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 |
|---|---|---|---|---|---|---|
Square $PQRS$ lies in the first quadrant. Points $(3,0), (5,0), (7,0),$ and $(13,0)$ lie on lines $SP, RQ, PQ$, and $SR$, respectively. What is the sum of the coordinates of the center of the square $PQRS$? | \frac{32}{5} | 1. **Identify the lines and their slopes**:
- Since $SP$ and $RQ$ are opposite sides of square $PQRS$, and points $(3,0)$ and $(5,0)$ lie on $SP$ and $RQ$ respectively, lines $SP$ and $RQ$ are parallel with some positive slope $m$.
- Similarly, points $(7,0)$ and $(13,0)$ lie on $PQ$ and $SR$ respectively, so li... | 0.0625 | 8,154.25 | 7,588 | 8,192 |
An infinite geometric series has common ratio $-1/5$ and sum $16.$ What is the first term of the series? | \frac{96}{5} | 1 | 1,846.8125 | 1,846.8125 | -1 | |
What is the result of subtracting eighty-seven from nine hundred forty-three? | 856 | Converting to a numerical expression, we obtain $943-87$ which equals 856. | 0.6875 | 417.0625 | 460.090909 | 322.4 |
Completely factor the following expression: \[(6a^3+92a^2-7)-(-7a^3+a^2-7)\] | 13a^2(a+7) | 1 | 1,717 | 1,717 | -1 | |
A square is inscribed in the ellipse
\[\frac{x^2}{5} + \frac{y^2}{10} = 1,\]
so that its sides are parallel to the coordinate axes. Find the area of the square. | \frac{40}{3} | 1 | 4,588.25 | 4,588.25 | -1 | |
Rewrite $\sqrt[3]{2^6\cdot3^3\cdot11^3}$ as an integer. | 132 | 1 | 1,927.25 | 1,927.25 | -1 | |
Given vectors \(\overrightarrow{O P}=\left(2 \cos \left(\frac{\pi}{2}+x\right),-1\right)\) and \(\overrightarrow{O Q}=\left(-\sin \left(\frac{\pi}{2}- x\right), \cos 2 x\right)\), and the function \(f(x)=\overrightarrow{O P} \cdot \overrightarrow{O Q}\). If \(a, b, c\) are the sides opposite angles \(A, B, C\) respecti... | 15/2 | 0.9375 | 5,569.9375 | 5,395.133333 | 8,192 | |
A plane intersects a right circular cylinder of radius $1$ forming an ellipse. If the major axis of the ellipse of $50\%$ longer than the minor axis, the length of the major axis is
$\textbf{(A)}\ 1\qquad \textbf{(B)}\ \frac{3}{2}\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ \frac{9}{4}\qquad \textbf{(E)}\ 3$
| 3 | 0 | 4,121.6875 | -1 | 4,121.6875 | |
It is given that there exists a unique triple of positive primes $(p,q,r)$ such that $p<q<r$ and \[\dfrac{p^3+q^3+r^3}{p+q+r} = 249.\] Find $r$ . | 19 | 0 | 8,192 | -1 | 8,192 | |
(1) Given that $0 < x < \dfrac{4}{3}$, find the maximum value of $x(4-3x)$.
(2) Point $(x,y)$ moves along the line $x+2y=3$. Find the minimum value of $2^{x}+4^{y}$. | 4 \sqrt{2} | 1 | 3,714.5 | 3,714.5 | -1 | |
Let $S$ be the set of complex numbers of the form $x + yi,$ where $x$ and $y$ are real numbers, such that
\[\frac{\sqrt{2}}{2} \le x \le \frac{\sqrt{3}}{2}.\]Find the smallest positive integer $m$ such that for all positive integers $n \ge m,$ there exists a complex number $z \in S$ such that $z^n = 1.$ | 16 | 0 | 8,170.3125 | -1 | 8,170.3125 | |
Emily has 8 blue marbles and 7 red marbles. She randomly selects a marble, notes its color, and returns it to the bag. She repeats this process 6 times. What is the probability that she selects exactly three blue marbles? | \frac{3512320}{11390625} | 0.0625 | 8,093.625 | 7,524 | 8,131.6 | |
Given that Anne, Cindy, and Ben repeatedly take turns tossing a die in the order Anne, Cindy, Ben, find the probability that Cindy will be the first one to toss a five. | \frac{30}{91} | 0.125 | 7,097.0625 | 6,565.5 | 7,173 | |
Consider the function $g(x)=3x-4$. For what value of $a$ is $g(a)=0$? | \frac{4}{3} | 1 | 1,564.6875 | 1,564.6875 | -1 | |
The set of points $\left(x_{1}, x_{2}, x_{3}, x_{4}\right)$ in $\mathbf{R}^{4}$ such that $x_{1} \geq x_{2} \geq x_{3} \geq x_{4}$ is a cone (or hypercone, if you insist). Into how many regions is this cone sliced by the hyperplanes $x_{i}-x_{j}=1$ for $1 \leq i<j \leq n$ ? | 14 | $C(4)=14$. | 0 | 7,797.625 | -1 | 7,797.625 |
