problem stringlengths 10 5.15k | answer stringlengths 0 1.22k | solution stringlengths 0 11.1k | reward float64 0 1 | length float64 172 8.19k | correct_length float64 -1 8.19k | incorrect_length float64 -1 8.19k |
|---|---|---|---|---|---|---|
A triangle has sides with lengths of $18$, $24$, and $30$. Calculate the length of the shortest altitude. | 18 | 0 | 2,647.5625 | -1 | 2,647.5625 | |
There are 6 class officers, among which there are 3 boys and 3 girls.
(1) Now, 3 people are randomly selected to participate in the school's voluntary labor. Calculate the probability that at least 2 of the selected people are girls.
(2) If these 6 people stand in a row for a photo, where boy A can only stand at the ... | 96 | 0.8125 | 6,573.125 | 6,199.538462 | 8,192 | |
A parametric graph is defined by:
\[
x = \cos t + \frac{t}{3}, \quad y = \sin t.
\]
Determine the number of times the graph intersects itself between \(x = 3\) and \(x = 45\). | 12 | 0 | 8,192 | -1 | 8,192 | |
Three distinct vertices are chosen at random from the vertices of a given regular polygon of $(2n+1)$ sides. If all such choices are equally likely, what is the probability that the center of the given polygon lies in the interior of the triangle determined by the three chosen random points? | \frac{n}{2n+1} | 1. **Counting Total Ways to Choose Vertices**:
The total number of ways to choose three vertices from a regular polygon with $(2n+1)$ sides is given by the combination formula:
\[
\binom{2n+1}{3} = \frac{(2n+1)(2n)(2n-1)}{6}
\]
2. **Counting Ways that Do Not Contain the Center**:
To count the number... | 0 | 7,791.375 | -1 | 7,791.375 |
In a bag of marbles, $\frac{3}{5}$ of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red? | \frac{4}{7} | 1. **Identify the fraction of blue and red marbles initially:**
Given that $\frac{3}{5}$ of the marbles are blue, the fraction of red marbles is the remainder when subtracted from 1 (since the total fraction must sum up to 1). Thus, the fraction of red marbles is:
\[
1 - \frac{3}{5} = \frac{5}{5} - \frac{3}{5}... | 0.8125 | 2,529.5625 | 2,064.923077 | 4,543 |
Each outcome on the spinner below has equal probability. If you spin the spinner four times and form a four-digit number from the four outcomes, such that the first outcome is the thousand digit, the second outcome is the hundreds digit, the third outcome is the tens digit, and the fourth outcome is the units digit, wh... | \frac{2}{3} | 0.3125 | 5,475.1875 | 5,464.8 | 5,479.909091 | |
In an after-school program for juniors and seniors, there is a debate team with an equal number of students from each class on the team. Among the $28$ students in the program, $25\%$ of the juniors and $10\%$ of the seniors are on the debate team. How many juniors are in the program? | 8 | 1. **Define Variables:**
Let $x$ be the number of juniors on the debate team and $y$ be the number of seniors on the debate team. Given that the number of juniors and seniors on the debate team is equal, we have $x = y$.
2. **Percentage Relationships:**
- $25\%$ of the juniors are on the debate team, which means... | 1 | 1,953.125 | 1,953.125 | -1 |
If the integers $m,n,k$ hold the equation $221m+247n+323k=2001$, find the smallest possible value of $k$ greater than 100. | 111 | 0.5625 | 6,707 | 5,552 | 8,192 | |
In triangle \(ABC\), the sides are known: \(AB = 6\), \(BC = 4\), and \(AC = 8\). The angle bisector of \(\angle C\) intersects side \(AB\) at point \(D\). A circle is drawn through points \(A\), \(D\), and \(C\), intersecting side \(BC\) at point \(E\). Find the area of triangle \(ADE\). | \frac{3 \sqrt{15}}{2} | 0 | 6,228.4375 | -1 | 6,228.4375 | |
Given \(2x^2 + 3xy + 2y^2 = 1\), find the minimum value of \(f(x, y) = x + y + xy\). | -\frac{9}{8} | 0.375 | 7,989.375 | 7,651.666667 | 8,192 | |
A bag has 4 red marbles and 6 white marbles. Three marbles are drawn from the bag without replacement. What is the probability that the sequence of marbles drawn is red, white, red? | \frac{1}{10} | 0.875 | 4,624.125 | 4,114.428571 | 8,192 | |
Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\frac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\frac{5}{8}$ times the radius of circle $D$. Furthermore, $AB = CD = 39$ and $PQ... | 192 | 1. **Understanding the Problem**: We are given four circles with centers at $A$, $B$, $C$, and $D$. Points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\frac{5}{8}$ times the radius of circle $B$, and similarly for circles $C$ and $D$. The distances $AB$ and $CD$ are both 39, and the length of segm... | 0.0625 | 8,083.9375 | 6,463 | 8,192 |
