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 certain cafeteria serves ham and cheese sandwiches, ham and tomato sandwiches, and tomato and cheese sandwiches. It is common for one meal to include multiple types of sandwiches. On a certain day, it was found that 80 customers had meals which contained both ham and cheese; 90 had meals containing both ham and tomat... | 230 | 230. Everyone who ate just one sandwich is included in exactly one of the first three counts, while everyone who ate more than one sandwich is included in all four counts. Thus, to count each customer exactly once, we must add the first three figures and subtract the fourth twice: $80+90+100-2 \cdot 20=230$. | 0.9375 | 3,884.5625 | 3,597.4 | 8,192 |
Find the volume of the three-dimensional solid given by the inequality $\sqrt{x^{2}+y^{2}}+$ $|z| \leq 1$. | 2 \pi / 3 | $2 \pi / 3$. The solid consists of two cones, one whose base is the circle $x^{2}+y^{2}=1$ in the $x y$-plane and whose vertex is $(0,0,1)$, and the other with the same base but vertex $(0,0,-1)$. Each cone has a base area of $\pi$ and a height of 1, for a volume of $\pi / 3$, so the answer is $2 \pi / 3$. | 0 | 4,887.75 | -1 | 4,887.75 |
We inscribed a regular hexagon $ABCDEF$ in a circle and then drew semicircles outward over the chords $AB$, $BD$, $DE$, and $EA$. Calculate the ratio of the combined area of the resulting 4 crescent-shaped regions (bounded by two arcs each) to the area of the hexagon. | 2:3 | 0 | 8,192 | -1 | 8,192 | |
Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run 100 meters. They next meet after Sally has run 150 meters past their first meeting point. Each girl runs at a constant speed. What is the length of the track in meters? | 400 | 1. **Define Variables:**
Let the length of the track be $x$ meters.
2. **Analyze First Meeting:**
Since Brenda and Sally start at diametrically opposite points, they start $x/2$ meters apart. When they first meet, Brenda has run 100 meters. Therefore, Sally must have run the remaining distance to complete half t... | 0 | 4,497.125 | -1 | 4,497.125 |
Square $ABCD$ has side length $1$ unit. Points $E$ and $F$ are on sides $AB$ and $CB$, respectively, with $AE = CF$. When the square is folded along the lines $DE$ and $DF$, sides $AD$ and $CD$ coincide and lie on diagonal $BD$. The length of segment $AE$ can be expressed in the form $\sqrt{k}-m$ units. What is the ... | 3 | 0.1875 | 7,603.0625 | 5,051 | 8,192 | |
$n$ balls are placed independently uniformly at random into $n$ boxes. One box is selected at random, and is found to contain $b$ balls. Let $e_n$ be the expected value of $b^4$ . Find $$ \lim_{n \to
\infty}e_n. $$ | 15 | 0.25 | 7,034.625 | 5,126 | 7,670.833333 | |
Find $c$ such that $\lfloor c \rfloor$ satisfies
\[3x^2 - 9x - 30 = 0\]
and $\{ c \} = c - \lfloor c \rfloor$ satisfies
\[4x^2 - 8x + 1 = 0.\] | 6 - \frac{\sqrt{3}}{2} | 0 | 7,565.25 | -1 | 7,565.25 | |
Find the product of all integer divisors of $105$ that also divide $14$. (Recall that the divisors of an integer may be positive or negative.) | 49 | 0.875 | 3,471.4375 | 2,797.071429 | 8,192 | |
Rectangle $ABCD$ has $AB = 8$ and $BC = 13$ . Points $P_1$ and $P_2$ lie on $AB$ and $CD$ with $P_1P_2 \parallel BC$ . Points $Q_1$ and $Q_2$ lie on $BC$ and $DA$ with $Q_1Q_2 \parallel AB$ . Find the area of quadrilateral $P_1Q_1P_2Q_2$ . | 52 | 0.8125 | 5,468.75 | 4,840.307692 | 8,192 | |
A parametric graph is given by
\begin{align*}
x &= \cos t + \frac{t}{2}, \\
y &= \sin t.
