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 |
|---|---|---|---|---|---|---|
Compute $\binom{12}{9}$ and then find the factorial of the result. | 220 | 0 | 2,667.875 | -1 | 2,667.875 | |
The sum of the first fifty positive odd integers subtracted from the sum of the first fifty positive even integers, each decreased by 3, calculate the result. | -100 | 0.1875 | 3,391.75 | 3,047.333333 | 3,471.230769 | |
What is the smallest integer $n$ , greater than one, for which the root-mean-square of the first $n$ positive integers is an integer?
$\mathbf{Note.}$ The root-mean-square of $n$ numbers $a_1, a_2, \cdots, a_n$ is defined to be \[\left[\frac{a_1^2 + a_2^2 + \cdots + a_n^2}n\right]^{1/2}\] | \(\boxed{337}\) | Let's first obtain an algebraic expression for the root mean square of the first $n$ integers, which we denote $I_n$ . By repeatedly using the identity $(x+1)^3 = x^3 + 3x^2 + 3x + 1$ , we can write \[1^3 + 3\cdot 1^2 + 3 \cdot 1 + 1 = 2^3,\] \[1^3 + 3 \cdot(1^2 + 2^2) + 3 \cdot (1 + 2) + 1 + 1 = 3^3,\] and \[1^3 + 3... | 0 | 8,072.4375 | -1 | 8,072.4375 |
Find the remainder when $x^4 + 2$ is divided by $(x - 2)^2.$ | 32x - 46 | 1 | 2,776.625 | 2,776.625 | -1 | |
Mom decides to take Xiaohua on a road trip to 10 cities during the vacation. After checking the map, Xiaohua is surprised to find that for any three cities among these 10, either all three pairs of cities are connected by highways, or exactly two pairs of cities among the three are not connected by highways. What is th... | 40 | 0.1875 | 6,693.25 | 6,639.666667 | 6,705.615385 | |
Consider triangle $A B C$ with side lengths $A B=4, B C=7$, and $A C=8$. Let $M$ be the midpoint of segment $A B$, and let $N$ be the point on the interior of segment $A C$ that also lies on the circumcircle of triangle $M B C$. Compute $B N$. | \frac{\sqrt{210}}{4} | Let $\angle B A C=\theta$. Then, $\cos \theta=\frac{4^{2}+8^{2}-7^{2}}{2 \cdot 4 \cdot 8}$. Since $A M=\frac{4}{2}=2$, and power of a point gives $A M \cdot A B=A N \cdot A C$, we have $A N=\frac{2 \cdot 4}{8}=1$, so $N C=8-1=7$. Law of cosines on triangle $B A N$ gives $$B N^{2}=4^{2}+1^{2}-2 \cdot 4 \cdot 1 \cdot \fr... | 0 | 7,154.8125 | -1 | 7,154.8125 |
The domain of the equation $y = \frac{3x}{2x^2 - 6x + 4}$ is the set of all real numbers with the exception of the values $x = A$ and $x = B$. What is $A + B$? | 3 | 1 | 1,394.0625 | 1,394.0625 | -1 | |
Three positive integers are each greater than $1$, have a product of $ 27000 $, and are pairwise relatively prime. What is their sum? | 160 | 1 | 3,535.75 | 3,535.75 | -1 | |
Determine the constants $\alpha$ and $\beta$ such that $\frac{x-\alpha}{x+\beta} = \frac{x^2 - 64x + 975}{x^2 + 99x - 2200}$. What is $\alpha+\beta$? | 138 | 0 | 7,461.9375 | -1 | 7,461.9375 | |
The line \(y = -\frac{1}{2}x + 8\) crosses the \(x\)-axis at \(P\) and the \(y\)-axis at \(Q\). Point \(T(r, s)\) is on line segment \(PQ\). If the area of \(\triangle POQ\) is twice the area of \(\triangle TOP\), then what is the value of \(r+s\)? | 12 | 1 | 2,695.8125 | 2,695.8125 | -1 | |
Express $0.6\overline{03}$ as a common fraction. | \frac{104}{165} | 0 | 6,420.1875 | -1 | 6,420.1875 | |
Positive integers $a$, $b$, $c$, and $d$ satisfy $a > b > c > d$, $a + b + c + d = 2010$, and $a^2 - b^2 + c^2 - d^2 = 2010$. Find the number of possible values of $a$. | 501 | Since $a+b$ must be greater than $1005$, it follows that the only possible value for $a-b$ is $1$ (otherwise the quantity $a^2 - b^2$ would be greater than $2010$). Therefore the only possible ordered pairs for $(a,b)$ are $(504, 503)$, $(505, 504)$, ... , $(1004, 1003)$, so $a$ has $\boxed{501}$ possible values. | 0.3125 | 7,478.375 | 5,908.4 | 8,192 |
The number \( A \) in decimal notation has the form \( A = \overline{7a631b} \), where \( a \) and \( b \) are non-zero digits. The number \( B \) is obtained by summing all the six-digit, distinct numbers derived from \( A \) by cyclic permutations of its digits (the first digit moves to the second place, the second t... | 796317 | 0.375 | 6,972.5 | 5,650.166667 | 7,765.9 | |
A circular piece of metal with a radius of 10 units has a maximum square cut out from it. Then, the largest possible circle is cut from this square. Calculate the total amount of metal wasted.
