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 |
|---|---|---|---|---|---|---|
Richard and Shreyas are arm wrestling against each other. They will play $10$ rounds, and in each round, there is exactly one winner. If the same person wins in consecutive rounds, these rounds are considered part of the same “streak”. How many possible outcomes are there in which there are strictly more than $3$ s... | 932 | 0.6875 | 6,448.6875 | 5,906.454545 | 7,641.6 | |
A farmer sold domestic rabbits. By the end of the market, he sold exactly one-tenth as many rabbits as the price per rabbit in forints. He then distributed the revenue between his two sons. Starting with the older son, the boys alternately received one-hundred forint bills, but at the end, the younger son received only... | 40 | 0 | 8,192 | -1 | 8,192 | |
In the diagram, what is the value of $x$? [asy]
size(120);
draw(Circle((0,0),1));
draw((0,0)--(.5,sqrt(3)/2));
draw((0,0)--(sqrt(3)/2,.5));
draw((0,0)--(sqrt(3)/2,-.5));
draw((0,0)--(-1,0));
label("$4x^\circ$",(0,0),NNW); label("$5x^\circ$",(0,0),SSW);
label("$2x^\circ$",(.3,0));label("$x^\circ$",(.3,.3));
[/asy] | 30 | 0.75 | 6,272.625 | 5,632.833333 | 8,192 | |
The bases \(AB\) and \(CD\) of trapezoid \(ABCD\) are equal to 41 and 24 respectively, and its diagonals are perpendicular to each other. Find the dot product of vectors \(\overrightarrow{AD}\) and \(\overrightarrow{BC}\). | 984 | 0.5 | 5,755.125 | 4,041.625 | 7,468.625 | |
When $x=-2$, what is the value of $(x+1)^{3}$? | -1 | When $x=-2$, we have $(x+1)^{3}=(-2+1)^{3}=(-1)^{3}=-1$. | 1 | 258.0625 | 258.0625 | -1 |
Eleven members of the Middle School Math Club each paid the same amount for a guest speaker to talk about problem solving at their math club meeting. They paid their guest speaker $\$1A2. What is the missing digit $A$ of this $3$-digit number? | 3 | 1. **Understanding the Problem**: The total amount paid to the guest speaker is $\textdollar\underline{1} \underline{A} \underline{2}$, where $A$ is the missing digit. Since the total amount is paid equally by 11 members, the total amount must be divisible by 11.
2. **Using the Divisibility Rule for 11**: A number is ... | 1 | 2,187.875 | 2,187.875 | -1 |
If the base $8$ representation of a perfect square is $ab3c$, where $a\ne 0$, then $c$ equals | 1 | 1. **Understanding the problem**: We are given a number in base $8$ represented as $ab3c$, which is a perfect square. We need to determine the value of $c$.
2. **Expressing the number in decimal**: The number $ab3c$ in base $8$ can be expressed in decimal as:
\[
a \cdot 8^3 + b \cdot 8^2 + 3 \cdot 8 + c = 512a +... | 0.5625 | 6,723.5 | 5,581.333333 | 8,192 |
Given three coplanar vectors $\overrightarrow{a}$, $\overrightarrow{b}$, and $\overrightarrow{c}$, where $\overrightarrow{a}=(\sqrt{2}, 2)$, $|\overrightarrow{b}|=2\sqrt{3}$, $|\overrightarrow{c}|=2\sqrt{6}$, and $\overrightarrow{a}$ is parallel to $\overrightarrow{c}$.
1. Find $|\overrightarrow{c}-\overrightarrow{a}|$... | -12 | 0 | 6,654.4375 | -1 | 6,654.4375 | |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $4a = \sqrt{5}c$ and $\cos C = \frac{3}{5}$.
$(Ⅰ)$ Find the value of $\sin A$.