\end{align*}How many times does the graph intersect itself between $x = 1$ and $x = 40$? | 12 | 0 | 8,192 | -1 | 8,192 | |
Let $N$ be the number of positive integers that are less than or equal to $2003$ and whose base-$2$ representation has more $1$'s than $0$'s. Find the remainder when $N$ is divided by $1000$.
| 155 | 0 | 8,192 | -1 | 8,192 | |
Given two lines $l_{1}$: $(3+m)x+4y=5-3m$ and $l_{2}$: $2x+(5+m)y=8$ are parallel, the value of the real number $m$ is ______. | -7 | 0.3125 | 6,987.3125 | 6,981.2 | 6,990.090909 | |
Suppose that $x^{2017} - 2x + 1 = 0$ and $x \neq 1.$ Find the value of
\[x^{2016} + x^{2015} + \dots + x + 1.\] | 2 | 0.9375 | 2,779.0625 | 2,418.2 | 8,192 | |
What is the least possible value of $(xy-1)^2+(x+y)^2$ for real numbers $x$ and $y$? | 1 | 1. **Expand and simplify the expression**: Start by expanding the given expression:
\[
(xy-1)^2 + (x+y)^2 = (xy-1)^2 + x^2 + 2xy + y^2.
\]
Expanding $(xy-1)^2$ gives:
\[
(xy-1)^2 = x^2y^2 - 2xy + 1.
\]
Substituting this into the expression, we get:
\[
x^2y^2 - 2xy + 1 + x^2 + 2xy + y^2 = x... | 0.5625 | 7,229.125 | 6,480.222222 | 8,192 |
We consider dissections of regular $n$-gons into $n - 2$ triangles by $n - 3$ diagonals which do not intersect inside the $n$-gon. A [i]bicoloured triangulation[/i] is such a dissection of an $n$-gon in which each triangle is coloured black or white and any two triangles which share an edge have different colours. We c... | 3\mid n |
To solve the problem, we need to determine which positive integers \( n \ge 4 \) allow a regular \( n \)-gon to be dissected into a bicoloured triangulation under the condition that, for each vertex \( A \), the number of black triangles having \( A \) as a vertex is greater than the number of white triangles having \... | 0 | 8,192 | -1 | 8,192 |
Given $0 \leq x \leq 2$, find the maximum and minimum values of the function $y = 4^{x- \frac {1}{2}} - 3 \times 2^{x} + 5$. | \frac {1}{2} | 0.9375 | 5,040.25 | 4,830.133333 | 8,192 | |
In a circle of radius $42$, two chords of length $78$ intersect at a point whose distance from the center is $18$. The two chords divide the interior of the circle into four regions. Two of these regions are bordered by segments of unequal lengths, and the area of either of them can be expressed uniquely in the form $m... | 378 | 0 | 8,192 | -1 | 8,192 | |
Express the quotient $2033_4 \div 22_4$ in base 4. | 11_4 | 0 | 7,718.125 | -1 | 7,718.125 | |
Find: $\frac{2}{7}+\frac{8}{10}$ | \frac{38}{35} | 0.9375 | 2,093.3125 | 2,145.333333 | 1,313 | |
Let $g: \mathbb{R} \to \mathbb{R}$ be a function such that
\[g((x + y)^2) = g(x)^2 - 2xg(y) + 2y^2\]
for all real numbers $x$ and $y.$ Find the number of possible values of $g(1)$ and the sum of all possible values of $g(1)$. | \sqrt{2} | 0 | 8,185.5625 | -1 | 8,185.5625 | |
Farmer Pythagoras has now expanded his field, which remains a right triangle. The lengths of the legs of this field are $5$ units and $12$ units, respectively. He leaves an unplanted rectangular area $R$ in the corner where the two legs meet at a right angle. This rectangle has dimensions such that its shorter side run... | \frac{151}{200} | 0 | 8,133.25 | -1 | 8,133.25 | |
Suppose $a$ is $150\%$ of $b$. What percent of $a$ is $3b$? | 200 | 1. **Understanding the problem**: We are given that $a$ is $150\%$ of $b$. This can be expressed as:
\[
a = 150\% \times b = 1.5b
\]
2. **Finding $3b$ in terms of $a$**: We need to find what percent $3b$ is of $a$. First, express $3b$ using the relationship between $a$ and $b$:
\[
3b = 3 \times b
\]
... | 1 | 1,637.125 | 1,637.125 | -1 |
In $\triangle ABC$, $A=\frac{\pi}{4}, B=\frac{\pi}{3}, BC=2$.