Alice and Ali each select a positive integer less than 250. Alice's number is a multiple of 25, and Ali's number is a multiple of 30. What is the probability that they selected the same number? Express your answer as a common fraction. | \frac{1}{80} | 0 | 3,386.9375 | -1 | 3,386.9375 | |
What is the range of the function $$G(x) = |x+1|-|x-1|~?$$Express your answer in interval notation. | [-2,2] | 1 | 2,641.1875 | 2,641.1875 | -1 | |
There are positive integers $x$ and $y$ that satisfy the system of equations\begin{align*} \log_{10} x + 2 \log_{10} (\text{gcd}(x,y)) &= 60\\ \log_{10} y + 2 \log_{10} (\text{lcm}(x,y)) &= 570. \end{align*}Let $m$ be the number of (not necessarily distinct) prime factors in the prime factorization of $x$, and let $n$ ... | 880 | 0.375 | 7,325.875 | 6,228.333333 | 7,984.4 | |
The circular region of the sign now has an area of 50 square inches. To decorate the edge with a ribbon, Vanessa plans to purchase 5 inches more than the circle’s circumference. How many inches of ribbon should she buy if she estimates \(\pi = \frac{22}{7}\)? | 30 | 0.3125 | 4,754.9375 | 1,843.4 | 6,078.363636 | |
What is the median of the following list of $4100$ numbers?
\[1, 2, 3, \ldots, 2050, 1^2, 2^2, 3^2, \ldots, 2050^2\]
A) $1977.5$
B) $2004.5$
C) $2005.5$
D) $2006.5$
E) $2025.5$ | 2005.5 | 0 | 7,958.3125 | -1 | 7,958.3125 | |
Evaluate $\left\lceil\sqrt{\frac{9}{4}}\right\rceil+\left\lceil\frac{9}{4}\right\rceil+\left\lceil\left(\frac{9}{4}\right)^2\right\rceil$. | 11 | 1 | 2,264.5625 | 2,264.5625 | -1 | |
Households A, B, and C plan to subscribe to newspapers. There are 5 different types of newspapers available. Each household subscribes to two different newspapers. It is known that each pair of households shares exactly one common newspaper. How many different subscription ways are there for the three households? | 180 | 0.0625 | 7,921.1875 | 7,676 | 7,937.533333 | |
Let $a_1,a_2,\cdots,a_{41}\in\mathbb{R},$ such that $a_{41}=a_1, \sum_{i=1}^{40}a_i=0,$ and for any $i=1,2,\cdots,40, |a_i-a_{i+1}|\leq 1.$ Determine the greatest possible value of
$(1)a_{10}+a_{20}+a_{30}+a_{40};$
$(2)a_{10}\cdot a_{20}+a_{30}\cdot a_{40}.$ | 10 |
Let \( a_1, a_2, \ldots, a_{41} \in \mathbb{R} \) such that \( a_{41} = a_1 \), \( \sum_{i=1}^{40} a_i = 0 \), and for any \( i = 1, 2, \ldots, 40 \), \( |a_i - a_{i+1}| \leq 1 \). We aim to determine the greatest possible values of:
1. \( a_{10} + a_{20} + a_{30} + a_{40} \)
2. \( a_{10} \cdot a_{20} + a_{30} \cdot a... | 0 | 8,192 | -1 | 8,192 |
Triangle $ABC$ is isosceles with $AB + AC$ and $BC = 65$ cm. $P$ is a point on $\overline{BC}$ such that the perpendicular distances from $P$ to $\overline{AB}$ and $\overline{AC}$ are $24$ cm and $36$ cm, respectively. The area of $\triangle ABC$, in cm $^2$, is | 2535 | 0.5 | 7,132.4375 | 6,072.875 | 8,192 | |
Find the least positive integer $N$ such that the set of $1000$ consecutive integers beginning with $1000\cdot N$ contains no square of an integer. | 282 | Let $x$ be the number being squared. Based on the reasoning above, we know that $N$ must be at least $250$, so $x$ has to be at least $500$. Let $k$ be $x-500$. We can write $x^2$ as $(500+k)^2$, or $250000+1000k+k^2$. We can disregard $250000$ and $1000k$, since they won't affect the last three digits, which determine... | 0 | 7,848 | -1 | 7,848 |
A number \( n \) has a sum of digits equal to 100, while \( 44n \) has a sum of digits equal to 800. Find the sum of the digits of \( 3n \). | 300 | 0.1875 | 8,030.5 | 7,330.666667 | 8,192 | |
It is known that, for all positive integers $k$,
$1^2+2^2+3^2+\ldots+k^{2}=\frac{k(k+1)(2k+1)}6$.
