $(x - y) = -... | 1 | 1,776 | 1,776 | -1 |
Inside a convex $n$-gon there are 100 points positioned in such a way that no three of these $n+100$ points are collinear. The polygon is divided into triangles, each having vertices among any 3 of the $n+100$ points. For what maximum value of $n$ can no more than 300 triangles be formed? | 102 | 0.8125 | 5,178.75 | 4,965.538462 | 6,102.666667 | |
Given that there is a point P (x, -1) on the terminal side of ∠Q (x ≠ 0), and $\tan\angle Q = -x$, find the value of $\sin\angle Q + \cos\angle Q$. | -\sqrt{2} | 0.25 | 8,124.75 | 8,142.25 | 8,118.916667 | |
There is a pentagon ABCDE. If the vertices A, B, C, D, E are colored with one of three colors: red, yellow, green, such that adjacent vertices are of different colors, then there are a total of different coloring methods. | 30 | 0.5 | 7,043.625 | 5,895.25 | 8,192 | |
Find the smallest natural number ending in the digit 6, which increases fourfold when its last digit is moved to the beginning of the number. | 153846 | 0.875 | 4,632.9375 | 4,124.5 | 8,192 | |
The curve $C$ is given by the equation $xy=1$. The curve $C'$ is the reflection of $C$ over the line $y=2x$ and can be written in the form $12x^2+bxy+cy^2+d=0$. Find the value of $bc$. | 84 | 0.75 | 6,013 | 5,286.666667 | 8,192 | |
In how many ways can the digits of $47,\!770$ be arranged to form a 5-digit number, ensuring the number does not begin with 0? | 16 | 0.4375 | 6,267.5625 | 3,793.285714 | 8,192 | |
A circle touches the longer leg of a right triangle, passes through the vertex of the opposite acute angle, and has its center on the hypotenuse of the triangle. What is the radius of the circle if the lengths of the legs are 5 and 12? | \frac{65}{18} | 0.4375 | 6,172.625 | 4,304.285714 | 7,625.777778 | |
Given an equilateral triangle with side length \( n \), divided into unit triangles as illustrated, let \( f(n) \) be the number of paths from the top-row triangle to the triangle in the center of the bottom row. The path must move through adjacent triangles sharing a common edge, never revisiting any triangle, and nev... | 2011! | 0 | 7,932.5 | -1 | 7,932.5 | |
Let $S = (1+i)^{17} - (1-i)^{17}$, where $i=\sqrt{-1}$. Find $|S|$.
| 512 | 0.75 | 5,582 | 4,712 | 8,192 | |
Find the volume of the solid generated by a rotation of the region enclosed by the curve $y=x^3-x$ and the line $y=x$ about the line $y=x$ as the axis of rotation. | \frac{64\pi}{105} | 0 | 8,028.5 | -1 | 8,028.5 | |
In the expansion of \((-xy + 2x + 3y - 6)^6\), what is the coefficient of \(x^4 y^3\)? (Answer with a specific number) | -21600 | 0.1875 | 7,665.25 | 6,746 | 7,877.384615 | |
The product $N$ of three positive integers is $6$ times their sum, and one of the integers is the sum of the other two. Find the sum of all possible values of $N$.
| 336 | 0.9375 | 4,018.875 | 3,740.666667 | 8,192 | |
If $\angle \text{CBD}$ is a right angle, then this protractor indicates that the measure of $\angle \text{ABC}$ is approximately | 20^{\circ} | 1. **Identify the given information**: We know that $\angle \text{CBD}$ is a right angle, which means $\angle \text{CBD} = 90^\circ$. We are also given that the sum of the angles around point B is $160^\circ$, which includes $\angle \text{ABC}$, $\angle \text{ABD}$, and $\angle \text{CBD}$.
2. **Set up the equation**:... | 0 | 4,706.25 | -1 | 4,706.25 |
An oreo shop sells $5$ different flavors of oreos and $3$ different flavors of milk. Alpha and Beta decide to purhcase some oreos. Since Alpha is picky, he will not order more than 1 of the same flavor. To be just as weird, Beta will only order oreos, but she will be willing to have repeats of flavors. How many ways co... | 351 | 0 | 7,762.5625 | -1 | 7,762.5625 | |
All three vertices of $\triangle ABC$ lie on the parabola defined by $y=x^2$, with $A$ at the origin and $\overline{BC}$ parallel to the $x$-axis. The area of the triangle is $64$. What is the length of $BC$? | 8 | 1. **Identify the Coordinates of Points**:
- Since $A$ is at the origin, $A = (0,0)$.