A) $50\pi - 200$
B) $200 - 50\pi$
C) $100\pi$
D) $50\pi$
E) None of these | 50\pi - 200 | 0 | 4,206.3125 | -1 | 4,206.3125 | |
A point is randomly dropped on the interval $[5 ; 7]$ and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}-3 k-4\right) x^{2}+(3 k-7) x+2=0$ satisfy the condition $x_{1} \leq 2 x_{2}$. | 1/3 | 0.1875 | 8,092.3125 | 7,660.333333 | 8,192 | |
Call a positive integer $n$ $k$-pretty if $n$ has exactly $k$ positive divisors and $n$ is divisible by $k$. For example, $18$ is $6$-pretty. Let $S$ be the sum of positive integers less than $2019$ that are $20$-pretty. Find $\tfrac{S}{20}$. | 472 | For $n$ to have exactly $20$ positive divisors, $n$ can only take on certain prime factorization forms: namely, $p^{19}, p^9q, p^4q^3, p^4qr$. No number that is a multiple of $20$ can be expressed in the first form, and the only integer divisible by $20$ that has the second form is $2^{9}5$, which is greater than $2019... | 0.3125 | 7,935.6875 | 7,663.2 | 8,059.545455 |
What is the 308th digit to the right of the decimal point when $\frac{12}{37}$ is expressed as a decimal? | 2 | 0.9375 | 2,069.0625 | 2,086.733333 | 1,804 | |
If the functions \( f(x) \) and \( g(x) \) are defined for all real numbers, and they satisfy the equation \( f(x-y) = f(x) g(y) - g(x) f(y) \), with \( f(-2) = f(1) \neq 0 \), then find \( g(1) + g(-1) \). | -1 | 0.6875 | 6,273.5 | 5,401.454545 | 8,192 | |
A cowboy is 4 miles south of a stream which flows due east. He is also 8 miles west and 7 miles north of his cabin. He wishes to water his horse at the stream and return home. The shortest distance (in miles) he can travel and accomplish this is | 17 | 1. **Identify the positions**: Let's denote the cowboy's initial position as point $C$, the stream as a horizontal line to the north of $C$, and the cabin as point $B$. According to the problem, $C$ is 4 miles south of the stream and 8 miles west and 7 miles north of $B$.
2. **Calculate the coordinates**:
- Assum... | 0.625 | 4,883.75 | 3,351.1 | 7,438.166667 |
A regular $n$-gon has $n$ diagonals, its perimeter is $p$, and the sum of the lengths of all the diagonals is $q$. What is $\frac{p}{q} + \frac{q}{p}$? | \sqrt{5} | 0.625 | 5,629.0625 | 4,091.3 | 8,192 | |
Let $ABCD$ be a cyclic quadrilateral, and $E$ be the intersection of its diagonals. If $m(\widehat{ADB}) = 22.5^\circ$ , $|BD|=6$ , and $|AD|\cdot|CE|=|DC|\cdot|AE|$ , find the area of the quadrilateral $ABCD$ . | 9\sqrt{2} | 0.125 | 7,975.9375 | 6,463.5 | 8,192 | |
Shelby drives her scooter at a speed of $30$ miles per hour if it is not raining, and $20$ miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of $16$ miles in $40$ minutes. How many minutes did she drive in the rain? | 24 | 1. **Define Variables:**
Let $x$ be the time (in minutes) Shelby drove in the rain. Therefore, the time she drove when it was not raining is $40 - x$ minutes.