Find the smallest positive integer $k$ such that $1^2+2^2+3^2+\ldots+k^2$ is a multiple of $200$.
| 112 | 0 | 8,192 | -1 | 8,192 | |
An $8\times8$ array consists of the numbers $1,2,...,64$. Consecutive numbers are adjacent along a row or a column. What is the minimum value of the sum of the numbers along the diagonal? | 88 |
We have an \(8 \times 8\) array filled with the numbers from 1 to 64, where consecutive numbers are adjacent either along a row or along a column. Our task is to find the minimum possible value of the sum of the numbers along a diagonal of this array.
### Analysis
Let's denote the elements of the array by \( a_{ij} ... | 0 | 8,000.3125 | -1 | 8,000.3125 |
A right circular cone is sliced into five pieces by planes parallel to its base. Each slice has the same height. What is the ratio of the volume of the second-largest piece to the volume of the largest piece? | \frac{37}{61} | 0.25 | 7,435.875 | 5,513.5 | 8,076.666667 | |
Petya's watch runs 5 minutes fast per hour, and Masha's watch runs 8 minutes slow per hour. At 12:00, they set their watches to the accurate school clock and agreed to meet at the skating rink at 6:30 PM according to their respective watches. How long will Petya wait for Masha if each arrives at the skating rink exactl... | 1.5 | 0.1875 | 6,389.25 | 5,275.333333 | 6,646.307692 | |
Let $\{a_{n}\}$ and $\{b_{n}\}$ be arithmetic sequences, and let $S_{n}$ and $T_{n}$ be the sums of the first $n$ terms of the sequences, respectively. If $\frac{S_{n}}{T_{n}}=\frac{3n-1}{n+3}$, then $\frac{a_{8}}{b_{5}+b_{11}}=\_\_\_\_\_\_$. | \frac{11}{9} | 0.75 | 4,923 | 4,501.916667 | 6,186.25 | |
The sum of the first $15$ positive even integers is also the sum of six consecutive even integers. What is the smallest of these six integers? | 35 | 0.125 | 7,614.625 | 3,573 | 8,192 | |
Let \( P \) be a regular 2006-sided polygon. If a diagonal of \( P \), whose endpoints divide the boundary of \( P \) into two parts each containing an odd number of sides, is called a "good diagonal". Note that each side of \( P \) is considered a "good diagonal". Given that 2003 non-intersecting diagonals within \( P... | 1003 | 0.125 | 8,131.8125 | 7,710.5 | 8,192 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $b\sin A=a\sin C$ and $c=1$, find the value of $b$ and the maximum area of $\triangle ABC$. | \frac{1}{2} | 0.625 | 5,572.6875 | 4,261.6 | 7,757.833333 | |
At a certain crosswalk, the pedestrian signal alternates between red and green lights, with the red light lasting for $40s$. If a pedestrian arrives at the crosswalk and encounters a red light, the probability that they need to wait at least $15s$ for the green light to appear is ______. | \dfrac{5}{8} | 0.9375 | 3,165.5625 | 2,830.466667 | 8,192 | |
Let $\mathbf{a}$ and $\mathbf{b}$ be vectors, and let $\mathbf{m}$ be the midpoint of $\mathbf{a}$ and $\mathbf{b}.$ Given $\mathbf{m} = \begin{pmatrix} 3 \\ 7 \end{pmatrix}$ and $\mathbf{a} \cdot \mathbf{b} = 6,$ find $\|\mathbf{a}\|^2 + \|\mathbf{b}\|^2.$ | 220 | 0.9375 | 2,597.5 | 2,224.533333 | 8,192 | |
From the first 539 positive integers, we select some such that their sum is at least one-third of the sum of the original numbers. What is the minimum number of integers we need to select for this condition to be satisfied? | 99 | 0.625 | 6,382.125 | 5,629.8 | 7,636 | |
Two people are flipping a coin: one flipped it 10 times, and the other 11 times.
