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 |
|---|---|---|---|---|---|---|
Two parabolas are the graphs of the equations $y=3x^2+4x-5$ and $y=x^2+11$. Give all points where they intersect. List the points in order of increasing $x$-coordinate, separated by semicolons. | (-4, 27);(2, 15) | 1 | 2,105.8125 | 2,105.8125 | -1 | |
Calculate the distance between the points $(5, -3)$ and $(-7, 4)$. Additionally, determine if the points $(5, -3)$, $(-7, 4)$, and $(5, 4)$ form a right triangle. | \sqrt{193} | 1 | 2,540.5 | 2,540.5 | -1 | |
Find the sum of all positive integers $n$ for which $n^2-19n+99$ is a perfect square.Find the sum of all integers $n$ such that $\dfrac{12}{n}$ is also an integer.
| 38 | 0.9375 | 4,692.625 | 4,459.333333 | 8,192 | |
How many positive divisors of 50! are either prime or the product of exactly two distinct primes? | 120 | 0.875 | 4,958.8125 | 4,496.928571 | 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 | 0.6875 | 5,058.9375 | 3,634.818182 | 8,192 | |
In cube \(ABCDA_1B_1C_1D_1\) with side length 1, a sphere is inscribed. Point \(E\) is located on edge \(CC_1\) such that \(C_1E = \frac{1}{8}\). From point \(E\), a tangent to the sphere intersects the face \(AA_1D_1D\) at point \(K\), with \(\angle KEC = \arccos \frac{1}{7}\). Find \(KE\). | \frac{7}{8} | 0 | 8,192 | -1 | 8,192 | |
For what value(s) of $k$ does the pair of equations $y=x^2$ and $y=3x+k$ have two identical solutions? | -\frac{9}{4} | 1. **Identify the condition for identical solutions**: The problem states that the equations $y = x^2$ and $y = 3x + k$ should have two identical solutions. This means that the quadratic equation formed by equating these two expressions should have exactly one unique solution, implying that it is a perfect square.
2. ... | 1 | 1,885.8125 | 1,885.8125 | -1 |
The set $\{3,6,9,10\}$ is augmented by a fifth element $n$, not equal to any of the other four. The median of the resulting set is equal to its mean. What is the sum of all possible values of $n$? | 26 | We are given the set $\{3,6,9,10\}$ and an additional element $n$, which is distinct from the other elements. We need to find the sum of all possible values of $n$ such that the median and the mean of the augmented set are equal.
#### Case 1: Median is $6$
For $6$ to be the median, $n$ must be less than or equal to $6... | 0.875 | 3,597.75 | 3,443.642857 | 4,676.5 |
A point $P$ is randomly selected from the rectangular region with vertices $(0,0), (2,0)$, $(2,1),(0,1)$. What is the probability that $P$ is closer to the origin than it is to the point $(3,1)$? | \frac{3}{4} | 0.25 | 7,588 | 5,776 | 8,192 | |
What is that largest positive integer $n$ for which $n^3+100$ is divisible by $n+10$?
| 890 | 1 | 3,237.1875 | 3,237.1875 | -1 | |
Let $S_{n}$ be the sum of the first $n$ terms of the sequence $\{a_{n}\}$, $a_{2}=5$, $S_{n+1}=S_{n}+a_{n}+4$; $\{b_{n}\}$ is a geometric sequence, $b_{2}=9$, $b_{1}+b_{3}=30$, with a common ratio $q \gt 1$.
$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$;
$(2)$ Let all terms of sequences ... | 660 | 0.125 | 7,822.25 | 6,231 | 8,049.571429 | |
How many positive integers less than $201$ are multiples of either $8$ or $11$, but not both at once? | 39 | 0.8125 | 4,748.625 | 3,954 | 8,192 | |
Find $a$ if $a$ and $b$ are integers such that $x^2 - x - 1$ is a factor of $ax^{17} + bx^{16} + 1$. | 987 | 0.8125 | 5,879.4375 | 5,345.769231 | 8,192 | |
Five distinct points are arranged on a plane such that the segments connecting them form lengths $a$, $a$, $a$, $a$, $a$, $b$, $b$, $2a$, and $c$. The shape formed by these points is no longer restricted to a simple polygon but could include one bend (not perfectly planar). What is the ratio of $c$ to $a$?