2. **Convert Speeds to Miles per Minute:**
- Shelby's speed in non-rainy conditions is $30$ miles per hour. Converting this to miles per minute:
\[
... | 1 | 2,064.75 | 2,064.75 | -1 |
What is the $111^{\text{st}}$ smallest positive integer which does not have $3$ and $4$ in its base- $5$ representation? | 755 | 0.0625 | 7,998.9375 | 5,103 | 8,192 | |
Determine the smallest odd prime factor of $2021^{10} + 1$. | 61 | 0.4375 | 7,548.6875 | 6,721.571429 | 8,192 | |
Four rectangular strips each measuring $4$ by $16$ inches are laid out with two vertical strips crossing two horizontal strips forming a single polygon which looks like a tic-tack-toe pattern. What is the perimeter of this polygon?
[asy]
size(100);
draw((1,0)--(2,0)--(2,1)--(3,1)--(3,0)--(4,0)--(4,1)--(5,1)--(5,2)... | 80 | 0 | 8,074.5 | -1 | 8,074.5 | |
Compute $\tan\left(\frac{\pi}{9}\right)\tan\left(\frac{2\pi}{9}\right)\tan\left(\frac{4\pi}{9}\right)$. | \sqrt{3} | 0.3125 | 7,376.4375 | 5,582.2 | 8,192 | |
A cone is inscribed in a regular quadrilateral pyramid. Find the ratio of the total surface area of the cone to the lateral surface area of the cone, given that the side length of the pyramid's base is 4 and the angle between the pyramid's height and the plane of its lateral face is $30^{\circ}$. | \frac{3 + \sqrt{3}}{3} | 0 | 6,149 | -1 | 6,149 | |
The English college entrance examination consists of two parts: listening and speaking test with a full score of $50$ points, and the English written test with a full score of $100$ points. The English listening and speaking test is conducted twice. If a student takes both tests, the higher score of the two tests will ... | \frac{7}{10} | 0 | 6,476.8125 | -1 | 6,476.8125 | |
Two numbers \( x \) and \( y \) satisfy the equation \( 280x^{2} - 61xy + 3y^{2} - 13 = 0 \) and are respectively the fourth and ninth terms of a decreasing arithmetic progression consisting of integers. Find the common difference of this progression. | -5 | 0.125 | 8,149.5 | 7,852 | 8,192 | |
What is the least four-digit whole number that is both a perfect square and a perfect cube? | 4096 | 1 | 2,379.75 | 2,379.75 | -1 | |
Suppose $a$, $b$, $c$ and $d$ are integers satisfying: $a-b+c=5$, $b-c+d=6$, $c-d+a=3$, and $d-a+b=2$. What is the value of $a+b+c+d$? | 16 | 1 | 3,124.9375 | 3,124.9375 | -1 | |
In the diagram, rectangle $PQRS$ is divided into three identical squares. If $PQRS$ has perimeter 120 cm, what is its area, in square centimeters? [asy]
size(4cm);
pair p = (0, 1); pair q = (3, 1); pair r = (3, 0); pair s = (0, 0);
draw(p--q--r--s--cycle);
draw(shift(1) * (p--s)); draw(shift(2) * (p--s));
label("$... | 675 | 1 | 1,710.9375 | 1,710.9375 | -1 | |
A self-employed individual plans to distribute two types of products, A and B. According to a survey, when the investment amount is $x$ (where $x \geq 0$) in ten thousand yuan, the profits obtained from distributing products A and B are $f(x)$ and $g(x)$ in ten thousand yuan, respectively, where $f(x) = a(x - 1) + 2$ (... | 12.6 | 0.75 | 4,546.25 | 3,541.583333 | 7,560.25 | |
Given $f(x)=\sin \omega x \cos \omega x - \sqrt{3} \cos^2 \omega x$, where $(\omega > 0)$, ${x}_{1}$ and ${x}_{2}$ are the two zeros of the function $y=f(x)+\frac{2+\sqrt{3}}{2}$, and $|x_{1}-x_{2}|_{\min }=\pi$. When $x\in[0,\frac{7\pi}{12}]$, the sum of the minimum and maximum values of $f(x)$ is ______. | \frac{2-3\sqrt{3}}{2} | 0 | 7,590 | -1 | 7,590 | |
The school plans to schedule six leaders to be on duty from May 1st to May 3rd, with each leader on duty for one day and two leaders scheduled each day. Given that Leader A cannot be on duty on May 2nd and Leader B cannot be on duty on May 3rd, determine the number of different ways to arrange the duty schedule. | 42 | 0 | 8,192 | -1 | 8,192 | |
Find
\[
\cos \left( 4 \arccos \frac{2}{5} \right).