Consider the quadratic equation $x^{2}-(r+7) x+r+87=0$ where $r$ is a real number. This equation has two distinct real solutions $x$ which are both negative exactly when $p<r<q$, for some real numbers $p$ and $q$. What is the value of $p^{2}+q^{2}$? | 8098 | A quadratic equation has two distinct real solutions exactly when its discriminant is positive. For the quadratic equation $x^{2}-(r+7) x+r+87=0$, the discriminant is $\Delta=(r+7)^{2}-4(1)(r+87)=r^{2}+14 r+49-4 r-348=r^{2}+10 r-299$. Since $\Delta=r^{2}+10 r-299=(r+23)(r-13)$ which has roots $r=-23$ and $r=13$, then $... | 0.6875 | 5,411.375 | 4,683.181818 | 7,013.4 |
How many rectangles can be formed by the vertices of a cube? (Note: square is also a special rectangle). | 12 | 0.25 | 7,652.1875 | 6,032.75 | 8,192 | |
Find all values of $x$ with $0 \le x < 2 \pi$ that satisfy $\sin x + \cos x = \sqrt{2}.$ Enter all the solutions, separated by commas. | \frac{\pi}{4} | 1 | 4,498.5625 | 4,498.5625 | -1 | |
Given that $a_1, a_2, b_1, b_2, b_3$ are real numbers, and $-1, a_1, a_2, -4$ form an arithmetic sequence, $-4, b_1, b_2, b_3, -1$ form a geometric sequence, calculate the value of $\left(\frac{a_2 - a_1}{b_2}\right)$. | \frac{1}{2} | 0.75 | 3,905.4375 | 2,977.333333 | 6,689.75 | |
Given that the line $x=\dfrac{\pi }{6}$ is the axis of symmetry of the graph of the function $f\left(x\right)=\sin \left(2x+\varphi \right)\left(|\varphi | \lt \dfrac{\pi }{2}\right)$, determine the horizontal shift required to transform the graph of the function $y=\sin 2x$ into the graph of $y=f\left(x\right)$. | \dfrac{\pi}{12} | 0.75 | 5,800.5625 | 5,003.416667 | 8,192 | |
Given vectors $\overrightarrow{a} = (4\cos \alpha, \sin \alpha)$, $\overrightarrow{b} = (\sin \beta, 4\cos \beta)$, and $\overrightarrow{c} = (\cos \beta, -4\sin \beta)$, where $\alpha, \beta \in \mathbb{R}$ and neither $\alpha$, $\beta$, nor $\alpha + \beta$ equals $\frac{\pi}{2} + k\pi, k \in \mathbb{Z}$:
1. Find th... | -30 | 0.75 | 6,452.75 | 5,873 | 8,192 | |
The product $\left(1-\frac{1}{2^{2}}\right)\left(1-\frac{1}{3^{2}}\right)\ldots\left(1-\frac{1}{9^{2}}\right)\left(1-\frac{1}{10^{2}}\right)$ equals | \frac{11}{20} | 1. **Factor each term as a difference of squares**:
Each term in the product can be written as $\left(1-\frac{1}{n^2}\right)$, which factors as $\left(1-\frac{1}{n}\right)\left(1+\frac{1}{n}\right)$.
2. **Write the product using the factored terms**:
\[
\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right... | 0.8125 | 4,880 | 4,115.692308 | 8,192 |
Let $n$ represent the smallest integer that satisfies the following conditions:
$\frac n2$ is a perfect square.
$\frac n3$ is a perfect cube.
$\frac n5$ is a perfect fifth.
How many divisors does $n$ have that are not multiples of 10?
| 242 | 0.5 | 6,779.1875 | 5,366.375 | 8,192 | |
On November 1, 2019, at exactly noon, two students, Gavrila and Glafira, set their watches (both have standard watches where the hands make a full rotation in 12 hours) accurately. It is known that Glafira's watch gains 12 seconds per day, and Gavrila's watch loses 18 seconds per day. After how many days will their wat... | 1440 | 0.1875 | 5,648.3125 | 3,267 | 6,197.846154 | |
(1) Point $P$ is any point on the curve $y=x^{2}-\ln x$. The minimum distance from point $P$ to the line $x-y-4=0$ is ______.
(2) If the tangent line to the curve $y=g(x)$ at the point $(1,g(1))$ is $y=2x+1$, then the equation of the tangent line to the curve $f(x)=g(x)+\ln x$ at the point $(1,f(1))$ is ______.
(3) G... | \frac{\sqrt{2}}{2} | 0 | 7,019.0625 | -1 | 7,019.0625 | |
Let $a$, $b$, and $c$ be positive integers such that $a + b + c = 30$ and $\gcd(a,b) + \gcd(b,c) + \gcd(c,a) = 11$. Determine the sum of all possible distinct values of $a^2 + b^2 + c^2$.
A) 302
B) 318
C) 620
D) 391
E) 419 | 620 | 0 | 8,192 | -1 | 8,192 | |
Three couples sit for a photograph in $2$ rows of three people each such that no couple is sitting in the same row next to each other or in the same column one behind the other. How many such arrangements are possible? | 96 | 0 | 8,066 | -1 | 8,066 | |
The main structure of the Chinese space station includes the Tianhe core module, the Wentian experiment module, and the Mengtian experiment module. Assuming that the space station needs to arrange 6 astronauts, including astronauts A and B, to conduct experiments, with each module having at least one person and at most... | 450 | 0.1875 | 7,480.6875 | 6,329.666667 | 7,746.307692 | |
Point $D$ is on side $AC$ of triangle $ABC$, $\angle ABD=15^{\circ}$ and $\angle DBC=50^{\circ}$. What is the measure of angle $BAD$, in degrees?