**A)** $\sqr... | 2\sqrt{3} | 0 | 8,192 | -1 | 8,192 | |
A construction company in Changsha has a total of 50 slag transport vehicles, including 20 Type A and 30 Type B vehicles. Now all these slag transport vehicles are allocated to the construction of the Changsha-Zhuzhou-Xiangtan intercity light rail, with 30 vehicles sent to Site A and 20 vehicles sent to Site B. The dai... | 80000 | 0.125 | 6,658.6875 | 5,271.5 | 6,856.857143 | |
Kevin writes a nonempty subset of $S = \{ 1, 2, \dots 41 \}$ on a board. Each day, Evan takes the set last written on the board and decreases each integer in it by $1.$ He calls the result $R.$ If $R$ does not contain $0$ he writes $R$ on the board. If $R$ contains $0$ he writes the set containing all e... | 94 | 0 | 8,040.125 | -1 | 8,040.125 | |
During a journey, the distance read on the odometer was 450 miles. On the return trip, using snow tires for the same distance, the reading was 440 miles. If the original wheel radius was 15 inches, find the increase in the wheel radius, correct to the nearest hundredth of an inch. | 0.34 | 0.6875 | 5,999.5625 | 5,003 | 8,192 | |
What is $\frac{2^2 \cdot 2^{-3}}{2^3 \cdot 2^{-2}}$? | \frac{1}{4} | 1 | 2,364.5 | 2,364.5 | -1 | |
What is the midpoint of the segment with endpoints (7,-6) and (-3,4)? | (2,-1) | 1 | 1,769.9375 | 1,769.9375 | -1 | |
Sixty cards are placed into a box, each bearing a number 1 through 15, with each number represented on four cards. Four cards are drawn from the box at random without replacement. Let \(p\) be the probability that all four cards bear the same number. Let \(q\) be the probability that three of the cards bear a number \(... | 224 | 0.9375 | 4,816.5 | 4,591.466667 | 8,192 | |
Point $P$ lies outside a circle, and two rays are drawn from $P$ that intersect the circle as shown. One ray intersects the circle at points $A$ and $B$ while the other ray intersects the circle at $M$ and $N$ . $AN$ and $MB$ intersect at $X$ . Given that $\angle AXB$ measures $127^{\circ}$ and the ... | 39 | 0.25 | 5,896.9375 | 3,890 | 6,565.916667 | |
Determine the greatest common divisor of all nine-digit integers formed by repeating a three-digit integer three times. For example, 256,256,256 or 691,691,691 are integers of this form. | 1001001 | 0.125 | 8,178.9375 | 8,087.5 | 8,192 | |
Solve for $x$ in the equation
\[2^{(16^x)} = 16^{(2^x)}.\] | \frac{2}{3} | 1 | 3,359.6875 | 3,359.6875 | -1 | |
In the sixth, seventh, eighth, and ninth basketball games of the season, a player scored $23$, $14$, $11$, and $20$ points, respectively. Her points-per-game average was higher after nine games than it was after the first five games. If her average after ten games was greater than $18$, what is the least number of poin... | 29 | 1. **Calculate the sum of the scores for games 6 through 9**:
The scores are $23$, $14$, $11$, and $20$. Therefore, the sum is:
\[
23 + 14 + 11 + 20 = 68
\]
2. **Calculate the average score for games 6 through 9**:
The average is:
\[
\frac{68}{4} = 17
\]
3. **Determine the total points for al... | 0.875 | 4,313.1875 | 3,759.071429 | 8,192 |
The sum of three numbers is $98$. The ratio of the first to the second is $\frac {2}{3}$,
and the ratio of the second to the third is $\frac {5}{8}$. The second number is: | 30 | 1. Let the three numbers be $a$, $b$, and $c$. According to the problem, the sum of these numbers is given by:
\[ a + b + c = 98 \]
2. The ratio of the first number to the second number is $\frac{2}{3}$, which can be expressed as:
\[ \frac{a}{b} = \frac{2}{3} \]
Multiplying both sides by $b$ gives:
\[ a = ... | 0.9375 | 1,396.8125 | 1,436.866667 | 796 |
Given that one of the children is a boy and the probability of having a boy or a girl is equal, calculate the probability that the other child is a girl. | \frac{2}{3} | 0.875 | 5,514.125 | 5,131.571429 | 8,192 | |
Let $f(x) = x^{2}(1-x)^{2}$. What is the value of the sum
\[f \left(\frac{1}{2019} \right)-f \left(\frac{2}{2019} \right)+f \left(\frac{3}{2019} \right)-f \left(\frac{4}{2019} \right)+\cdots + f \left(\frac{2017}{2019} \right) - f \left(\frac{2018}{2019} \right)?\] | 0 | 1. **Identify Symmetry in Function**:
We start by examining the function $f(x) = x^2(1-x)^2$. We need to check if there is any symmetry that might simplify the problem. Let's consider $f(1-x)$:
\[
f(1-x) = (1-x)^2 x^2 = x^2(1-x)^2 = f(x)
\]
This shows that $f(x) = f(1-x)$, which means the function is sym... | 0.1875 | 7,818.625 | 6,200.666667 | 8,192 |
Find the smallest positive integer $k$ such that $
z^{10} + z^9 + z^6+z^5+z^4+z+1
$ divides $z^k-1$. | 84 | 0 | 7,071.8125 | -1 | 7,071.8125 | |
If point \( P \) is on the curve \( y=\frac{1}{2} e^{x} \) and point \( Q \) is on the curve \( y=\ln (2 x) \), then the minimum value of \( |PQ| \) is \( \qquad \). | \sqrt{2}(1 - \ln 2) | 0.3125 | 7,460.1875 | 5,850.2 | 8,192 | |