\] | -\frac{47}{625} | 0.625 | 6,057.5625 | 4,776.9 | 8,192 | |
What is the largest multiple of $9$ whose negation is greater than $-100$? | 99 | 1 | 2,521.5 | 2,521.5 | -1 | |
Given the sequence $\{a_n\}$, $a_1=1$, $a_{n+1} > a_n$, $(a_n+a_{n+1}-1)^2=4a_na_{n+1}$ $(n\in\mathbb{N}^*)$.
$(1)$ Find the general formula for the sequence $\{a_n\}$;
$(2)$ Let $b_n=\frac{1}{\sqrt[4]{a_n}}$, $T_n=b_1+b_2+\ldots+b_n$, estimate the integer part of $T_{2017}$.
(Reference data: $1.41 < \sqrt{2} < 1.42... | 88 | 0.1875 | 7,966.5 | 7,412.666667 | 8,094.307692 | |
Find the smallest positive number $\lambda$ such that for any triangle with side lengths $a, b, c$, if $a \geqslant \frac{b+c}{3}$, then the following inequality holds:
$$
ac + bc - c^2 \leqslant \lambda \left( a^2 + b^2 + 3c^2 + 2ab - 4bc \right).
$$ | \frac{2\sqrt{2} + 1}{7} | 0 | 8,157.375 | -1 | 8,157.375 | |
Seven distinct points are identified on the circumference of a circle. How many different triangles can be formed if each vertex must be one of these 7 points? | 35 | 1 | 1,793.875 | 1,793.875 | -1 | |
Find the smallest natural number $n$ such that $\sin n^{\circ} = \sin (2016n^{\circ})$. | 72 | 0.1875 | 7,846.6875 | 6,350.333333 | 8,192 | |
What is the greatest divisor of 360 that is smaller than 60 and also a factor of 90? | 30 | 0 | 3,200.625 | -1 | 3,200.625 | |
Find the value of \( k \) such that, for all real numbers \( a, b, \) and \( c \),
$$
(a+b)(b+c)(c+a) = (a+b+c)(ab + bc + ca) + k \cdot abc
$$ | -2 | 0 | 3,821.6875 | -1 | 3,821.6875 | |
Bag $A$ contains two 10 yuan bills and three 1 yuan bills, and Bag $B$ contains four 5 yuan bills and three 1 yuan bills. Two bills are randomly drawn from each bag. What is the probability that the total value of the bills remaining in Bag $A$ is greater than the total value of the bills remaining in Bag $B$? | 9/35 | 0.3125 | 7,189.3125 | 4,983.4 | 8,192 | |
Simplify $15\cdot\frac{16}{9}\cdot\frac{-45}{32}$. | -\frac{25}{6} | 0 | 4,754.6875 | -1 | 4,754.6875 | |
Cagney can frost a cupcake every 15 seconds, Lacey can frost a cupcake every 25 seconds, and Hardy can frost a cupcake every 50 seconds. Calculate the number of cupcakes that Cagney, Lacey, and Hardy can frost together in 6 minutes. | 45 | 0.9375 | 3,760.875 | 3,465.466667 | 8,192 | |
Below is a portion of the graph of a quadratic function, $y=p(x)=dx^2 + ex + f$:
The value of $p(12)$ is an integer. The graph's axis of symmetry is $x = 10.5$, and the graph passes through the point $(3, -5)$. Based on this, what is the value of $p(12)$? | -5 | 0.0625 | 7,928.6875 | 8,192 | 7,911.133333 | |
There are 99 positive integers, and their sum is 101101. Find the greatest possible value of the greatest common divisor of these 99 positive integers. | 101 | 0 | 7,692.0625 | -1 | 7,692.0625 | |
Given that $\alpha$ is an angle in the third quadrant, and $f(\alpha)=\frac{{\sin({\frac{\pi}{2}-\alpha})\cos(-\alpha)\tan(\pi+\alpha)}}{{\cos(\pi-\alpha)}}$.