- The line $\overline{BC}$ is parallel to the $x$-axis, and since all vertices lie on the parabola $y = x^2$, the coordinates of $B$ and $C$ must be symmetric about the $y$-axis. Let's denote the $x$-coordinates of $B$ and $C$ ... | 1 | 2,729.75 | 2,729.75 | -1 |
Find the leading coefficient in the polynomial $-3(x^4 - x^3 + x) + 7(x^4 + 2) - 4(2x^4 + 2x^2 + 1)$ after it is simplified. | -4 | 1 | 1,960.375 | 1,960.375 | -1 | |
A rectangle has a perimeter of 64 inches and each side has an integer length. How many non-congruent rectangles meet these criteria? | 16 | 0.9375 | 4,939.5625 | 4,722.733333 | 8,192 | |
Find $n$ such that $2^6 \cdot 3^3 \cdot n = 10!$. | 1050 | 0 | 3,305.5 | -1 | 3,305.5 | |
If $\frac{1}{4}$ of $2^{30}$ is $4^x$, then what is the value of $x$ ? | 14 | 1 | 1,758.375 | 1,758.375 | -1 | |
$Q$ is the point of intersection of the diagonals of one face of a cube whose edges have length 2 units. The length of $Q R$ is | $\sqrt{6}$ | 0 | 6,680.3125 | -1 | 6,680.3125 | |
Each block on the grid shown in the Figure is 1 unit by 1 unit. Suppose we wish to walk from $A$ to $B$ via a 7 unit path, but we have to stay on the grid -- no cutting across blocks. How many different paths can we take?[asy]size(3cm,3cm);int w=5;int h=4;int i;for (i=0; i<h; ++i){draw((0,i) -- (w-1,i));}for (i=0; i<... | 35 | 0.875 | 4,058.375 | 3,467.857143 | 8,192 | |
A point \((x, y)\) is randomly picked from inside the rectangle with vertices \((0,0)\), \((5,0)\), \((5,2)\), and \((0,2)\). What is the probability that \(x < 2y\)? | \frac{2}{5} | 0.5 | 6,538.4375 | 4,997.875 | 8,079 | |
The square with vertices $(-a, -a), (a, -a), (-a, a), (a, a)$ is cut by the line $y = x/2$ into congruent quadrilaterals. The perimeter of one of these congruent quadrilaterals divided by $a$ equals what? Express your answer in simplified radical form. | 4+\sqrt{5} | 0.4375 | 6,598.3125 | 4,902.571429 | 7,917.222222 | |
Given that $a > 0$, $b > 0$, and $a + b = 1$, find the maximum value of $(-\frac{1}{2a} - \frac{2}{b})$. | -\frac{9}{2} | 0.9375 | 4,291.3125 | 4,125.066667 | 6,785 | |
With all angles measured in degrees, determine the value of $\prod_{k=1}^{30} \csc^2(3k)^\circ \sec^2 (6k)^\circ = p^q$, where $p$ and $q$ are integers greater than 1. Find $p+q$. | 62 | 0 | 8,192 | -1 | 8,192 | |
Evaluate $|7-24i|$. | 25 | 1 | 1,066.5 | 1,066.5 | -1 | |
In how many ways can you select two letters from the word "УЧЕБНИК" such that one of the letters is a consonant and the other is a vowel? | 12 | 0.9375 | 2,548.5625 | 2,526.866667 | 2,874 | |
Given the inequality $(|x|-1)^2+(|y|-1)^2<2$, determine the number of lattice points $(x, y)$ that satisfy it. | 16 | 0.4375 | 7,321.375 | 6,202 | 8,192 | |
Calculate using your preferred method!
100 - 54 - 46
234 - (134 + 45)
125 × 7 × 8
15 × (61 - 45)
318 ÷ 6 + 165. | 218 | 1 | 721 | 721 | -1 | |
The difference between two perfect squares is 133. What is the smallest possible sum of the two perfect squares? | 205 | 0.9375 | 3,285.5625 | 2,958.466667 | 8,192 | |
For the ellipse shown below, find the distance between the foci.
