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 |
|---|---|---|---|---|---|---|
Let $\\((2-x)^5 = a_0 + a_1x + a_2x^2 + \ldots + a_5x^5\\)$. Evaluate the value of $\dfrac{a_0 + a_2 + a_4}{a_1 + a_3}$. | -\dfrac{122}{121} | 0.0625 | 5,925.6875 | 2,531 | 6,152 | |
Given a tower with a height of $8$ cubes, where a blue cube must always be at the top, determine the number of different towers the child can build using $2$ red cubes, $4$ blue cubes, and $3$ green cubes. | 210 | 0.1875 | 6,574.9375 | 5,384.666667 | 6,849.615385 | |
Four balls of radius $1$ are mutually tangent, three resting on the floor and the fourth resting on the others.
A tetrahedron, each of whose edges have length $s$, is circumscribed around the balls. Then $s$ equals | 2+2\sqrt{6} | To solve this problem, we need to find the edge length $s$ of the tetrahedron that circumscribes four mutually tangent balls of radius $1$. The configuration described is such that three balls are resting on the floor and the fourth ball is resting on top of these three.
1. **Understanding the Configuration**:
- Th... | 0.0625 | 7,371.125 | 6,113 | 7,455 |
Assume an even function $f(x)$ satisfies $f(x+6) = f(x) + f(3)$ for any $x \in \mathbb{R}$, and $f(x) = 5x$ when $x \in (-3, -2)$. Calculate $f(201.2)$. | -16 | 0.0625 | 7,781.25 | 7,597 | 7,793.533333 | |
The number \( C \) is defined as the sum of all the positive integers \( n \) such that \( n-6 \) is the second largest factor of \( n \). What is the value of \( 11C \)? | 308 | 0.5 | 6,083.625 | 5,112.5 | 7,054.75 | |
Consider equations of the form $x^2 + bx + c = 0$. How many such equations have real roots and have coefficients $b$ and $c$ selected from the set of integers $\{1,2,3, 4, 5,6\}$? | 19 | To determine how many quadratic equations of the form $x^2 + bx + c = 0$ have real roots with $b$ and $c$ chosen from the set $\{1, 2, 3, 4, 5, 6\}$, we need to ensure that the discriminant of the quadratic equation is non-negative. The discriminant $\Delta$ for a quadratic equation $ax^2 + bx + c = 0$ is given by:
\[
... | 1 | 4,333.5625 | 4,333.5625 | -1 |
Given circle $C_1$: $(x-1)^2+(y-2)^2=1$
(1) Find the equation of the tangent line to circle $C_1$ passing through point $P(2,4)$.
(2) If circle $C_1$ intersects with circle $C_2$: $(x+1)^2+(y-1)^2=4$ at points $A$ and $B$, find the length of segment $AB$. | \frac {4 \sqrt {5}}{5} | 0 | 7,744.875 | -1 | 7,744.875 | |
Given $f(x)=\frac{1}{2}\cos^{2}x-\frac{1}{2}\sin^{2}x+1-\sqrt{3}\sin x \cos x$.
$(1)$ Find the period and the interval where $f(x)$ is monotonically decreasing.
$(2)$ Find the minimum value of $f(x)$ on $[0,\frac{\pi}{2}]$ and the corresponding set of independent variables. | \left\{\frac{\pi}{3}\right\} | 0 | 6,835.5625 | -1 | 6,835.5625 | |
When a bucket is two-thirds full of water, the bucket and water weigh $a$ kilograms. When the bucket is one-half full of water the total weight is $b$ kilograms. In terms of $a$ and $b$, what is the total weight in kilograms when the bucket is full of water? | 3a - 2b |
Let's denote:
- $x$ as the weight of the empty bucket.
- $y$ as the weight of the water when the bucket is full.
From the problem, we have two equations based on the given conditions:
1. When the bucket is two-thirds full, the total weight is $a$ kilograms:
\[
x + \frac{2}{3}y = a
\]
2. When the bucket is on... | 0.8125 | 5,949.625 | 5,828.615385 | 6,474 |
Petya's bank account contains 500 dollars. The bank allows only two types of transactions: withdrawing 300 dollars or adding 198 dollars.
