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 |
|---|---|---|---|---|---|---|
Given that the ellipse $C\_2$ passes through the two foci and the two endpoints of the minor axis of the ellipse $C\_1$: $\frac{x^{2}}{14} + \frac{y^{2}}{9} = 1$, find the eccentricity of the ellipse $C\_2$. | \frac{2}{3} | 0.75 | 4,421.5 | 3,544.666667 | 7,052 | |
Three vertices of a cube are $P=(7,12,10)$, $Q=(8,8,1)$, and $R=(11,3,9)$. What is the surface area of the cube? | 294 | $PQ=\sqrt{(8-7)^2+(8-12)^2+(1-10)^2}=\sqrt{98}$
$PR=\sqrt{(11-7)^2+(3-12)^2+(9-10)^2}=\sqrt{98}$
$QR=\sqrt{(11-8)^2+(3-8)^2+(9-1)^2}=\sqrt{98}$
So, $PQR$ is an equilateral triangle. Let the side of the cube be $a$.
$a\sqrt{2}=\sqrt{98}$
So, $a=7$, and hence the surface area is $6a^2=\boxed{294}$. | 0.6875 | 6,381.25 | 5,558.181818 | 8,192 |
Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $3, n$, and $n+1$ cents, $115$ cents is the greatest postage that cannot be formed. | 59 | 0 | 8,192 | -1 | 8,192 | |
For every $a \in \mathbb N$ denote by $M(a)$ the number of elements of the set
\[ \{ b \in \mathbb N | a + b \text{ is a divisor of } ab \}.\]
Find $\max_{a\leq 1983} M(a).$ | 121 |
To solve the problem, we need to analyze the set \( S(a) = \{ b \in \mathbb{N} \mid a + b \text{ is a divisor of } ab \} \) for a given \( a \) in the natural numbers, and we need to find the maximum number of elements \( M(a) \) in this set for \( a \leq 1983 \).
### Step 1: Understand the Condition
For \( a + b \m... | 0 | 8,192 | -1 | 8,192 |
Teams A and B each have 7 players who will compete in a Go tournament in a predetermined order. The match starts with player 1 from each team competing against each other. The loser is eliminated, and the winner next competes against the loser’s teammate. This process continues until all players of one team are elimina... | 3432 | 0 | 8,181.375 | -1 | 8,181.375 | |
Given vectors $\overrightarrow{a}=(2\cos x,1)$, $\overrightarrow{b}=(\sqrt{3}\sin x+\cos x,-1)$, and the function $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}$.
1. Find the maximum and minimum values of $f(x)$ in the interval $[0,\frac{\pi}{4}]$.
2. If $f(x_{0})=\frac{6}{5}$, $x_{0}\in[\frac{\pi}{4},\frac{\pi}{2}]$... | \frac{1}{4} | 0 | 7,777.25 | -1 | 7,777.25 | |
Calculate $7 \cdot 7! + 5 \cdot 5! + 3 \cdot 3! + 3!$. | 35904 | 1 | 4,403.8125 | 4,403.8125 | -1 | |
Find the numerical value of $k$ for which
\[\frac{7}{x + y} = \frac{k}{x + z} = \frac{11}{z - y}.\] | 18 | 1 | 3,069.9375 | 3,069.9375 | -1 | |
Let $S_n$ be the sum of the reciprocals of the non-zero digits of the integers from $1$ to $10^n$ inclusive. Find the smallest positive integer $n$ for which $S_n$ is an integer.
| 63 | 0.125 | 8,182.375 | 8,115 | 8,192 | |
Evaluate $(\sqrt[6]{4})^9$. | 8 | 1 | 2,360.0625 | 2,360.0625 | -1 | |
How many positive integers which divide $5n^{11}-2n^5-3n$ for all positive integers $n$ are there? | 12 | 0.4375 | 6,881.5 | 5,712.428571 | 7,790.777778 | |
A certain organism starts with 4 cells. Each cell splits into two cells at the end of three days. However, at the end of each 3-day period, 10% of the cells die immediately after splitting. This process continues for a total of 9 days. How many cells are there at the end of the $9^\text{th}$ day? | 23 | 0.375 | 6,554.75 | 6,550.666667 | 6,557.2 | |
In the following image, there is a hexagon $ABEFGD$. Quadrilaterals $ABCD$ and $EFGC$ are congruent rectangles, and quadrilateral $BEGD$ is also a rectangle. Determine the ratio of the areas of the white and shaded parts of the hexagon, given that $|AB| = 5 \text{ cm}$ and triangle $BEC$ is equilateral. | 2:1 | 0 | 8,192 | -1 | 8,192 | |
Given the family of curves
$$
2(2 \sin \theta - \cos \theta + 3) x^{2} - (8 \sin \theta + \cos \theta + 1) y = 0,
$$