$(Ⅱ)$ If $b = 11$, find the area of $\triangle ABC$. | 22 | 0.8125 | 5,021.8125 | 4,290.230769 | 8,192 | |
The vertices of a regular nonagon are colored such that 1) adjacent vertices are different colors and 2) if 3 vertices form an equilateral triangle, they are all different colors. Let m be the minimum number of colors needed for a valid coloring, and n be the total number of colorings using m colors. Determine mn. (Ass... | 54 | It's clear that m is more than 2 since it's impossible to alternate the color of the vertices without having two of the same color adjacent (since the graph is not bipartite). However, it's possible to use 3 colors. Number the vertices 1 through 9 in order and let the colors be A, B, C. Coloring the vertices in the ord... | 0 | 8,192 | -1 | 8,192 |
Let $x$, $y$, and $z$ be real numbers such that
\[\cos x + \cos y + \cos z = \sin x + \sin y + \sin z = 0.\]Find the sum of all possible values of $\cos 2x + \cos 2y + \cos 2z.$ | 0 | 0.5625 | 6,565.625 | 5,300.666667 | 8,192 | |
What is the reciprocal of the common fraction form of $.\overline{2}$? | \frac92 | 1 | 892.125 | 892.125 | -1 | |
In triangle $ABC$, the sides opposite to angles A, B, and C are denoted by $a$, $b$, and $c$ respectively, with $A+C=\frac{2\pi}{3}$ and $b=1$.
(1) If we let angle A be $x$ and define $f(x)=a+c$, find the range of $f(x)$ when triangle $ABC$ is an acute triangle;
(2) Determine the maximum area of triangle $ABC$. | \frac{\sqrt{3}}{4} | 0 | 7,571.9375 | -1 | 7,571.9375 | |
Given triangle \( \triangle ABC \), point \( P \) is an internal point such that \( \angle PBC = \angle PCB = 24^\circ \). If \( \angle ABP = 30^\circ \) and \( \angle ACP = 54^\circ \), find the measure of \( \angle BAP \). | 18 | 0.75 | 6,134.1875 | 5,448.25 | 8,192 | |
A rectangular prism has dimensions of 1 by 1 by 2. Calculate the sum of the areas of all triangles whose vertices are also vertices of this rectangular prism, and express the sum in the form $m + \sqrt{n} + \sqrt{p}$, where $m, n,$ and $p$ are integers. Find $m + n + p$. | 40 | 0 | 8,192 | -1 | 8,192 | |
The diagonal of square $I$ is $a+b$. The area of square $II$ with twice the area of $I$ is: | $(a+b)^2$ | 1. **Identify the side length of square $I$**: Given that the diagonal of square $I$ is $a+b$, we can use the relationship between the side length $s$ and the diagonal $d$ of a square, which is $d = s\sqrt{2}$. Therefore, the side length $s$ of square $I$ is:
\[
s = \frac{d}{\sqrt{2}} = \frac{a+b}{\sqrt{2}}
\]... | 0 | 2,691.125 | -1 | 2,691.125 |
A polynomial $p(x)$ leaves a remainder of $2$ when divided by $x - 3,$ a remainder of 1 when divided by $x - 4,$ and a remainder of 5 when divided by $x + 4.$ Let $r(x)$ be the remainder when $p(x)$ is divided by $(x - 3)(x - 4)(x + 4).$ Find $r(5).$ | \frac{1}{13} | 0 | 6,186.125 | -1 | 6,186.125 | |
Let $x$ and $y$ be positive real numbers such that $4x + 9y = 60.$ Find the maximum value of $xy.$ | 25 | 1 | 3,690.3125 | 3,690.3125 | -1 | |
Calculate the sum of all four-digit numbers that can be formed using the digits 0, 1, 2, 3, and 4, with no repeated digits. | 259980 | 0 | 8,192 | -1 | 8,192 | |
Sector $OAB$ is a quarter of a circle with a radius of 6 cm. A circle is inscribed within this sector, tangent to the two radii and the arc at three points. Determine the radius of the inscribed circle, expressed in simplest radical form. | 6\sqrt{2} - 6 | 0.0625 | 4,772.1875 | 5,529 | 4,721.733333 | |