(I) Find the length of $AC$;
(II) Find the length of $AB$. | 1+ \sqrt{3} | 1 | 4,016.25 | 4,016.25 | -1 | |
Several positive integers are written on a blackboard. The sum of any two of them is some power of two (for example, $2, 4, 8,...$). What is the maximal possible number of different integers on the blackboard? | 2 |
To determine the maximal possible number of different positive integers on the blackboard, given the condition that the sum of any two of them must be a power of two, we proceed as follows:
First, recall that a power of two can be expressed as \(2^k\) for some integer \(k\). The integers on the blackboard must sum to... | 0 | 8,192 | -1 | 8,192 |
The coefficient of the $x$ term in the expansion of $(x^{2}-x-2)^{3}$ is what value? | -12 | 0.875 | 5,470.75 | 5,082 | 8,192 | |
On Tony's map, the distance from Saint John, NB to St. John's, NL is $21 \mathrm{~cm}$. The actual distance between these two cities is $1050 \mathrm{~km}$. What is the scale of Tony's map? | 1:5 000 000 | 0 | 493 | -1 | 493 | |
Let $n$ be a positive integer. What is the largest $k$ for which there exist $n \times n$ matrices $M_1, \dots, M_k$ and $N_1, \dots, N_k$ with real entries such that for all $i$ and $j$, the matrix product $M_i N_j$ has a zero entry somewhere on its diagonal if and only if $i \neq j$? | n^n | The largest such $k$ is $n^n$. We first show that this value can be achieved by an explicit construction. Let $e_1,\dots,e_n$ be the standard basis of $\RR^n$. For $i_1,\dots,i_n \in \{1,\dots,n\}$, let $M_{i_1,\dots,i_n}$ be the matrix with row vectors $e_{i_1},\dots,e_{i_n}$, and let $N_{i_1,\dots,i_n}$ be the transp... | 0 | 8,169.6875 | -1 | 8,169.6875 |
Let $T = (1+i)^{19} - (1-i)^{19}$, where $i=\sqrt{-1}$. Determine $|T|$. | 512\sqrt{2} | 0 | 5,426.875 | -1 | 5,426.875 | |
A fair coin is tossed 4 times. What is the probability of getting at least two consecutive heads? | \frac{5}{8} | 0 | 7,587.5625 | -1 | 7,587.5625 | |
The difference between two perfect squares is 221. What is the smallest possible sum of the two perfect squares? | 24421 | 0 | 4,733.9375 | -1 | 4,733.9375 | |
An ant starts at the origin of a coordinate plane. Each minute, it either walks one unit to the right or one unit up, but it will never move in the same direction more than twice in the row. In how many different ways can it get to the point $(5,5)$ ? | 84 | We can change the ant's sequence of moves to a sequence $a_{1}, a_{2}, \ldots, a_{10}$, with $a_{i}=0$ if the $i$-th step is up, and $a_{i}=1$ if the $i$-th step is right. We define a subsequence of moves $a_{i}, a_{i+1}, \ldots, a_{j}$, ( $i \leq j$ ) as an up run if all terms of the subsequence are equal to 0 , and $... | 0 | 8,192 | -1 | 8,192 |
The distance from \(A\) to \(B\) is 999 km. Along the road, there are kilometer markers indicating the distances to \(A\) and \(B\): 0।999, 1।998, \(\ldots, 999।0. How many of these markers have only two different digits? | 40 | 0 | 8,192 | -1 | 8,192 | |
Given that the function $f(x)$ is an odd function defined on $\mathbb{R}$ and satisfies $f(x+2)=f(x)$ for all $x \in \mathbb{R}$, and when $x \in (-1, 0)$, $f(x)=2^x$, find the value of $f(\log_2 5)$. | -\frac{4}{5} | 0.9375 | 5,087.3125 | 5,050.466667 | 5,640 | |
Determine one of the symmetry axes of the function $y = \cos 2x - \sin 2x$. | -\frac{\pi}{8} | 0.4375 | 7,218 | 7,623.142857 | 6,902.888889 | |
In the $xy$-plane, consider the L-shaped region bounded by horizontal and vertical segments with vertices at $(0,0)$, $(0,3)$, $(3,3)$, $(3,1)$, $(5,1)$ and $(5,0)$. The slope of the line through the origin that divides the area of this region exactly in half is | \frac{7}{9} | 1. **Identify the vertices and shape of the region**: The vertices of the L-shaped region are given as $A=(0,0)$, $B=(0,3)$, $C=(3,3)$, $D=(3,1)$, $E=(5,1)$, and $F=(5,0)$. The region is composed of two rectangles, one with vertices $A, B, C, D$ and the other with vertices $D, E, F, C$.