$(1)$ Simplify $f(\alpha)$;
$(2)$ If $f(\alpha)=\frac{{2\sqrt{5}}}{5}$, find the value of $\cos \alpha$. | -\frac{\sqrt{5}}{5} | 0 | 2,985.4375 | -1 | 2,985.4375 | |
In the diagram, $\triangle ABE$, $\triangle BCE$, and $\triangle CDE$ are right-angled with $\angle AEB = 30^\circ, \angle BEC = 45^\circ$, and $\angle CED = 45^\circ$, and $AE=30$. Find the length of $CE$. | 15 | 0 | 7,870.8125 | -1 | 7,870.8125 | |
If $(w+13)^2=(3w+7)(2w+4)$, find $w^2$. Express your answer as a decimal. | 28.2 | 1 | 3,610.6875 | 3,610.6875 | -1 | |
Add $26_7 + 245_7.$ Express your answer in base 7. | 304_7 | 0.9375 | 2,506.8125 | 2,127.8 | 8,192 | |
Calculate the definite integral:
$$
\int_{\frac{\pi}{2}}^{2 \operatorname{arctg} 2} \frac{d x}{\sin ^{2} x(1+\cos x)}
$$ | 29/24 | 0.0625 | 6,700.375 | 7,031 | 6,678.333333 | |
For real numbers $a,$ $b,$ and $c,$ and a real scalar $\lambda,$ consider the matrix
\[\begin{pmatrix} a + \lambda & b & c \\ b & c + \lambda & a \\ c & a & b + \lambda \end{pmatrix}.\] Determine all possible values of \[\frac{a}{b + c} + \frac{b}{a + c} + \frac{c}{a + b}\] assuming the matrix is not invertible. | \frac{3}{2} | 0 | 8,192 | -1 | 8,192 | |
Add $537_{8} + 246_{8}$. Express your answer in base $8$, and then convert your answer to base $16$, using alphabetic representation for numbers $10$ to $15$ (e.g., $A$ for $10$, $B$ for $11$, etc.). | 205_{16} | 0.1875 | 5,403 | 5,679.666667 | 5,339.153846 | |
Given that $40\%$ of the birds were geese, $20\%$ were swans, $15\%$ were herons, and $25\%$ were ducks, calculate the percentage of the birds that were not ducks and were geese. | 53.33\% | 0 | 3,208 | -1 | 3,208 | |
A rectangular room measures 12-feet by 6-feet. How many square yards of carpet are needed to cover the floor of the room? | 8 | 1 | 1,491.5625 | 1,491.5625 | -1 | |
Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $2500$ are not factorial tails? | 500 | 0 | 8,109.375 | -1 | 8,109.375 | |
A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is preparing to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered. | 8.8\% | 0 | 4,186.125 | -1 | 4,186.125 | |
Let $m=n^{4}+x$, where $n \in \mathbf{N}$ and $x$ is a two-digit positive integer. Which value of $x$ ensures that $m$ is always a composite number? | 64 | 0 | 8,041.9375 | -1 | 8,041.9375 | |
Charlyn walks completely around the boundary of a square whose sides are each 5 km long. From any point on her path she can see exactly 1 km horizontally in all directions. What is the area of the region consisting of all points Charlyn can see during her walk, expressed in square kilometers and rounded to the nearest ... | 39 | 0 | 6,830.375 | -1 | 6,830.375 | |
Two mathematicians, Kelly and Jason, play a cooperative game. The computer selects some secret positive integer $n<60$ (both Kelly and Jason know that $n<60$, but that they don't know what the value of $n$ is). The computer tells Kelly the unit digit of $n$, and it tells Jason the number of divisors of $n$. Then, Kelly... | 10 | The only way in which Kelly can know that $n$ is divisible by at least two different primes is if she is given 0 as the unit digit of $n$, since if she received anything else, then there is some number with that unit digit and not divisible by two primes (i.e., $1,2,3,4,5,16,7,8,9$ ). Then, after Kelly says the first l... | 0 | 8,192 | -1 | 8,192 |
What is the value of $x^2+y^2-z^2+2xy$ when $x=-3$, $y=5$, and $z=-4$? | -12 | 1 | 3,083.875 | 3,083.875 | -1 | |
Each segment with endpoints at the vertices of a regular 100-sided polygon is colored red if there is an even number of vertices between the endpoints, and blue otherwise (in particular, all sides of the 100-sided polygon are red). Numbers are placed at the vertices such that the sum of their squares equals 1, and the ... | 1/2 | 0 | 7,978.3125 | -1 | 7,978.3125 | |
The constant term in the expansion of $(x^2-2)\left(x-\frac{2}{\sqrt{x}}\right)^{6}$ is ______. | -480 | 0.6875 | 6,220.0625 | 5,748.818182 | 7,256.8 | |
Hooligan Vasily tore out an entire chapter from a book, with the first page numbered 231, and the number of the last page consisted of the same digits. How many sheets did Vasily tear out of the book? | 41 | 0.3125 | 6,776 | 4,691.2 | 7,723.636364 | |
Given points $P(-2,7)$ and $Q(4,y)$ in a coordinate plane, for what value of $y$ is the slope of the line through $P$ and $Q$ equal to $\frac{-3}{2}$? | -2 | 1 | 1,301 | 1,301 | -1 | |
A three-wheeled vehicle travels 100 km. Two spare wheels are available. Each of the five wheels is used for the same distance during the trip. For how many kilometers is each wheel used? | 60 | 0.6875 | 4,422.1875 | 3,427.181818 | 6,611.2 | |
Faces $ABC$ and $BCD$ of tetrahedron $ABCD$ meet at an angle of $30^\circ$. The area of face $ABC$ is $120$, the area of face $BCD$ is $80$, and $BC=10$. Find the volume of the tetrahedron.