What is the maximum amount Petya can withdraw from the account if he does not have any other money? | 498 | 0 | 8,174.125 | -1 | 8,174.125 | |
Let $g$ be a function defined by $g(x) = \frac{px + q}{rx + s}$, where $p$, $q$, $r$ and $s$ are nonzero real numbers, and the function has the properties $g(31)=31$, $g(41)=41$, and $g(g(x))=x$ for all values except $\frac{-s}{r}$. Determine the unique number that is not in the range of $g$. | 36 | 0.8125 | 5,708.625 | 5,135.538462 | 8,192 | |
If real numbers \( x \) and \( y \) satisfy the relation \( xy - x - y = 1 \), calculate the minimum value of \( x^{2} + y^{2} \). | 6 - 4\sqrt{2} | 0.625 | 7,642.75 | 7,313.2 | 8,192 | |
Suppose $a$ and $b$ are positive integers. Isabella and Vidur both fill up an $a \times b$ table. Isabella fills it up with numbers $1,2, \ldots, a b$, putting the numbers $1,2, \ldots, b$ in the first row, $b+1, b+2, \ldots, 2 b$ in the second row, and so on. Vidur fills it up like a multiplication table, putting $i j... | 21 | Using the formula $1+2+\cdots+n=\frac{n(n+1)}{2}$, we get $$\begin{aligned} \frac{a b(a b+1)}{2}-\frac{a(a+1)}{2} \cdot \frac{b(b+1)}{2} & =\frac{a b(2(a b+1)-(a+1)(b+1))}{4} \\ & =\frac{a b(a b-a-b+1)}{4} \\ & =\frac{a b(a-1)(b-1)}{4} \\ & =\frac{a(a-1)}{2} \cdot \frac{b(b-1)}{2} \end{aligned}$$ This means we can writ... | 0.0625 | 8,084.4375 | 6,471 | 8,192 |
Angles $A$ and $B$ are supplementary. If the measure of angle $A$ is $8$ times angle $B$, what is the measure of angle A? | 160 | 1 | 1,060.25 | 1,060.25 | -1 | |
A notebook with 75 pages numbered from 1 to 75 is renumbered in reverse, from 75 to 1. Determine how many pages have the same units digit in both the old and new numbering systems. | 15 | 0.8125 | 6,242.9375 | 5,793.153846 | 8,192 | |
Four friends initially plan a road trip and decide to split the fuel cost equally. However, 3 more friends decide to join at the last minute. Due to the increase in the number of people sharing the cost, the amount each of the original four has to pay decreases by $\$$8. What was the total cost of the fuel? | 74.67 | 0 | 3,858.6875 | -1 | 3,858.6875 | |
What is the smallest positive integer with exactly 20 positive divisors? | 432 | 0 | 6,376.9375 | -1 | 6,376.9375 | |
Given vectors $\overrightarrow {a}$=(2,6) and $\overrightarrow {b}$=(m,-1), find the value of m when $\overrightarrow {a}$ is perpendicular to $\overrightarrow {b}$ and when $\overrightarrow {a}$ is parallel to $\overrightarrow {b}$. | -\frac{1}{3} | 1 | 1,680.75 | 1,680.75 | -1 | |
Find the minimum value of the expression \(\frac{5 x^{2}-8 x y+5 y^{2}-10 x+14 y+55}{\left(9-25 x^{2}+10 x y-y^{2}\right)^{5 / 2}}\). If necessary, round the answer to hundredths. | 0.19 | 0.125 | 8,172 | 8,032 | 8,192 | |
A magician has $5432_{9}$ tricks for his magical performances. How many tricks are there in base 10? | 3998 | 0.5 | 2,601.125 | 2,698 | 2,504.25 | |
From the numbers 2, 3, 4, 5, 6, 7, 8, 9, two different numbers are selected to be the base and the exponent of a logarithm, respectively. How many different logarithmic values can be formed? | 52 | 0 | 8,192 | -1 | 8,192 | |
The sum of the first and third of three consecutive integers is 118. What is the value of the second integer? | 59 | 1 | 1,558.5 | 1,558.5 | -1 | |
The expression $\frac{x-3}{4x}$ is equal to zero for what value of $x$? | 3 | 1 | 1,108.5625 | 1,108.5625 | -1 | |
If $y<0$, find the range of all possible values of $y$ such that $\lceil{y}\rceil\cdot\lfloor{y}\rfloor=110$. Express your answer using interval notation. | (-11, -10) | 0.4375 | 5,459.1875 | 4,857.142857 | 5,927.444444 | |
$ f(x)$ is a given polynomial whose degree at least 2. Define the following polynomial-sequence: $ g_1(x)\equal{}f(x), g_{n\plus{}1}(x)\equal{}f(g_n(x))$ , for all $ n \in N$ . Let $ r_n$ be the average of $ g_n(x)$ 's roots. If $ r_{19}\equal{}99$ , find $ r_{99}$ . | 99 | 0.25 | 7,821.9375 | 6,711.75 | 8,192 | |
Given non-zero plane vectors $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ satisfy $|\overrightarrow{a}|=2$, $|\overrightarrow{b}-\overrightarrow{c}|=1$. If the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{π}{3}$, calculate the minimum value of $|\overrightarrow{a}-\overright... | \sqrt{3} - 1 | 0.3125 | 7,728.6875 | 6,709.4 | 8,192 | |