2. **Calculate the total area o... | 0 | 8,192 | -1 | 8,192 |
The number of increasing sequences of positive integers $a_1 \le a_2 \le a_3 \le \cdots \le a_{10} \le 2007$ such that $a_i-i$ is even for $1\le i \le 10$ can be expressed as ${m \choose n}$ for some positive integers $m > n$. Compute the remainder when $m$ is divided by 1000.
| 8 | 0.375 | 6,628.1875 | 4,707.5 | 7,780.6 | |
Find the sum of the first five terms in the geometric sequence $\frac13,\frac19,\frac1{27},\dots$. Express your answer as a common fraction. | \frac{121}{243} | 1 | 3,688.8125 | 3,688.8125 | -1 | |
For each integer \( n \geq 2 \), let \( A(n) \) be the area of the region in the coordinate plane defined by the inequalities \( 1 \leq x \leq n \) and \( 0 \leq y \leq x \left\lfloor \log_2{x} \right\rfloor \), where \( \left\lfloor \log_2{x} \right\rfloor \) is the greatest integer not exceeding \( \log_2{x} \). Find... | 99 | 0 | 8,192 | -1 | 8,192 | |
Let $a,$ $b,$ and $c$ be the roots of the equation $x^3 - 24x^2 + 50x - 42 = 0.$ Find the value of $\frac{a}{\frac{1}{a}+bc} + \frac{b}{\frac{1}{b}+ca} + \frac{c}{\frac{1}{c}+ab}.$ | \frac{476}{43} | 0.9375 | 4,707.5625 | 4,475.266667 | 8,192 | |
Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length $1$. The polygons meet at a point $A$ in such a way that the sum of the three interior angles at $A$ is $360^{\circ}$. Thus the three polygons form a new polygon with $A$ as an interior point. What is the largest... | 21 | 1. **Identify the Problem Requirements:**
We need to find three non-overlapping regular polygons, at least two of which are congruent, that meet at a point $A$ such that the sum of their interior angles at $A$ is $360^\circ$. The sides of each polygon are of length $1$. We aim to maximize the perimeter of the new po... | 0 | 8,161.0625 | -1 | 8,161.0625 |
Compute the sum of $302^2 - 298^2$ and $152^2 - 148^2$. | 3600 | 0.875 | 3,867.1875 | 3,249.357143 | 8,192 | |
Points $A$ and $B$ are selected on the graph of $y = -\frac{1}{2}x^2$ so that triangle $ABO$ is equilateral. Find the length of one side of triangle $ABO$. [asy]
size(150);
draw( (-4, -8) -- (-3.4641, -6)-- (-3, -9/2)-- (-5/2, -25/8)-- (-2,-2)-- (-3/2, -9/8) -- (-1, -1/2) -- (-3/4, -9/32) -- (-1/2, -1/8) -- (-1/4, -1/3... | 4\sqrt{3} | 0.9375 | 4,317 | 4,058.666667 | 8,192 | |
An equilateral triangle $ABC$ is divided by nine lines parallel to $BC$ into ten bands that are equally wide. We colour the bands alternately red and blue, with the smallest band coloured red. The difference between the total area in red and the total area in blue is $20$ $\text{cm}^2$ .
What is the area of tri... | 200 | 0.375 | 6,020 | 4,278 | 7,065.2 | |
Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(10,2)$, respectively. Calculate its area. | 52\sqrt{3} | 0.5 | 7,862.625 | 7,533.25 | 8,192 | |
Given a point P is 9 units away from the center of a circle with a radius of 15 units, find the number of chords passing through point P that have integer lengths. | 12 | 0.1875 | 7,742.9375 | 6,671.666667 | 7,990.153846 | |
How many distinct triangles can be drawn using three of the dots below as vertices?
[asy]
dot(origin^^(1,0)^^(2,0)^^(0,1)^^(1,1)^^(2,1));
[/asy] | 18 | To solve this problem, we need to count the number of distinct triangles that can be formed using three of the given dots as vertices. The dots form a grid of $2 \times 3$ (2 rows and 3 columns).