| 320 | 0.5 | 7,253.3125 | 6,314.625 | 8,192 | |
Real numbers between 0 and 1, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is 0 if the second flip is heads, and 1 if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen unifor... | \frac{7}{16} | To solve the problem, we need to calculate the probability that the absolute difference between two independently chosen numbers $x$ and $y$ is greater than $\frac{1}{2}$. We can express this as $P(|x-y| > \frac{1}{2}) = 2P(x-y > \frac{1}{2})$ due to symmetry (the situation for $x-y > \frac{1}{2}$ is symmetric to $y-x ... | 0.0625 | 8,090.6875 | 8,192 | 8,083.933333 |
Given the function $f(x)=\sqrt{3}\sin \omega x+\cos \omega x (\omega > 0)$, where the x-coordinates of the points where the graph of $f(x)$ intersects the x-axis form an arithmetic sequence with a common difference of $\frac{\pi}{2}$, determine the probability of the event "$g(x) \geqslant \sqrt{3}$" occurring, where $... | \frac{1}{6} | 0.875 | 5,127.125 | 4,689.285714 | 8,192 | |
Given the function $f(x)=2\cos^2\frac{x}{2}+\sin x-1$. Find:
- $(Ⅰ)$ The minimum positive period, monotonic decreasing interval, and symmetry center of $f(x)$.
- $(Ⅱ)$ When $x\in \left[-\pi ,0\right]$, find the minimum value of $f(x)$ and the corresponding value of $x$. | -\frac{3\pi}{4} | 0.9375 | 5,685.125 | 5,518 | 8,192 | |
The diagram shows a karting track circuit. The start and finish are at point $A$, and the kart driver can return to point $A$ and continue on the circuit as many times as desired.
The time taken to travel from $A$ to $B$ or from $B$ to $A$ is one minute. The time taken to travel around the loop is also one minute. The... | 34 | 0 | 7,886.9375 | -1 | 7,886.9375 | |
Consider a square ABCD with side length 8 units. On side AB, semicircles are constructed inside the square both with diameter AB. Inside the square and tangent to AB at its midpoint, another quarter circle with its center at the midpoint of AB is also constructed pointing inward. Calculate the ratio of the shaded area ... | \frac{3}{4} | 0.25 | 6,949.375 | 6,566.25 | 7,077.083333 | |
The center of the circle inscribed in a trapezoid is at distances of 5 and 12 from the ends of one of the non-parallel sides. Find the length of this side. | 13 | 0 | 7,935.75 | -1 | 7,935.75 | |
In a rectangular coordinate system, what is the number of units in the distance from the origin to the point $(-15, 8)$? | 17 | 1 | 1,371.0625 | 1,371.0625 | -1 | |
In $\triangle ABC$, $\angle A = 60^{\circ}$ and $AB > AC$. Point $O$ is the circumcenter, and the altitudes $BE$ and $CF$ intersect at point $H$. Points $M$ and $N$ lie on segments $BH$ and $HF$, respectively, such that $BM = CN$. Find the value of $\frac{MH + NH}{OH}$. | \sqrt{3} | 0.3125 | 7,790.4375 | 6,907 | 8,192 | |
The graph of the equation $x+y=\left\lfloor x^{2}+y^{2}\right\rfloor$ consists of several line segments. Compute the sum of their lengths. | 4+\sqrt{6}-\sqrt{2} | We split into cases on the integer $k=\left\lfloor x^{2}+y^{2}\right\rfloor$. Note that $x+y=k$ but $x^{2}+y^{2} \geq$ $\frac{1}{2}(x+y)^{2}=\frac{1}{2} k^{2}$ and $x^{2}+y^{2}<k+1$, which forces $k \leq 2$. If $k=0$, the region defined by $0 \leq x^{2}+y^{2}<1$ and $x+y=0$ is the diameter from $\left(\frac{\sqrt{2}}{2... | 0 | 7,827.6875 | -1 | 7,827.6875 |
Let \( x, y, z, u, v \in \mathbf{R}_{+} \). The maximum value of
\[
f = \frac{x y + y z + z u + u v}{2 x^{2} + y^{2} + 2 z^{2} + u^{2} + 2 v^{2}}
\]
is $\qquad$ . | \frac{\sqrt{6}}{4} | 0 | 8,192 | -1 | 8,192 | |