A rock is dropped off a cliff of height $ h $ As it falls, a camera takes several photographs, at random intervals. At each picture, I measure the distance the rock has fallen. Let the average (expected value) of all of these distances be $ kh $ . If the number of photographs taken is huge, find $ k $ . That is: wh... | $\dfrac{1}{3}$ | 0 | 3,275.9375 | -1 | 3,275.9375 | |
The time right now is exactly midnight. What time will it be in 1234 minutes? | 8\!:\!34 \text{ p.m.} | 0.625 | 4,362.5625 | 4,202.1 | 4,630 | |
What is the area, in square units, of the square with the four vertices at $A\ (0, 0)$, $B\ (-5, -1)$, $C\ (-4, -6)$ and $D\ (1, -5)$? | 26 | 1 | 3,030.125 | 3,030.125 | -1 | |
The distance from home to work is $s = 6$ km. At the moment Ivan left work, his favorite dog dashed out of the house and ran to meet him. They met at a distance of one-third of the total route from work. The dog immediately turned back and ran home. Upon reaching home, the dog turned around instantly and ran back towar... | 12 | 0.0625 | 7,083.1875 | 7,268 | 7,070.866667 | |
You are given 16 pieces of paper numbered $16,15, \ldots, 2,1$ in that order. You want to put them in the order $1,2, \ldots, 15,16$ switching only two adjacent pieces of paper at a time. What is the minimum number of switches necessary? | 120 | Piece 16 has to move to the back 15 times, piece 15 has to move to the back 14 times, ..., piece 2 has to move to the back 1 time, piece 1 has to move to the back 0 times. Since only one piece can move back in each switch, we must have at least $15+14+\ldots+1=\mathbf{120}$ switches. | 0.625 | 5,096.6875 | 3,356.9 | 7,996.333333 |
Natasha and Inna each bought the same box of tea bags. It is known that one tea bag is enough for either two or three cups of tea. This box lasted Natasha for 41 cups of tea, and Inna for 58 cups of tea. How many tea bags were in the box? | 20 | 0.3125 | 1,515.1875 | 2,627.2 | 1,009.727273 | |
Miki has a dozen oranges of the same size and a dozen pears of the same size. Miki uses her juicer to extract 8 ounces of pear juice from 3 pears and 8 ounces of orange juice from 2 oranges. She makes a pear-orange juice blend from an equal number of pears and oranges. What percent of the blend is pear juice? | 40 | 1. **Determine the amount of juice from pears and oranges:**
- Miki extracts 8 ounces of pear juice from 3 pears. Therefore, the amount of pear juice per pear is:
\[
\frac{8 \text{ ounces}}{3 \text{ pears}} = \frac{8}{3} \text{ ounces per pear}
\]
- Miki extracts 8 ounces of orange juice from 2 ora... | 1 | 2,315.625 | 2,315.625 | -1 |
In the triangle below, find $XY$. Triangle $XYZ$ is a right triangle with $XZ = 18$ and $Z$ as the right angle. Angle $Y = 60^\circ$.
[asy]
unitsize(1inch);
pair P,Q,R;
P = (0,0);
Q= (1,0);
R = (0.5,sqrt(3)/2);
draw (P--Q--R--P,linewidth(0.9));
draw(rightanglemark(R,P,Q,3));
label("$X$",P,S);
label("$Y$",Q,S);
label("... | 36 | 0.125 | 4,607.5 | 6,707.5 | 4,307.5 | |
A $6$ -inch-wide rectangle is rotated $90$ degrees about one of its corners, sweeping out an area of $45\pi$ square inches, excluding the area enclosed by the rectangle in its starting position. Find the rectangle’s length in inches. | 12 | 0.75 | 5,482 | 4,860.75 | 7,345.75 | |
An arithmetic sequence {a_n} has a sum of the first n terms as S_n, and S_6/S_3 = 4. Find the value of S_9/S_6. | \dfrac{9}{4} | 1 | 3,935.4375 | 3,935.4375 | -1 | |
Find the number of different complex numbers $z$ such that $|z|=1$ and $z^{7!}-z^{6!}$ is a real number. | 7200 | 0 | 8,192 | -1 | 8,192 | |
What is the smallest positive integer \( n \) such that \( 5n \equiv 105 \pmod{24} \)? | 21 | 1 | 3,331.4375 | 3,331.4375 | -1 | |
As shown in the diagram, square ABCD and square EFGH have their corresponding sides parallel to each other. Line CG is extended to intersect with line BD at point I. Given that BD = 10, the area of triangle BFC is 3, and the area of triangle CHD is 5, what is the length of BI? | 15/4 | 0.375 | 7,438.5625 | 6,182.833333 | 8,192 | |
Two prime numbers that differ by exactly 2 are called twin primes. For example, 3 and 5 are twin primes, as are 29 and 31. In number theory research, twin primes are one of the most popular topics. If a pair of twin primes both do not exceed 200, what is the maximum sum of these two prime numbers? | 396 | 0.9375 | 2,353.875 | 2,334.6 | 2,643 | |