[asy]
unitsize(0.3 cm);
int i, n = 10;
for (i = -n; i <= n; ++i) {
draw((i,-n)--(i,n),gray(0.7));
draw((-n,i)--(n,i),gray(0.7));
}
draw((0,-n)--(0,n));
draw((-n,0)--(n,0));
draw(shift((1,1))*xscale(2)*yscale(6)*Circle((0,0),1),red);
dot((1,1));
... | 8 \sqrt{2} | 0.875 | 2,112.875 | 2,070.285714 | 2,411 | |
On a rectangular table of size \( x \) cm \(\times 80\) cm, identical sheets of paper of size 5 cm \(\times 8\) cm are placed. The first sheet is placed in the bottom left corner, and each subsequent sheet is placed one centimeter higher and one centimeter to the right of the previous one. The last sheet is placed in t... | 77 | 0.375 | 6,206.3125 | 3,695.666667 | 7,712.7 | |
Given that the complex number $z\_1$ satisfies $((z\_1-2)(1+i)=1-i)$, the imaginary part of the complex number $z\_2$ is $2$, and $z\_1z\_2$ is a real number, find $z\_2$ and $|z\_2|$. | 2 \sqrt {5} | 0 | 2,028.0625 | -1 | 2,028.0625 | |
Given that -1, a, b, -4 form an arithmetic sequence, and -1, c, d, e, -4 form a geometric sequence, calculate the value of $$\frac{b-a}{d}$$. | \frac{1}{2} | 1 | 2,935.8125 | 2,935.8125 | -1 | |
Given that the boat is leaking water at the rate of 15 gallons per minute, the maximum time to reach the shore is 50/15 minutes. If Amy rows towards the shore at a speed of 2 mph, then every 30 minutes she increases her speed by 1 mph, find the maximum rate at which Boris can bail water in gallons per minute so that th... | 14 | 0 | 7,142.625 | -1 | 7,142.625 | |
Malcolm wants to visit Isabella after school today and knows the street where she lives but doesn't know her house number. She tells him, "My house number has two digits, and exactly three of the following four statements about it are true."
(1) It is prime.
(2) It is even.
(3) It is divisible by 7.
(4) One of its dig... | 8 | 1. **Analyze the given statements**: Isabella's house number is a two-digit number, and exactly three out of the four statements about it are true:
- (1) It is prime.
- (2) It is even.
- (3) It is divisible by 7.
- (4) One of its digits is 9.
2. **Determine the false statement**:
- If (1) is true (the ... | 0.5625 | 6,171.5 | 4,600 | 8,192 |
In isosceles triangle $\triangle ABC$, $A$ is located at the origin and $B$ is located at $(20,0)$. Point $C$ is in the first quadrant with $AC = BC$ and angle $BAC = 75^{\circ}$. If triangle $ABC$ is rotated counterclockwise about point $A$ until the image of $C$ lies on the positive $y$-axis, the area of the region c... | 875 | Call the points of the intersections of the triangles $D$, $E$, and $F$ as noted in the diagram (the points are different from those in the diagram for solution 1). $\overline{AD}$ bisects $\angle EDE'$.
Through HL congruency, we can find that $\triangle AED$ is congruent to $\triangle AE'D$. This divides the region $... | 0 | 8,192 | -1 | 8,192 |
Find all solutions to the equation $\displaystyle\sqrt[3]{2 - \frac{x}{2}} = -3$. | 58 | 1 | 1,717.5625 | 1,717.5625 | -1 | |
Susan wants to determine the average and median number of candies in a carton. She buys 9 cartons of candies, opens them, and counts the number of candies in each one. She finds that the cartons contain 5, 7, 8, 10, 12, 14, 16, 18, and 20 candies. What are the average and median number of candies per carton? | 12 | 0.9375 | 2,207.5625 | 2,156.733333 | 2,970 | |
Find the possible value of $x + y$ given that $x^3 + 6x^2 + 16x = -15$ and $y^3 + 6y^2 + 16y = -17$. | -4 | 0.5625 | 5,833 | 3,998.222222 | 8,192 | |
For how many positive integers $n$ does $\frac{1}{n}$ yield a terminating decimal with a non-zero hundredths digit? | 11 | 0 | 8,192 | -1 | 8,192 | |
In a right triangle JKL, the hypotenuse KL measures 13 units, and side JK measures 5 units. Determine $\tan L$ and $\sin L$. | \frac{5}{13} | 0.75 | 2,673.125 | 2,540.083333 | 3,072.25 | |
Let \( x[n] \) denote \( x \) raised to the power of \( x \), repeated \( n \) times. What is the minimum value of \( n \) such that \( 9[9] < 3[n] \)?