where $\theta$ is a parameter. Find the maximum length of the chord that these curves cut on the line $y = 2 x$. | 8\sqrt{5} | 0.125 | 7,874.0625 | 6,000.5 | 8,141.714286 | |
Determine the minimum value of the function $$y = \frac {4x^{2}+2x+5}{x^{2}+x+1}$$ for \(x > 1\). | \frac{16 - 2\sqrt{7}}{3} | 0 | 6,370.9375 | -1 | 6,370.9375 | |
For positive real numbers $a,$ $b,$ $c,$ and $d,$ compute the maximum value of
\[\frac{abcd(a + b + c + d)}{(a + b)^2 (c + d)^2}.\] | \frac{1}{4} | 0 | 8,027.75 | -1 | 8,027.75 | |
In city "N", there are 10 horizontal and 12 vertical streets. A pair of horizontal and a pair of vertical streets form the rectangular boundary of the city, while the rest divide it into blocks shaped like squares with a side length of 100 meters. Each block has an address consisting of two integers \((i, j)\), \(i = 1... | 14 | 0.0625 | 7,848.375 | 7,245 | 7,888.6 | |
Given that Luis wants to arrange his sticker collection in rows with exactly 4 stickers in each row, and he has 29 stickers initially, find the minimum number of additional stickers Luis must purchase so that the total number of stickers can be exactly split into 5 equal groups without any stickers left over. | 11 | 0.3125 | 576 | 682 | 527.818182 | |
Given $M=\{1,2,x\}$, we call the set $M$, where $1$, $2$, $x$ are elements of set $M$. The elements in the set have definiteness (such as $x$ must exist), distinctiveness (such as $x\neq 1, x\neq 2$), and unorderedness (i.e., changing the order of elements does not change the set). If set $N=\{x,1,2\}$, we say $M=N$. I... | \frac{1}{2} | 0.75 | 6,234.8125 | 5,582.416667 | 8,192 | |
One of Euler's conjectures was disproved in the 1960s by three American mathematicians when they showed there was a positive integer such that \[133^5+110^5+84^5+27^5=n^{5}.\] Find the value of $n$. | 144 | Note that $n$ is even, since the LHS consists of two odd and two even numbers. By Fermat's Little Theorem, we know $n^5\equiv n\pmod{5}.$ Hence, \[n\equiv3+0+4+2\equiv4\pmod{5}.\] Continuing, we examine the equation modulo $3,$ \[n\equiv1-1+0+0\equiv0\pmod{3}.\] Thus, $n$ is divisible by three and leaves a remainder of... | 0.5 | 7,301.9375 | 6,411.875 | 8,192 |
Given the function $f(x) = \sqrt{2}\cos(2x - \frac{\pi}{4})$, where $x \in \mathbb{R}$,
1. Find the smallest positive period of the function $f(x)$ and its intervals of monotonically increasing values.
2. Find the minimum and maximum values of the function $f(x)$ on the interval $\left[-\frac{\pi}{8}, \frac{\pi}{2}\rig... | -1 | 0 | 4,855.125 | -1 | 4,855.125 | |
In the following list of numbers, the integer $n$ appears $n$ times in the list for $1 \leq n \leq 200$.
\[1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \ldots, 200, 200, \ldots , 200\]What is the median of the numbers in this list? | 142 | 1. **Understanding the Problem**: We are given a list where each integer $n$ from 1 to 200 appears exactly $n$ times. We need to find the median of this list.
2. **Calculating the Total Number of Elements**: The total number of elements in the list is the sum of the first 200 natural numbers, since $n$ appears $n$ tim... | 1 | 5,405 | 5,405 | -1 |
Find the equation of the directrix of the parabola $y = -2x^2 + 4x - 8.$ | y = -\frac{47}{8} | 1 | 4,759.25 | 4,759.25 | -1 | |
A four-digit positive integer is called [i]virtual[/i] if it has the form $\overline{abab}$, where $a$ and $b$ are digits and $a \neq 0$. For example 2020, 2121 and 2222 are virtual numbers, while 2002 and 0202 are not. Find all virtual numbers of the form $n^2+1$, for some positive integer $n$. | 8282 |
To solve the problem of finding all virtual numbers of the form \( n^2 + 1 \), we need to express a virtual number in the required form and establish conditions for \( n \).
A virtual number \(\overline{abab}\) can be expressed mathematically as:
\[
101a + 10b + 10a + b = 110a + 11b.