Using the numbers $2, 4, 12, 40$ each exactly once, you can perform operations to obtain 24. | 40 \div 4 + 12 + 2 | 0 | 7,495.3125 | -1 | 7,495.3125 | |
What number is a multiple of every integer? | 0 | 0.6875 | 4,431.375 | 2,722 | 8,192 | |
A certain type of alloy steel production company needs to ensure that the carbon content percentage of the alloy steel is within a specified range. Under the same test conditions, the inspector randomly samples $10$ times each day and measures the carbon content (unit: $\%$). It is known that the carbon content of thei... | 0.026 | 0 | 7,441.5625 | -1 | 7,441.5625 | |
Adding two dots above the decimal 0.142857 makes it a repeating decimal. The 2020th digit after the decimal point is 5. What is the repeating cycle? $\quad$ . | 142857 | 0.0625 | 8,104.25 | 6,788 | 8,192 | |
In triangle $ABC$, $\angle A$ is a right angle, and $\sin B$ is given as $\frac{3}{5}$. Calculate $\cos C$. | \frac{3}{5} | 0.9375 | 2,994.75 | 2,648.266667 | 8,192 | |
My frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score. The game ends after one of the two teams scores three points (total, not necessarily consecutive). If every possible sequence of scores is equally likely, what is the expected score of the losing team? | 3/2 | 0.3125 | 6,930.375 | 5,506.2 | 7,577.727273 | |
Given that the area of $\triangle ABC$ is 360, and point $P$ is a point on the plane of the triangle, with $\overrightarrow {AP}= \frac {1}{4} \overrightarrow {AB}+ \frac {1}{4} \overrightarrow {AC}$, then the area of $\triangle PAB$ is \_\_\_\_\_\_. | 90 | 0.875 | 5,730.875 | 5,379.285714 | 8,192 | |
What is the value of the sum $-1 + 2 - 3 + 4 - 5 + 6 - 7 +\dots+ 10,\!000$? | 5000 | 0.9375 | 3,805.125 | 3,512.666667 | 8,192 | |
Calculate the following expression:
\[\left( 1 - \frac{1}{\cos 30^\circ} \right) \left( 1 + \frac{1}{\sin 60^\circ} \right) \left( 1 - \frac{1}{\sin 30^\circ} \right) \left( 1 + \frac{1}{\cos 60^\circ} \right).\] | -1 | 0 | 3,765.6875 | -1 | 3,765.6875 | |
Points $A,$ $B,$ $C,$ and $D$ are equally spaced along a line such that $AB = BC = CD.$ A point $P$ is located so that $\cos \angle APC = \frac{4}{5}$ and $\cos \angle BPD = \frac{3}{5}.$ Determine $\sin (2 \angle BPC).$ | \frac{18}{25} | 0 | 8,192 | -1 | 8,192 | |
Egor wrote a number on the board and encoded it according to the rules of letter puzzles (different letters correspond to different digits, and identical letters correspond to identical digits). The word "GUATEMALA" was the result. How many different numbers could Egor have originally written, if his number was divisib... | 20160 | 0 | 7,713.1875 | -1 | 7,713.1875 | |
Joe and JoAnn each bought 12 ounces of coffee in a 16-ounce cup. Joe drank 2 ounces of his coffee and then added 2 ounces of cream. JoAnn added 2 ounces of cream, stirred the coffee well, and then drank 2 ounces. What is the resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee? Express y... | \frac{7}{6} | 1 | 2,324.4375 | 2,324.4375 | -1 | |
Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E\left[ X \right] = 1$, $E\left[ X^2 \right] = 2$, and $E \left[ X^3 \right] = 5$. Determine the smallest possible value of the probability of the event $X=0$. | \frac{1}{3} | The answer is $\frac{1}{3}$. Let $a_n = P(X=n)$; we want the minimum value for $a_0$. If we write $S_k = \sum_{n=1}^\infty n^k a_n$, then the given expectation values imply that $S_1 = 1$, $S_2 = 2$, $S_3 = 5$. Now define $f(n) = 11n-6n^2+n^3$, and note that $f(0) = 0$, $f(1)=f(2)=f(3)=6$, and $f(n)>6$ for $n\geq 4$; t... | 0.125 | 8,189.1875 | 8,169.5 | 8,192 |
The grade received on a certain teacher's 100-point test varies in direct proportion to the amount of time a student spends preparing for the test. If a student receives 72 points on a test for which she spent 3 hours preparing, what score would she receive on the next test if she spent 4 hours preparing? | 96 | 1 | 636.125 | 636.125 | -1 | |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(f(x) - y) = f(x) + f(f(y) - f(-x)) + x\]for all real numbers $x$ and $y.$
Let $n$ be the number of possible values of $f(3),$ and let $s$ be the sum of all possible values of $f(3).$ Find $n \times s.$ | -3 | 0.3125 | 7,894.4375 | 7,239.8 | 8,192 | |
Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is $8: 7$. Find the minimum possible value of their common perimeter.