What is the probability that the second person gets more heads than the first person? | \frac{1}{2} | 0 | 8,108.75 | -1 | 8,108.75 | |
If for any real numbers $u,v$, the inequality ${{(u+5-2v)}^{2}}+{{(u-{{v}^{2}})}^{2}}\geqslant {{t}^{2}}(t > 0)$ always holds, then the maximum value of $t$ is | 2 \sqrt{2} | 1 | 4,714.0625 | 4,714.0625 | -1 | |
Perpendiculars \( B E \) and \( D F \), dropped from the vertices \( B \) and \( D \) of parallelogram \( A B C D \) onto sides \( A D \) and \( B C \) respectively, divide the parallelogram into three parts of equal area. On the extension of diagonal \( B D \) past vertex \( D \), a segment \( D G \) is laid off equal... | 1:1 | 0 | 8,192 | -1 | 8,192 | |
On the ray $(0,+\infty)$ of the number line, there are several (more than two) segments of length 1. For any two different segments, you can select one number from each so that these numbers differ exactly by a factor of 2. The left end of the leftmost segment is the number $a$, and the right end of the rightmost segme... | 5.5 | 0 | 8,192 | -1 | 8,192 | |
A state requires that all boat licenses consist of the letter A or M followed by any five digits. What is the number of groups of letters and numbers available for boat licenses? | 200000 | 1 | 1,347.4375 | 1,347.4375 | -1 | |
Given that the two distinct square roots of a positive number $x$ are $a+3$ and $2a-15$, and $\sqrt [3] {x+y-2}=4$. Find the value of $x-2y+2$. | 17 | 0.9375 | 1,944.5625 | 1,980.733333 | 1,402 | |
What percent of the prime numbers less than 12 are divisible by 2? | 20\% | 1 | 1,214.125 | 1,214.125 | -1 | |
A certain department store sells suits and ties, with each suit priced at $1000$ yuan and each tie priced at $200 yuan. During the "National Day" period, the store decided to launch a promotion offering two discount options to customers.<br/>Option 1: Buy one suit and get one tie for free;<br/>Option 2: Pay 90% of the ... | 21800 | 0 | 7,260.625 | -1 | 7,260.625 | |
Given a lawn with a rectangular shape of $120$ feet by $200$ feet, a mower with a $32$-inch swath width, and a $6$-inch overlap between each cut, and a walking speed of $4000$ feet per hour, calculate the time it will take John to mow the entire lawn. | 2.8 | 0.0625 | 8,048.9375 | 8,192 | 8,039.4 | |
In $\triangle PQR,$ where $PQ=PR=17$ and $QR=15.$ Points $G,H,$ and $I$ are on sides $\overline{PQ},$ $\overline{QR},$ and $\overline{PR},$ respectively, such that $\overline{GH}$ and $\overline{HI}$ are parallel to $\overline{PR}$ and $\overline{PQ},$ respectively. What is the perimeter of parallelogram $PGHI$? | 34 | 0.5625 | 7,618.3125 | 7,172.111111 | 8,192 | |
Given the function $f(x) = \sqrt[3]{x}$, calculate the value of $\lim\limits_{\Delta x \to 0} \frac{f(1-\Delta x)-f(1)}{\Delta x}$. | -\frac{1}{3} | 0.6875 | 5,288.1875 | 3,968.272727 | 8,192 | |
Given a right triangle with sides of length $5$, $12$, and $13$, and a square with side length $x$ inscribed in it so that one vertex of the square coincides with the right-angle vertex of the triangle, and another square with side length $y$ inscribed in a different right triangle with sides of length $5$, $12$, and $... | \frac{39}{51} | 0 | 7,946.375 | -1 | 7,946.375 | |
Class 2-5 planted 142 trees. Class 2-3 planted 18 fewer trees than Class 2-5. How many trees did Class 2-3 plant? How many trees did the two classes plant in total? | 266 | 0.6875 | 367.5625 | 380.727273 | 338.6 | |
Among the four students A, B, C, and D participating in competitions in mathematics, writing, and English, each subject must have at least one participant (and each participant can only choose one subject). If students A and B cannot participate in the same competition, the total number of different participation schem... | 30 | 0.1875 | 7,205.6875 | 5,622.333333 | 7,571.076923 | |
Let $f(x)=x^{2}-2$, and let $f^{n}$ denote the function $f$ applied $n$ times. Compute the remainder when $f^{24}(18)$ is divided by 89. | 47 | Let $L_{n}$ denote the Lucas numbers given by $L_{0}=2, L_{1}=1$, and $L_{n+2}=L_{n+1}+L_{n}$. Note that $L_{n}^{2}-2=L_{2 n}$ when $n$ is even (one can show this by induction, or explicitly using $L_{n}=$ $\left.\left(\frac{1+\sqrt{5}}{2}\right)^{n}+\left(\frac{1-\sqrt{5}}{2}\right)^{n}\right)$.So, $f^{24}\left(L_{6}\... | 0.6875 | 5,745.6875 | 5,679.636364 | 5,891 |
A mouse has a wheel of cheese which is cut into $2018$ slices. The mouse also has a $2019$ -sided die, with faces labeled $0,1,2,\ldots, 2018$ , and with each face equally likely to come up. Every second, the mouse rolls the dice. If the dice lands on $k$ , and the mouse has at least $k$ slices of cheese remaini... | 2019 | 0 | 7,953.125 | -1 | 7,953.125 | |
Let $P_1P_2\ldots P_{24}$ be a regular $24$-sided polygon inscribed in a circle $\omega$ with circumference $24$. Determine the number of ways to choose sets of eight distinct vertices from these $24$ such that none of the arcs has length $3$ or $8$. | 258 |
Let \( P_1P_2\ldots P_{24} \) be a regular 24-sided polygon inscribed in a circle \(\omega\) with circumference 24. We aim to determine the number of ways to choose sets of eight distinct vertices from these 24 such that none of the arcs has length 3 or 8.