\]
We are tasked with finding \( ... | 1 | 6,159.75 | 6,159.75 | -1 |
Let $a,$ $b,$ and $c$ be distinct real numbers such that
\[\frac{a^3 + 6}{a} = \frac{b^3 + 6}{b} = \frac{c^3 + 6}{c}.\]Find $a^3 + b^3 + c^3.$ | -18 | 1 | 3,402.5625 | 3,402.5625 | -1 | |
Given the function $f(x)=(a\sin x+b\cos x)\cdot e^{x}$ has an extremum at $x= \frac {\pi}{3}$, determine the value of $\frac {a}{b}$. | 2- \sqrt {3} | 0 | 3,752.5 | -1 | 3,752.5 | |
In the triangular prism \(A-BCD\), the side edges \(AB, AC, AD\) are mutually perpendicular. The areas of triangles \(\triangle ABC\), \(\triangle ACD\), and \(\triangle ADB\) are \(\frac{\sqrt{2}}{2}\), \(\frac{\sqrt{3}}{2}\), and \(\frac{\sqrt{6}}{2}\) respectively. Find the volume of the circumscribed sphere of the ... | \sqrt{6}\pi | 0.3125 | 7,243.75 | 6,476.8 | 7,592.363636 | |
If $\{a_n\}$ is an arithmetic sequence, with the first term $a_1 > 0$, $a_{2011} + a_{2012} > 0$, and $a_{2011} \cdot a_{2012} < 0$, determine the natural number $n$ that maximizes the sum of the first $n$ terms $S_n$. | 2011 | 0.625 | 7,172.9375 | 6,580.7 | 8,160 | |
Given a deck of cards consisting of four red cards numbered 1, 2, 3, 4; four green cards numbered 1, 2, 3, 4; and four yellow cards numbered 1, 2, 3, 4, calculate the probability of drawing a winning pair, where a winning pair consists of two cards of the same color or two cards with the same number. | \frac{5}{11} | 0.6875 | 6,051 | 5,349.454545 | 7,594.4 | |
We are given $n$ coins of different weights and $n$ balances, $n>2$. On each turn one can choose one balance, put one coin on the right pan and one on the left pan, and then delete these coins out of the balance. It's known that one balance is wrong (but it's not known ehich exactly), and it shows an arbitrary result o... | 2n - 1 |
Given are \( n \) coins of different weights and \( n \) balances, where \( n > 2 \). One of these balances is faulty and provides arbitrary results on each turn. Our goal is to determine the smallest number of turns required to find the heaviest coin.
### Strategy
1. **Initial Understanding**: We need to find which... | 0 | 7,988.1875 | -1 | 7,988.1875 |
Find the difference between $1234_8$ and $765_8$ in base $8$. | 225_8 | 0 | 6,629.8125 | -1 | 6,629.8125 | |
Rhombus $PQRS^{}_{}$ is inscribed in rectangle $ABCD^{}_{}$ so that vertices $P^{}_{}$, $Q^{}_{}$, $R^{}_{}$, and $S^{}_{}$ are interior points on sides $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, and $\overline{DA}$, respectively. It is given that $PB^{}_{}=15$, $BQ^{}_{}=20$, $PR^{}_{}=30$, and $QS^{}_{}=40$. ... | 677 | We can just use areas. Let $AP = b$ and $AS = a$. $a^2 + b^2 = 625$. Also, we can add up the areas of all 8 right triangles and let that equal the total area of the rectangle, $(a+20)(b+15)$. This gives $3a + 4b = 120$. Solving this system of equation gives $\frac{44}{5} = a$, $\frac{117}{5} = b$, from which it is stra... | 0.4375 | 6,620.625 | 5,532.857143 | 7,466.666667 |
In the Cartesian coordinate system $xOy$, with the origin $O$ as the pole and the non-negative half-axis of the $x$-axis as the polar axis, a polar coordinate system is established. It is known that the polar equation of curve $C$ is $\rho^{2}= \dfrac {16}{1+3\sin ^{2}\theta }$, and $P$ is a moving point on curve $C$, ... | 2 \sqrt {2}+4 | 0 | 7,867.875 | -1 | 7,867.875 | |
An element is randomly chosen from among the first $20$ rows of Pascal’s Triangle. What is the probability that the value of the element chosen is $1$? | \frac{39}{210} | 0 | 3,607.3125 | -1 | 3,607.3125 | |
In the rectangular coordinate system, a pole coordinate system is established with the origin as the pole and the positive semi-axis of the $x$-axis as the polar axis. Given the curve $C$: ${p}^{2}=\frac{12}{2+{\mathrm{cos}}^{}θ}$ and the line $l$: $2p\mathrm{cos}\left(θ-\frac{π}{6}\right)=\sqrt{3}$.
1. Write the rect... | \frac{4\sqrt{10}}{3} | 0 | 4,750.125 | -1 | 4,750.125 | |
Given that \(x\) is a positive real, find the maximum possible value of \(\sin \left(\tan ^{-1}\left(\frac{x}{9}\right)-\tan ^{-1}\left(\frac{x}{16}\right)\right)\). | \frac{7}{25} | Consider a right triangle \(A O C\) with right angle at \(O, A O=16\) and \(C O=x\). Moreover, let \(B\) be on \(A O\) such that \(B O=9\). Then \(\tan ^{-1} \frac{x}{9}=\angle C B O\) and \(\tan ^{-1} \frac{x}{16}=\angle C A O\), so their difference is equal to \(\angle A C B\). Note that the locus of all possible poi... | 0.875 | 6,611.5625 | 6,385.785714 | 8,192 |
The number 119 has the following properties:
(a) Division by 2 leaves a remainder of 1;
(b) Division by 3 leaves a remainder of 2;
(c) Division by 4 leaves a remainder of 3;
(d) Division by 5 leaves a remainder of 4;
(e) Division by 6 leaves a remainder of 5.