Two parabolas are the graphs of the equations $y=2x^2-10x-10$ and $y=x^2-4x+6$. Find all points where they intersect. List the points in order of increasing $x$-coordinate, separated by semicolons. | (8,38) | 0 | 2,344.8125 | -1 | 2,344.8125 | |
The average age of 8 people in a room is 25 years. A 20-year-old person leaves the room. Calculate the average age of the seven remaining people. | \frac{180}{7} | 0 | 387.5625 | -1 | 387.5625 | |
Given complex numbers $\mathrm{z}_{\mathrm{i}} (\mathrm{i} = 1, 2, 3, 4, 5)$ satisfying:
$$\left\{\begin{array}{c}
\left|z_{1}\right| \leq 1, \quad \left|z_{2}\right| \leq 1 \\
\left|2 z_{3}-(\mathrm{z}_{1}+\mathrm{z}_{2})\right| \leq \left|\mathrm{z}_{1}-\mathrm{z}_{2}\right| \\
\left|2 \mathrm{z}_{4}-(\mathrm{z}_{1}... | \sqrt{3} | 0 | 8,192 | -1 | 8,192 | |
Let $D$ be the determinant of the matrix whose column vectors are $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}.$ Find the determinant of the matrix whose column vectors are $\mathbf{a} + \mathbf{b},$ $\mathbf{b} + \mathbf{c},$ and $\mathbf{c} + \mathbf{a},$ in terms of $D.$ | 2D | 0.4375 | 6,863.875 | 5,156.285714 | 8,192 | |
In a certain country, the airline system is arranged so that each city is connected by airlines to no more than three other cities, and from any city, it's possible to reach any other city with no more than one transfer. What is the maximum number of cities that can exist in this country? | 10 | 0.875 | 5,525.9375 | 5,145.071429 | 8,192 | |
A cooperative farm can purchase two types of feed mixtures from a neighboring farm to feed its animals. The Type I feed costs $30 per sack and contains 10 kg of component A and 10 kg of component B. The Type II feed costs $50 per sack and contains 10 kg of component A, 20 kg of component B, and 5 kg of component C. It ... | 165 | 0 | 7,669 | -1 | 7,669 | |
In triangle $XYZ,$ angle bisectors $\overline{XU}$ and $\overline{YV}$ intersect at $Q.$ If $XY = 8,$ $XZ = 6,$ and $YZ = 4,$ find $\frac{YQ}{QV}.$ | 1.5 | 0 | 6,053.5 | -1 | 6,053.5 | |
Wendy eats sushi for lunch. She wants to eat six pieces of sushi arranged in a $2 \times 3$ rectangular grid, but sushi is sticky, and Wendy can only eat a piece if it is adjacent to (not counting diagonally) at most two other pieces. In how many orders can Wendy eat the six pieces of sushi, assuming that the pieces of... | 360 | Call the sushi pieces $A, B, C$ in the top row and $D, E, F$ in the bottom row of the grid. Note that Wendy must first eat either $A, C, D$, or $F$. Due to the symmetry of the grid, all of these choices are equivalent. Without loss of generality, suppose Wendy eats piece $A$. Now, note that Wendy cannot eat piece $E$, ... | 0 | 8,163.25 | -1 | 8,163.25 |
Find all functions $f$ defined on the set of positive reals which take positive real values and satisfy: $f(xf(y))=yf(x)$ for all $x,y$; and $f(x)\to0$ as $x\to\infty$. | f(x)=\frac1x |
To tackle this problem, we want to find all functions \( f: \mathbb{R}^+ \to \mathbb{R}^+ \) that satisfy:
1. \( f(xf(y)) = yf(x) \) for all \( x, y \in \mathbb{R}^+ \).
2. \( \lim_{x \to \infty} f(x) = 0 \).
### Step-by-step Solution:
1. **Substitute Special Values**:
- Let \( y = 1 \) in the functional equati... | 0 | 6,203.5 | -1 | 6,203.5 |
Consider a city grid with intersections labeled A, B, C, and D. Assume a student walks from intersection A to intersection B every morning, always walking along the designated paths and only heading east or south. The student passes through intersections C and D along the way. The intersections are placed such that A t... | \frac{15}{77} | 0 | 7,994.25 | -1 | 7,994.25 | |
Consider the following flowchart: INPUT $\rightarrow$ Subtract $8 \rightarrow \square \rightarrow$ Divide by $2 \rightarrow \square$ Add $16 \rightarrow$ OUTPUT. If the OUTPUT is 32, what was the INPUT? | 40 | We start from the OUTPUT and work back to the INPUT. Since the OUTPUT 32 is obtained from adding 16 to the previous number, then the previous number is $32 - 16 = 16$. Since 16 is obtained by dividing the previous number by 2, then the previous number is $2 \times 16$ or 32. Since 32 is obtained by subtracting 8 from t... | 0.875 | 460.25 | 456.928571 | 483.5 |
Consider the parabola consisting of the points $(x, y)$ in the real plane satisfying $(y+x)=(y-x)^{2}+3(y-x)+3$. Find the minimum possible value of $y$. | -\frac{1}{2} | Let $w=y-x$. Adding $w$ to both sides and dividing by two gives $$y=\frac{w^{2}+4w+3}{2}=\frac{(w+2)^{2}-1}{2}$$ which is minimized when $w=-2$. This yields $y=-\frac{1}{2}$. | 0.6875 | 5,344.75 | 4,050.545455 | 8,192 |
Let $a, b, c$ be positive real numbers such that $3a + 4b + 2c = 3$. Find the minimum value of
\[
\frac{1}{2a + b} + \frac{1}{a + 3c} + \frac{1}{4b + c}.