[asy]draw((-43,0)--(43,0)--(43,40)--cycle);
draw((-4,0)--(43,40));
draw((39,4)--(39,0));
draw((39,4)--(43,4));
draw((-1,36)--(22.5,26),Arrow);
label("$15^{\circ}$",(-1,36),... | 25^\circ | 0 | 7,382.875 | -1 | 7,382.875 | |
Below is the graph of \( y = a \sin(bx + c) \) for some constants \( a > 0 \), \( b > 0 \), and \( c \). The graph reaches its maximum value at \( 3 \) and completes one full cycle by \( 2\pi \). There is a phase shift where the maximum first occurs at \( \pi/6 \). Find the values of \( a \), \( b \), and \( c \). | \frac{\pi}{3} | 0.875 | 4,013.9375 | 3,417.071429 | 8,192 | |
Given $f(x)=x^{2005}+ax^{3}- \frac {b}{x}-8$, and $f(-2)=10$, find $f(2)$. | -26 | 0.9375 | 2,891.3125 | 2,671.666667 | 6,186 | |
A cone and a sphere are placed inside a cylinder without overlapping. All bases are aligned and rest on the same flat surface. The cylinder has a radius of 6 cm and a height of 15 cm. The cone has a radius of 6 cm and a height of 7.5 cm. The sphere has a radius of 6 cm. Find the ratio of the volume of the cone to the t... | \frac{15}{138} | 0 | 3,526.5625 | -1 | 3,526.5625 | |
The centers of two circles are $41$ inches apart. The smaller circle has a radius of $4$ inches and the larger one has a radius of $5$ inches.
The length of the common internal tangent is: | 40 \text{ inches} | 1. **Identify the centers and radii of the circles**: Let $A$ be the center of the circle with radius $5$ inches, and $B$ be the center of the circle with radius $4$ inches. The distance between the centers $A$ and $B$ is given as $41$ inches.
2. **Understanding the geometry**: Consider the common internal tangent $\o... | 0.5625 | 5,048.6875 | 5,370.333333 | 4,635.142857 |
At each of the sixteen circles in the network below stands a student. A total of $3360$ coins are distributed among the sixteen students. All at once, all students give away all their coins by passing an equal number of coins to each of their neighbors in the network. After the trade, all students have the same number ... | 280 | Since the students give the same number of gifts of coins as they receive and still end up the same number of coins, we can assume that every gift of coins has the same number of coins. Let $x$ be the number of coins in each gift of coins. There $10$ people who give $4$ gifts of coins, $5$ people who give $3$ gifts of ... | 0 | 8,192 | -1 | 8,192 |
$BL$ is the angle bisector of triangle $ABC$. Find its area, given that $|AL| = 2$, $|BL| = 3\sqrt{10}$, and $|CL| = 3$. | \frac{15 \sqrt{15}}{4} | 0 | 5,278.0625 | -1 | 5,278.0625 | |
Niki usually leaves her cell phone on. If her cell phone is on but she is not actually using it, the battery will last for $24$ hours. If she is using it constantly, the battery will last for only $3$ hours. Since the last recharge, her phone has been on $9$ hours, and during that time she has used it for $60$ minutes.... | 8 | 1. **Calculate the battery consumption rate**:
- When the phone is not in use, it consumes \(\frac{1}{24}\) of its battery per hour.
- When the phone is in use, it consumes \(\frac{1}{3}\) of its battery per hour.