How many positive multiples of 6 that are less than 150 have a units digit of 6? | 25 | 0 | 2,487.6875 | -1 | 2,487.6875 | |
$\sqrt{53+20\sqrt{7}}$ can be written in the form $a+b\sqrt{c}$, where $a,$ $b,$ and $c$ are integers and $c$ has no factors which is a perfect square of any positive integer other than 1. Find $a+b+c$. | 14 | 1 | 1,988.0625 | 1,988.0625 | -1 | |
In the expansion of $(1+x)^3+(1+x)^4+\ldots+(1+x)^{19}$, the coefficient of the $x^2$ term is \_\_\_\_\_\_. | 1139 | 0.6875 | 6,057.3125 | 5,087 | 8,192 | |
Given that \( P \) is a point on the hyperbola \( C: \frac{x^{2}}{4} - \frac{y^{2}}{12} = 1 \), and \( F_{1} \) and \( F_{2} \) are the left and right foci of the hyperbola \( C \), and \( M \) and \( I \) are the centroid and incenter of the triangle \( \triangle P F_{1} F_{2} \) respectively. If \( M I \perp x \)-axi... | \sqrt{6} | 0.25 | 7,657.8125 | 6,093.75 | 8,179.166667 | |
What is the smallest positive integer $n$ such that $\frac{n}{n+150}$ is equal to a terminating decimal? | 10 | 0.4375 | 7,306.5 | 6,168 | 8,192 | |
How many distinct, positive factors does $1320$ have? | 24 | 0 | 3,701.0625 | -1 | 3,701.0625 | |
Given that the location of the military camp is $A(1,1)$, and the general sets off from point $B(4,4)$ at the foot of the mountain, with the equation of the riverbank line $l$ being $x-y+1=0$, find the shortest total distance of the "General Drinking Horse" problem. | 2\sqrt{5} | 0.9375 | 4,377.625 | 4,123.333333 | 8,192 | |
Given the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $e = \frac{\sqrt{2}}{2}$, and one of its vertices is at $(0, -1)$.
(Ⅰ) Find the equation of the ellipse $C$.
(Ⅱ) If there exist two distinct points $A$ and $B$ on the ellipse $C$ that are symmetric about the line $y = -\frac{... | \frac{\sqrt{2}}{2} | 0 | 8,192 | -1 | 8,192 | |
Given a pair of concentric circles, chords $AB,BC,CD,\dots$ of the outer circle are drawn such that they all touch the inner circle. If $\angle ABC = 75^{\circ}$ , how many chords can be drawn before returning to the starting point ?
 | 24 | 0.4375 | 6,459.5 | 5,219.571429 | 7,423.888889 | |
Mia is designing a rectangular flower bed against her house on one side, using 450 feet of fencing to enclose the remaining three sides. If 150 feet of it is planned to be used along the house, what is the maximum area she can achieve for her garden? | 22500 | 0 | 5,891.6875 | -1 | 5,891.6875 | |
It is given that $\log_{6}a + \log_{6}b + \log_{6}c = 6,$ where $a,$ $b,$ and $c$ are positive integers that form an increasing geometric sequence and $b - a$ is the square of an integer. Find $a + b + c.$ | 111 | $abc=6^6$. Since they form an increasing geometric sequence, $b$ is the geometric mean of the product $abc$. $b=\sqrt[3]{abc}=6^2=36$.
Since $b-a$ is the square of an integer, we can find a few values of $a$ that work: $11, 20, 27, 32,$ and $35$. Out of these, the only value of $a$ that works is $a=27$, from which we ... | 0.5625 | 6,707.1875 | 5,716.777778 | 7,980.571429 |
A sled loaded with children starts from rest and slides down a snowy $25^\circ$ (with respect to the horizontal) incline traveling $85$ meters in $17$ seconds. Ignore air resistance. What is the coefficient of kinetic friction between the sled and the slope? | 0.40 | 0.5625 | 5,639.875 | 5,744.555556 | 5,505.285714 | |
If 7 knicks = 2 knacks and 3 knacks = 4 knocks, how many knicks are equal to 24 knocks? | 63 | 1 | 2,990.5 | 2,990.5 | -1 | |
Compute
\[
\sum_{n = 1}^\infty \frac{1}{(n+1)(n + 3)}.