Tac is dressing his cat to go outside. He has four indistinguishable socks, four indistinguishable shoes, and 4 indistinguishable show-shoes. In a hurry, Tac randomly pulls pieces of clothing out of a door and tries to put them on a random one of his cat's legs; however, Tac never tries to put more than one of each typ... | $\frac{1}{1296}$ | On each leg, Tac's cat will get a shoe, a sock, and a snow-shoe in a random order. Thus, the probability that they will be put on in order for any given leg is $\frac{1}{3!}=\frac{1}{6}$. Thus, the probability that this will occur for all 4 legs is $\left(\frac{1}{6}\right)^{4}=\frac{1}{1296}$. | 0 | 8,124.8125 | -1 | 8,124.8125 |
A rectangular sheet of paper is folded so that two diagonally opposite corners come together. If the crease formed is the same length as the longer side of the sheet, what is the ratio of the longer side of the sheet to the shorter side? | \sqrt{\frac{2}{\sqrt{5} - 1}} | 0 | 7,493.0625 | -1 | 7,493.0625 | |
If a three-digit number \( n = \overline{abc} \) is a perfect square and its digit sum \( a+b+c \) is also a perfect square, then \( n \) is called a super square number. Let the set of super square numbers be \( A \), and let the sum of the elements in set \( A \) be denoted as \( S(A) \). The integer closest to \( \f... | 384 | 0.375 | 6,695.875 | 6,485.5 | 6,822.1 | |
Given that $\{a_n\}$ is a geometric sequence with a common ratio of $q$, and $a_m$, $a_{m+2}$, $a_{m+1}$ form an arithmetic sequence.
(Ⅰ) Find the value of $q$;
(Ⅱ) Let the sum of the first $n$ terms of the sequence $\{a_n\}$ be $S_n$. Determine whether $S_m$, $S_{m+2}$, $S_{m+1}$ form an arithmetic sequence and exp... | -\frac{1}{2} | 0.0625 | 6,595.25 | 8,192 | 6,488.8 | |
The sum of one hundred natural numbers $x, x+1, x+2, \cdots, x+99$ is denoted as $a$. If the sum of the digits of $a$ is 50, then what is the smallest value of $x$? | 99950 | 0 | 8,192 | -1 | 8,192 | |
Masha and the Bear ate a basket of raspberries and 60 pies, starting and finishing at the same time. Initially, Masha ate raspberries while the Bear ate pies, and then they switched at some point. The Bear ate raspberries 6 times faster than Masha and pies 3 times faster. How many pies did the Bear eat if the Bear ate ... | 54 | 0.1875 | 6,951.125 | 4,565.666667 | 7,501.615385 | |
Given the function \( f(x) = |x-1| + |x-3| + \mathrm{e}^x \) (where \( x \in \mathbf{R} \)), find the minimum value of the function. | 6-2\ln 2 | 0.9375 | 3,541.6875 | 3,380.2 | 5,964 | |
If the line passing through the point $P(2,1)$ intersects the graph of the function $f(x)= \frac{2x+3}{2x-4}$ at points $A$ and $B$, and $O$ is the origin, calculate the value of $( \overrightarrow{OA}+ \overrightarrow{OB})\cdot \overrightarrow{OP}$. | 10 | 0.75 | 5,909.0625 | 5,148.083333 | 8,192 | |
How many functions $f:\{1,2, \ldots, 10\} \rightarrow\{1,2, \ldots, 10\}$ satisfy the property that $f(i)+f(j)=11$ for all values of $i$ and $j$ such that $i+j=11$. | 100000 | To construct such a function $f$, we just need to choose a value for $f(x)$ from $\{1,2, \ldots, 10\}$ for each $x \in\{1,2, \ldots, 10\}$. But the condition that $f(i)+f(j)=11$ whenever $i+j=11$ means that $$\begin{aligned} f(10) & =11-f(1) \\ f(9) & =11-f(2) \\ \vdots & \\ f(6) & =11-f(5) \end{aligned}$$ This means t... | 0.5625 | 6,226 | 5,938.222222 | 6,596 |
Find the sum of all positive divisors of $50$ that are also divisors of $15$. | 6 | 1 | 1,523.375 | 1,523.375 | -1 | |
What is the remainder when $99^{36}$ is divided by 100? | 1 | 1 | 2,237.1875 | 2,237.1875 | -1 | |
The surface of a 3 x 3 x 3 Rubik's Cube consists of 54 cells. What is the maximum number of cells you can mark such that the marked cells do not share any vertices? | 14 | 0.0625 | 8,119.375 | 7,104 | 8,187.066667 | |