We generalize the problem by considering a regular polygon wi... | 0 | 8,192 | -1 | 8,192 |
Emily's quiz scores so far are: 92, 95, 87, 89 and 100. What score does she need to get on the sixth quiz to make the arithmetic mean of the six scores equal 93? | 95 | 1 | 2,567.375 | 2,567.375 | -1 | |
We say a triple of real numbers $ (a_1,a_2,a_3)$ is [b]better[/b] than another triple $ (b_1,b_2,b_3)$ when exactly two out of the three following inequalities hold: $ a_1 > b_1$, $ a_2 > b_2$, $ a_3 > b_3$. We call a triple of real numbers [b]special[/b] when they are nonnegative and their sum is $ 1$.
For which natu... | n\geq4 |
To solve this problem, we need to determine for which natural numbers \( n \) there exists a set \( S \) of special triples, with \( |S| = n \), such that any special triple is bettered by at least one element of \( S \).
### Understanding the Definitions
A **special triple** \((a_1, a_2, a_3)\) is defined as a trip... | 0 | 8,167.75 | -1 | 8,167.75 |
Let $f$ be a function for which $f\left(\dfrac{x}{3}\right) = x^2 + x + 1$. Find the sum of all values of $z$ for which $f(3z) = 7$. | -1/9 | 1. **Identify the function and equation:** Given the function $f\left(\frac{x}{3}\right) = x^2 + x + 1$, we need to find the sum of all values of $z$ for which $f(3z) = 7$.
2. **Relate $f(3z)$ to the given function:** Since $f\left(\frac{x}{3}\right) = x^2 + x + 1$, substituting $x = 9z$ (because $\frac{9z}{3} = 3z$) ... | 0 | 2,193.6875 | -1 | 2,193.6875 |
Mayar and Rosie are 90 metres apart. Starting at the same time, they run towards each other. Mayar runs twice as fast as Rosie. How far has Mayar run when they meet? | 60 | Suppose that Rosie runs \(x\) metres from the time that they start running until the time that they meet. Since Mayar runs twice as fast as Rosie, then Mayar runs \(2x\) metres in this time. When Mayar and Rosie meet, they will have run a total of 90 m, since between the two of them, they have covered the full 90 m. Th... | 1 | 1,278.75 | 1,278.75 | -1 |
A finite non-empty set of integers is called $3$ -*good* if the sum of its elements is divisible by $3$ . Find the number of $3$ -good subsets of $\{0,1,2,\ldots,9\}$ . | 351 | 0.1875 | 7,924.6875 | 6,766.333333 | 8,192 | |
If $\sum_{n = 0}^{\infty}\cos^{2n}\theta = 5$, what is the value of $\cos{2\theta}$? | \frac{3}{5} | 0.9375 | 1,953.5625 | 1,537.666667 | 8,192 | |
Compute $(5+7)^3+(5^3+7^3)$. | 2196 | 0.9375 | 3,651.875 | 3,349.2 | 8,192 | |
In the Cartesian coordinate system, there are points $P_0$, $P_1$, $P_2$, $P_3$, ..., $P_{n-1}$, $P_n$. Let the coordinates of point $P_k$ be $(x_k,y_k)$ $(k\in\mathbb{N},k\leqslant n)$, where $x_k$, $y_k\in\mathbb{Z}$. Denote $\Delta x_k=x_k-x_{k-1}$, $\Delta y_k=y_k-y_{k-1}$, and it satisfies $|\Delta x_k|\cdot|\Delt... | 4066272 | 0 | 8,192 | -1 | 8,192 | |
Approximate the number $0.00356$ to the nearest ten-thousandth: $0.00356 \approx$____. | 0.0036 | 0.9375 | 604.6875 | 618.933333 | 391 | |
Let $a$ and $b$ be positive integers such that all but $2009$ positive integers are expressible in the form $ma + nb$ , where $m$ and $n$ are nonnegative integers. If $1776 $ is one of the numbers that is not expressible, find $a + b$ . | 133 | 0.1875 | 7,677.625 | 5,448.666667 | 8,192 | |
The sum of the first and the third of three consecutive odd integers is 152. What is the value of the second integer? | 76 | 0.3125 | 7,959.75 | 7,448.8 | 8,192 | |
What is the minimum number of connections required to organize a wired communication network of 10 nodes, so that if any two nodes fail, it still remains possible to transmit information between any two remaining nodes (at least through a chain via other nodes)? | 15 | 0.0625 | 6,386.5625 | 2,834 | 6,623.4 | |
