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 |
|---|---|---|---|---|---|---|
Call a positive integer $N$ a 7-10 double if the digits of the base-$7$ representation of $N$ form a base-$10$ number that is twice $N$. For example, $51$ is a 7-10 double because its base-$7$ representation is $102$. What is the largest 7-10 double? | 315 | Since this is an AIME problem, the maximum number of digits the 7-10 double can have is 3. Let the number be \[abc\] in base 7. Then the number in expanded form is \[49a+7b+c\] in base 7 and \[100a+10b+c\] in base 10. Since the number in base 7 is half the number in base 10, we get the following equation. \[98b+14b+2c=... | 0.125 | 7,897.8125 | 5,838.5 | 8,192 |
In a rectangular parallelepiped with dimensions AB = 4, BC = 2, and CG = 5, point M is the midpoint of EF, calculate the volume of the rectangular pyramid with base BDFE and apex M. | \frac{40}{3} | 0 | 8,128.8125 | -1 | 8,128.8125 | |
What is the value of $ { \sum_{1 \le i< j \le 10}(i+j)}_{i+j=odd} $ $ - { \sum_{1 \le i< j \le 10}(i+j)}_{i+j=even} $ | 55 | 0.125 | 7,914.8125 | 6,186.5 | 8,161.714286 | |
Given a regular pentagon \(ABCDE\). Point \(K\) is marked on side \(AE\), and point \(L\) is marked on side \(CD\). It is known that \(\angle LAE + \angle KCD = 108^\circ\) and \(AK: KE = 3:7\). Find \(CL: AB\).
A regular pentagon is a pentagon where all sides and all angles are equal. | 0.7 | 0 | 8,192 | -1 | 8,192 | |
At a party, each man danced with exactly three women and each woman danced with exactly two men. Twelve men attended the party. How many women attended the party? | 18 | 1 | 1,203.875 | 1,203.875 | -1 | |
Blind boxes are a new type of product. Merchants package different styles of products from the same series in boxes with the same appearance, so that consumers do not know which style of product they are buying. A merchant has designed three types of dolls, $A$, $B$, and $C$, in the same series, and sells them in blind... | 0.216 | 0 | 7,700.4375 | -1 | 7,700.4375 | |
A convex quadrilateral is divided by its diagonals into four triangles; the areas of three of them are \(10 \, \text{cm}^2\), \(20 \, \text{cm}^2\), and \(30 \, \text{cm}^2\), and each is less than the area of the fourth triangle. Find the area of the given quadrilateral. | 120 | 0.5 | 5,908.8125 | 4,626.75 | 7,190.875 | |
The triangle \( \triangle ABC \) has side \( AC \) with length \( 24 \text{ cm} \) and a height from vertex \( B \) with length \( 25 \text{ cm} \). Side \( AB \) is divided into five equal parts, with division points labeled \( K, L, M, N \) from \( A \) to \( B \). Each of these points has a parallel line drawn to si... | 120 | 0.0625 | 7,975.125 | 4,722 | 8,192 | |
$ABCDEF$ is a regular hexagon. Let $R$ be the overlap between $\vartriangle ACE$ and $\vartriangle BDF$ . What is the area of $R$ divided by the area of $ABCDEF$ ? | 1/3 | 0.25 | 8,160.8125 | 8,067.25 | 8,192 | |
Three young brother-sister pairs from different families need to take a trip in a van. These six children will occupy the second and third rows in the van, each of which has three seats. To avoid disruptions, siblings may not sit right next to each other in the same row, and no child may sit directly in front of his or... | 96 | 1. **Identify the constraints**: Each row has three seats, and siblings must not sit next to each other in the same row or directly in front of each other. This implies that each sibling pair must be split between the two rows.
2. **Assign siblings to rows**: Let's denote the siblings from the first family as $a_1$ an... | 0.0625 | 8,115.625 | 6,970 | 8,192 |
At noon on a certain day, Minneapolis is $N$ degrees warmer than St. Louis. At $4{:}00$ the temperature in Minneapolis has fallen by $5$ degrees while the temperature in St. Louis has risen by $3$ degrees, at which time the temperatures in the two cities differ by $2$ degrees. What is the product of all possible values... | 60 | 1. **Define the variables:**
Let $M$ represent the temperature in Minneapolis at noon, and $L$ represent the temperature in St. Louis at the same time. Given that Minneapolis is $N$ degrees warmer than St. Louis, we can express this relationship as:
\[
M = L + N
\]
2. **Temperature change by 4:00 PM:**
... | 1 | 1,273.8125 | 1,273.8125 | -1 |
A regular octahedron is formed by joining the centers of adjoining faces of a cube. The ratio of the volume of the octahedron to the volume of the cube is
$\mathrm{(A) \frac{\sqrt{3}}{12} } \qquad \mathrm{(B) \frac{\sqrt{6}}{16} } \qquad \mathrm{(C) \frac{1}{6} } \qquad \mathrm{(D) \frac{\sqrt{2}}{8} } \qquad \mathrm{(... | \frac{1}{6} | 0 | 5,980.3125 | -1 | 5,980.3125 | |
The average of the numbers $1, 2, 3,\dots, 148, 149,$ and $x$ is $50x$. What is $x$? | \frac{11175}{7499} | 0.0625 | 8,192 | 8,192 | 8,192 | |
Given $sin(x+\frac{π}{12})=-\frac{1}{4}$, find the value of $cos(\frac{5π}{6}-2x)$. | -\frac{7}{8} | 0.75 | 6,487.4375 | 5,919.25 | 8,192 | |
10.25 people are lined up in a row, each of whom either tells the truth or lies.