\] | \frac{5}{12} | 0.5 | 4,994.5625 | 3,693.125 | 6,296 | |
Given the equations \(3x - 4y - 2z = 0\) and \(x - 2y + 5z = 0\) with \(z \neq 0\), find the numerical value of \(\frac{2x^2 - xy}{y^2 + 4z^2}\). | \frac{744}{305} | 0.5625 | 7,169.375 | 7,445.222222 | 6,814.714286 | |
Let $x,$ $y,$ and $z$ be real numbers such that $x + y + z = 7$ and $x, y, z \geq 2.$ Find the maximum value of
\[\sqrt{2x + 3} + \sqrt{2y + 3} + \sqrt{2z + 3}.\] | \sqrt{69} | 0.5 | 7,254.875 | 6,594.875 | 7,914.875 | |
Given that $2^x+ 2^x+ 2^x+ 2^x= 128$, what is the value of $(x + 1)(x - 1)$? | 24 | 1 | 1,260.875 | 1,260.875 | -1 | |
What is the greatest possible number of digits in the product of a 4-digit whole number and a 3-digit whole number? | 7 | 0.5625 | 6,002.0625 | 4,801.666667 | 7,545.428571 | |
Call a triangle nice if the plane can be tiled using congruent copies of this triangle so that any two triangles that share an edge (or part of an edge) are reflections of each other via the shared edge. How many dissimilar nice triangles are there? | 4 | The triangles are 60-60-60, 45-45-90, 30-60-90, and 30-30-120. We make two observations. - By reflecting "around" the same point, any angle of the triangle must be an integer divisor of $360^{\circ}$. - if any angle is an odd divisor of $360^{\circ}$, i.e equals \frac{360}{k}$ for odd $k$, then the two adjacent sides m... | 0 | 6,978.625 | -1 | 6,978.625 |
Convex quadrilateral $ABCD$ has $AB = 18$, $\angle A = 60^\circ$, and $\overline{AB} \parallel \overline{CD}$. In some order, the lengths of the four sides form an arithmetic progression, and side $\overline{AB}$ is a side of maximum length. The length of another side is $a$. What is the sum of all possible values of $... | 66 | 1. **Identify the properties of the quadrilateral**: Given that $ABCD$ is a convex quadrilateral with $AB = 18$, $\angle A = 60^\circ$, and $\overline{AB} \parallel \overline{CD}$. The lengths of the sides form an arithmetic progression, and $AB$ is the longest side.
2. **Introduce a point $E$ on $\overline{AB}$**: Co... | 0 | 8,192 | -1 | 8,192 |
Xiao Liang went to the science museum. The science museum is located 400 meters northeast of Xiao Liang's home. When Xiao Liang left home, he walked 600 meters in the northwest direction by mistake. At this time, he did not see the science museum. He asked a lady, and she didn't know where the science museum was, so sh... | 600 | 0.6875 | 4,465.5 | 4,408.454545 | 4,591 | |
At a certain meeting, a total of \(12k\) people attended. Each person greeted exactly \(3k+6\) other people. For any two individuals, the number of people who greeted both of them is the same. How many people attended the meeting? | 36 | 0 | 8,192 | -1 | 8,192 | |
Five positive integers (not necessarily all different) are written on five cards. Boris calculates the sum of the numbers on every pair of cards. He obtains only three different totals: 57, 70, and 83. What is the largest integer on any card? | 48 | 0 | 8,192 | -1 | 8,192 | |
What is the digit in the tens place when $7^{2005}$ is expressed in decimal notation? | 0 | 1 | 3,730.875 | 3,730.875 | -1 | |
Suppose that $f(x) = ax+b$ and $g(x) = -3x+5$. If $h(x) = f(g(x))$ and $h^{-1}(x) = x+7$, find $a-b$. | 5 | 0.9375 | 3,535.8125 | 3,225.4 | 8,192 | |
Given that all edges of a tetrahedron have a length of $\sqrt{2}$, and the four vertices are on the same sphere, calculate the surface area of this sphere. | 3 \pi | 0.875 | 3,915 | 3,304 | 8,192 | |
In quadrilateral $ABCD,$ $AB = 5,$ $BC = 8$, and $CD = 20$ units. Angle $B$ and angle $C$ are both right angles. What is the length of segment $AD$? | 17 | 0.875 | 4,983.1875 | 4,885 | 5,670.5 | |
What is the value of $\displaystyle\frac{109^2-100^2}{9}$? | 209 | 1 | 2,152.4375 | 2,152.4375 | -1 | |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(x^2 + yf(z)) = xf(x) + zf(y)\]for all real numbers $x,$ $y,$ and $z.$