#### Step 1: Counting all possible combinations of three points
We start by calculating the total number of ways to choose ... | 0.3125 | 7,500.8125 | 6,438 | 7,983.909091 |
Compute the value of \[N = 100^2 + 99^2 - 98^2 - 97^2 + 96^2 + \cdots + 4^2 + 3^2 - 2^2 - 1^2,\]where the additions and subtractions alternate in pairs. | 10100 | 0.5625 | 6,750.3125 | 5,629 | 8,192 | |
How many ways are there to put 6 balls in 4 boxes if the balls are not distinguishable but the boxes are? | 84 | 1 | 2,424.8125 | 2,424.8125 | -1 | |
Given that z and w are complex numbers with a modulus of 1, and 1 ≤ |z + w| ≤ √2, find the minimum value of |z - w|. | \sqrt{2} | 0.9375 | 6,370.8125 | 6,249.4 | 8,192 | |
How many integers $-11 \leq n \leq 11$ satisfy $(n-2)(n+4)(n + 8)<0$? | 8 | 1 | 3,518 | 3,518 | -1 | |
In a 24-hour format digital watch that displays hours and minutes, calculate the largest possible sum of the digits in the display if the sum of the hour digits must be even. | 22 | 0 | 7,539.625 | -1 | 7,539.625 | |
In a football championship with 16 teams, each team played with every other team exactly once. A win was awarded 3 points, a draw 1 point, and a loss 0 points. A team is considered successful if it scored at least half of the maximum possible points. What is the maximum number of successful teams that could have partic... | 15 | 0 | 8,192 | -1 | 8,192 | |
An equilateral triangle is subdivided into 4 smaller equilateral triangles. Using red and yellow to paint the vertices of the triangles, each vertex must be colored and only one color can be used per vertex. If two colorings are considered the same when they can be made identical by rotation, how many different colorin... | 24 | 0.5625 | 5,508.25 | 4,239.222222 | 7,139.857143 | |
A pyramid is intersected by a plane parallel to its base, dividing its lateral surface into two parts of equal area. In what ratio does this plane divide the lateral edges of the pyramid? | \frac{1}{\sqrt{2}} | 0 | 7,520.6875 | -1 | 7,520.6875 | |
Given a sphere, a cylinder with a square axial section, and a cone. The cylinder and cone have identical bases, and their heights equal the diameter of the sphere. How do the volumes of the cylinder, sphere, and cone compare to each other? | 3 : 2 : 1 | 0.0625 | 4,236.375 | 2,195 | 4,372.466667 | |
The Benton Youth Soccer Team has 20 players on the team, including reserves. Of these, three are goalies. Today, the team is having a contest to see which goalie can block the most number of penalty kicks. For each penalty kick, a goalie stands in the net while the rest of the team (including other goalies) takes a sho... | 57 | 0 | 7,452 | -1 | 7,452 | |
Consider a kite-shaped field with the following measurements and angles: sides AB = 120 m, BC = CD = 80 m, DA = 120 m. The angle between sides AB and BC is 120°. The angle between sides CD and DA is also 120°. The wheat harvested from any location in the field is brought to the nearest point on the field's perimeter. W... | \frac{1}{2} | 0 | 8,192 | -1 | 8,192 | |
What percent of $x$ is equal to $40\%$ of $50\%$ of $x$? | 20 | 1 | 1,555.25 | 1,555.25 | -1 | |
Given the function $f(x)=ax+b\sin x\ (0 < x < \frac {π}{2})$, if $a\neq b$ and $a, b\in \{-2,0,1,2\}$, the probability that the slope of the tangent line at any point on the graph of $f(x)$ is non-negative is ___. | \frac {7}{12} | 0.25 | 7,760.625 | 6,568.5 | 8,158 | |
Calculate the area of the parallelogram formed by the vectors \( a \) and \( b \).