2. **Convert usage time to hours**:
- Niki used her phone for \(60\) minutes, which is equivalen... | 0.9375 | 3,627.1875 | 3,322.866667 | 8,192 |
Using the numbers 3, 0, 4, 8, and a decimal point to form decimals, the largest three-digit decimal is \_\_\_\_\_\_, the smallest decimal is \_\_\_\_\_\_, and their difference is \_\_\_\_\_\_. | 8.082 | 0.0625 | 574.6875 | 577 | 574.533333 | |
Among four people, A, B, C, and D, they pass a ball to each other. The first pass is from A to either B, C, or D, and the second pass is from the receiver to any of the other three. This process continues for several passes. Calculate the number of ways the ball can be passed such that it returns to A on the fourth pas... | 21 | 0.25 | 7,538.625 | 6,957.75 | 7,732.25 | |
Given three forces in space, $\overrightarrow {F_{1}}$, $\overrightarrow {F_{2}}$, and $\overrightarrow {F_{3}}$, each with a magnitude of 2, and the angle between any two of them is 60Β°, the magnitude of their resultant force $\overrightarrow {F}$ is ______. | 2 \sqrt {6} | 0 | 4,827.8125 | -1 | 4,827.8125 | |
The coefficient of $x^{3}$ in the expansion of $(2x^{2}+x-1)^{5}$ is _______. | -30 | 0.4375 | 6,893.3125 | 5,223.571429 | 8,192 | |
What is the area of the smallest square that will contain a circle of radius 7? | 196 | 0.875 | 2,374.1875 | 1,724.214286 | 6,924 | |
For any real number $x$, $\lfloor x \rfloor$ represents the largest integer not exceeding $x$, for example: $\lfloor 2 \rfloor = 2$, $\lfloor 3.2 \rfloor = 3$. Then $\lfloor \log_{2}1 \rfloor + \lfloor \log_{2}2 \rfloor + \lfloor \log_{2}3 \rfloor + \ldots + \lfloor \log_{2}64 \rfloor =$ ? | 264 | 0.6875 | 6,453.1875 | 5,662.818182 | 8,192 | |
A whole number is said to be ''9-heavy'' if the remainder when the number is divided by 9 is greater than 5. What is the least three-digit 9-heavy whole number? | 105 | 1 | 2,586.125 | 2,586.125 | -1 | |
The base of a rectangular parallelepiped is a square with a side length of \(2 \sqrt{3}\). The diagonal of a lateral face forms an angle of \(30^\circ\) with the plane of an adjacent lateral face. Find the volume of the parallelepiped. | 72 | 0.5 | 5,922.9375 | 5,678.75 | 6,167.125 | |
Express, as concisely as possible, the value of the product $$\left(0^{3}-350\right)\left(1^{3}-349\right)\left(2^{3}-348\right)\left(3^{3}-347\right) \cdots\left(349^{3}-1\right)\left(350^{3}-0\right)$$ | 0 | 0. One of the factors is $7^{3}-343=0$, so the whole product is zero. | 0.9375 | 3,077.375 | 2,736.4 | 8,192 |
$\mathbf{7 3 8 , 8 2 6}$. This can be arrived at by stepping down, starting with finding how many combinations are there that begin with a letter other than V or W , and so forth. The answer is $\frac{8 \cdot 9!}{2 \cdot 2}+\frac{4 \cdot 7!}{2}+4 \cdot 6!+4 \cdot 4!+3!+2!+2!=738826$. | 738826 | The number of combinations is 738826. | 0.3125 | 7,793.375 | 6,916.4 | 8,192 |
A local community group sells 180 event tickets for a total of $2652. Some tickets are sold at full price, while others are sold at a discounted rate of half price. Determine the total revenue generated from the full-price tickets.
A) $960
B) $984
C) $1008
D) $1032 | 984 | 0 | 1,257.9375 | -1 | 1,257.9375 | |
**How many different positive integers can be represented as a difference of two distinct members of the set $\{1, 3, 6, \ldots, 45\}$, where the numbers form an arithmetic sequence? Already given that the common difference between successive elements is 3.** | 14 | 0.125 | 7,837.5 | 6,425 | 8,039.285714 | |
Let
\[T=\frac{1}{3-\sqrt{8}}-\frac{1}{\sqrt{8}-\sqrt{7}}+\frac{1}{\sqrt{7}-\sqrt{6}}-\frac{1}{\sqrt{6}-\sqrt{5}}+\frac{1}{\sqrt{5}-2}.\]
Then | T>2 | 1. **Rationalize each term**: We start by rationalizing each term in the expression for $T$. For a general term of the form $\frac{1}{\sqrt{n+1}-\sqrt{n}}$, we multiply the numerator and the denominator by the conjugate of the denominator, $\sqrt{n+1}+\sqrt{n}$:
\[
\frac{1}{\sqrt{n+1}-\sqrt{n}} = \frac{\sqrt{n+1}... | 0 | 3,039.8125 | -1 | 3,039.8125 |
An equilateral pentagon $AMNPQ$ is inscribed in triangle $ABC$ such that $M\in\overline{AB}$ , $Q\in\overline{AC}$ , and $N,P\in\overline{BC}$ .
Suppose that $ABC$ is an equilateral triangle of side length $2$ , and that $AMNPQ$ has a line of symmetry perpendicular to $BC$ . Then the area of $AMNPQ$ i... | 5073 | 0.125 | 8,105.625 | 7,501 | 8,192 | |
In rectangle $ABCD$, $AB = 10$ cm, $BC = 14$ cm, and $DE = DF$. The area of triangle $DEF$ is one-fifth the area of rectangle $ABCD$. What is the length in centimeters of segment $EF$? Express your answer in simplest radical form. | 4\sqrt{7} | 0.5625 | 6,121.625 | 4,511.333333 | 8,192 | |
Suppose that $x$ is a positive multiple of $3$. If $x$ cubed is less than $1000$, what is the greatest possible value of $x$? | 9 | 0.9375 | 2,637.5 | 2,741.8 | 1,073 | |
Given \\(a > b\\), the quadratic trinomial \\(a{x}^{2}+2x+b \geqslant 0 \\) holds for all real numbers, and there exists \\(x_{0} \in \mathbb{R}\\), such that \\(ax_{0}^{2}+2{x_{0}}+b=0\\), then the minimum value of \\(\dfrac{a^{2}+b^{2}}{a-b}\\) is \_\_\_\_\_\_\_\_\_. | 2 \sqrt{2} | 0.9375 | 5,265.1875 | 5,242.266667 | 5,609 | |
The admission fee for an exhibition is $ \$25$ per adult and $ \$12$ per child. Last Tuesday, the exhibition collected $ \$1950$ in admission fees from at least one adult and at least one child. Of all the possible ratios of adults to children at the exhibition last Tuesday, which one is closest to $ 1$? | \frac{27}{25} | 0.6875 | 7,074.25 | 6,923.727273 | 7,405.4 | |
There are two ${\bf positive}$ integers $c$ for which the equation $$5x^2+11x+c=0$$has rational solutions. What is the product of those two values of $c$? | 12 | 1 | 2,551.75 | 2,551.75 | -1 | |
Three frogs in a swamp jumped one after another. Each one landed exactly in the middle of the segment between the other two. The jump length of the second frog is 60 cm. Find the jump length of the third frog. | 30 | 0.75 | 5,763.75 | 4,954.333333 | 8,192 | |
Many television screens are rectangles that are measured by the length of their diagonals. The ratio of the horizontal length to the height in a standard television screen is $4:3$. What is the horizontal length (in inches) of a ``27-inch'' television screen?