The person at the front of the line says: "Everyone behind me is lying."
Everyone else says: "The person in front of me (the one directly next to the speaker) is lying."
Among these 25 people, there are ___ people who are lying. | 13 | 0.75 | 5,478.25 | 4,573.666667 | 8,192 | |
Given the function $y=\left[x\right]$, which is called the Gaussian function and represents the greatest integer not exceeding $x$, such as $\left[\pi \right]=3$, $\left[-2.5\right]=-3$. The solution set of the inequality $\frac{[x]}{[x]-4}＜0$ is ______; when $x \gt 0$, the maximum value of $\frac{[x]}{[x]^2+4}$ is ___... | \frac{1}{4} | 1 | 3,254.6875 | 3,254.6875 | -1 | |
Completely factor the following expression: \[(15x^3+80x-5)-(-4x^3+4x-5).\] | 19x(x^2+4) | 1 | 1,747.1875 | 1,747.1875 | -1 | |
In February 1983, $789$ millimeters of rain fell in Jorhat, India. What was the average rainfall in millimeters per hour during that particular month?
A) $\frac{789}{672}$
B) $\frac{789 \times 28}{24}$
C) $\frac{789 \times 24}{28}$
D) $\frac{28 \times 24}{789}$
E) $789 \times 28 \times 24$ | \frac{789}{672} | 0 | 1,453.1875 | -1 | 1,453.1875 | |
Given that positive real numbers $a$ and $b$ satisfy $a \gt b$ and $ab=\frac{1}{2}$, find the minimum value of $\frac{4{a}^{2}+{b}^{2}+3}{2a-b}$. | 2\sqrt{5} | 0.625 | 6,782.3125 | 6,067.6 | 7,973.5 | |
Given a geometric sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, and $a_1=2$, if $\frac {S_{6}}{S_{2}}=21$, then the sum of the first five terms of the sequence $\{\frac {1}{a_n}\}$ is
A) $\frac {1}{2}$ or $\frac {11}{32}$
B) $\frac {1}{2}$ or $\frac {31}{32}$
C) $\frac {11}{32}$ or $\frac {31}{32}$
... | \frac {31}{32} | 0 | 5,018.8125 | -1 | 5,018.8125 | |
A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0), (2020, 0), (2020, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\frac{1}{2}$. (A point $(x, y)$ is a lattice point if $x$ and $y$ are both integers.) What is $d$ to the ... | 0.4 |
We are given a square with vertices at $(0, 0), (2020, 0), (2020, 2020),$ and $(0, 2020)$, and we need to find the radius $d$ such that the probability a randomly chosen point within the square is within $d$ units of a lattice point is $\frac{1}{2}$.
#### Step 1: Understanding the Problem
A lattice point is a point ... | 0.5 | 7,065.6875 | 6,046.25 | 8,085.125 |
Interior numbers begin in the third row of Pascal's Triangle. The sum of the interior numbers in the fourth row is 6. The sum of the interior numbers of the fifth row is 14. What is the sum of the interior numbers of the seventh row? | 62 | 0.4375 | 4,263.4375 | 2,425 | 5,693.333333 | |
Find the smallest possible sum of two perfect squares such that their difference is 175 and both squares are greater or equal to 36. | 625 | 0 | 4,978.1875 | -1 | 4,978.1875 | |
Determine the exact value of
\[
\sqrt{\left( 2 - \sin^2 \frac{\pi}{9} \right) \left( 2 - \sin^2 \frac{2 \pi}{9} \right) \left( 2 - \sin^2 \frac{4 \pi}{9} \right)}.