$$
\begin{aligned}
& a = p - 4q \\
& b = 3p + q \\
& |p| = 1 \\
& |q| = 2 \\
& \angle(p, q) = \frac{\pi}{6}
\end{aligned}
$$ | 13 | 0.9375 | 3,514.8125 | 3,203 | 8,192 | |
Place 6 balls, labeled from 1 to 6, into 3 different boxes with each box containing 2 balls. If the balls labeled 1 and 2 cannot be placed in the same box, the total number of different ways to do this is \_\_\_\_\_\_. | 72 | 0.25 | 7,966.125 | 7,422.25 | 8,147.416667 | |
What is the sum of all two-digit positive integers whose squares end with the digits 25? | 495 | 0.9375 | 3,518.1875 | 3,206.6 | 8,192 | |
In how many ways can 9 distinct items be distributed into three boxes so that one box contains 3 items, another contains 2 items, and the third contains 4 items? | 7560 | 0.125 | 3,476.3125 | 5,634.5 | 3,168 | |
Find four positive integers that are divisors of each number in the list $$45, 90, -15, 135, 180.$$ Calculate the sum of these four integers. | 24 | 0.75 | 3,975.5625 | 2,570.083333 | 8,192 | |
An underground line has $26$ stops, including the first and the final one, and all the stops are numbered from $1$ to $26$ according to their order. Inside the train, for each pair $(x,y)$ with $1\leq x < y \leq 26$ there is exactly one passenger that goes from the $x$ -th stop to the $y$ -th one. If every ... | 25 | 0.1875 | 7,404.75 | 6,332.666667 | 7,652.153846 | |
If $x\%$ of five-digit numbers have at least one repeated digit, then what is $x$? Express your answer as a decimal to the nearest tenth. | 69.8 | 0.6875 | 4,477.625 | 3,745.090909 | 6,089.2 | |
How many positive divisors does $24$ have? | 8 | 1 | 1,897.1875 | 1,897.1875 | -1 | |
George purchases a sack of apples, a bunch of bananas, a cantaloupe, and a carton of dates for $ \$ 20$. If a carton of dates costs twice as much as a sack of apples and the price of a cantaloupe is equal to the price of a sack of apples minus a bunch of bananas, how much would it cost George to purchase a bunch of ban... | \$ 5 | 1 | 1,793.8125 | 1,793.8125 | -1 | |
Find the digits left and right of the decimal point in the decimal form of the number \[ (\sqrt{2} + \sqrt{3})^{1980}. \] | 7.9 | 0 | 8,192 | -1 | 8,192 | |
Simplify the expression $\dfrac{20}{21} \cdot \dfrac{35}{54} \cdot \dfrac{63}{50}$. | \frac{7}{9} | 0.9375 | 4,497.375 | 4,251.066667 | 8,192 | |
In triangle $XYZ,$ points $G,$ $H,$ and $I$ are on sides $\overline{YZ},$ $\overline{XZ},$ and $\overline{XY},$ respectively, such that $YG:GZ = XH:HZ = XI:IY = 2:3.$ Line segments $\overline{XG},$ $\overline{YH},$ and $\overline{ZI}$ intersect at points $S,$ $T,$ and $U,$ respectively. Compute $\frac{[STU]}{[XYZ]}.$ | \frac{9}{55} | 0 | 8,192 | -1 | 8,192 | |
Alec must purchase 14 identical shirts and only has $\$130$. There is a flat $\$2$ entrance fee for shopping at the warehouse store where he plans to buy the shirts. The price of each shirt is the same whole-dollar amount. Assuming a $5\%$ sales tax is added to the price of each shirt, what is the greatest possible ... | 8 | 0.9375 | 5,509.625 | 5,330.8 | 8,192 | |
A number of trucks with the same capacity were requested to transport cargo from one place to another. Due to road issues, each truck had to carry 0.5 tons less than planned, which required 4 additional trucks. The mass of the transported cargo was at least 55 tons but did not exceed 64 tons. How many tons of cargo wer... | 2.5 | 0.125 | 5,138.5 | 4,484 | 5,232 | |
If \\(f(x)\\) is an odd function defined on \\(R\\) and satisfies \\(f(x+1)=f(x-1)\\), and when \\(x \in (0,1)\\), \\(f(x)=2^{x}-2\\), then the value of \\(f(\log_{\frac{1}{2}}24)\\) is \_\_\_\_\_\_. | \frac{1}{2} | 0.8125 | 6,071.6875 | 5,582.384615 | 8,192 | |