The diagram shows a rectangle $AEFJ$ inside a regular decagon $ABCDEFGHIJ$. What is the ratio of the area of the rectangle to the area of the decagon? | $2:5$ | 0 | 7,207.625 | -1 | 7,207.625 | |
A dark room contains 120 red socks, 100 green socks, 70 blue socks, 50 yellow socks, and 30 black socks. A person randomly selects socks from the room without the ability to see their colors. What is the smallest number of socks that must be selected to guarantee that the selection contains at least 15 pairs? | 146 | 0 | 7,275.75 | -1 | 7,275.75 | |
Determine the number of arrangements of the letters a, b, c, d, e in a sequence such that neither a nor b is adjacent to c. | 36 | 0.5625 | 6,323.125 | 5,143.111111 | 7,840.285714 | |
For the odd function $f(x)$ defined on domain $\mathbb{R}$ that satisfies $f(4 - x) + f(x) = 0$, given that $f(x) = 2^x$ for $-2 < x < 0$, calculate $f(\log_2 20)$. | -\frac{4}{5} | 0.3125 | 7,380.8125 | 6,380 | 7,835.727273 | |
A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he increases the number of members in each row by... | 98 | 1 | 4,167.0625 | 4,167.0625 | -1 | |
Given a square \( ABCD \). Point \( N \) lies on side \( AD \) such that \( AN : ND = 2 : 3 \), point \( F \) lies on side \( CD \) such that \( DF : FC = 1 : 4 \), and point \( K \) lies on side \( AB \) such that \( AK : KB = 1 : 4 \). Find the angle \( \angle KNF \). | 135 | 0.5 | 6,247.0625 | 4,355.625 | 8,138.5 | |
Given the parabola $C: x^{2}=2py\left(p \gt 0\right)$ with focus $F$, and the minimum distance between $F$ and a point on the circle $M: x^{2}+\left(y+4\right)^{2}=1$ is $4$.<br/>$(1)$ Find $p$;<br/>$(2)$ If point $P$ lies on $M$, $PA$ and $PB$ are two tangents to $C$ with points $A$ and $B$ as the points of tangency, ... | 20\sqrt{5} | 0.0625 | 8,140.875 | 7,374 | 8,192 | |
The number of points equidistant from a circle and two parallel tangents to the circle is: | 3 | To solve this problem, we need to understand the geometric configuration and the properties of points equidistant from a circle and two parallel tangents.
1. **Understanding the Configuration**:
- Consider a circle with center $O$ and radius $r$.
- Let there be two parallel tangents to the circle, and without lo... | 0.25 | 7,167.875 | 5,863.25 | 7,602.75 |
A certain shopping mall sells a batch of brand-name shirts, with an average daily sales of 20 pieces and a profit of $40 per piece. In order to expand sales and reduce inventory quickly, the mall decides to take appropriate price reduction measures. After investigation, it was found that if the price of each shirt is r... | 20 | 0.375 | 5,534 | 6,903.5 | 4,712.3 | |
A ticket to a school play cost $x$ dollars, where $x$ is a whole number. A group of 9th graders buys tickets costing a total of $48, and a group of 10th graders buys tickets costing a total of $64. How many values for $x$ are possible? | 5 | To determine the possible values of $x$, the cost of each ticket, we need to consider the conditions given in the problem:
1. The total cost for the 9th graders is $48$ dollars.
2. The total cost for the 10th graders is $64$ dollars.
3. $x$ must be a whole number.
We start by noting that $x$ must be a common divisor ... | 1 | 1,839.1875 | 1,839.1875 | -1 |
Two ferries leave simultaneously from opposite shores of a river and cross it perpendicularly to the banks. The speeds of the ferries are constant. The ferries meet each other 720 meters from the nearest shore. Upon reaching the shore, they immediately depart back. On the return trip, they meet 400 meters from the othe... | 1280 | 0.0625 | 7,764.875 | 5,827 | 7,894.066667 | |
In $\triangle ABC$, let the sides opposite to angles $A$, $B$, and $C$ be $a$, $b$, and $c$ respectively. If $\sin A = \sin B = -\cos C$.