\] | 1.5 | 0 | 8,192 | -1 | 8,192 | |
Two students, A and B, each choose 2 out of 6 extracurricular reading materials. Calculate the number of ways in which the two students choose extracurricular reading materials such that they have exactly 1 material in common. | 60 | 0.1875 | 6,134.625 | 6,460.666667 | 6,059.384615 | |
Triangle ABC is an equilateral triangle and O is the center of its inscribed circle. If the area of the circle is $4\pi$ sq cm, what is the area, in square centimeters, of triangle ABC? Express your answer in simplest radical form.
Note: The area of a circle with radius $r$ is $\pi r^2.$ | 12\sqrt 3 | 0 | 1,808.8125 | -1 | 1,808.8125 | |
Given the parametric equation of line C1 as $$\begin{cases} x=2+t \\ y=t \end{cases}$$ (where t is the parameter), and the polar coordinate equation of the ellipse C2 as ρ²cos²θ + 9ρ²sin²θ = 9. Establish a rectangular coordinate system with the origin O as the pole and the positive semi-axis of the x-axis as the polar ... | \frac{6\sqrt{3}}{5} | 0 | 6,377.125 | -1 | 6,377.125 | |
A drawer contains a mixture of red socks and blue socks, at most $1991$ in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly $\frac{1}{2}$ that both are red or both are blue. What is the largest possible number of red socks in the drawer that is consist... | 990 | Let $r$ and $b$ denote the number of red and blue socks such that $r+b\le1991$. Then by complementary counting, the number of ways to get a red and a blue sock must be equal to $1-\frac12=\frac12=\frac{2rb}{(r+b)(r+b-1)}\implies4rb=(r+b)(r+b-1)$ $=(r+b)^2-(r+b)\implies r^2+2rb+b^2-r-b=4rb\implies r^2-2rb+b^2$ $=(r-b)^2... | 0.875 | 6,049.125 | 5,743 | 8,192 |
For a positive integer $p$, define the positive integer $n$ to be $p$-safe if $n$ differs in absolute value by at least $3$ from all multiples of $p$. Considering the prime numbers $5$, $7$, and $11$, find the number of positive integers less than or equal to $15,000$ which are simultaneously $5$-safe, $7$-safe, and $1... | 975 | 0 | 8,155.8125 | -1 | 8,155.8125 | |
Find the greatest common factor of 8! and 9!. | 40320 | 1 | 2,156.3125 | 2,156.3125 | -1 | |
Find the number of cubic polynomials $p(x) = x^3 + ax^2 + bx + c,$ where $a, b,$ and $c$ are integers in $\{-20,-19,-18,\ldots,18,19,20\},$ such that there is a unique integer $m \not= 2$ with $p(m) = p(2).$ | 738 | $p(x)-p(2)$ is a cubic with at least two integral real roots, therefore it has three real roots, which are all integers.
There are exactly two distinct roots, so either $p(x)=p(2)+(x-2)^2(x-m)$ or $p(x)=p(2)+(x-2)(x-m)^2$, with $m\neq 2$.