\] | \frac{\sqrt{619}}{16} | 0 | 7,658.3125 | -1 | 7,658.3125 | |
In a rhombus $P Q R S$ with $P Q=Q R=R S=S P=S Q=6$ and $P T=R T=14$, what is the length of $S T$? | 10 | First, we note that $\triangle P Q S$ and $\triangle R Q S$ are equilateral. Join $P$ to $R$. Since $P Q R S$ is a rhombus, then $P R$ and $Q S$ bisect each other at their point of intersection, $M$, and are perpendicular. Note that $Q M=M S=\frac{1}{2} Q S=3$. Since $\angle P S Q=60^{\circ}$, then $P M=P S \sin (\angl... | 0.0625 | 8,053.5625 | 5,977 | 8,192 |
A school offers 7 courses for students to choose from, among which courses A, B, and C cannot be taken together due to scheduling conflicts, allowing at most one of them to be chosen. The school requires each student to choose 3 courses. How many different combinations of courses are there? (Solve using mathematics) | 22 | 0 | 4,928.6875 | -1 | 4,928.6875 | |
There are $N$ natural numbers written on a board, where $N \geq 5$. It is known that the sum of all the numbers is 80, and the sum of any five of them is no more than 19. What is the smallest possible value of $N$? | 26 | 0 | 7,877.125 | -1 | 7,877.125 | |
Given rectangle $R_1$ with one side $2$ inches and area $12$ square inches. Rectangle $R_2$ with diagonal $15$ inches is similar to $R_1$. Expressed in square inches the area of $R_2$ is: | \frac{135}{2} | 1. **Identify the dimensions of $R_1$:**
Given that one side of rectangle $R_1$ is $2$ inches and its area is $12$ square inches, we can find the other side by dividing the area by the given side:
\[
\text{Other side} = \frac{\text{Area}}{\text{Given side}} = \frac{12}{2} = 6 \text{ inches}
\]
Thus, the ... | 1 | 3,580.8125 | 3,580.8125 | -1 |
Let $O$ be the center of the square $ABCD$. If $3$ points are chosen from $O$, $A$, $B$, $C$, and $D$ at random, calculate the probability that the $3$ points are collinear. | \frac{1}{5} | 0.625 | 6,263.125 | 5,498.6 | 7,537.333333 | |
Below is a portion of the graph of a function, $y=h(x)$:
[asy]
import graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.25,xmax=4.25,ymin=-7.25,ymax=6.25;
pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 2"); real gx=1,gy... | 4 | 1 | 3,496.6875 | 3,496.6875 | -1 | |
Let $x$ be a real number such that $x^{3}+4 x=8$. Determine the value of $x^{7}+64 x^{2}$. | 128 | For any integer $n \geq 0$, the given implies $x^{n+3}=-4 x^{n+1}+8 x^{n}$, so we can rewrite any such power of $x$ in terms of lower powers. Carrying out this process iteratively gives $$\begin{aligned} x^{7} & =-4 x^{5}+8 x^{4} \\ & =8 x^{4}+16 x^{3}-32 x^{2} \\ & =16 x^{3}-64 x^{2}+64 x \\ & =-64 x^{2}+128 . \end{al... | 0.9375 | 3,643.8125 | 3,340.6 | 8,192 |
It is known that 9 cups of tea cost less than 10 rubles, and 10 cups of tea cost more than 11 rubles. How much does one cup of tea cost? | 111 | 0 | 2,357.875 | -1 | 2,357.875 | |
In an isosceles triangle \(ABC \) (\(AB = BC\)), a point \(D\) is taken on the side \(BC\) such that \(BD : DC = 1 : 4\).
In what ratio does the line \(AD\) divide the height \(BE\) of the triangle \(ABC\), counted from the vertex \(B\)? | 1:2 | 0.75 | 6,454.375 | 6,183.916667 | 7,265.75 | |
Except for the first two terms, each term of the sequence $1000, x, 1000 - x,\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encountered. What positive integer $x$ produces a sequence of maximum length? | 618 | 0 | 8,192 | -1 | 8,192 | |
In an isosceles triangle $\triangle ABC$ with vertex angle $A = \frac{2\pi}{3}$ and base $BC = 2\sqrt{3}$, find the dot product $\vec{BA} \cdot \vec{AC}$. | -2 | 0 | 4,668.5 | -1 | 4,668.5 | |
Erika, who is 14 years old, flips a fair coin whose sides are labeled 10 and 20, and then she adds the number on the top of the flipped coin to the number she rolls on a standard die. What is the probability that the sum equals her age in years? Express your answer as a common fraction. | \frac{1}{12} | 1 | 1,877.9375 | 1,877.9375 | -1 | |
Kolya, an excellent student in the 7th-8th grade, found the sum of the digits of all the numbers from 0 to 2012 and added them all together. What number did he get? | 28077 | 0.0625 | 7,988.0625 | 7,680 | 8,008.6 | |
Given that in a mathematics test, $20\%$ of the students scored $60$ points, $25\%$ scored $75$ points, $20\%$ scored $85$ points, $25\%$ scored $95$ points, and the rest scored $100$ points, calculate the difference between the mean and the median score of the students' scores on this test. | 6.5 | 0 | 4,452.5625 | -1 | 4,452.5625 | |
A piece of string fits exactly once around the perimeter of a square whose area is 144. Rounded to the nearest whole number, what is the area of the largest circle that can be formed from the piece of string? | 183 | 0.9375 | 4,208.375 | 3,942.8 | 8,192 | |
Among the following propositions, the correct ones are __________.