How many positive integers less than 2007 satisfy thes... | 33 | 0.9375 | 3,336.75 | 3,013.066667 | 8,192 | |
Consider a geometric sequence where the first term is $\frac{5}{8}$, and the second term is $25$. What is the smallest $n$ for which the $n$th term of the sequence, multiplied by $n!$, is divisible by one billion (i.e., $10^9$)? | 10 | 0 | 6,838.0625 | -1 | 6,838.0625 | |
The constant term in the expansion of (1+x)(e^(-2x)-e^x)^9. | 84 | 0.5625 | 6,865.5 | 5,833.777778 | 8,192 | |
A certain shopping mall sells two types of products, A and B. The profit margin for each unit of product A is $40\%$, and for each unit of product B is $50\%$. When the quantity of product A sold is $150\%$ of the quantity of product B sold, the total profit margin for selling these two products in the mall is $45\%$. ... | 47.5\% | 0.5625 | 6,103.3125 | 4,689 | 7,921.714286 | |
Given triangle ABC, where sides $a$, $b$, and $c$ correspond to angles A, B, and C respectively, and $a=4$, $\cos{B}=\frac{4}{5}$.
(1) If $b=6$, find the value of $\sin{A}$;
(2) If the area of triangle ABC, $S=12$, find the values of $b$ and $c$. | 2\sqrt{13} | 0.0625 | 5,071.875 | 7,250 | 4,926.666667 | |
A fair 6-sided die and a fair 8-sided die are rolled once each. Calculate the probability that the sum of the numbers rolled is greater than 10. | \frac{3}{16} | 0 | 4,268.3125 | -1 | 4,268.3125 | |
A bag contains $5$ small balls of the same shape and size, with $2$ red balls and $3$ white balls. Three balls are randomly drawn from the bag.<br/>$(1)$ Find the probability that exactly one red ball is drawn;<br/>$(2)$ Let the random variable $X$ represent the number of red balls drawn. Find the distribution of the r... | \frac{3}{10} | 0.375 | 2,918.375 | 3,429.166667 | 2,611.9 | |
Consider the cube whose vertices are the eight points $(x, y, z)$ for which each of $x, y$, and $z$ is either 0 or 1 . How many ways are there to color its vertices black or white such that, for any vertex, if all of its neighbors are the same color then it is also that color? Two vertices are neighbors if they are the... | 118 | Divide the 8 vertices of the cube into two sets $A$ and $B$ such that each set contains 4 vertices, any two of which are diagonally adjacent across a face of the cube. We do casework based on the number of vertices of each color in set $A$. - Case 1: 4 black. Then all the vertices in $B$ must be black, for 1 possible c... | 0 | 8,147.875 | -1 | 8,147.875 |
Let $A B C$ be an acute isosceles triangle with orthocenter $H$. Let $M$ and $N$ be the midpoints of sides $\overline{A B}$ and $\overline{A C}$, respectively. The circumcircle of triangle $M H N$ intersects line $B C$ at two points $X$ and $Y$. Given $X Y=A B=A C=2$, compute $B C^{2}$. | 2(\sqrt{17}-1) | Let $D$ be the foot from $A$ to $B C$, also the midpoint of $B C$. Note that $D X=D Y=M A=M B=M D=N A=N C=N D=1$. Thus, $M N X Y$ is cyclic with circumcenter $D$ and circumradius 1. $H$ lies on this circle too, hence $D H=1$. If we let $D B=D C=x$, then since $\triangle H B D \sim \triangle B D A$, $$B D^{2}=H D \cdot ... | 0 | 8,189.125 | -1 | 8,189.125 |
Vaccination is one of the important means to protect one's own and others' health and lives. In order to test the immune effect of a certain vaccine on the $C$ virus, researchers used white rabbits as experimental subjects and conducted the following experiments:<br/>Experiment 1: Select 10 healthy white rabbits, numbe... | 80\% | 0 | 7,909.5625 | -1 | 7,909.5625 | |
Given that $\sin \alpha$ is a root of the equation $5x^{2}-7x-6=0$, find:
$(1)$ The value of $\frac {\cos (2\pi-\alpha)\cos (\pi+\alpha)\tan ^{2}(2\pi-\alpha)}{\cos ( \frac {\pi}{2}+\alpha)\sin (2\pi-\alpha)\cot ^{2}(\pi-\alpha)}$.