(For example, \( 3[2] = 3^3 = 27 \); \( 2[3] = 2^{2^2} = 16 \).) | 10 | 0.125 | 8,090 | 7,665.5 | 8,150.642857 | |
Unit circle $\Omega$ has points $X, Y, Z$ on its circumference so that $X Y Z$ is an equilateral triangle. Let $W$ be a point other than $X$ in the plane such that triangle $W Y Z$ is also equilateral. Determine the area of the region inside triangle $W Y Z$ that lies outside circle $\Omega$. | $\frac{3 \sqrt{3}-\pi}{3}$ | Let $O$ be the center of the circle. Then, we note that since $\angle W Y Z=60^{\circ}=\angle Y X Z$, that $Y W$ is tangent to $\Omega$. Similarly, $W Z$ is tangent to $\Omega$. Now, we note that the circular segment corresponding to $Y Z$ is equal to $\frac{1}{3}$ the area of $\Omega$ less the area of triangle $O Y Z$... | 0 | 7,267.5 | -1 | 7,267.5 |
Find the least positive integer $n$ such that there are at least $1000$ unordered pairs of diagonals in a regular polygon with $n$ vertices that intersect at a right angle in the interior of the polygon. | 28 | 0 | 8,192 | -1 | 8,192 | |
Let $L$ be the line with slope $\frac{5}{12}$ that contains the point $A=(24,-1)$, and let $M$ be the line perpendicular to line $L$ that contains the point $B=(5,6)$. The original coordinate axes are erased, and line $L$ is made the $x$-axis and line $M$ the $y$-axis. In the new coordinate system, point $A$ is on the ... | 31 | Given that $L$ has slope $\frac{5}{12}$ and contains the point $A=(24,-1)$, we may write the point-slope equation for $L$ as $y+1=\frac{5}{12}(x-24)$. Since $M$ is perpendicular to $L$ and contains the point $B=(5,6)$, we have that the slope of $M$ is $-\frac{12}{5}$, and consequently that the point-slope equation for ... | 0.1875 | 7,998.875 | 7,162 | 8,192 |
Given that Lucas makes a batch of lemonade using 200 grams of lemon juice, 100 grams of sugar, and 300 grams of water. If there are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar, and water has no calories, determine the total number of calories in 200 grams of this lemonade. | 145 | 0 | 558.75 | -1 | 558.75 | |
Given that the hyperbola C passes through the point (4, 3) and shares the same asymptotes with $$\frac {x^{2}}{4}$$ - $$\frac {y^{2}}{9}$$ = 1, determine the equation of C and its eccentricity. | \frac {\sqrt {13}} {2} | 0 | 4,006.875 | -1 | 4,006.875 | |
Given \(a\) and \(b\) are real numbers, satisfying:
\[
\sqrt[3]{a} - \sqrt[3]{b} = 12, \quad ab = \left( \frac{a + b + 8}{6} \right)^3.