| 676 | 0.4375 | 7,733 | 7,142.857143 | 8,192 | |
2 diagonals of a regular decagon (a 10-sided polygon) are chosen. What is the probability that their intersection lies inside the decagon? | \dfrac{42}{119} | 0 | 6,499.25 | -1 | 6,499.25 | |
The sum of all digits used in the numbers 1, 2, 3, ..., 999 is . | 13500 | 0.1875 | 7,906.875 | 6,671.333333 | 8,192 | |
A rectangular prism has dimensions 10 inches by 3 inches by 30 inches. If a cube has the same volume as this prism, what is the surface area of the cube, in square inches? | 6 \times 900^{2/3} | 0 | 7,784 | -1 | 7,784 | |
Can you multiply 993 and 879 in your head? Interestingly, if we have two two-digit numbers containing the same number of tens, and the sum of the digits of their units place equals 10, then such numbers can always be multiplied mentally as follows:
Suppose we need to multiply 97 by 93. Multiply 7 by 3 and write down t... | 872847 | 0.3125 | 7,603.875 | 6,413.6 | 8,144.909091 | |
A hare is jumping in one direction on a strip divided into cells. In one jump, it can move either one cell or two cells. How many ways can the hare get from the 1st cell to the 12th cell? | 144 | 0.25 | 5,692.8125 | 4,251 | 6,173.416667 | |
Given that in the expansion of $\left(1+x\right)^{n}$, the coefficient of $x^{3}$ is the largest, then the sum of the coefficients of $\left(1+x\right)^{n}$ is ____. | 64 | 0.875 | 4,709.5 | 4,544.428571 | 5,865 | |
Find \(x\) if
\[2 + 7x + 12x^2 + 17x^3 + \dotsb = 100.\] | 0.6 | 0 | 6,002.6875 | -1 | 6,002.6875 | |
Given two integers \( m \) and \( n \) which are coprime, calculate the GCD of \( 5^m + 7^m \) and \( 5^n + 7^n \). | 12 | 0 | 8,192 | -1 | 8,192 | |
In an opaque bag, there are four identical balls labeled with numbers $3$, $4$, $5$, and $6$ respectively. Outside the bag, there are two balls labeled with numbers $3$ and $6$. Determine the probability that a triangle with the drawn ball and the numbers on the two balls outside the bag forms an isosceles triangle. | \frac{1}{4} | 0.125 | 7,361.1875 | 7,227 | 7,380.357143 | |
What is the smallest integer $n$, greater than $1$, such that $n^{-1}\pmod{1050}$ is defined? | 11 | 1 | 2,416.9375 | 2,416.9375 | -1 | |
In triangle $ABC$, $\tan \angle CAB = 22/7$, and the altitude from $A$ divides $BC$ into segments of length 3 and 17. What is the area of triangle $ABC$? | 110 | Let $D$ be the intersection of the altitude with $\overline{BC}$, and $h$ be the length of the altitude. Without loss of generality, let $BD = 17$ and $CD = 3$. Then $\tan \angle DAB = \frac{17}{h}$ and $\tan \angle CAD = \frac{3}{h}$. Using the tangent sum formula,
\begin{align*} \tan CAB &= \tan (DAB + CAD)\\ \frac{2... | 0.75 | 4,372.625 | 3,677.416667 | 6,458.25 |
Find $\sum_{k=0}^{\infty}\left\lfloor\frac{1+\sqrt{\frac{2000000}{4^{k}}}}{2}\right\rfloor$ where $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$. | 1414 | The $k$ th floor (for $k \geq 0$) counts the number of positive integer solutions to $4^{k}(2 x-1)^{2} \leq 2 \cdot 10^{6}$. So summing over all $k$, we want the number of integer solutions to $4^{k}(2 x-1)^{2} \leq 2 \cdot 10^{6}$ with $k \geq 0$ and $x \geq 1$. But each positive integer can be uniquely represented as... | 0.0625 | 8,143.8125 | 8,192 | 8,140.6 |
Given that \( PQ = 4 \), \( QR = 8 \), \( RS = 8 \), and \( ST = 3 \), if \( PQ \) is perpendicular to \( QR \), \( QR \) is perpendicular to \( RS \), and \( RS \) is perpendicular to \( ST \), calculate the distance from \( P \) to \( T \). | 13 | 0.1875 | 6,823.875 | 6,214.666667 | 6,964.461538 | |