[asy]
fill((0,0)--(8,0)--(8,6)--cycle,gray(0.7));
draw((0,0... | 21.6 | 0.875 | 1,394 | 1,501.357143 | 642.5 | |
Given the set $M$ consisting of all functions $f(x)$ that satisfy the property: there exist real numbers $a$ and $k$ ($k \neq 0$) such that for all $x$ in the domain of $f$, $f(a+x) = kf(a-x)$. The pair $(a,k)$ is referred to as the "companion pair" of the function $f(x)$.
1. Determine whether the function $f(x) = x^2... | 2016 | 0.125 | 7,939.3125 | 7,648.5 | 7,980.857143 | |
Let\[S=\sqrt{1+\dfrac1{1^2}+\dfrac1{2^2}}+\sqrt{1+\dfrac1{2^2}+\dfrac1{3^2}}+\cdots+\sqrt{1+\dfrac1{2007^2}+\dfrac1{2008^2}}.\]Compute $\lfloor S^2\rfloor$.
| 4032062 | 0.8125 | 6,421.25 | 6,012.615385 | 8,192 | |
Fix a sequence $ a_1,a_2,a_3,... $ of integers satisfying the following condition:for all prime numbers $ p $ and all positive integers $ k $ , we have $ a_{pk+1}=pa_k-3a_p+13 $ .Determine all possible values of $ a_{2013} $ . | 2016 | 0.125 | 7,919.1875 | 6,009.5 | 8,192 | |
Using the digits $0$, $1$, $2$, $3$, $4$ to form a five-digit number without repeating any digit, the probability that the number is even and the digits $1$, $2$ are adjacent is ______. | 0.25 | 0 | 7,716.75 | -1 | 7,716.75 | |
For a $k$-element subset $T$ of the set $\{1,2,\cdots,242\}$, every pair of elements (which may be the same) in $T$ has a sum that is not an integer power of 3. Find the maximum value of $k$. | 121 | 0 | 8,192 | -1 | 8,192 | |
In the diagram, \(BD\) is perpendicular to \(BC\) and to \(AD\). If \(AB = 52\), \(BC = 21\), and \(AD = 48\), what is the length of \(DC\)? | 29 | 0.625 | 4,279.1875 | 3,635.5 | 5,352 | |
When the base-16 number $66666_{16}$ is written in base 2, how many base-2 digits (bits) does it have? | 19 | 0.375 | 6,973.9375 | 5,869 | 7,636.9 | |
Suppose that $x$ and $y$ are positive real numbers such that $x^{2}-xy+2y^{2}=8$. Find the maximum possible value of $x^{2}+xy+2y^{2}$. | \frac{72+32 \sqrt{2}}{7} | Let $u=x^{2}+2y^{2}$. By AM-GM, $u \geq \sqrt{8}xy$, so $xy \leq \frac{u}{\sqrt{8}}$. If we let $xy=ku$ where $k \leq \frac{1}{\sqrt{8}}$, then we have $u(1-k)=8$ and $u(1+k)=x^{2}+xy+2y^{2}$, that is, $u(1+k)=8 \cdot \frac{1+k}{1-k}$. It is not hard to see that the maximum value of this expression occurs at $k=\frac{1... | 0 | 7,543.8125 | -1 | 7,543.8125 |
In Rad's garden there are exactly 30 red roses, exactly 19 yellow roses, and no other roses. How many of the yellow roses does Rad need to remove so that $\frac{2}{7}$ of the roses in the garden are yellow? | 7 | If $\frac{2}{7}$ of the roses are to be yellow, then the remaining $\frac{5}{7}$ of the roses are to be red. Since there are 30 red roses and these are to be $\frac{5}{7}$ of the roses, then $\frac{1}{7}$ of the total number of roses would be $30 \div 5 = 6$, which means that there would be $6 \times 7 = 42$ roses in t... | 1 | 1,975.5 | 1,975.5 | -1 |
Let the function \( f:(0,1) \rightarrow \mathbf{R} \) be defined as
$$
f(x)=\left\{\begin{array}{l}
x, \text{ if } x \text{ is irrational; } \\
\frac{p+1}{q}, \text{ if } x=\frac{p}{q}, \text{ where } (p, q)=1 \text{ and } 0<p<q.