Consider a rectangle $ABCD$ containing three squares. Two smaller squares each occupy a part of rectangle $ABCD$, and each smaller square has an area of 1 square inch. A larger square, also inside rectangle $ABCD$ and not overlapping with the smaller squares, has a side length three times that of one of the smaller squ... | 11 | 0.0625 | 7,553.375 | 6,093 | 7,650.733333 | |
Twelve chairs are evenly spaced around a round table and numbered clockwise from $1$ through $12$. Six married couples are to sit in the chairs with men and women alternating, and no one is to sit either next to or across from his/her spouse or next to someone of the same profession. Determine the number of seating arr... | 2880 | 0 | 8,192 | -1 | 8,192 | |
Using the digits 0, 2, 3, 5, 7, how many four-digit numbers divisible by 5 can be formed if:
(1) Digits do not repeat;
(2) Digits can repeat. | 200 | 0.1875 | 4,246.375 | 5,234.333333 | 4,018.384615 | |
What is the ratio of the numerical value of the area, in square units, of an equilateral triangle of side length 8 units to the numerical value of its perimeter, in units? Express your answer as a common fraction in simplest radical form. | \frac{2\sqrt{3}}{3} | 0 | 1,190.5625 | -1 | 1,190.5625 | |
For real numbers $x$, let
\[P(x)=1+\cos(x)+i\sin(x)-\cos(2x)-i\sin(2x)+\cos(3x)+i\sin(3x)\]
where $i = \sqrt{-1}$. For how many values of $x$ with $0\leq x<2\pi$ does
\[P(x)=0?\] | 0 | 1. **Express $P(x)$ using Euler's formula**: Euler's formula states that $e^{i\theta} = \cos(\theta) + i\sin(\theta)$. Using this, we can rewrite $P(x)$ as:
\[
P(x) = 1 + e^{ix} - e^{2ix} + e^{3ix}
\]
where $e^{ix} = \cos(x) + i\sin(x)$, $e^{2ix} = \cos(2x) + i\sin(2x)$, and $e^{3ix} = \cos(3x) + i\sin(3x)$... | 0 | 8,192 | -1 | 8,192 |
Let $ABCD$ be a square. Let $E, F, G$ and $H$ be the centers, respectively, of equilateral triangles with bases $\overline{AB}, \overline{BC}, \overline{CD},$ and $\overline{DA},$ each exterior to the square. What is the ratio of the area of square $EFGH$ to the area of square $ABCD$? | \frac{2+\sqrt{3}}{3} | 1. **Assign Coordinates to Square $ABCD$**:
Assume the side length of square $ABCD$ is $s$. Without loss of generality, let $s = 6$ for simplicity. Place $ABCD$ in the coordinate plane with $A = (0, 0)$, $B = (6, 0)$, $C = (6, 6)$, and $D = (0, 6)$.
2. **Locate Points $E, F, G, H$**:
- **$E$** is the center of a... | 0 | 8,192 | -1 | 8,192 |
In the cartesian coordinate system $(xOy)$, curve $({C}_{1})$ is defined by the parametric equations $\begin{cases}x=t+1,\ y=1-2t\end{cases}$ and curve $({C}_{2})$ is defined by the parametric equations $\begin{cases}x=a\cos θ,\ y=3\sin θ\end{cases}$ where $a > 0$.
1. If curve $({C}_{1})$ and curve $({C}_{2})$ have a c... | \frac{12\sqrt{5}}{5} | 0 | 4,850.125 | -1 | 4,850.125 | |
Given that $a$, $b$, $c \in R^{+}$ and $a + b + c = 1$, find the maximum value of $\sqrt{4a + 1} + \sqrt{4b + 1} + \sqrt{4c + 1}$. | \sqrt{21} | 0.875 | 4,853.4375 | 4,376.5 | 8,192 | |
Given the function $f(x) = x^3 - ax^2 + 3x$, and $x=3$ is an extremum of $f(x)$.
(Ⅰ) Determine the value of the real number $a$;
(Ⅱ) Find the equation of the tangent line $l$ to the graph of $y=f(x)$ at point $P(1, f(1))$;
(Ⅲ) Find the minimum and maximum values of $f(x)$ on the interval $[1, 5]$. | 15 | 1 | 2,930 | 2,930 | -1 | |
Given the function $f(x)=2\sin ωx\cos ωx-2\sqrt{3} \cos ^{2}ωx+\sqrt{3} (ω > 0)$, and the distance between two adjacent symmetry axes of the graph of $y=f(x)$ is $\frac{π}{2}$.