(1) The regression line $\hat{y}=\hat{b}x+\hat{a}$ always passes through the center of the sample points $(\bar{x}, \bar{y})$, and at least through one sample point;
(2) After adding the same constant to each data point in a set of data, the variance ... | (2)(6)(7) | 0 | 1,940.0625 | -1 | 1,940.0625 | |
Let $\Omega$ be a circle of radius 8 centered at point $O$, and let $M$ be a point on $\Omega$. Let $S$ be the set of points $P$ such that $P$ is contained within $\Omega$, or such that there exists some rectangle $A B C D$ containing $P$ whose center is on $\Omega$ with $A B=4, B C=5$, and $B C \| O M$. Find the area ... | 164+64 \pi | We wish to consider the union of all rectangles $A B C D$ with $A B=4, B C=5$, and $B C \| O M$, with center $X$ on $\Omega$. Consider translating rectangle $A B C D$ along the radius $X O$ to a rectangle $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ now centered at $O$. It is now clear that that every point inside $A ... | 0 | 7,979.5 | -1 | 7,979.5 |
Given the expression $(xy - \frac{1}{2})^2 + (x - y)^2$ for real numbers $x$ and $y$, find the least possible value. | \frac{1}{4} | 0 | 6,844 | -1 | 6,844 | |
Below is the graph of $y = a \sin bx$ for some constants $a < 0$ and $b > 0.$ Find $a.$
[asy]import TrigMacros;
size(400);
real g(real x)
{
return (-2*sin(x/3));
}
draw(graph(g,-3*pi,3*pi,n=700,join=operator ..),red);
trig_axes(-3*pi,3*pi,-3,3,pi/2,1);
layer();
rm_trig_labels(-5, 5, 2);
label("$1$", (0,1), E);
l... | -2 | 1 | 2,651.3125 | 2,651.3125 | -1 | |
Let $\triangle PQR$ be a right triangle with angle $Q$ as the right angle. A circle with diameter $QR$ intersects side $PR$ at point $S$. If the area of $\triangle PQR$ is $192$ and $PR = 32$, what is the length of $QS$? | 12 | 0.75 | 6,215.9375 | 5,557.25 | 8,192 | |
Evaluate the infinite sum $$\sum_{n=2}^{\infty} \log _{2}\left(\frac{1-\frac{1}{n}}{1-\frac{1}{n+1}}\right)$$ | -1 | Using the identity $\log _{2}\left(\frac{a}{b}\right)=\log _{2} a-\log _{2} b$, the sum becomes $$\sum_{n=2}^{\infty} \log _{2}\left(\frac{n-1}{n}\right)-\sum_{n=2}^{\infty} \log _{2}\left(\frac{n}{n+1}\right)$$ Most of the terms cancel out, except the $\log _{2}\left(\frac{1}{2}\right)$ term from the first sum. Theref... | 0.5 | 6,721.875 | 5,251.75 | 8,192 |
Calculate the probability of a contestant winning the quiz given that they answer at least 3 out of 4 questions correctly, assuming random guesses for each question with 3 possible answer options. | \frac{1}{9} | 0.8125 | 4,136.9375 | 3,655.769231 | 6,222 | |
On a windless day, a polar bear found itself on a small ice floe that broke off from an iceberg, floating in still water. Rescuers from a helicopter hovering above the floe noted that the animal was walking in a circle with a diameter of 9.5 meters. They were surprised when later, in a photograph, they saw the bear's t... | 11400 | 0 | 7,993.3125 | -1 | 7,993.3125 | |
In a survey of 500 students at a different school, it was found that 75 students own cats and 125 students own dogs. What percent of the students own cats? Also, what percent of the students own dogs? | 25\% | 0.5625 | 529.4375 | 574.555556 | 471.428571 | |
Find the smallest positive integer whose cube ends in $888$. | 192 | Let $x^3 = 1000a + 888$. We factor an $8$ out of the right hand side, and we note that $x$ must be of the form $x = 2y$, where $y$ is a positive integer. Then, this becomes $y^3 = 125a + 111$. Taking mod $5$, $25$, and $125$, we get $y^3 \equiv 1\pmod 5$, $y^3 \equiv 11\pmod{25}$, and $y^3 \equiv 111\pmod{125}$.