$(2)$ In $\triangle ABC$, $\sin A+ \cos A= \frac { \sqrt {2}}{2}$, $AC=2$, $AB=3$, find ... | -2- \sqrt {3} | 0 | 7,412.25 | -1 | 7,412.25 | |
Larry can swim from Harvard to MIT (with the current of the Charles River) in $40$ minutes, or back (against the current) in $45$ minutes. How long does it take him to row from Harvard to MIT, if he rows the return trip in $15$ minutes? (Assume that the speed of the current and Larry’s swimming and rowing speeds ... | 14:24 | 0.6875 | 4,580.0625 | 3,973.909091 | 5,913.6 | |
The famous skater Tony Hawk is riding a skateboard (segment $A B$) in a ramp, which is a semicircle with a diameter $P Q$. Point $M$ is the midpoint of the skateboard, and $C$ is the foot of the perpendicular dropped from point $A$ to the diameter $P Q$. What values can the angle $\angle A C M$ take, if it is known tha... | 12 | 0.25 | 7,925.875 | 7,127.5 | 8,192 | |
Chris received a mark of $50 \%$ on a recent test. Chris answered 13 of the first 20 questions correctly. Chris also answered $25 \%$ of the remaining questions on the test correctly. If each question on the test was worth one mark, how many questions in total were on the test? | 32 | Suppose that there were $n$ questions on the test. Since Chris received a mark of $50 \%$ on the test, then he answered $\frac{1}{2} n$ of the questions correctly. We know that Chris answered 13 of the first 20 questions correctly and then $25 \%$ of the remaining questions. Since the test has $n$ questions, then after... | 0.8125 | 785.875 | 749 | 945.666667 |
Let $\overline{AB}$ be a chord of a circle $\omega$, and let $P$ be a point on the chord $\overline{AB}$. Circle $\omega_1$ passes through $A$ and $P$ and is internally tangent to $\omega$. Circle $\omega_2$ passes through $B$ and $P$ and is internally tangent to $\omega$. Circles $\omega_1$ and $\omega_2$ intersect at... | 65 | Connect $AQ,QB$, since $\angle{AO_1P}=\angle{AOB}=\angle{BO_2P}$, so $\angle{AQP}=\frac{\angle{AO_1P}}{2}=\angle{BQP}=\frac{\angle{BO_2P}}{2}, \angle{AQB}=\angle{AOB}$ then, so $A,O,Q,B$ are concyclic
We let $\angle{AO_1P}=\angle{AOB}=\angle{BO_2P}=2\alpha$, it is clear that $\angle{BQP}=\alpha, \angle{O_1AP}=90^{\cir... | 0 | 8,192 | -1 | 8,192 |
A jacket and a shirt originally sold for $80$ dollars and $40$ dollars, respectively. During a sale Chris bought the $80$ dollar jacket at a $40\%$ discount and the $40$ dollar shirt at a $55\%$ discount. The total amount saved was what percent of the total of the original prices? | 45\% | 1. **Calculate the original total cost**: The jacket was originally priced at $80$ dollars and the shirt at $40$ dollars. Therefore, the total original cost is:
\[
80 + 40 = 120 \text{ dollars}
\]
2. **Calculate the savings on each item**:
- **Jacket**: The discount on the jacket is $40\%$. Therefore, the... | 1 | 1,573.5625 | 1,573.5625 | -1 |
How many positive multiples of 5 that are less than 100 have a units digit of 5? | 10 | 0.8125 | 526.75 | 561.384615 | 376.666667 | |
Based on the definition of the derivative, find $f^{\prime}(0)$:
$$
f(x)=\left\{\begin{array}{c}
\frac{2^{\operatorname{tg} x}-2^{\sin x}}{x^{2}}, x \neq 0 \\
0, x=0
\end{array}\right.
$$ | \ln \sqrt{2} | 0 | 8,088.4375 | -1 | 8,088.4375 | |
Given that the power function $y=x^{m}$ is an even function and is a decreasing function when $x \in (0,+\infty)$, determine the possible value of the real number $m$. | -2 | 0.8125 | 6,903.6875 | 6,606.384615 | 8,192 | |
Given a right triangle \(ABC\). On the extension of the hypotenuse \(BC\), a point \(D\) is chosen such that the line \(AD\) is tangent to the circumscribed circle \(\omega\) of triangle \(ABC\). The line \(AC\) intersects the circumscribed circle of triangle \(ABD\) at point \(E\). It turns out that the angle bisector... | 1:2 | 0 | 7,995.125 | -1 | 7,995.125 | |
Caroline starts with the number 1, and every second she flips a fair coin; if it lands heads, she adds 1 to her number, and if it lands tails she multiplies her number by 2. Compute the expected number of seconds it takes for her number to become a multiple of 2021. | 4040 | Consider this as a Markov chain on $\mathbb{Z} / 2021 \mathbb{Z}$. This Markov chain is aperiodic (since 0 can go to 0) and any number can be reached from any other number (by adding 1), so it has a unique stationary distribution $\pi$, which is uniform (since the uniform distribution is stationary). It is a well-known... | 0 | 8,192 | -1 | 8,192 |
What is the remainder when $6x^3-15x^2+21x-23$ is divided by $3x-6$? | 7 | 1 | 2,494.5625 | 2,494.5625 | -1 | |
How many times does the digit 9 appear in the list of all integers from 1 to 1000? | 300 | 0.4375 | 6,281.125 | 3,824.285714 | 8,192 | |
Given a triangle $\triangle ABC$ with its three interior angles $A$, $B$, and $C$ satisfying: $$A+C=2B, \frac {1}{\cos A}+ \frac {1}{\cos C}=- \frac { \sqrt {2}}{\cos B}$$, find the value of $$\cos \frac {A-C}{2}$$. | \frac { \sqrt {2}}{2} | 0 | 6,548.125 | -1 | 6,548.125 | |
Through how many squares does the diagonal of a 1983 × 999 chessboard pass? | 2979 | 0.8125 | 4,126.5 | 3,188.307692 | 8,192 | |
What is the area enclosed by the region defined by the equation $x^2+y^2+6x+8y=0$? | 25\pi | 1 | 1,480.875 | 1,480.875 | -1 | |
Ice-cream-o-rama is eager to advertise how many flavors it has. But it really only has three basic flavors: chocolate, vanilla, and strawberry. However, they can make "new" flavors by taking four scoops of ice cream of those basic flavors and blending them together. Different proportions of the basic flavors give diffe... | 15 | 1 | 2,716.25 | 2,716.25 | -1 | |
Given the function $f(x)=2 \sqrt {3}\sin \frac {x}{3}\cos \frac {x}{3}-2\sin ^{2} \frac {x}{3}$.