\]
Find \(a - b\). | 468 | 0.6875 | 5,080.5625 | 3,666.272727 | 8,192 | |
The isosceles trapezoid has base lengths of 24 units (bottom) and 12 units (top), and the non-parallel sides are each 12 units long. How long is the diagonal of the trapezoid? | 12\sqrt{3} | 0.9375 | 4,409.5 | 4,157.333333 | 8,192 | |
A factory's annual fixed cost for producing a certain product is 2.5 million yuan. For every $x$ thousand units produced, an additional cost $C(x)$ is incurred. When the annual output is less than 80 thousand units, $C(x)=\frac{1}{3}x^2+10x$ (in million yuan). When the annual output is not less than 80 thousand units, ... | 100 | 0 | 5,155.5625 | -1 | 5,155.5625 | |
After a gymnastics meet, each gymnast shook hands once with every gymnast on every team (except herself). Afterwards, a coach came down and only shook hands with each gymnast from her own team. There were a total of 281 handshakes. What is the fewest number of handshakes the coach could have participated in? | 5 | 0.1875 | 7,822.8125 | 6,223 | 8,192 | |
Find the remainder when $x^4 +x + 2$ is divided by $x-3$. | 86 | 1 | 2,194.1875 | 2,194.1875 | -1 | |
Let $a, b, c$ be positive integers such that $\frac{a}{77}+\frac{b}{91}+\frac{c}{143}=1$. What is the smallest possible value of $a+b+c$? | 79 | We need $13 a+11 b+7 c=1001$, which implies $13(a+b+c-77)=2 b+6 c$. Then $2 b+6 c$ must be divisible by both 2 and 13, so it is minimized at 26 (e.g. with $b=10, c=1$). This gives $a+b+c=79$. | 0.0625 | 8,183.625 | 8,058 | 8,192 |
Given the set \( A = \{1, 2, 3, \ldots, 1002\} \), Petya and Vasya play a game. Petya names a number \( n \), and Vasya then selects a subset of \( A \) consisting of \( n \) elements. Vasya wins if there are no two coprime numbers in the subset he selects; otherwise, Petya wins. What is the minimum \( n \) that Petya... | 502 | 0.4375 | 6,707.375 | 5,439 | 7,693.888889 | |
Find the product $abc$ for the polynomial $Q(x) = x^3 + ax^2 + bx + c$ if its roots are $\sin \frac{\pi}{6}, \sin \frac{\pi}{3},$ and $\sin \frac{5\pi}{6}$. | \frac{\sqrt{3}}{2} | 0 | 6,925.25 | -1 | 6,925.25 | |
Given the parabola $C: x^{2}=2py\left(p \gt 0\right)$ with focus $F$, and the minimum distance between $F$ and a point on the circle $M: x^{2}+\left(y+4\right)^{2}=1$ is $4$.
$(1)$ Find $p$;
$(2)$ If point $P$ lies on $M$, $PA$ and $PB$ are two tangents to $C$ with points $A$ and $B$ as the points of tangency, find... | 20\sqrt{5} | 0 | 8,192 | -1 | 8,192 | |
For how many odd integers $k$ between 0 and 100 does the equation $2^{4m^{2}}+2^{m^{2}-n^{2}+4}=2^{k+4}+2^{3m^{2}+n^{2}+k}$ have exactly two pairs of positive integers $(m, n)$ that are solutions? | 18 | Step 1: Using parity and properties of powers of 2 to simplify the equation. We note that if $2^{x}=2^{y}$ for some real numbers $x$ and $y$, then $x=y$. We examine equations of the form $2^{a}+2^{b}=2^{c}+2^{d}$ where $a, b, c$, and $d$ are integers. We may assume without loss of generality that $a \leq b$ and $c \leq... | 0.0625 | 8,093.875 | 7,202 | 8,153.333333 |
The car engine operates with a power of \( P = 60 \text{ kW} \). Determine the car's speed \( v_0 \) if, after turning off the engine, it stops after traveling a distance of \( s = 450 \text{ m} \). The force resisting the car's motion is proportional to its speed. The mass of the car is \( m = 1000 \text{ kg} \). | 30 | 0.0625 | 7,881.4375 | 3,223 | 8,192 | |
Find the remainder when $5^{2021}$ is divided by $17$. | 14 | 1 | 3,287.1875 | 3,287.1875 | -1 | |
Edward stopped to rest at a place 1,875 feet from the prison and was spotted by a guard with a crossbow. The guard fired an arrow with an initial velocity of \( 100 \, \mathrm{ft/s} \). At the same time, Edward started running away with an acceleration of \( 1 \, \mathrm{ft/s^2} \). Assuming that air resistance causes ... | 75 | 0.375 | 6,998 | 5,335 | 7,995.8 | |
In $\Delta XYZ$, $XZ = YZ$, $m\angle DYZ = 50^{\circ}$, and $DY \parallel XZ$. Determine the number of degrees in $m\angle FDY$.