We ca... | 0.5 | 6,946.875 | 5,914.75 | 7,979 |
Let $A B C$ be a right triangle with $\angle A=90^{\circ}$. Let $D$ be the midpoint of $A B$ and let $E$ be a point on segment $A C$ such that $A D=A E$. Let $B E$ meet $C D$ at $F$. If $\angle B F C=135^{\circ}$, determine $B C / A B$. | \frac{\sqrt{13}}{2} | Let $\alpha=\angle A D C$ and $\beta=\angle A B E$. By exterior angle theorem, $\alpha=\angle B F D+\beta=$ $45^{\circ}+\beta$. Also, note that $\tan \beta=A E / A B=A D / A B=1 / 2$. Thus, $$1=\tan 45^{\circ}=\tan (\alpha-\beta)=\frac{\tan \alpha-\tan \beta}{1+\tan \alpha \tan \beta}=\frac{\tan \alpha-\frac{1}{2}}{1+\... | 0 | 7,632.8125 | -1 | 7,632.8125 |
Given 50 feet of fencing, where 5 feet is used for a gate that does not contribute to the enclosure area, what is the greatest possible number of square feet in the area of a rectangular pen enclosed by the remaining fencing? | 126.5625 | 0 | 4,723.4375 | -1 | 4,723.4375 | |
Given the polar equation of circle $C$ is $\rho=2\cos \theta$, and the parametric equation of line $l$ is $\begin{cases}x= \frac{1}{2}+ \frac{ \sqrt{3}}{2}t \\ y= \frac{1}{2}+ \frac{1}{2}t\end{cases}$ (where $t$ is the parameter), and the polar coordinates of point $A$ are $\left( \frac{ \sqrt{2}}{2}, \frac{\pi}{4}\rig... | \frac{1}{2} | 0.875 | 5,909 | 5,582.857143 | 8,192 | |
If the inequality $\cos \alpha_{1} \cos \alpha_{2} \cdots \cos \alpha_{n} + \sin \alpha_{1} \sin \alpha_{2} \cdots \sin \alpha_{n} \leqslant M$ always holds, then what is the minimum value of $M$? | \sqrt{2} | 0.4375 | 7,726.4375 | 7,127.857143 | 8,192 | |
Given the regression equation $y = 0.849x - 85.712$, where $x$ represents the height in cm and $y$ represents the weight in kg, determine the predicted weight of a female student who is 172 cm tall. | 60.316 | 0.0625 | 1,232.5 | 449 | 1,284.733333 | |
Let $a$, $n$, and $l$ be real numbers, and suppose that the roots of the equation \[x^4 - 10x^3 + ax^2 - nx + l = 0\] are four distinct positive integers. Compute $a + n + l.$ | 109 | 1 | 2,452.0625 | 2,452.0625 | -1 | |
Positive integers $a$, $b$, and $c$ are randomly and independently selected with replacement from the set $\{1, 2, 3, \dots, 2020\}$. What is the probability that $abc + ab + a$ is divisible by $4$?
A) $\frac{1}{4}$
B) $\frac{1}{32}$
C) $\frac{8}{32}$
D) $\frac{9}{32}$
E) $\frac{1}{16}$ | \frac{9}{32} | 0 | 8,192 | -1 | 8,192 | |
The hypotenuse of a right triangle measures $8\sqrt{2}$ inches and one angle is $45^{\circ}$. Calculate both the area and the perimeter of the triangle. | 16 + 8\sqrt{2} | 1 | 1,464.75 | 1,464.75 | -1 | |
Given the function $$f(x)=4\sin(x- \frac {π}{6})\cos x+1$$.