(1) Find the range of the function $f(x)$;
(2) In $\triangle ABC$, angles $A$, $B$, $C$ correspond to sides $a$, $b$, $c$ respectively. If $f(C)=1$ and $b^{2}=ac$, find the value of $\sin A$. | \frac {\sqrt {5}-1}{2} | 0 | 5,284.5625 | -1 | 5,284.5625 | |
Evaluate: $(12345679^2 \times 81 - 1) \div 11111111 \div 10 \times 9 - 8$ in billions. (Answer in billions) | 10 | 0 | 8,192 | -1 | 8,192 | |
Suppose $w$ is a complex number such that $w^2 = 45-21i$. Find $|w|$. | \sqrt[4]{2466} | 0.25 | 8,004.4375 | 7,441.75 | 8,192 | |
Eleanor is making chocolate chip cookies for her friends. If she divides the cookies equally among $11$ of her friends, she'll have $4$ cookies left over. If she divides the cookies equally among $7$ of her friends, she'll have $1$ cookie left over. Assuming that Eleanor made fewer than $100$ cookies, what is the sum o... | 107 | 1 | 2,140.6875 | 2,140.6875 | -1 | |
A box contains tiles, numbered 1, 2, 3,..., 49, 50. Only tiles which are marked with a number congruent to $2 \pmod{5}$ are blue. One tile is chosen randomly from the box. What is the probability that the tile is blue? | \frac{1}{5} | 1 | 1,672.875 | 1,672.875 | -1 | |
Given the sequence 1, 1+2, 2+3+4, 3+4+5+6, ..., the value of the 8th term in this sequence is: ______. | 84 | 0.6875 | 2,273.875 | 2,274.363636 | 2,272.8 | |
The number of terms in an A.P. (Arithmetic Progression) is even. The sum of the odd and even-numbered terms are 24 and 30, respectively. If the last term exceeds the first by 10.5, the number of terms in the A.P. is | 8 | 1. **Define the variables:**
Let $a$ be the first term, $n$ be the number of terms, and $d$ be the common difference in the arithmetic progression (A.P.). The last term of the A.P. can be expressed as $a + (n-1)d$.
2. **Set up the equation for the difference between the last and first term:**
Given that the last... | 1 | 3,031.25 | 3,031.25 | -1 |
There is more than one integer greater than $1$ which, when divided by any integer $k$ such that $2 \le k \le 11$, has a remainder of $1$.
What is the difference between the two smallest such integers? | 27720 | 1. **Identify the Problem Requirements:**
We need to find integers greater than $1$ that, when divided by any integer $k$ such that $2 \le k \le 11$, leave a remainder of $1$. We are asked to find the difference between the two smallest such integers.
2. **Set Up the Congruences:**
Let $n$ be such an integer. Th... | 0.8125 | 929.6875 | 932.384615 | 918 |
If $(1-2)^{9}=a_{9}x^{9}+a_{8}x^{8}+\ldots+a_{1}x+a_{0}$, then the sum of $a_1+a_2+\ldots+a$ is \_\_\_\_\_\_. | -2 | 0.625 | 3,224.375 | 1,853.1 | 5,509.833333 | |
Of the numbers $\frac{8}{12}, \frac{5}{6}$, and $\frac{9}{12}$, which number is the arithmetic mean of the other two? | \frac{3}{4} | 0.3125 | 2,790.4375 | 2,455.8 | 2,942.545455 | |
The number $25^{64} \cdot 64^{25}$ is the square of a positive integer $N$. In decimal representation, the sum of the digits of $N$ is | 14 | 1. **Identify the expression for $N$:**
Given that $N^2 = 25^{64} \cdot 64^{25}$, we find $N$ by taking the square root:
\[
N = \sqrt{25^{64} \cdot 64^{25}}
\]
2. **Simplify the expression using properties of exponents:**
Recognize that $25 = 5^2$ and $64 = 2^6$, and substitute:
\[
N = \sqrt{(... | 0.8125 | 4,235 | 4,008.923077 | 5,214.666667 |
In the BIG N, a middle school football conference, each team plays every other team exactly once. If a total of 21 conference games were played during the 2012 season, how many teams were members of the BIG N conference? | 7 | To determine the number of teams in the BIG N conference, we need to consider that each team plays every other team exactly once. This setup is a classic example of a round-robin tournament, where the number of games played can be calculated using the combination formula for choosing 2 teams out of $n$ total teams. The... | 1 | 1,402.5 | 1,402.5 | -1 |
Annie and Bonnie are running laps around a $400$-meter oval track. They started together, but Annie has pulled ahead, because she runs $25\%$ faster than Bonnie. How many laps will Annie have run when she first passes Bonnie? | 5 |
#### Step-by-step Analysis:
1. **Understanding the Problem:**
Annie and Bonnie start running together on a 400-meter track. Annie runs 25% faster than Bonnie. We need to determine how many laps Annie will have run when she first passes Bonnie.