The coordinates of the vertices of isosceles trapezoid $ABCD$ are all integers, with $A=(20,100)$ and $D=(21,107)$. The trapezoid has no horizontal or vertical sides, and $\overline{AB}$ and $\overline{CD}$ are the only parallel sides. The sum of the absolute values of all possible slopes for $\overline{AB}$ is $m/n$, ... | 131 | A very natural solution: . Shift $A$ to the origin. Suppose point $B$ was $(x, kx)$. Note $k$ is the slope we're looking for. Note that point $C$ must be of the form: $(x \pm 1, kx \pm 7)$ or $(x \pm 7, kx \pm 1)$ or $(x \pm 5, kx \pm 5)$. Note that we want the slope of the line connecting $D$ and $C$ so also be $k$, s... | 0 | 8,116.9375 | -1 | 8,116.9375 |
A trapezoid $ABCD$ has bases $AD$ and $BC$. If $BC = 60$ units, and altitudes from $B$ and $C$ to line $AD$ divide it into segments of lengths $AP = 20$ units and $DQ = 19$ units, with the length of the altitude itself being $30$ units, what is the perimeter of trapezoid $ABCD$?
**A)** $\sqrt{1300} + 159$
**B)** $\sqrt... | \sqrt{1300} + \sqrt{1261} + 159 | 0 | 4,481.75 | -1 | 4,481.75 | |
In the diagram, $AB$ is perpendicular to $BC$, and $CD$ is perpendicular to $AD$. Also, $AC = 625$ and $AD = 600$. If $\angle BAC = 2 \angle DAC$, what is the length of $BC$? | 336 | 0.6875 | 4,893.3125 | 3,868.090909 | 7,148.8 | |
Calculate the definite integral:
$$
\int_{0}^{\pi} 2^{4} \cdot \cos ^{8}\left(\frac{x}{2}\right) dx
$$ | \frac{35\pi}{8} | 0.5 | 6,736.1875 | 5,280.375 | 8,192 | |
Three numbers, $a_1, a_2, a_3$, are drawn randomly and without replacement from the set $\{1, 2, 3,\ldots, 1000\}$. Three other numbers, $b_1, b_2, b_3$, are then drawn randomly and without replacement from the remaining set of $997$ numbers. Let $p$ be the probability that, after suitable rotation, a brick of dimensio... | 5 | There is a total of $P(1000,6)$ possible ordered $6$-tuples $(a_1,a_2,a_3,b_1,b_2,b_3).$
There are $C(1000,6)$ possible sets $\{a_1,a_2,a_3,b_1,b_2,b_3\}.$ We have five valid cases for the increasing order of these six elements:
$aaabbb$
$aababb$
$aabbab$
$abaabb$
$ababab$
Note that the $a$'s are different from each ... | 0.0625 | 8,078.4375 | 8,141 | 8,074.266667 |
Let $\triangle A B C$ be an acute triangle, with $M$ being the midpoint of $\overline{B C}$, such that $A M=B C$. Let $D$ and $E$ be the intersection of the internal angle bisectors of $\angle A M B$ and $\angle A M C$ with $A B$ and $A C$, respectively. Find the ratio of the area of $\triangle D M E$ to the area of $\... | \frac{2}{9} | Let $[X Y Z]$ denote the area of $\triangle X Y Z$. Solution 1: Let $A M=\ell$, let $D E=d$, and let the midpoint of $\overline{D E}$ be $F$. Since $\frac{A D}{A B}=\frac{A E}{A C}=\frac{2}{3}$ by the angle bisector theorem, $F$ lies on $\overline{A M}$ and $\triangle A D E$ is similar to $\triangle A B C$. Note that $... | 0.625 | 6,694.9375 | 5,873.3 | 8,064.333333 |
A regular octagon has a side length of 8 cm. What is the number of square centimeters in the area of the shaded region formed by diagonals connecting alternate vertices (forming a square in the center)? | 192 + 128\sqrt{2} | 0 | 8,180.0625 | -1 | 8,180.0625 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and satisfy the vectors $\overrightarrow{m}=(\cos A,\cos B)$, $\overrightarrow{n}=(a,2c-b)$, and $\overrightarrow{m} \parallel \overrightarrow{n}$.