In the first case $p(x)=x^3-(4+m)x^2+(4+4m)x-4m+p(2)$, with $|4+4m|\leq 20$ (wh... | 0 | 8,192 | -1 | 8,192 |
Given a tetrahedron $ABCD$, each edge is colored red or blue with equal probability. What is the probability that point $A$ can reach point $B$ through red edges within the tetrahedron $ABCD$? | \frac{3}{4} | 0 | 7,867 | -1 | 7,867 | |
What is the smallest positive integer $n$ such that $\sqrt{n}-\sqrt{n-1}<.01$? | 2501 | 1. **Starting from the inequality:** We are given that $\sqrt{n} - \sqrt{n-1} < 0.01$. To isolate $\sqrt{n}$, we add $\sqrt{n-1}$ to both sides:
\[
\sqrt{n} < \sqrt{n-1} + 0.01
\]
2. **Squaring both sides:** To eliminate the square roots, we square both sides of the inequality:
\[
(\sqrt{n})^2 < (\sqrt{... | 0.3125 | 7,332 | 6,752 | 7,595.636364 |
Let $\mathbf{v}$ and $\mathbf{w}$ be the vectors such that $\mathbf{v} \cdot \mathbf{w} = -3$ and $\|\mathbf{w}\| = 5.$ Find the magnitude of $\operatorname{proj}_{\mathbf{w}} \mathbf{v}.$ | \frac{3}{5} | 1 | 1,209.9375 | 1,209.9375 | -1 | |
Find the area of the triangle with vertices $(0,7,10),$ $(-1,6,6),$ and $(-4,9,6).$ | 9 | 1 | 3,618.625 | 3,618.625 | -1 | |
Two teachers are taking a photo with 3 male students and 3 female students lined up in a row. The teachers can only stand at the ends, and the male students cannot be adjacent. How many different ways are there to arrange the photo? | 288 | 0.5625 | 6,535.5625 | 5,442 | 7,941.571429 | |
A natural number \( x \) in a base \( r \) system (\( r \leq 36 \)) is represented as \( \overline{ppqq} \), where \( 2q = 5p \). It turns out that the base-\( r \) representation of \( x^2 \) is a seven-digit palindrome with a middle digit of zero. (A palindrome is a number that reads the same from left to right and f... | 36 | 0 | 8,192 | -1 | 8,192 | |
In a regular tetrahedron with edge length $2\sqrt{6}$, the total length of the intersection between the sphere with center $O$ and radius $\sqrt{3}$ and the surface of the tetrahedron is ______. | 8\sqrt{2}\pi | 0.4375 | 7,199.9375 | 5,924.428571 | 8,192 | |
Given two lines \\({{l}\_{1}}:(a-1)x+2y+3=0\\) and \\({{l}\_{2}}:x+ay+3=0\\) are parallel, then \\(a=\\)_______. | -1 | 0.375 | 6,227.125 | 6,364.666667 | 6,144.6 | |
Omkar, \mathrm{Krit}_{1}, \mathrm{Krit}_{2}, and \mathrm{Krit}_{3} are sharing $x>0$ pints of soup for dinner. Omkar always takes 1 pint of soup (unless the amount left is less than one pint, in which case he simply takes all the remaining soup). Krit $_{1}$ always takes \frac{1}{6}$ of what is left, Krit ${ }_{2}$ alw... | \frac{49}{3} | The main observation is that if $x>1$ pints of soup are left, then in one round, Omkar gets 1 and each Krit $_{n}$ gets \frac{x-1}{6}$, with \frac{x-1}{2}$ soup left. Thus it is evident that each Krit ${ }_{n}$ gets the same amount of soup, which means it suffices to find $x$ for which Omkar gets \frac{x}{4}$. Omkar ge... | 0 | 8,186.8125 | -1 | 8,186.8125 |
Let $\Delta A_{1} B_{1} C$ be a triangle with $\angle A_{1} B_{1} C=90^{\circ}$ and $\frac{C A_{1}}{C B_{1}}=\sqrt{5}+2$. For any $i \geq 2$, define $A_{i}$ to be the point on the line $A_{1} C$ such that $A_{i} B_{i-1} \perp A_{1} C$ and define $B_{i}$ to be the point on the line $B_{1} C$ such that $A_{i} B_{i} \perp... | 4030 | We claim that $\Gamma_{2}$ is the incircle of $\triangle B_{1} A_{2} C$. This is because $\triangle B_{1} A_{2} C$ is similar to $A_{1} B_{1} C$ with dilation factor $\sqrt{5}-2$, and by simple trigonometry, one can prove that $\Gamma_{2}$ is similar to $\Gamma_{1}$ with the same dilation factor. By similarities, we ca... | 0 | 8,160.3125 | -1 | 8,160.3125 |
Meghal is playing a game with 2016 rounds $1,2, \cdots, 2016$. In round $n$, two rectangular double-sided mirrors are arranged such that they share a common edge and the angle between the faces is $\frac{2 \pi}{n+2}$. Meghal shoots a laser at these mirrors and her score for the round is the number of points on the two ... | 1019088 | Let points $O, A_{1}, A_{2}$ lie in a plane such that $\angle A_{1} O A_{2}=\frac{2 \pi}{n+2}$. We represent the mirrors as line segments extending between $O$ and $A_{1}$, and $O$ and $A_{2}$. Also let points $A_{3}, A_{4}, \cdots, A_{n+2}$ lie in the plane such that $A_{i+1}$ is the reflection of $A_{i-1}$ over $O A_... | 0 | 8,026.6875 | -1 | 8,026.6875 |
For how many values of $k$ is $12^{12}$ the least common multiple of the positive integers $6^6$, $8^8$, and $k$?
| 25 | 0.9375 | 4,283 | 4,220.333333 | 5,223 | |
Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a half dollar. What is the probability that at least 25 cents worth of coins come up heads? | \dfrac{13}{16} | 0 | 7,938.0625 | -1 | 7,938.0625 | |
Let \[f(x) =
\begin{cases}
2x + 9 &\text{if }x<-2, \\
5-2x&\text{if }x\ge -2.