Let $n$ be the number of possible values of $f(5),$ and let $s$ be the sum of all possible values of $f(5).$ Find $n \times s.$ | 10 | 0.75 | 6,382.9375 | 5,779.916667 | 8,192 | |
For a natural number $N$, if at least five out of the nine natural numbers $1-9$ can divide $N$, then $N$ is called a "five-divisible number". What is the smallest "five-divisible number" greater than 2000? | 2004 | 0.1875 | 7,788.75 | 6,245.666667 | 8,144.846154 | |
Find all prime numbers $p$ such that there exists a unique $a \in \mathbb{Z}_p$ for which $a^3 - 3a + 1 = 0.$ | 3 |
To find all prime numbers \( p \) such that there exists a unique \( a \in \mathbb{Z}_p \) for which the equation \( a^3 - 3a + 1 = 0 \) holds, we proceed as follows:
The equation in question is \( a^3 - 3a + 1 = 0 \). We need to determine under what conditions (i.e., for which primes \( p \)) this polynomial has exa... | 0.25 | 7,971.0625 | 7,308.25 | 8,192 |
Find the minimum value of
\[9y + \frac{1}{y^6}\] for \(y > 0\). | 10 | 0 | 8,192 | -1 | 8,192 | |
Jessica works at a bakery, and every day she makes 30 pie crusts that each use up $\frac16$ cup of flour. One day, she decides that she instead wants to make 20 larger pie crusts using the same total amount of flour. How many cups of flour will go into each of the new crusts? Express your answer in simplest fractional ... | \frac14 | 0.875 | 2,311 | 1,470.857143 | 8,192 | |
Positive integers $a$, $b$, $c$, and $d$ satisfy $a > b > c > d$, $a + b + c + d = 2020$, and $a^2 - b^2 + c^2 - d^2 = 2024$. Find the number of possible values of $a$. | 503 | 0 | 8,192 | -1 | 8,192 | |
Given that \(\sin (x + \sin x) = \cos (x - \cos x)\), where \(x \in [0, \pi]\). Find \(x\). | \frac{\pi}{4} | 0.4375 | 7,511.1875 | 6,635.857143 | 8,192 | |
A trirectangular tetrahedron $M-ABC$ has three pairs of adjacent edges that are perpendicular, and a point $N$ inside the base triangle $ABC$ is at distances of $2\sqrt{2}$, $4$, and $5$ from the three faces respectively. Find the surface area of the smallest sphere that passes through both points $M$ and $N$. | 49\pi | 0.125 | 7,896.4375 | 5,827.5 | 8,192 | |
A positive integer whose digits are the same when read forwards or backwards is called a palindrome. An example of a palindrome is 13931. What is the sum of the digits of the next palindrome greater than 13931? | 10 | 0.8125 | 4,928.3125 | 4,175.153846 | 8,192 | |
Compute the sum of the geometric series $-1 + 2 - 4 + 8 - \cdots + 512$. | 341 | 1 | 2,915.375 | 2,915.375 | -1 | |
A bag contains ten balls, some of which are red and the rest of which are yellow. When two balls are drawn at random at the same time, the probability that both balls are red is $\frac{1}{15}$. How many balls in the bag are red? | 3 | 1 | 1,731.0625 | 1,731.0625 | -1 | |
How many positive integers less than 100 are relatively prime to 200? | 40 | $401 \leq n<100$ is relatively prime to 200 if and only if it's relatively prime to 100 (200, 100 have the same prime factors). Thus our answer is $\phi(100)=100 \frac{1}{2} \frac{4}{5}=40$. | 0.8125 | 4,456.8125 | 3,980.615385 | 6,520.333333 |
In trapezoid \(ABCD\) with \(BC \parallel AD\), it is known that \(AD = 3 \cdot BC\). A line intersects the non-parallel sides of the trapezoid at points \(M\) and \(N\) such that \(AM:MB = 3:5\) and \(CN:ND = 2:7\). Find the ratio of the areas of quadrilaterals \(MBCN\) and \(AMND\). | 9/23 | 0.1875 | 7,925.5 | 6,770.666667 | 8,192 | |
In square ABCD, an isosceles triangle AEF is inscribed; point E lies on side BC, point F lies on side CD, and AE = AF. The tangent of angle AEF is 3. Find the cosine of angle FAD. | \frac{2\sqrt{5}}{5} | 0 | 4,074.75 | -1 | 4,074.75 | |
In tetrahedron $ABCD$, edge $AB$ has length 3 cm. The area of face $ABC$ is $15\mbox{cm}^2$ and the area of face $ABD$ is $12 \mbox { cm}^2$. These two faces meet each other at a $30^\circ$ angle. Find the volume of the tetrahedron in $\mbox{cm}^3$.