2. **Setting Up the Relationship:**
Let's denote Bonnie's speed as ... | 1 | 2,295.6875 | 2,295.6875 | -1 |
If $x=2y$ and $y \neq 0$, what is the value of $(x+2y)-(2x+y)$? | -y | We simplify first, then substitute $x=2y$: $(x+2y)-(2x+y)=x+2y-2x-y=y-x=y-2y=-y$. Alternatively, we could substitute first, then simplify: $(x+2y)-(2x+y)=(2y+2y)-(2(2y)+y)=4y-5y=-y$. | 1 | 674.5625 | 674.5625 | -1 |
All the complex roots of $(z + 1)^4 = 16z^4,$ when plotted in the complex plane, lie on a circle. Find the radius of this circle. | \frac{2}{3} | 0.8125 | 6,234.625 | 5,782.923077 | 8,192 | |
Suppose $p$ and $q$ are both real numbers, and $\sin \alpha$ and $\cos \alpha$ are the two real roots of the equation $x^{2}+px+q=0$ with respect to $x$. Find the minimum value of $p+q$. | -1 | 1 | 5,058.1875 | 5,058.1875 | -1 | |
A large cube is formed by stacking 27 unit cubes. A plane is perpendicular to one of the internal diagonals of the large cube and bisects that diagonal. The number of unit cubes that the plane intersects is | 19 | 1. **Understanding the Problem:**
- We have a large cube formed by stacking 27 smaller unit cubes (3x3x3).
- A plane is perpendicular to one of the internal diagonals of the large cube and bisects that diagonal.
- We need to find the number of unit cubes that the plane intersects.
2. **Visualizing the Cube an... | 0.0625 | 8,158.625 | 7,658 | 8,192 |
If angle $A$ lies in the second quadrant and $\sin A = \frac{3}{4},$ find $\cos A.$ | -\frac{\sqrt{7}}{4} | 0 | 1,293.875 | -1 | 1,293.875 | |
Choose one digit from the set {0, 2} and two different digits from the set {1, 3, 5} to form a three-digit number without any repeating digits. The total number of such odd three-digit numbers is _________. | 18 | 0.125 | 6,740.25 | 5,239.5 | 6,954.642857 | |
Suppose that the angles of $\triangle ABC$ satisfy $\cos(3A)+\cos(3B)+\cos(3C)=1.$ Two sides of the triangle have lengths 10 and 13. There is a positive integer $m$ so that the maximum possible length for the remaining side of $\triangle ABC$ is $\sqrt{m}.$ Find $m.$ | 399 | \[\cos3A+\cos3B=1-\cos(3C)=1+\cos(3A+3B)\] \[2\cos\frac{3}{2}(A+B)\cos\frac{3}{2}(A-B)=2\cos^2\frac{3}{2}(A+B)\] If $\cos\frac{3}{2}(A+B) = 0$, then $\frac{3}{2}(A+B)=90$, $A+B=60$, so $C=120$; otherwise, \[2\cos\frac{3}{2}(A-B)=2cos\frac{3}{2}(A+B)\] \[\sin\frac{3}{2}A\sin\frac{3}{2}B=0\] so either $\sin\frac{3}{2}A=0... | 0.0625 | 8,188.625 | 8,138 | 8,192 |
Two identical cylindrical containers are connected at the bottom by a small tube with a tap. While the tap was closed, water was poured into the first container, and oil was poured into the second one, so that the liquid levels were the same and equal to $h = 40$ cm. At what level will the water in the first container ... | 16.47 | 0 | 7,804.4375 | -1 | 7,804.4375 | |
Calculate $6!-5\cdot5!-5!$. | 0 | 1 | 2,222.125 | 2,222.125 | -1 | |
Let $p(x)$ be a polynomial of degree strictly less than $100$ and such that it does not have $(x^3-x)$ as a factor. If $$ \frac{d^{100}}{dx^{100}}\bigg(\frac{p(x)}{x^3-x}\bigg)=\frac{f(x)}{g(x)} $$ for some polynomials $f(x)$ and $g(x)$ then find the smallest possible degree of $f(x)$ . | 200 | 0 | 7,722.3125 | -1 | 7,722.3125 | |
It took $4$ days for $75$ workers, all working together at the same rate, to build an embankment. If only $50$ workers had been available, how many total days would it have taken to build the embankment? | 6 | 0.9375 | 2,936.125 | 2,585.733333 | 8,192 | |
In a store, there are four types of nuts: hazelnuts, almonds, cashews, and pistachios. Stepan wants to buy 1 kilogram of nuts of one type and 1 kilogram of nuts of another type. He has calculated the cost of such a purchase depending on which two types of nuts he chooses. Five of Stepan's six possible purchases would c... | 2290 | 0.375 | 7,606.6875 | 6,631.166667 | 8,192 | |