(Ⅰ) Find the smallest positive period of f(x);
(Ⅱ) Find the maximum and minimum values of f(x) in the interval $$\[-\frac {π}{4}, \frac {π}{4}\]$$ . | -2 | 0.6875 | 4,927.5 | 5,222.909091 | 4,277.6 | |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f((x - y)^2) = f(x)^2 - 2xf(y) + y^2\]for all real numbers $x$ and $y.$
Let $n$ be the number of possible values of $f(1),$ and let $s$ be the sum of all possible values of $f(1).$ Find $n \times s.$ | 6 | 0.8125 | 6,038.75 | 5,715.307692 | 7,440.333333 | |
On New Year's Eve, Santa Claus gave the children the following task: by using all nine digits from 1 to 9 exactly once, insert either "+" or "-" between each pair of adjacent digits so that the result yields all possible two-digit prime numbers. How many such numbers can be obtained? | 10 | 0.0625 | 8,169.4375 | 7,831 | 8,192 | |
In the coordinate plane, a square $K$ with vertices at points $(0,0)$ and $(10,10)$ is given. Inside this square, illustrate the set $M$ of points $(x, y)$ whose coordinates satisfy the equation
$$
[x] < [y]
$$
where $[a]$ denotes the integer part of the number $a$ (i.e., the largest integer not exceeding $a$; for e... | 0.45 | 0 | 6,226.125 | -1 | 6,226.125 | |
Given the numbers 2, 3, 4, 3, 1, 6, 3, 7, determine the sum of the mean, median, and mode of these numbers. | 9.625 | 0.625 | 967.125 | 759.6 | 1,313 | |
The number $\sqrt{104\sqrt{6}+468\sqrt{10}+144\sqrt{15}+2006}$ can be written as $a\sqrt{2}+b\sqrt{3}+c\sqrt{5},$ where $a, b,$ and $c$ are positive integers. Find $abc$. | 936 | We begin by equating the two expressions:
\[a\sqrt{2}+b\sqrt{3}+c\sqrt{5} = \sqrt{104\sqrt{6}+468\sqrt{10}+144\sqrt{15}+2006}\]
Squaring both sides yields:
\[2ab\sqrt{6} + 2ac\sqrt{10} + 2bc\sqrt{15} + 2a^2 + 3b^2 + 5c^2 = 104\sqrt{6}+468\sqrt{10}+144\sqrt{15}+2006\]
Since $a$, $b$, and $c$ are integers, we can match... | 1 | 2,637.8125 | 2,637.8125 | -1 |
Find the number of ordered triples of integers $(a, b, c)$ with $1 \leq a, b, c \leq 100$ and $a^{2} b+b^{2} c+c^{2} a=a b^{2}+b c^{2}+c a^{2}$ | 29800 | This factors as $(a-b)(b-c)(c-a)=0$. By the inclusion-exclusion principle, we get $3 \cdot 100^{2}-3 \cdot 100+100=29800$. | 0.625 | 6,558.875 | 5,643.8 | 8,084 |
Suppose that $\sec x+\tan x=\frac{22}7$ and that $\csc x+\cot x=\frac mn,$ where $\frac mn$ is in lowest terms. Find $m+n.$
| 44 | 0.9375 | 4,283.5625 | 4,023 | 8,192 | |
There exist real numbers $t$ and $s$ such that
\[\begin{pmatrix} 2 \\ 0 \end{pmatrix} + t \begin{pmatrix} 7 \\ -5 \end{pmatrix} = \begin{pmatrix} 1 \\ -1 \end{pmatrix} + s \begin{pmatrix} -2 \\ 3 \end{pmatrix}.\]Enter the ordered pair $(t,s).$ | \left( -\frac{5}{11}, \frac{12}{11} \right) | 1 | 2,810.5625 | 2,810.5625 | -1 | |
For how many values of $n$ in the set $\{101, 102, 103, ..., 200\}$ is the tens digit of $n^2$ even? | 60 | 0 | 8,192 | -1 | 8,192 | |
Around the outside of a $6$ by $6$ square, construct four semicircles with the four sides of the square as their diameters. Another square, $ABCD$, has its sides parallel to the corresponding sides of the original square, and each side of $ABCD$ is tangent to one of the semicircles. Calculate the area of square $ABCD$.... | 144 | 0 | 6,446.875 | -1 | 6,446.875 | |
Given that square PQRS has dimensions 3 × 3, points T and U are located on side QR such that QT = TU = UR = 1, and points V and W are positioned on side RS such that RV = VW = WS = 1, find the ratio of the shaded area to the unshaded area. | 2:1 | 0 | 7,944.625 | -1 | 7,944.625 | |
There is a magical tree with 58 fruits. On the first day, 1 fruit falls from the tree. From the second day onwards, the number of fruits falling each day increases by 1 compared to the previous day. However, if on any given day the number of fruits on the tree is less than the number of fruits that should fall on that ... | 12 | 0.0625 | 6,624.25 | 7,435 | 6,570.2 | |