| 20 | 0.5625 | 6,628.5625 | 5,412.555556 | 8,192 | |
Jing purchased eight identical items. If the total cost was $\$ 26$, what is the cost per item, in dollars? | \frac{26}{8} | Since Jing purchased 8 identical items and the total cost was $\$ 26$, then to obtain the cost per item, she divides the total cost by the number of items. Thus, the answer is $26 \div 8$. | 0 | 216 | -1 | 216 |
How many of the natural numbers from 1 to 1000, inclusive, contain the digit 5 at least once? | 270 | 0 | 6,910.6875 | -1 | 6,910.6875 | |
If a sequence $\{a_n\}$ satisfies $$\frac {1}{a_{n+1}}- \frac {1}{a_{n}}=d$$ (where $n\in\mathbb{N}^*$, $d$ is a constant), then the sequence $\{a_n\}$ is called a "harmonic sequence". Given that the sequence $\{\frac {1}{x_{n}}\}$ is a "harmonic sequence", and $x_1+x_2+\ldots+x_{20}=200$, the maximum value of $x_3x_{1... | 100 | 0.125 | 7,756.125 | 4,705 | 8,192 | |
Three tanks contain water. The number of litres in each is shown in the table: Tank A: 3600 L, Tank B: 1600 L, Tank C: 3800 L. Water is moved from each of Tank A and Tank C into Tank B so that each tank contains the same volume of water. How many litres of water are moved from Tank A to Tank B? | 600 | In total, the three tanks contain $3600 \mathrm{~L} + 1600 \mathrm{~L} + 3800 \mathrm{~L} = 9000 \mathrm{~L}$. If the water is divided equally between the three tanks, each will contain $\frac{1}{3} \cdot 9000 \mathrm{~L} = 3000 \mathrm{~L}$. Therefore, $3600 \mathrm{~L} - 3000 \mathrm{~L} = 600 \mathrm{~L}$ needs to b... | 0.875 | 2,049.4375 | 2,206.285714 | 951.5 |
Given the function $f(x)=\sqrt{3}\sin x\cos x+\cos^{2}x$.
(1) Find the value of $f(\frac{\pi }{24})$;
(2) If the function $f(x)$ is monotonically increasing in the interval $[-m,m]$, find the maximum value of the real number $m$. | \frac{\pi}{6} | 0.8125 | 6,579.9375 | 6,252.769231 | 7,997.666667 | |
What is the value of the product \[
\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?
\] | \frac{50}{99} | 1. **Rewrite the Product**: We start by rewriting the given product:
\[
\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)
\]
2. **Simplify Each Term**: Each term in... | 0.6875 | 5,107.5625 | 4,014.272727 | 7,512.8 |
The diagram shows the miles traveled by cyclists Clara and David. After five hours, how many more miles has Clara cycled than David?