[asy] draw(circle((0,6sqrt(2)),2sqrt(2)),black+linewidth(.75)); draw(circle((0,3sqrt(2)),sqrt(2)),black+linewidth(.75)); draw((-8/3,16sqrt(2)/3)--(-4/3,8sqrt(2)/3)--(0,0)--(4/3,8sqrt(2)/3)--(8/3,16sqrt(2)/3),dot); MP("B",(-8/3,16*sqrt(2)/3),W);MP("B'",(8/3,16*sqrt(2)/3),E); MP("A",(-4/3,8*sqrt(2)/3),W);MP("A'",(4/3,8*s... | 2\pi | 0 | 7,651.5 | -1 | 7,651.5 | |
Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth. | 0.138 | 0 | 7,140.1875 | -1 | 7,140.1875 | |
Evaluate the expression $\left(b^b - b(b-1)^b\right)^b$ when $b=4$. | 21381376 | 0.8125 | 3,713.0625 | 3,748.846154 | 3,558 | |
How many positive integer multiples of $77$ (product of $7$ and $11$) can be expressed in the form $10^{j}-10^{i}$, where $i$ and $j$ are integers and $0 \leq i < j \leq 99$? | 784 | 0.5625 | 6,693.1875 | 6,311.111111 | 7,184.428571 | |
Quadratic equations of the form $ax^2 + bx + c = 0$ have real roots with coefficients $a$, $b$, and $c$ selected from the set of integers $\{1, 2, 3, 4\}$, where $a \neq b$. Calculate the total number of such equations. | 17 | 0 | 7,179.9375 | -1 | 7,179.9375 | |
A paper triangle with sides of lengths $3,4,$ and $5$ inches, as shown, is folded so that point $A$ falls on point $B$. What is the length in inches of the crease? | $\frac{15}{8}$ | 1. **Identify the Triangle Type**: Given the side lengths $3, 4, 5$, we recognize $\triangle ABC$ as a right triangle with $AB = 3$, $BC = 4$, and $AC = 5$.
2. **Determine the Slope of $AB$**: The slope of line $AB$ is calculated as $\frac{\text{rise}}{\text{run}} = \frac{BC}{AC} = \frac{3}{4}$.
3. **Slope of the Cre... | 0 | 5,991.5625 | -1 | 5,991.5625 |
Let $ABCD$ be a rectangle such that $\overline{AB}=\overline{CD}=30$, $\overline{BC}=\overline{DA}=50$ and point $E$ lies on line $AB$, 20 units from $A$. Find the area of triangle $BEC$. | 1000 | 0 | 3,763 | -1 | 3,763 | |
Three identical rectangles are put together to form rectangle $ABCD$, as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is 5 feet, what is the area in square feet of rectangle $ABCD$? | 150 |
1. **Identify the dimensions of the smaller rectangles**:
Each smaller rectangle has a shorter side of 5 feet. Since the problem suggests that the rectangles are placed such that two smaller rectangles' shorter sides are aligned vertically, the longer side of each smaller rectangle must be twice the shorter side (... | 0.5625 | 6,519.5 | 5,903.222222 | 7,311.857143 |
A parabolic arch has a height of $16$ inches and a span of $40$ inches. The height, in inches, of the arch at the point $5$ inches from the center $M$ is: | 15 | 1. **Identify the shape and equation of the arch**: Given that the arch is parabolic, we can model it using a quadratic equation of the form:
\[
y = ax^2 + k
\]
where \(y\) is the height of the arch at a horizontal distance \(x\) from the center, \(a\) is a constant that determines the curvature of the para... | 0.875 | 4,362 | 3,814.857143 | 8,192 |
If
\[\frac{\sin^4 \theta}{a} + \frac{\cos^4 \theta}{b} = \frac{1}{a + b},\]then find the value of
\[\frac{\sin^8 \theta}{a^3} + \frac{\cos^8 \theta}{b^3}\]in terms of $a$ and $b.$ | \frac{1}{(a + b)^3} | 0.75 | 5,272 | 4,298.666667 | 8,192 | |
The sequence $\{a_n\}$ satisfies $a_{n+1}=(2|\sin \frac{n\pi}{2}|-1)a_{n}+n$, then the sum of the first $100$ terms of the sequence $\{a_n\}$ is __________. | 2550 | 0.125 | 7,850.3125 | 5,978 | 8,117.785714 | |
The graph of the rational function $\frac{1}{q(x)}$ is shown below. If $q(x)$ is a quadratic and $q(2) = 6$, find $q(x).$
[asy]
size(8cm);
import graph;
Label f;
f.p=fontsize(6);
real f(real x) {return 1/(2*(x+1)*(x-1));}
int gridsize = 5;
draw((-gridsize,0)--(gridsize,0), black+1bp, Arrows(8));
draw((0,-gridsize... | 2x^2 - 2 | 1 | 2,244.6875 | 2,244.6875 | -1 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.