(I) Find the period of the function $f(x)$;
(II) In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ res... | 3\sqrt{3} | 0.625 | 6,926.0625 | 6,430.8 | 7,751.5 | |
There are 6 boxes, each containing a key that cannot be used interchangeably. If one key is placed in each box and all the boxes are locked, and it is required that after breaking open one box, the remaining 5 boxes can still be opened with the keys, then the number of ways to place the keys is ______. | 120 | 0.0625 | 6,606.9375 | 6,526 | 6,612.333333 | |
In the rectangular coordinate system on a plane, establish a polar coordinate system with $O$ as the pole and the positive semi-axis of $x$ as the polar axis. The parametric equations of the curve $C$ are $\begin{cases} x=1+\cos \alpha \\ y=\sin \alpha \end{cases} (\alpha \text{ is the parameter, } \alpha \in \left[ 0,... | \frac{5 \sqrt{2}}{2}-1 | 0 | 7,116.5 | -1 | 7,116.5 | |
Given two lines $l_{1}$: $mx+2y-2=0$ and $l_{2}$: $5x+(m+3)y-5=0$, if $l_{1}$ is parallel to $l_{2}$, find the value of $m$. | -5 | 0.125 | 5,965.1875 | 6,741 | 5,854.357143 | |
Convert $\sqrt{2} e^{11 \pi i/4}$ to rectangular form. | -1 + i | 1 | 2,730.4375 | 2,730.4375 | -1 | |
Given that $m$ and $n$ are two non-coincident lines, and $\alpha$, $\beta$, $\gamma$ are three pairwise non-coincident planes, consider the following four propositions:
$(1)$ If $m \perp \alpha$ and $m \perp \beta$, then $\alpha \parallel \beta$;
$(2)$ If $\alpha \perp \gamma$ and $\beta \perp \gamma$, then $\alpha... | (2)(3)(4) | 0 | 3,146.4375 | -1 | 3,146.4375 | |
12. If $p$ is the smallest positive prime number such that there exists an integer $n$ for which $p$ divides $n^{2}+5n+23$, then $p=$ ______ | 13 | 0 | 6,749.8125 | -1 | 6,749.8125 | |
How many times do you have to subtract 8 from 792 to get 0? | 99 | 1 | 261.25 | 261.25 | -1 | |
Given the vertices of a pentagon at coordinates $(1, 1)$, $(4, 1)$, $(5, 3)$, $(3, 5)$, and $(1, 4)$, calculate the area of this pentagon. | 12 | 0.75 | 6,154.0625 | 5,474.75 | 8,192 | |
Determine the binomial coefficient and the coefficient of the 4th term in the expansion of $\left( \left. x^{2}- \frac{1}{2x} \right. \right)^{9}$. | - \frac{21}{2} | 1 | 2,699.375 | 2,699.375 | -1 | |
What is the distance between the center of the circle with equation $x^2+y^2=2x+4y-1$ and the point $(13,7)$? | 13 | 1 | 1,696.0625 | 1,696.0625 | -1 | |
Compute $\dbinom{25}{2}$. | 300 | 1 | 1,801.625 | 1,801.625 | -1 | |
Five fair ten-sided dice are rolled. Calculate the probability that at least four of the five dice show the same value. | \frac{23}{5000} | 0.4375 | 5,670.75 | 5,112.428571 | 6,105 | |
Find the integer $n,$ $-90 \le n \le 90,$ such that $\sin n^\circ = \sin 604^\circ.$ | -64 | 1 | 3,317.0625 | 3,317.0625 | -1 | |
Given a square ABCD with a side length of 2, points M and N are the midpoints of sides BC and CD, respectively. If vector $\overrightarrow {MN}$ = x $\overrightarrow {AB}$ + y $\overrightarrow {AD}$, find the values of xy and $\overrightarrow {AM}$ • $\overrightarrow {MN}$. | -1 | 1 | 2,452.125 | 2,452.125 | -1 | |
Suppose two distinct integers are chosen from between 1 and 29, inclusive. What is the probability that their product is neither a multiple of 2 nor 3? | \dfrac{45}{406} | 0.4375 | 7,148.5 | 5,806.857143 | 8,192 | |
If $\frac{137}{a}=0.1 \dot{2} 3 \dot{4}$, find the value of $a$. | 1110 | 0.25 | 6,945.125 | 3,204.5 | 8,192 |
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