\end{array}\right.
$$
Find the maximum value of \( f(x) \) on the interval \( \left(\frac... | \frac{16}{17} | 0 | 7,941.5625 | -1 | 7,941.5625 | |
The set of all solutions of the system $$
\begin{cases}
x+y\leq 3 \\
2x+y\geq 2 \\
x\geq 0 \\
y\geq 0
\end{cases}
$$ is a quadrilateral region. Find the number of units in the length of the longest side. Express your answer in simplest radical form. | 3\sqrt{2} | 0.875 | 4,188.6875 | 3,972.285714 | 5,703.5 | |
Determine the number of zeros in the quotient $Q = R_{30}/R_6$, where $R_k$ is a number consisting of $k$ repeated digits of 1 in base-ten. | 25 | 0 | 8,035.3125 | -1 | 8,035.3125 | |
Parallelogram $PQRS$ has vertices $P(4,4)$, $Q(-2,-2)$, $R(-8,-2)$, and $S(2,4)$. If a point is selected at random from the region determined by the parallelogram, what is the probability that the point lies below the $x$-axis? | \frac{1}{3} | 0.0625 | 8,019.4375 | 8,192 | 8,007.933333 | |
\(ABCD\) is a square and \(X\) is a point on the side \(DA\) such that the semicircle with diameter \(CX\) touches the side \(AB\). Find the ratio \(AX: XD\). | 1 : 3 | 0.625 | 4,825.5 | 4,212.4 | 5,847.333333 | |
Azmi has four blocks, each in the shape of a rectangular prism and each with dimensions $2 imes 3 imes 6$. She carefully stacks these four blocks on a flat table to form a tower that is four blocks high. What is the number of possible heights for this tower? | 14 | The height of each block is 2, 3 or 6. Thus, the total height of the tower of four blocks is the sum of the four heights, each of which equals 2, 3 or 6. If 4 blocks have height 6, the total height equals $4 imes 6=24$. If 3 blocks have height 6, the fourth block has height 3 or 2. Therefore, the possible heights are ... | 0.4375 | 7,575.5625 | 6,783 | 8,192 |
In tetrahedron \(ABCD\), it is known that \(\angle ADB = \angle BDC = \angle CDA = 60^\circ\), \(AD = BD = 3\), and \(CD = 2\). Find the radius of the circumscribed sphere of tetrahedron \(ABCD\). | \sqrt{3} | 0.4375 | 7,366 | 6,832.142857 | 7,781.222222 | |
Suppose that a positive integer $N$ can be expressed as the sum of $k$ consecutive positive integers \[ N = a + (a+1) +(a+2) + \cdots + (a+k-1) \] for $k=2017$ but for no other values of $k>1$. Considering all positive integers $N$ with this property, what is the smallest positive integer $a$ that occurs in any of thes... | 16 | We prove that the smallest value of $a$ is 16. Note that the expression for $N$ can be rewritten as $k(2a+k-1)/2$, so that $2N = k(2a+k-1)$. In this expression, $k>1$ by requirement; $k < 2a+k-1$ because $a>1$; and obviously $k$ and $2a+k-1$ have opposite parity. Conversely, for any factorization $2N = mn$ with $1<m<n$... | 0.3125 | 7,379.125 | 5,590.8 | 8,192 |
Trapezoid $ABCD$ has $AD||BC$, $BD = 1$, $\angle DBA = 23^{\circ}$, and $\angle BDC = 46^{\circ}$. The ratio $BC: AD$ is $9: 5$. What is $CD$? | \frac{4}{5} |
#### Step 1: Extend Lines and Identify Angles
Extend $\overline{AB}$ and $\overline{DC}$ to meet at point $E$. Since $\angle DBA = 23^\circ$ and $\angle BDC = 46^\circ$, we can calculate $\angle EDB$ and $\angle DBE$:
- $\angle EDB = 180^\circ - \angle BDC = 180^\circ - 46^\circ = 134^\circ$.