$(1)$ Find the sizes of angles $A$, $B$, and $C$;
$(2)$ If the length of the median $AM$ on side $BC$ is $\sqrt{7}$, find the area of $\triangle ABC$. | \sqrt{3} | 0.75 | 4,851.6875 | 4,083.666667 | 7,155.75 | |
Point $D$ lies on side $AC$ of equilateral triangle $ABC$ such that the measure of angle $DBC$ is $45$ degrees. What is the ratio of the area of triangle $ADB$ to the area of triangle $CDB$? Express your answer as a common fraction in simplest radical form. | \frac{\sqrt{3}- 1}{2} | 0 | 7,715.3125 | -1 | 7,715.3125 | |
Find the sum of all positive integers such that their expression in base $5$ digits is the reverse of their expression in base $11$ digits. Express your answer in base $10$. | 10 | 0.1875 | 8,033.6875 | 7,629.666667 | 8,126.923077 | |
A convex polyhedron S has vertices U1, U2, …, Um, and 120 edges. This polyhedron is intersected by planes Q1, Q2, …, Qm, where each plane Qk intersects only those edges that are connected to vertex Uk. No two planes intersect within the volume or on the surface of S. As a result, m pyramids are formed along with a new ... | 360 | 0.375 | 7,091.25 | 5,546 | 8,018.4 | |
Let $f(x)=x^{4}+14 x^{3}+52 x^{2}+56 x+16$. Let $z_{1}, z_{2}, z_{3}, z_{4}$ be the four roots of $f$. Find the smallest possible value of $|z_{a} z_{b}+z_{c} z_{d}|$ where $\{a, b, c, d\}=\{1,2,3,4\}$. | 8 | Note that $\frac{1}{16} f(2 x)=x^{4}+7 x^{3}+13 x^{2}+7 x+1$. Because the coefficients of this polynomial are symmetric, if $r$ is a root of $f(x)$ then $\frac{4}{r}$ is as well. Further, $f(-1)=-1$ and $f(-2)=16$ so $f(x)$ has two distinct roots on $(-2,0)$ and two more roots on $(-\infty,-2)$. Now, if $\sigma$ is a p... | 0.0625 | 8,092.375 | 8,192 | 8,085.733333 |
Jamie has a jar of coins containing the same number of nickels, dimes and quarters. The total value of the coins in the jar is $\$$13.20. How many nickels does Jamie have? | 33 | 1 | 1,298.8125 | 1,298.8125 | -1 | |
Given the function $f(x)=\cos^4x-2\sin x\cos x-\sin^4x.$
$(1)$ Find the smallest positive period of $f(x)$.
$(2)$ When $x\in\left[0, \frac{\pi}{2}\right]$, find the minimum value of $f(x)$ and the set of $x$ values for which this minimum is achieved. | \frac{3\pi}{8} | 0.8125 | 5,331.6875 | 4,671.615385 | 8,192 | |
Let $a,b,c$ be three distinct positive integers such that the sum of any two of them is a perfect square and having minimal sum $a + b + c$ . Find this sum. | 55 | 0.0625 | 8,192 | 8,192 | 8,192 | |
Let $ABCD$ be a square, and let $E$ and $F$ be points on $\overline{AB}$ and $\overline{BC},$ respectively. The line through $E$ parallel to $\overline{BC}$ and the line through $F$ parallel to $\overline{AB}$ divide $ABCD$ into two squares and two nonsquare rectangles. The sum of the areas of the two squares is $\frac... | 18 | After we get the polynomial $x^2 - 18x + 1,$ we want to find $x + \frac 1 {x}.$ Since the product of the roots of the polynomial is 1, the roots of the polynomial are simply $x, \frac 1 {x}.$ Hence $x + \frac 1 {x}$ is just $18$ by Vieta's formula, or $\boxed{018}$ | 0.3125 | 7,464.75 | 5,864.8 | 8,192 |
Person A and Person B started working on the same day. The company policy states that Person A works for 3 days and then rests for 1 day, while Person B works for 7 days and then rests for 3 consecutive days. How many days do Person A and Person B rest on the same day within the first 1000 days? | 100 | 0.1875 | 7,181.3125 | 5,831.666667 | 7,492.769231 | |
Given the two-digit integer $MM$, where both digits are equal, when multiplied by the one-digit integer $K$ (different from $M$ and only $1\leq K \leq 9$), it results in a three-digit number $NPK$. Identify the digit pairs $(M, K)$ that yield the highest value of $NPK$. | 891 | 0 | 8,192 | -1 | 8,192 | |
Andrea and Lauren are $20$ kilometers apart. They bike toward one another with Andrea traveling three times as fast as Lauren, and the distance between them decreasing at a rate of $1$ kilometer per minute. After $5$ minutes, Andrea stops biking because of a flat tire and waits for Lauren. After how many minutes from t... | 65 | 1. **Define Variables:**
Let $v_A$ be Andrea's speed and $v_L$ be Lauren's speed, both in kilometers per hour.