\end{cases}
\]Find $f(-7).$ | -5 | 1 | 1,815.625 | 1,815.625 | -1 | |
Bob's password consists of a positive single-digit number followed by a letter and another positive single-digit number. What is the probability that Bob's password consists of an even single-digit number followed by a vowel (from A, E, I, O, U) and a number greater than 5? | \frac{40}{1053} | 0.625 | 2,592.8125 | 2,690.8 | 2,429.5 | |
Given that the circumferences of the two bases of a cylinder lie on the surface of a sphere $O$ with a volume of $\frac{{32π}}{3}$, the maximum value of the lateral surface area of the cylinder is ______. | 8\pi | 0.75 | 5,189.875 | 4,609.583333 | 6,930.75 | |
Given point A (-2, 1) and circle C: $(x-2)^2+(y-2)^2=1$, a ray of light is emitted from point A to the x-axis and then reflects in the direction of the tangent to the circle. The distance traveled by the ray of light from point A to the tangent point is ______. | 2\sqrt{6} | 0.0625 | 8,118.3125 | 8,192 | 8,113.4 | |
Given that the odd function $f(x)$ and the even function $g(x)$ defined on $\mathbb{R}$ satisfy $f(x) + g(x) = a^x - a^{-x} + 2$, and $g(2) = a$, find the value of $f(2)$. | \frac{15}{4} | 1 | 2,924.9375 | 2,924.9375 | -1 | |
Find the smallest three-digit number \(n\) such that if the three digits are \(a\), \(b\), and \(c\), then
\[ n = a + b + c + ab + bc + ac + abc. \] | 199 | 0.125 | 7,565.9375 | 5,841 | 7,812.357143 | |
Eli, Joy, Paul, and Sam want to form a company; the company will have 16 shares to split among the 4 people. The following constraints are imposed: - Every person must get a positive integer number of shares, and all 16 shares must be given out. - No one person can have more shares than the other three people combined.... | 315 | We are finding the number of integer solutions to $a+b+c+d=16$ with $1 \leq a, b, c, d \leq 8$. We count the number of solutions to $a+b+c+d=16$ over positive integers, and subtract the number of solutions in which at least one variable is larger than 8. If at least one variable is larger than 8, exactly one of the var... | 0.375 | 7,353.75 | 6,552.666667 | 7,834.4 |
There are 11 of the number 1, 22 of the number 2, 33 of the number 3, and 44 of the number 4 on the blackboard. The following operation is performed: each time, three different numbers are erased, and the fourth number, which is not erased, is written 2 extra times. For example, if 1 of 1, 1 of 2, and 1 of 3 are erased... | 12 | 0 | 8,192 | -1 | 8,192 | |
On every kilometer marker along the highway between the villages of Yolkino and Palkino, there is a post with a sign. On one side of the sign, the distance to Yolkino is indicated, and on the other side, the distance to Palkino is indicated. Borya noticed that the sum of all the digits on each sign equals 13. What is t... | 49 | 0.25 | 7,330 | 6,319.5 | 7,666.833333 | |
Suppose $z$ is a complex number such that $z^2 = 24-32i$. Find $|z|$. | 2\sqrt{10} | 1 | 2,030.8125 | 2,030.8125 | -1 | |
The random variable $X$ follows a normal distribution $N(1, 4)$, where the mean $\mu = 1$ and the variance $\sigma^2 = 4$. Given that $P(X \geq 2) = 0.2$, calculate the probability $P(0 \leq X \leq 1)$. | 0.3 | 0.1875 | 7,655.375 | 7,140.666667 | 7,774.153846 | |
A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display? | 23 | 1. **Understanding the Display Format**: The digital watch displays time in a 12-hour format with AM and PM, showing hours and minutes. The hours can range from 01 to 12, and the minutes from 00 to 59.
2. **Maximizing the Hour Digits**:
- The hours are displayed as either 01, 02, ..., 12. We need to find the hour ... | 0.0625 | 8,192 | 8,192 | 8,192 |
The dollar is now worth $\frac{1}{980}$ ounce of gold. After the $n^{th}$ 7001 billion dollars bailout package passed by congress, the dollar gains $\frac{1}{2{}^2{}^{n-1}}$ of its $(n-1)^{th}$ value in gold. After four bank bailouts, the dollar is worth $\frac{1}{b}(1-\frac{1}{2^c})$ in gold, where $b, c$ ... | 506 | 0.0625 | 7,527.0625 | 3,929 | 7,766.933333 | |
From the numbers $1, 2, \cdots, 2005$, choose $n$ different numbers. If it is always possible to find three numbers among these $n$ numbers that can form the side lengths of a triangle, determine the minimum value of $n$. | 17 | 0.1875 | 7,538.9375 | 5,756 | 7,950.384615 | |
Triangle $PAB$ is formed by three tangents to circle $O$ and $\angle APB = 40^\circ$. Find $\angle AOB$.