[asy] pair X,Y,Z,D,F; Y = dir(-50); X = dir(-130); D = (.5,0); F = .4 * dir(50);
draw(Z--Y--X--F,EndArrow); draw(Z--D,EndArrow);
label("$X$",X,W); label("$Z$",Z,NW);label("$Y$",Y,E);label("$... | 50 | 0.1875 | 7,367.5625 | 5,224.333333 | 7,862.153846 | |
Six test scores have a mean of 85, a median of 88, and a mode of 90. The highest score exceeds the second highest by 5 points. Find the sum of the three highest scores. | 275 | 0.1875 | 7,819.8125 | 6,207 | 8,192 | |
Suppose $x$ is a real number such that $\sin \left(1+\cos ^{2} x+\sin ^{4} x\right)=\frac{13}{14}$. Compute $\cos \left(1+\sin ^{2} x+\cos ^{4} x\right)$. | -\frac{3 \sqrt{3}}{14} | We first claim that $\alpha:=1+\cos ^{2} x+\sin ^{4} x=1+\sin ^{2} x+\cos ^{4} x$. Indeed, note that $$\sin ^{4} x-\cos ^{4} x=\left(\sin ^{2} x+\cos ^{2} x\right)\left(\sin ^{2} x-\cos ^{2} x\right)=\sin ^{2} x-\cos ^{2} x$$ which is the desired after adding $1+\cos ^{2} x+\cos ^{4} x$ to both sides. Hence, since $\si... | 0 | 4,787 | -1 | 4,787 |
The polynomial
\[ax^4 + bx^3 + cx^2 + dx + e = 0\]has coefficients that are all integers, and has roots $-2,$ $5,$ $9,$ and $-1/3.$ If $e$ is a positive integer, then find its smallest possible value. | 90 | 0.625 | 6,207.6875 | 5,365.8 | 7,610.833333 | |
What is the arithmetic mean of 14, 22 and 36? | 24 | 1 | 653.0625 | 653.0625 | -1 | |
Let $a$ and $b$ be the roots of the equation $x^2-mx+2=0$. Suppose that $a + \frac{1}{b}$ and $b + \frac{1}{a}$ are the roots of the equation $x^2-px+q=0$. What is $q$? | \frac{9}{2} | 1 | 3,189.1875 | 3,189.1875 | -1 | |
On a number line, there are three points A, B, and C which represent the numbers -24, -10, and 10, respectively. Two electronic ants, named Alpha and Beta, start moving towards each other from points A and C, respectively. Alpha moves at a speed of 4 units per second, while Beta moves at a speed of 6 units per second.
... | -44 | 0.125 | 6,959.3125 | 5,378.5 | 7,185.142857 | |
Let $s$ be the set of all rational numbers $r$ that satisfy the following conditions: $0<r<1$, and $r$ can be represented as a repeating decimal of the form $0.abcabcabc\cdots = 0.\dot{a}b\dot{c}$, where the digits $a, b, c$ do not necessarily have to be distinct. How many different numerators are there when the elemen... | 660 | 0 | 8,192 | -1 | 8,192 | |
There are three piles containing 22, 14, and 12 nuts. It is necessary to equalize the number of nuts in all piles by making three moves, while adhering to the following condition: it is only allowed to move as many nuts from one pile to another as there are in the pile to which the nuts are being moved.
| 16 | 0.5625 | 6,940.0625 | 5,966.333333 | 8,192 | |
The product of all the positive integer divisors of \( 6^{16} \) equals \( 6^k \) for some integer \( k \). Determine the value of \( k \). | 2312 | 0.8125 | 5,004.875 | 4,269.384615 | 8,192 | |
How many different ways can 6 different books be distributed according to the following requirements?
(1) Among three people, A, B, and C, one person gets 1 book, another gets 2 books, and the last one gets 3 books;
(2) The books are evenly distributed to A, B, and C, with each person getting 2 books;
(3) The boo... | 30 | 0.5 | 6,190.25 | 4,589.75 | 7,790.75 | |
How many positive odd integers greater than 1 and less than $200$ are square-free? | 79 | 0.0625 | 7,548.3125 | 7,477 | 7,553.066667 |
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