2. **Set Up Equations:**
Given that Andrea travels at three times the speed of Lauren, we have:
\[ v_A = 3v_L \]
Also, the rate at which the distance between them decreases is $1$ kilometer per mi... | 0.8125 | 4,018 | 3,054.769231 | 8,192 |
On a plane, points are colored in the following way:
1. Choose any positive integer \( m \), and let \( K_{1}, K_{2}, \cdots, K_{m} \) be circles with different non-zero radii such that \( K_{i} \subset K_{j} \) or \( K_{j} \subset K_{i} \) for \( i \neq j \).
2. Points chosen inside the circles are colored differently... | 2019 | 0.3125 | 7,980.5625 | 7,515.4 | 8,192 | |
Compute
\[\csc \frac{\pi}{14} - 4 \cos \frac{2 \pi}{7}.\] | 2 | 0.125 | 8,102.3125 | 7,474.5 | 8,192 | |
A certain store sells a batch of thermal shirts, with an average daily sales of 20 pieces and a profit of $40 per piece. In order to increase sales and profits, the store has taken appropriate price reduction measures. After investigation, it was found that within a certain range, for every $1 decrease in the unit pric... | 20 | 0.375 | 5,064.5625 | 5,528 | 4,786.5 | |
How many six-digit numbers exist in which each subsequent digit is less than the previous one? | 210 | 0.8125 | 4,375.25 | 3,945.923077 | 6,235.666667 | |
For a positive integer $n$, let $p(n)$ denote the product of the positive integer factors of $n$. Determine the number of factors $n$ of 2310 for which $p(n)$ is a perfect square. | 27 | Note that $2310=2 \times 3 \times 5 \times 7 \times 11$. In general, we see that if $n$ has $d(n)$ positive integer factors, then $p(n)=n^{\frac{d}{2}}$ since we can pair factors $\left(d, \frac{n}{d}\right)$ which multiply to $n$. As a result, $p(n)$ is a square if and only if $n$ is a square or $d$ is a multiple of 4... | 0.125 | 8,068 | 7,715 | 8,118.428571 |
How many different triangles can be formed having a perimeter of 7 units if each side must have integral length? | 2 | 0.875 | 5,940.5 | 5,618.857143 | 8,192 | |
A parabola with equation $y = x^2 + bx + c$ passes through the points $(2,3)$ and $(4,3)$. What is $c$? | 11 | 1 | 2,236.0625 | 2,236.0625 | -1 | |
Suppose a real number \(x>1\) satisfies \(\log _{2}\left(\log _{4} x\right)+\log _{4}\left(\log _{16} x\right)+\log _{16}\left(\log _{2} x\right)=0\). Compute \(\log _{2}\left(\log _{16} x\right)+\log _{16}\left(\log _{4} x\right)+\log _{4}\left(\log _{2} x\right)\). | -\frac{1}{4} | Let \(A\) and \(B\) be these sums, respectively. Then \(B-A =\log _{2}\left(\frac{\log _{16} x}{\log _{4} x}\right)+\log _{4}\left(\frac{\log _{2} x}{\log _{16} x}\right)+\log _{16}\left(\frac{\log _{4} x}{\log _{2} x}\right) =\log _{2}\left(\log _{16} 4\right)+\log _{4}\left(\log _{2} 16\right)+\log _{16}\left(\log _{... | 0.6875 | 6,245.3125 | 5,360.454545 | 8,192 |
What is the difference between the sum of the first $2003$ even counting numbers and the sum of the first $2003$ odd counting numbers? | 2003 | 1. **Identify the sequences**:
- The first $2003$ odd counting numbers form the sequence $O = 1, 3, 5, \ldots, 4005$.
- The first $2003$ even counting numbers can be considered in two cases:
- Including $0$: $E_1 = 0, 2, 4, \ldots, 4004$
- Excluding $0$: $E_2 = 2, 4, 6, \ldots, 4006$
2. **Calculate th... | 1 | 1,804 | 1,804 | -1 |
Let $a_1,$ $a_2,$ $\dots,$ $a_{2018}$ be the roots of the polynomial
\[x^{2018} + x^{2017} + \dots + x^2 + x - 1345 = 0.\]Compute
\[\sum_{n = 1}^{2018} \frac{1}{1 - a_n}.\] | 3027 | 0.625 | 4,314.0625 | 3,466.1 | 5,727.333333 | |
If the function \( y = \sin(w x) \) with \( w > 0 \) attains its maximum value at least 50 times in the interval \([0,1]\), what is the minimum value of \( w \)? | 100 \pi | 0 | 7,037.1875 | -1 | 7,037.1875 |
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