Let $P(x)$ be a nonzero polynomial such that $(x-1)P(x+1)=(x+2)P(x)$ for every real $x$, and $\left(P(2)\right)^2 = P(3)$. Then $P(\tfrac72)=\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 109 | Substituting $x=2$ into the given equation, we find that $P(3)=4P(2)=P(2)^2$. Therefore, either $P(2)=0$ or $P(2)=4$. Now for integers $n\ge 2$, we know that \[P(n+1)=\frac{n+2}{n-1}P(n).\] Applying this repeatedly, we find that \[P(n+1)=\frac{(n+2)!/3!}{(n-1)!}P(2).\] If $P(2)=0$, this shows that $P(x)$ has infinitely... | 0.5 | 6,743.875 | 5,295.75 | 8,192 |
The decimal number corresponding to the binary number $111011001001_2$ is to be found. | 3785 | 0.8125 | 6,851.8125 | 6,542.538462 | 8,192 | |
If $3n=9+9+9$, what is the value of $n$? | 9 | Since $3n=9+9+9=3 imes 9$, then $n=9$. Alternatively, we could note that $9+9+9=27$ and so $3n=27$ which gives $n= rac{27}{3}=9$. | 1 | 358.1875 | 358.1875 | -1 |
Let \( g(x) = 3x^4 + 2x^3 - x^2 - 4x + s \). Find the value of \( s \) such that \( g(-1) = 0 \). | -4 | 1 | 1,723.125 | 1,723.125 | -1 | |
The sequence $a_1,$ $a_2,$ $a_3,$ $\dots$ satisfies $a_1 = 19,$ $a_9 = 99,$ and for all $n \ge 3,$ $a_n$ is the arithmetic mean of the first $n - 1$ terms. Find $a_2.$ | 179 | 0.625 | 5,356.5625 | 4,213.9 | 7,261 | |
If $\tan \alpha= \sqrt {2}$, then $2\sin ^{2}\alpha-\sin \alpha\cos \alpha+\cos ^{2}\alpha=$ \_\_\_\_\_\_ . | \frac {5- \sqrt {2}}{3} | 0 | 5,336.5625 | -1 | 5,336.5625 | |
Two cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a spade and the second card is a king? | \frac{1}{52} | 0.8125 | 6,170.3125 | 5,703.769231 | 8,192 | |
In a box, there are 3 red, 4 gold, and 5 silver stars. Stars are randomly drawn one by one from the box and placed on a Christmas tree. What is the probability that a red star is placed on the top of the tree, no more red stars are on the tree, and there are exactly 3 gold stars on the tree, if a total of 6 stars are ... | 5/231 | 0.25 | 7,202.5625 | 6,545.75 | 7,421.5 | |
Given that $F_{1}$ and $F_{2}$ are two foci of ellipse $C$, $P$ is a point on $C$, and $\angle F_{1}PF_{2}=60^{\circ}$, $|PF_{1}|=3|PF_{2}|$, calculate the eccentricity of $C$. | \frac{\sqrt{7}}{4} | 0 | 4,209.3125 | -1 | 4,209.3125 | |
Find the maximum number of elements in a set $S$ that satisfies the following conditions:
(1) Every element in $S$ is a positive integer not exceeding 100.
(2) For any two distinct elements $a$ and $b$ in $S$, there exists another element $c$ in $S$ such that the greatest common divisor (gcd) of $a + b$ and $c$ is 1.
(... | 50 | 0 | 8,192 | -1 | 8,192 | |
For each pair of real numbers $a \ne b$, define the operation $\star$ as \[
(a \star b) = \frac{a + b}{a - b}.
\]What is the value of $((1 \star 2) \star 4)$? | -\frac{1}{7} | 0.9375 | 3,145.375 | 2,808.933333 | 8,192 | |
On an $8 \times 8$ chessboard, 6 black rooks and $k$ white rooks are placed on different cells so that each rook only attacks rooks of the opposite color. Compute the maximum possible value of $k$. | 14 | The answer is $k=14$. For a valid construction, place the black rooks on cells $(a, a)$ for $2 \leq a \leq 7$ and the white rooks on cells $(a, a+1)$ and $(a+1, a)$ for $1 \leq a \leq 7$. Now, we prove the optimality. As rooks can only attack opposite color rooks, the color of rooks in each row is alternating. The diff... | 0 | 8,078.25 | -1 | 8,078.25 |
Let $A,B$ be the points on the coordinate plane with coordinates $(t-4,-1)$ and $(-2,t+3)$, respectively. The square of the distance between the midpoint of $\overline{AB}$ and an endpoint of $\overline{AB}$ is equal to $t^2/2$. What is the value of $t$? | -5 | 0.9375 | 3,725 | 3,427.2 | 8,192 | |
Let $x$ be a real number, $x > 1.$ Compute
\[\sum_{n = 0}^\infty \frac{1}{x^{2^n} - x^{-2^n}}.\] | \frac{1}{x - 1} | 0.375 | 6,441.0625 | 4,148.333333 | 7,816.7 | |
If $6a^2 + 5a + 4 = 3,$ then what is the smallest possible value of $2a + 1$? | 0 | 1 | 3,459.75 | 3,459.75 | -1 | |
A train is made up of 18 carriages. There are 700 passengers traveling on the train. In any block of five adjacent carriages, there are 199 passengers in total. How many passengers in total are in the middle two carriages of the train? | 96 | 0.625 | 5,701.1875 | 4,206.7 | 8,192 | |
For positive integers $N$ and $k$ define $N$ to be $k$-nice if there exists a positive integer $a$ such that $a^k$ has exactly $N$ positive divisors. Determine the quantity of positive integers smaller than $1500$ that are neither $9$-nice nor $10$-nice. | 1199 | 0.0625 | 7,973.5625 | 7,781 | 7,986.4 | |