- $\angle DBE = \angle DB... | 0 | 8,192 | -1 | 8,192 |
The segment connecting the centers of two intersecting circles is divided by their common chord into segments equal to 5 and 2. Find the common chord, given that the radius of one circle is twice the radius of the other. | 2 \sqrt{3} | 0.625 | 5,737.375 | 4,367.9 | 8,019.833333 | |
The perimeter of triangle $BQN$ is $180$, and the angle $QBN$ is a right angle. A circle of radius $15$ with center $O$ on $\overline{BQ}$ is drawn such that it is tangent to $\overline{BN}$ and $\overline{QN}$. Given that $OQ=p/q$ where $p$ and $q$ are relatively prime positive integers, find $p+q$. | 79 | 0 | 8,192 | -1 | 8,192 | |
In an experiment, a certain constant \( c \) is measured to be 2.43865 with an error range of \(\pm 0.00312\). The experimenter wants to publish the value of \( c \), with each digit being significant. This means that regardless of how large \( c \) is, the announced value of \( c \) (with \( n \) digits) must match th... | 2.44 | 0.0625 | 8,171.8125 | 7,869 | 8,192 | |
The region shown is bounded by the arcs of circles having radius 4 units, having a central angle measure of 60 degrees and intersecting at points of tangency. The area of the region can be expressed in the form $a\sqrt{b}+c\pi$ square units, where $\sqrt{b}$ is a radical in simplest form. What is the value of $a + b + ... | 11 | 0 | 7,400.75 | -1 | 7,400.75 | |
In right triangle $DEF$ with $\angle D = 90^\circ$, we have $DE = 8$ and $EF = 17$. Find $\cos F$. | \frac{8}{17} | 0 | 3,155.1875 | -1 | 3,155.1875 | |
There are $n$ mathematicians attending a conference. Each mathematician has exactly 3 friends (friendship is mutual). If they are seated around a circular table such that each person has their friends sitting next to them on both sides, the number of people at the table is at least 7. Find the minimum possible value of... | 24 | 0.0625 | 7,431.125 | 5,358 | 7,569.333333 | |
Three congruent isosceles triangles are constructed inside an equilateral triangle with a side length of $\sqrt{2}$. Each base of the isosceles triangle is placed on one side of the equilateral triangle. If the total area of the isosceles triangles equals $\frac{1}{2}$ the area of the equilateral triangle, find the len... | \frac{1}{2} | 0 | 8,192 | -1 | 8,192 | |
From a group of boys and girls, 15 girls leave. There are then left two boys for each girl. After this 45 boys leave. There are then 5 girls for each boy. The number of girls in the beginning was: | 40 | 1. **Define Variables:**
Let $b$ represent the number of boys and $g$ represent the number of girls initially.
2. **Translate the Problem into Equations:**
- After 15 girls leave, the number of girls becomes $g - 15$. According to the problem, there are then two boys for each girl, so we have the equation:
... | 1 | 1,760.375 | 1,760.375 | -1 |
A cone has a volume of $2592\pi$ cubic inches and the vertex angle of the vertical cross section is 90 degrees. What is the height of the cone? Express your answer as a decimal to the nearest tenth. | 20.0 | 0 | 7,326.5 | -1 | 7,326.5 | |
Two lines with slopes $\dfrac{1}{3}$ and $3$ intersect at $(3,3)$. Find the area of the triangle enclosed by these two lines and the line $x+y=12$. | 8.625 | 0 | 4,756.3125 | -1 | 4,756.3125 | |
$(1)$ Calculate: $\sqrt{9}+2\sin30{}Β°-(Ο-3){}Β°$;<br/>$(2)$ Solve the equation: $\left(2x-3\right)^{2}=2\left(2x-3\right)$. | \frac{5}{2} | 0.8125 | 6,018.6875 | 5,517.153846 | 8,192 | |
Let $X Y Z$ be an equilateral triangle, and let $K, L, M$ be points on sides $X Y, Y Z, Z X$, respectively, such that $X K / K Y=B, Y L / L Z=1 / C$, and $Z M / M X=1$. Determine the ratio of the area of triangle $K L M$ to the area of triangle $X Y Z$. | $\frac{1}{5}$ | First, we note that $$[K L M]=[X Y Z]-[X K M]-[Y L K]-[Z M L]$$ Then, note that $$\begin{gathered} {[X K M]=\frac{X K}{X Y} \cdot \frac{X M}{X Z} \cdot[X Y Z]=\frac{B}{B+1} \cdot \frac{1}{2} \cdot[X Y Z]} \\ {[Y L K]=\frac{Y L}{Y Z} \cdot \frac{Y K}{Y X} \cdot[X Y Z]=\frac{1}{C+1} \cdot \frac{1}{B+1} \cdot[X Y Z]} \\ {... | 0 | 7,881.5 | -1 | 7,881.5 |
Determine how many positive integer multiples of $2002$ can be represented in the form $10^{j} - 10^{i}$, where $i$ and $j$ are integers and $0 \leq i < j \leq 150$. | 1825 | 0.25 | 7,914 | 7,466.75 | 8,063.083333 | |
In the rectangular coordinate system $(xOy)$, a line $l_{1}$ is given by the equation $y = \tan \alpha \cdot x \ (0 \leqslant \alpha < \pi, \alpha \neq \frac{\pi}{2})$, and a parabola $C$ is given by the parametric equations $\begin{cases} x = t^{2} \\ y = -2t \end{cases} \ (t \text{ is a parameter})$. Establish a pola... | 16 | 0.5625 | 7,122.5 | 6,290.666667 | 8,192 | |
Find the mathematical expectation of the area of the projection of a cube with edge of length $1$ onto a plane with an isotropically distributed random direction of projection. | \frac{3}{2} | 0.125 | 7,824.4375 | 5,251.5 | 8,192 | |