(I) Find the measure of angle $A$;
(II) If $a=2 \sqrt {5}$, find the maximum ... | 5 \sqrt {3} | 0 | 6,851.5625 | -1 | 6,851.5625 | |
A company needs to renovate its new office building. If the renovation is done solely by Team A, it would take 18 weeks, and if done solely by Team B, it would take 12 weeks. The result of the bidding is that Team A will work alone for the first 3 weeks, and then both Team A and Team B will work together. The total ren... | 2000 | 0.25 | 5,638.5 | 3,785 | 6,256.333333 | |
Given a circle C that passes through points A(1, 4) and B(3, -2), and the distance from the center of the circle C to line AB is $\sqrt{10}$, find the equation of circle C. | 20 | 0 | 4,013.5625 | -1 | 4,013.5625 | |
What is the sum of all positive integer values of $n$ such that $\frac{n+18}{n}$ is an integer? | 39 | 1 | 1,891.1875 | 1,891.1875 | -1 | |
a) In how many ways can a rectangle $8 \times 2$ be divided into $1 \times 2$ rectangles?
b) Imagine and describe a shape that can be divided into $1 \times 2$ rectangles in exactly 555 ways. | 34 | 0.125 | 8,020.625 | 6,821 | 8,192 | |
Mrs. Thompson gives extra points on tests to her students with test grades that exceed the class median. Given that 101 students take the same test, what is the largest number of students who can be awarded extra points? | 50 | 0.875 | 4,559.75 | 4,040.857143 | 8,192 | |
Let $S$ be a list of positive integers--not necessarily distinct--in which the number $68$ appears. The average (arithmetic mean) of the numbers in $S$ is $56$. However, if $68$ is removed, the average of the remaining numbers drops to $55$. What is the largest number that can appear in $S$? | 649 | Suppose that $S$ has $n$ numbers other than the $68.$ We have the following table: \[\begin{array}{c|c|c|c} & & & \\ [-2.5ex] & \textbf{Count} & \textbf{Arithmetic Mean} & \textbf{Sum} \\ \hline & & & \\ [-2.5ex] \textbf{Initial} & n+1 & 56 & 56(n+1) \\ \hline & & & \\ [-2.5ex] \textbf{Final} & n & 55 & 55n \end{array}... | 0.625 | 6,549.5625 | 5,583.6 | 8,159.5 |
A cuboid has dimensions of 2 units by 2 units by 2 units. It has vertices $P_1, P_2, P_3, P_4, P_1', P_2', P_3', P_4'.$ Vertices $P_2, P_3,$ and $P_4$ are adjacent to $P_1$, and vertices $P_i' (i = 1,2,3,4)$ are opposite to $P_i$. A regular octahedron has one vertex in each of the segments $\overline{P_1P_2}, \overline... | \frac{4\sqrt{2}}{3} | 0 | 8,192 | -1 | 8,192 | |
If $R_n=\frac{1}{2}(a^n+b^n)$ where $a=3+2\sqrt{2}$ and $b=3-2\sqrt{2}$, and $n=0,1,2,\cdots,$ then $R_{12345}$ is an integer. Its units digit is | 9 | 1. **Identify the recurrence relation**: Given $R_n = \frac{1}{2}(a^n + b^n)$ where $a = 3 + 2\sqrt{2}$ and $b = 3 - 2\sqrt{2}$, we start by expressing $R_{n+1}$ and $R_{n-1}$ in terms of $R_n$:
\[
(a+b)R_n = \frac{1}{2}(a+b)(a^n + b^n) = \frac{1}{2}(a^{n+1} + ab^n + ba^n + b^{n+1})
\]
\[
= \frac{1}{2}(a... | 0.5625 | 6,291.625 | 5,275.555556 | 7,598 |
John has five children. What is the probability that at least half of them are girls? (We can assume a boy is equally likely to be born as is a girl, and vice-versa.) | \frac{1}{2} | 0.9375 | 3,382 | 3,061.333333 | 8,192 | |
What is the volume of tetrahedron $ABCD$ with edge lengths $AB = 2$, $AC = 3$, $AD = 4$, $BC = \sqrt{13}$, $BD = 2\sqrt{5}$, and $CD = 5$? | 4 | 1. **Positioning the Tetrahedron in the Coordinate System**:
We place tetrahedron $ABCD$ in the $xyz$-plane. Assume $A=(0,0,0)$, $C=(3,0,0)$, and $D=(0,4,0)$. This positioning is based on the edge lengths $AC = 3$ and $AD = 4$. We also know that $CD = 5$, and by checking the Pythagorean theorem for $\triangle ACD$, ... | 0.25 | 7,337.6875 | 4,774.75 | 8,192 |