In a certain hyperbola, the center is at $(-2,0),$ one focus is at $(-2 + \sqrt{34},0),$ and one vertex is at $(-5,0).$ The equation of this hyperbola can be written as
\[\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1.\]Find $h + k + a + b.$ | 6 | 1 | 1,413.75 | 1,413.75 | -1 | |
A nine-digit number is formed by repeating a three-digit number three times. For example, 123,123,123 or 456,456,456 are numbers of this form. What is the greatest common divisor of all nine-digit numbers of this form? | 1001001 | 0.0625 | 8,192 | 8,192 | 8,192 | |
Given that 2 students exercised 0 days, 4 students exercised 1 day, 5 students exercised 2 days, 3 students exercised 4 days, 7 students exercised 5 days, and 2 students exercised 6 days, calculate the average number of days exercised last week by the students in Ms. Brown's class. | 3.17 | 0.3125 | 558.1875 | 497.6 | 585.727273 | |
How many of the first $500$ positive integers can be expressed in the form
\[\lfloor 3x \rfloor + \lfloor 6x \rfloor + \lfloor 9x \rfloor + \lfloor 12x \rfloor\]
where \( x \) is a real number? | 300 | 0 | 8,192 | -1 | 8,192 | |
We define \( a @ b = a \times (a + 1) \times \ldots \times (a + b - 1) \). Given \( x @ y @ 2 = 420 \), then \( y @ x = \) ? | 20 | 0.0625 | 5,571.9375 | 6,484 | 5,511.133333 | |
One day, School A bought 56 kilograms of fruit candy at 8.06 yuan per kilogram. A few days later, School B also needed to buy the same 56 kilograms of fruit candy, but it happened that there was a promotional event, and the price of fruit candy was reduced by 0.56 yuan per kilogram. Additionally, they received 5% extra... | 51.36 | 0 | 706.4375 | -1 | 706.4375 | |
The table shows the vertical drops of six roller coasters in Fibonacci Fun Park:
\begin{tabular}{|l|c|}
\hline
Speed Demon & 150 feet \\
\hline
Looper & 230 feet \\
\hline
Dare Devil & 160 feet \\
\hline
Giant Drop & 190 feet \\
\hline
Sky Scream & 210 feet \\
\hline
Hell Spiral & 180 feet \\
\hline
\end{tabular}
... | 1.67 | 0.4375 | 903.4375 | 675 | 1,081.111111 | |
Three faucets fill a 100-gallon tub in 6 minutes. How long, in seconds, does it take six faucets to fill a 25-gallon tub? Assume that all faucets dispense water at the same rate. | 45 | 0.75 | 5,082.4375 | 4,045.916667 | 8,192 | |
Find
\[\cos \left( 6 \arccos \frac{1}{3} \right).\] | \frac{329}{729} | 0.5625 | 7,191.25 | 6,412.888889 | 8,192 | |
Find the minimum value of the expression \(\left\lfloor \frac{8(a+b)}{c} \right\rfloor + \left\lfloor \frac{8(a+c)}{b} \right\rfloor + \left\lfloor \frac{8(b+c)}{a} \right\rfloor\), where \(a\), \(b\), and \(c\) are arbitrary natural numbers. | 46 | 0 | 8,032.25 | -1 | 8,032.25 | |
Calculate
$$
\frac{1 \times 2 \times 4+2 \times 4 \times 8+3 \times 6 \times 12+4 \times 8 \times 16}{1 \times 3 \times 9+2 \times 6 \times 18+3 \times 9 \times 27+4 \times 12 \times 36}
$$
Only a numerical answer is expected here. The answer must be given in the form of an irreducible fraction (i.e., in the form $\f... | 8/27 | 1 | 2,931.75 | 2,931.75 | -1 | |
Given the parametric equation of line $l$ is $\begin{cases} & x=1+3t \\ & y=2-4t \end{cases}$ (where $t$ is the parameter), calculate the cosine of the inclination angle of line $l$. | -\frac{3}{5} | 0.625 | 4,408.1875 | 4,001.3 | 5,086.333333 | |
Starting with some gold coins and some empty treasure chests, I tried to put $9$ gold coins in each treasure chest, but that left $2$ treasure chests empty. So instead I put $6$ gold coins in each treasure chest, but then I had $3$ gold coins left over. How many gold coins did I have? | 45 | Let $n$ be the number of gold coins and $c$ be the number of treasure chests.
1. **Analyzing the first condition:**
- If we try to put $9$ gold coins in each chest, but $2$ chests remain empty, then the number of chests that actually contain coins is $c - 2$. The total number of coins is then $9(c - 2)$.
- This ... | 1 | 1,544.4375 | 1,544.4375 | -1 |
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