A student typed out several circles on the computer as follows: βββββββββββββββββββββ¦ If this pattern continues, forming a series of circles, then the number of β in the first 120 circles is ______. | 14 | 0.1875 | 2,889.375 | 4,414 | 2,537.538462 | |
Two numbers are independently selected from the set of positive integers less than or equal to 6. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction. | \frac{2}{3} | 0.375 | 7,475 | 6,280 | 8,192 | |
In rectangle $ABCD$, $AB = 4$ and $BC = 8$. The rectangle is folded so that points $A$ and $C$ coincide, forming the pentagon $ABEFD$. What is the length of segment $EF$? Express your answer in simplest radical form. | 2\sqrt{5} | 0.8125 | 5,479.25 | 5,163.461538 | 6,847.666667 | |
Which one of the following is not equivalent to $0.000000375$? | $\frac{3}{8} \times 10^{-7}$ | To find which option is not equivalent to $0.000000375$, we first convert $0.000000375$ into scientific notation:
1. **Convert to Scientific Notation:**
\[
0.000000375 = 3.75 \times 10^{-7}
\]
2. **Evaluate Each Option:**
- **Option (A) $3.75 \times 10^{-7}$:**
\[
3.75 \times 10^{-7} = 3.75 \tim... | 0 | 8,102 | -1 | 8,102 |
Consider $9$ points in space, no four of which are coplanar. Each pair of points is joined by an edge (that is, a line segment) and each edge is either colored blue or red or left uncolored. Find the smallest value of $\,n\,$ such that whenever exactly $\,n\,$ edges are colored, the set of colored edges necessarily co... | 33 |
Consider a configuration where you have 9 points in space, with each pair of points joined by an edge, for a total of \(\binom{9}{2} = 36\) edges. We want to find the smallest \( n \) such that if exactly \( n \) edges are colored (either blue or red), there must exist a monochromatic triangle (a triangle with all edg... | 0.0625 | 7,972.3125 | 7,584 | 7,998.2 |
Bev is driving from Waterloo, ON to Marathon, ON. She has driven 312 km and has 858 km still to drive. How much farther must she drive in order to be halfway from Waterloo to Marathon? | 273 \mathrm{~km} | Since Bev has driven 312 km and still has 858 km left to drive, the distance from Waterloo to Marathon is $312 \mathrm{~km} + 858 \mathrm{~km} = 1170 \mathrm{~km}$. The halfway point of the drive is $\frac{1}{2}(1170 \mathrm{~km}) = 585 \mathrm{~km}$ from Waterloo. To reach this point, she still needs to drive $585 \ma... | 0 | 1,693.4375 | -1 | 1,693.4375 |
If the integer $k^{}_{}$ is added to each of the numbers $36^{}_{}$, $300^{}_{}$, and $596^{}_{}$, one obtains the squares of three consecutive terms of an arithmetic series. Find $k^{}_{}$. | 925 | 1 | 3,323.8125 | 3,323.8125 | -1 | |
Five points are chosen on a sphere of radius 1. What is the maximum possible volume of their convex hull? | \frac{\sqrt{3}}{2} | 0 | 8,192 | -1 | 8,192 | |
When two distinct digits are randomly chosen in $N=123456789$ and their places are swapped, one gets a new number $N'$ (for example, if 2 and 4 are swapped, then $N'=143256789$ ). The expected value of $N'$ is equal to $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute the rem... | 555556 | 0 | 8,076.3125 | -1 | 8,076.3125 | |
A sequence $a_{1}, a_{2}, a_{3}, \ldots$ of positive reals satisfies $a_{n+1}=\sqrt{\frac{1+a_{n}}{2}}$. Determine all $a_{1}$ such that $a_{i}=\frac{\sqrt{6}+\sqrt{2}}{4}$ for some positive integer $i$. | \frac{\sqrt{2}+\sqrt{6}}{2}, \frac{\sqrt{3}}{2}, \frac{1}{2} | Clearly $a_{1}<1$, or else $1 \leq a_{1} \leq a_{2} \leq a_{3} \leq \ldots$ We can therefore write $a_{1}=\cos \theta$ for some $0<\theta<90^{\circ}$. Note that $\cos \frac{\theta}{2}=\sqrt{\frac{1+\cos \theta}{2}}$, and $\cos 15^{\circ}=$ $\frac{\sqrt{6}+\sqrt{2}}{4}$. Hence, the possibilities for $a_{1}$ are $\cos 15... | 0 | 8,192 | -1 | 8,192 |
The perpendicular bisectors of the sides of triangle $DEF$ meet its circumcircle at points $D'$, $E'$, and $F'$, respectively. If the perimeter of triangle $DEF$ is 42 and the radius of the circumcircle is 10, find the area of hexagon $DE'F'D'E'F$. | 105 | 0 | 8,192 | -1 | 8,192 | |
Given the function $f(x)=- \sqrt {3}\sin ^{2}x+\sin x\cos x$.
(1) Find the value of $f( \frac {25Ο}{6})$.
(2) Find the smallest positive period of the function $f(x)$ and its maximum and minimum values in the interval $[0, \frac {Ο}{2}]$. | -\sqrt{3} | 1 | 5,386 | 5,386 | -1 | |
Kate bakes a $20$-inch by $18$-inch pan of cornbread. The cornbread is cut into pieces that measure $2$ inches by $2$ inches. How many pieces of cornbread does the pan contain? | 90 | 1. **Calculate the area of the pan**:
The pan has dimensions $20$ inches by $18$ inches. Therefore, the area of the pan is calculated by multiplying these dimensions:
\[
\text{Area of the pan} = 20 \times 18 = 360 \text{ square inches}
\]
2. **Calculate the area of each piece of cornbread**:
Each piece... | 1 | 1,546.4375 | 1,546.4375 | -1 |
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