20 balls numbered 1 through 20 are placed in a bin. In how many ways can 4 balls be drawn, in order, from the bin, if each ball remains outside the bin after it is drawn and each drawn ball has a consecutive number as the previous one? | 17 | 0 | 7,898.0625 | -1 | 7,898.0625 | |
Find the area of quadrilateral $EFGH$, given that $m\angle F = m \angle G = 135^{\circ}$, $EF=4$, $FG=6$, and $GH=8$. | 18\sqrt{2} | 0.1875 | 7,741.875 | 6,314.333333 | 8,071.307692 | |
If $x=2$ and $y=3$, express the value of the following as a common fraction: $$
\frac
{~\frac{1}{y}~}
{\frac{1}{x}}
$$ | \frac{2}{3} | 1 | 1,766.875 | 1,766.875 | -1 | |
Solve the inequality
\[\frac{8x^2 + 16x - 51}{(2x - 3)(x + 4)} < 3.\] | (-4,-3) \cup \left( \frac{3}{2}, \frac{5}{2} \right) | 0.75 | 4,583.375 | 4,298.5 | 5,438 | |
Calculate the value of \(\sin \left(-\frac{5 \pi}{3}\right) + \cos \left(-\frac{5 \pi}{4}\right) + \tan \left(-\frac{11 \pi}{6}\right) + \cot \left(-\frac{4 \pi}{3}\right)\). | \frac{\sqrt{3} - \sqrt{2}}{2} | 0 | 4,300.1875 | -1 | 4,300.1875 | |
In convex quadrilateral \(WXYZ\), \(\angle W = \angle Y\), \(WZ = YX = 150\), and \(WX \ne ZY\). The perimeter of \(WXYZ\) is 520. Find \(\cos W\). | \frac{11}{15} | 0 | 8,101.4375 | -1 | 8,101.4375 | |
Given that $\sin \alpha = 2 \cos \alpha$, find the value of $\cos ( \frac {2015\pi}{2}-2\alpha)$. | - \frac {4}{5} | 0.6875 | 5,952.875 | 4,935.090909 | 8,192 | |
How many ways can a student schedule $3$ mathematics courses -- algebra, geometry, and number theory -- in a $6$-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other $3$ periods is of no concern here.) | 24 | To solve this problem, we need to determine the number of ways to schedule 3 mathematics courses (algebra, geometry, and number theory) in a 6-period day such that no two mathematics courses are taken in consecutive periods.
#### Step 1: Calculate the total number of unrestricted arrangements
First, we calculate the t... | 0.5625 | 6,582 | 5,825 | 7,555.285714 |
Determine all six-digit numbers \( p \) that satisfy the following properties:
(1) \( p, 2p, 3p, 4p, 5p, 6p \) are all six-digit numbers;
(2) Each of the six-digit numbers' digits is a permutation of \( p \)'s six digits. | 142857 | 0.25 | 7,935.125 | 7,164.5 | 8,192 | |
A solid box is 20 cm by 15 cm by 12 cm. A new solid is formed by removing a cube 4 cm on a side from each of the top four corners of this box. After that, four cubes, 2 cm on a side, are placed on each lower corner of the box. What percent of the original volume has been altered (either lost or gained)? | 6.22\% | 0.8125 | 5,439.3125 | 5,191.230769 | 6,514.333333 | |
If the larger base of an isosceles trapezoid equals a diagonal and the smaller base equals the altitude, then the ratio of the smaller base to the larger base is: | \frac{3}{5} | 1. **Assign Variables:**
Let $ABCD$ be an isosceles trapezoid with $AB$ as the smaller base and $CD$ as the larger base. Let the length of $AB$ be $a$ and the length of $CD$ be $1$. The ratio of the smaller base to the larger base is $\frac{a}{1} = a$.
2. **Identify Key Points and Relationships:**
Let $E$ be the... | 0.75 | 5,595 | 5,511 | 5,847 |
If the graph of the function $f(x) = \sin \omega x \cos \omega x + \sqrt{3} \sin^2 \omega x - \frac{\sqrt{3}}{2}$ ($\omega > 0$) is tangent to the line $y = m$ ($m$ is a constant), and the abscissas of the tangent points form an arithmetic sequence with a common difference of $\pi$.
(Ⅰ) Find the values of $\omega$ and... | \frac{11\pi}{3} | 0.0625 | 8,061.125 | 7,421 | 8,103.8 | |
Class 5(2) has 28 female students, which is 6 more than the male students. The ratio of female to male students is ____, and the percentage of male students in the whole class is ____. | \frac{11}{25} | 0 | 617 | -1 | 617 |
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