[asy]
import graph;
unitsize(1.5 cm);
pair A, B, O, P, R, S, T;
R = dir(115);
S = dir(230);
T = dir(270);
P = extension(R, R + rotate(90)*(R), T, T + rotate(90)*(T));
A = extension(S, S + rotate(90)*(S), T, T +... | 70^\circ | 0 | 4,860.5625 | -1 | 4,860.5625 | |
In racing over a distance $d$ at uniform speed, $A$ can beat $B$ by $20$ yards, $B$ can beat $C$ by $10$ yards, and $A$ can beat $C$ by $28$ yards. Then $d$, in yards, equals: | 100 | 1. **Define Variables:**
Let $a$, $b$, and $c$ be the speeds of $A$, $B$, and $C$ respectively.
2. **Set Up Equations:**
- When $A$ finishes the race, $B$ is 20 yards behind. Thus, in the time $\frac{d}{a}$ it takes for $A$ to finish, $B$ covers $d - 20$ yards. The equation is:
\[
\frac{d}{a} = \frac{d... | 0.875 | 4,581.625 | 4,065.857143 | 8,192 |
In how many ways can you form 5 quartets from 5 violinists, 5 violists, 5 cellists, and 5 pianists? | (5!)^3 | 0 | 6,629.8125 | -1 | 6,629.8125 | |
Given the equation concerning $x$, $(m-1)x^{m^{2}+1}+2x-3=0$, the value of $m$ is ________. | -1 | 0.5 | 4,926.375 | 3,616 | 6,236.75 | |
For each real number $a$ with $0 \leq a \leq 1$, let numbers $x$ and $y$ be chosen independently at random from the intervals $[0, a]$ and $[0, 1]$, respectively, and let $P(a)$ be the probability that
$\sin^2{(\pi x)} + \sin^2{(\pi y)} > 1$
What is the maximum value of $P(a)?$ | 2 - \sqrt{2} | To solve for the maximum value of $P(a)$, we start by analyzing the condition given:
\[\sin^2{(\pi x)} + \sin^2{(\pi y)} > 1.\]
We can rewrite this inequality using trigonometric identities:
\[\sin^2{(\pi x)} + \sin^2{(\pi y)} = 1 - \cos^2{(\pi x)} + 1 - \cos^2{(\pi y)} = 2 - (\cos^2{(\pi x)} + \cos^2{(\pi y)}).\]
Th... | 0.1875 | 7,815.4375 | 6,183.666667 | 8,192 |
Given 40 feet of fencing, what is the greatest possible number of square feet in the area of a rectangular pen enclosed by the fencing? | 100 | 1 | 2,057.75 | 2,057.75 | -1 | |
The product of two 2-digit numbers is $3774$. What is the smaller of the two numbers? | 51 | 1 | 3,310.5625 | 3,310.5625 | -1 | |
Square ABCD has its center at $(8,-8)$ and has an area of 4 square units. The top side of the square is horizontal. The square is then dilated with the dilation center at (0,0) and a scale factor of 2. What are the coordinates of the vertex of the image of square ABCD that is farthest from the origin? Give your answer ... | (18, -18) | 0.75 | 4,641.6875 | 3,638.75 | 7,650.5 | |
What is the value of the expression $\frac{20+16 \times 20}{20 \times 16}$? | \frac{17}{16} | Evaluating, $\frac{20+16 \times 20}{20 \times 16}=\frac{20+320}{320}=\frac{340}{320}=\frac{17}{16}$. Alternatively, we could notice that each of the numerator and denominator is a multiple of 20, and so $\frac{20+16 \times 20}{20 \times 16}=\frac{20(1+16)}{20 \times 16}=\frac{1+16}{16}=\frac{17}{16}$. | 0.8125 | 1,941.0625 | 2,305.692308 | 361 |
In a new diagram, the grid is composed of squares. The grid is segmented into various levels that step upwards as you move to the right. Determine the area of the shaded region in the following configuration:
- The grid dimensions are 15 units wide and 5 units tall.