Lines in the $xy$-plane are drawn through the point $(3,4)$ and the trisection points of the line segment joining the points $(-4,5)$ and $(5,-1)$. One of these lines has the equation | x-4y+13=0 | 1. **Finding the Trisection Points:**
The trisection points of the line segment joining $(-4, 5)$ and $(5, -1)$ are calculated by dividing the segment into three equal parts. We start by finding the differences in the x-coordinates and y-coordinates:
- Difference in x-coordinates: $5 - (-4) = 9$
- Difference i... | 0 | 6,925 | -1 | 6,925 |
In the diagram, $P$ is on $RS$ so that $QP$ bisects $\angle SQR$. Also, $PQ=PR$, $\angle RSQ=2y^\circ$, and $\angle RPQ=3y^\circ$. What is the measure, in degrees, of $\angle RPQ$? [asy]
// C14
import olympiad;
size(7cm);
real x = 50; real y = 20;
pair q = (1, 0);
pair r = (0, 0);
pair p = intersectionpoints((10 * ... | 108 | 0.1875 | 7,368.6875 | 3,876 | 8,174.692308 | |
Given that \( x, y, z \in \mathbf{R}_{+} \) and \( x^{2} + y^{2} + z^{2} = 1 \), find the value of \( z \) when \(\frac{(z+1)^{2}}{x y z} \) reaches its minimum. | \sqrt{2} - 1 | 0.6875 | 7,182.75 | 6,724 | 8,192 | |
In a tetrahedron \(ABCD\), \(AD = \sqrt{2}\) and all other edge lengths are 1. Find the shortest path distance from the midpoint \(M\) of edge \(AB\) to the midpoint \(N\) of edge \(CD\) along the surface of the tetrahedron. | \frac{\sqrt{3}}{2} | 0 | 8,192 | -1 | 8,192 | |
A merchant offers a large group of items at $30\%$ off. Later, the merchant takes $20\%$ off these sale prices and claims that the final price of these items is $50\%$ off the original price. As a percentage of the original price, what is the difference between the true discount and the merchant's claimed discount? (... | 6\% | 1 | 2,039.8125 | 2,039.8125 | -1 | |
Six boys stood equally spaced on a circle of radius 40 feet. Each boy walked to all of the other non-adjacent persons on the circle, shook their hands and then returned to his original spot on the circle before the next boy started his trip to shake hands with all of the other non-adjacent boys on the circle. After all... | 480 + 480\sqrt{3}\text{ feet} | 0 | 8,042.9375 | -1 | 8,042.9375 | |
Determine all the functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[ f(x^2 + f(y)) = f(f(x)) + f(y^2) + 2f(xy) \]
for all real numbers $x$ and $y$. | $f(x) = x^2 \text{ and } f(x) = 0$ |
To determine all functions \( f : \mathbb{R} \to \mathbb{R} \) satisfying the functional equation:
\[
f(x^2 + f(y)) = f(f(x)) + f(y^2) + 2f(xy)
\]
for all real numbers \( x \) and \( y \), we will go through the following steps:
### Step 1: Substitution and Initial Analysis
First, consider substituting special val... | 0 | 8,192 | -1 | 8,192 |
Find the minimum value of
\[f(x) = x + \frac{x}{x^2 + 1} + \frac{x(x + 4)}{x^2 + 2} + \frac{2(x + 2)}{x(x^2 + 2)}\]for $x > 0.$ | 5 | 0 | 8,192 | -1 | 8,192 | |
Let $S$ be the set of positive integers $N$ with the property that the last four digits of $N$ are $2020,$ and when the last four digits are removed, the result is a divisor of $N.$ For example, $42,020$ is in $S$ because $4$ is a divisor of $42,020.$ Find the sum of all the digits of all the numbers in $S.$ For exampl... | 93 | We note that any number in $S$ can be expressed as $a(10,000) + 2,020$ for some integer $a$. The problem requires that $a$ divides this number, and since we know $a$ divides $a(10,000)$, we need that $a$ divides 2020. Each number contributes the sum of the digits of $a$, as well as $2 + 0 + 2 +0 = 4$. Since $2020$ can ... | 0.9375 | 4,748 | 4,518.4 | 8,192 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.