- The shaded region fills up from the bottom to a hei... | 37.5 | 0 | 7,978.875 | -1 | 7,978.875 | |
A single-elimination ping-pong tournament has $2^{2013}$ players, seeded in order of ability. If the player with seed $x$ plays the player with seed $y$, then it is possible for $x$ to win if and only if $x \leq y+3$. For how many players $P$ it is possible for $P$ to win? (In each round of a single elimination tournam... | 6038 | We calculate the highest seed $n$ that can win. Below, we say that a player $x$ vicariously defeats a player $y$ if $x$ defeats $y$ directly or indirectly through some chain (i.e. $x$ defeats $x_{1}$, who defeated $x_{2}, \ldots$, who defeated $x_{n}$, who defeated $y$ for some players $\left.x_{1}, \ldots, x_{n}\right... | 0 | 8,192 | -1 | 8,192 |
Let $i=\sqrt{-1}$. The product of the real parts of the roots of $z^2-z=5-5i$ is | -6 | 1. **Identify the quadratic equation and apply the quadratic formula**:
The given equation is $z^2 - z = 5 - 5i$. Rearranging, we have:
\[
z^2 - z - (5 - 5i) = 0
\]
Applying the quadratic formula, $z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 1$, $b = -1$, and $c = -(5 - 5i)$, we get:
\[
z =... | 0.6875 | 6,543.875 | 6,185.727273 | 7,331.8 |
For each integer $i=0,1,2, \dots$ , there are eight balls each weighing $2^i$ grams. We may place balls as much as we desire into given $n$ boxes. If the total weight of balls in each box is same, what is the largest possible value of $n$ ? | 15 | 0 | 7,913.75 | -1 | 7,913.75 | |
(1) Given a point P(-4, 3) on the terminal side of angle $\alpha$, calculate the value of $$\frac {\cos( \frac {\pi}{2}+\alpha)\sin(-\pi-\alpha)}{\cos( \frac {11\pi}{2}-\alpha )\sin( \frac {9\pi}{2}+\alpha )}$$.
(2) If $\sin x= \frac {m-3}{m+5}$ and $\cos x= \frac {4-2m}{m+5}$, where $x$ is in the interval ($\frac {\pi... | -\frac {5}{12} | 0.9375 | 4,081 | 4,084.8 | 4,024 | |
Stephan has been busy at work writing all the possible rearrangements of the letters in his name. He is one bored boy. If he can write twelve rearrangements of his name every minute, how many hours does it take to write all the possible rearrangements of his name? | 7\text{ hours}. | 0.9375 | 646.3125 | 655.4 | 510 | |
When $\sqrt[4]{2^6 \cdot 3^5 \cdot 5^2}$ is fully simplified, the result is $x\sqrt[4]{y}$, where $x$ and $y$ are positive integers. What is $x+y$? | 306 | 0.75 | 5,021.25 | 4,245.5 | 7,348.5 | |
Let $M$ denote the number of $9$-digit positive integers in which the digits are in increasing order, given that repeated digits are allowed and the digit ‘0’ is permissible. Determine the remainder when $M$ is divided by $1000$. | 620 | 0.0625 | 6,175.4375 | 3,148 | 6,377.266667 | |
Given three sequences $\{F_n\}$, $\{k_n\}$, $\{r_n\}$ satisfying: $F_1=F_2=1$, $F_{n+2}=F_{n+1}+F_n$ ($n\in\mathbb{N}^*$), $r_n=F_n-3k_n$, $k_n\in\mathbb{N}$, $0\leq r_n<3$, calculate the sum $r_1+r_3+r_5+\ldots+r_{2011}$. | 1509 | 0.25 | 7,055.5625 | 4,922 | 7,766.75 | |
In trapezoid $ABCD$, sides $AB$ and $CD$ are parallel with lengths of 10 and 24 units respectively, and the altitude is 15 units. Points $G$ and $H$ are the midpoints of sides $AD$ and $BC$, respectively. Determine the area of quadrilateral $GHCD$. | 153.75 | 0.0625 | 7,012.6875 | 8,192 | 6,934.066667 | |
Given that $\sec x - \tan x = \frac{5}{4},$ find all possible values of $\sin x.$ | \frac{1}{4} | 0 | 4,134.9375 | -1 | 4,134.9375 | |
A group with 7 young men and 7 young women was divided into pairs randomly. Find the probability that at least one pair consists of two women. Round the answer to two decimal places. | 0.96 | 0 | 8,192 | -1 | 8,192 | |
Convert the point $\left( 5, \frac{3 \pi}{2}, 4 \right)$ in cylindrical coordinates to rectangular coordinates. | (0, -5, 4) | 1 | 1,351 | 1,351 | -1 | |
Square $A B C D$ has side length 1. A dilation is performed about point $A$, creating square $A B^{\prime} C^{\prime} D^{\prime}$. If $B C^{\prime}=29$, determine the area of triangle $B D C^{\prime}$. | 420 | $29^{2}-2 \cdot \frac{1}{2}(29)\left(\frac{29}{2}\right)-\frac{1}{2}=420$. | 0 | 4,705.375 | -1 | 4,705.375 |
In $\triangle ABC$, let the sides opposite to angles $A$, $B$, and $C$ be $a$, $b$, and $c$ respectively. Given that $a\cos B=3$ and $b\sin A=4$.
(I) Find $\tan B$ and the value of side $a$;
(II) If the area of $\triangle ABC$ is $S=10$, find the perimeter $l$ of $\triangle ABC$. | 10 + 2\sqrt{5} | 0.8125 | 5,001.75 | 4,265.538462 | 8,192 |
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