[asy]
/* Modified AMC8 1999 #4 Problem */
draw((0,0)--(6,0)--(6,4.5)--(0,4.5)--cycle);
for(int x=0; x <= 6; ++x) {
for(real y=0; y <=4.5; y+=0.9) {
dot((x, y));
}
}
draw((... | 18 | 0.625 | 5,960.875 | 4,672.7 | 8,107.833333 | |
Find the sum of the even positive divisors of 1000. | 2184 | Notice that $2 k$ is a divisor of 1000 iff $k$ is a divisor of 500, so we need only find the sum of the divisors of 500 and multiply by 2. This can be done by enumerating the divisors individually, or simply by using the formula: $\sigma\left(2^{2} \cdot 5^{3}\right)=\left(1+2+2^{2}\right)(1+5+5^{2}+5^{3}\right)=1092$,... | 0.8125 | 4,839.3125 | 4,065.615385 | 8,192 |
Define $a$ ? $=(a-1) /(a+1)$ for $a \neq-1$. Determine all real values $N$ for which $(N ?)$ ?=\tan 15. | -2-\sqrt{3} | Let $x=N$ ?. Then $(x-1) \cos 15=(x+1) \sin 15$. Squaring and rearranging terms, and using the fact that $\cos ^{2} 15-\sin ^{2} 15=\cos 30=\frac{\sqrt{3}}{2}$, we have $3 x^{2}-4 \sqrt{3} x+3=0$. Solving, we find that $x=\sqrt{3}$ or \frac{\sqrt{3}}{3}$. However, we may reject the second root because it yields a negat... | 1 | 4,459.25 | 4,459.25 | -1 |
Kevin starts with the vectors \((1,0)\) and \((0,1)\) and at each time step, he replaces one of the vectors with their sum. Find the cotangent of the minimum possible angle between the vectors after 8 time steps. | 987 | Say that the vectors Kevin has at some step are \((a, b)\) and \((c, d)\). Notice that regardless of which vector he replaces with \((a+c, b+d)\), the area of the triangle with vertices \((0,0),(a, b)\), and \((c, d)\) is preserved with the new coordinates. We can see this geometrically: the parallelogram with vertices... | 0 | 8,192 | -1 | 8,192 |
Given $( \sqrt {x}+ \dfrac {2}{x^{2}})^{n}$, the ratio of the coefficient of the fifth term to the coefficient of the third term in its expansion is $56:3$.
(Ⅰ) Find the constant term in the expansion;
(Ⅱ) When $x=4$, find the term with the maximum binomial coefficient in the expansion. | \dfrac {63}{256} | 0.4375 | 6,561.4375 | 5,747.714286 | 7,194.333333 | |
In the game of Galactic Dominion, players compete to amass cards, each of which is worth a certain number of points. Say you are playing a version of this game with only two kinds of cards, planet cards and hegemon cards. Each planet card is worth 2010 points, and each hegemon card is worth four points per planet card ... | 503 | If you have $P$ planets and $H$ hegemons, buying a planet gives you $2010+4 H$ points while buying a hegemon gives you $4 P$ points. Thus you buy a hegemon whenever $P-H \geq 502.5$, and you buy a planet whenever $P-H \leq 502.5$. Therefore $a_{i}=1$ for $1 \leq i \leq 503$. Starting at $i=504$ (at which point you have... | 0.125 | 8,028.25 | 6,882 | 8,192 |
Let \( A B C \) be an isosceles triangle with \( B \) as the vertex of the equal angles. Let \( F \) be a point on the bisector of \( \angle A B C \) such that \( (A F) \) is parallel to \( (B C) \). Let \( E \) be the midpoint of \([B C]\), and let \( D \) be the symmetric point of \( A \) with respect to \( F \). Cal... | 1/2 | 0.6875 | 5,649.9375 | 4,494.454545 | 8,192 | |
How many numbers in the list $43$, $4343$, $434343$, $\dots$, are prime? | 1 | 0.75 | 5,897.6875 | 5,132.916667 | 8,192 | |
On a table, there are 2020 boxes. Some of them contain candies, while others are empty. The first box has a label that reads: "All boxes are empty." The second box reads: "At least 2019 boxes are empty." The third box reads: "At least 2018 boxes are empty," and so on, up to the 2020th box, which reads: "At least one bo... | 1010 | 0.1875 | 7,646.1875 | 7,240.333333 | 7,739.846154 | |
At an elementary school, the students in third grade, fourth grade, and fifth grade run an average of $12$, $15$, and $10$ minutes per day, respectively. There are twice as many third graders as fourth graders, and twice as many fourth graders as fifth graders. What is the average number of minutes run per day by these... | \frac{88}{7} | 1. **Define the number of students in each grade:**
Let the number of fifth graders be $f$. According to the problem, there are twice as many fourth graders as fifth graders, and twice as many third graders as fourth graders. Therefore, the number of fourth graders is $2f$ and the number of third graders is $4f$.
2... | 1 | 2,294.1875 | 2,294.1875 | -1 |
Compute $55^2 - 45^2$ in your head. | 1000 | 1 | 2,295.5 | 2,295.5 | -1 | |
In triangle ABC, $a-b=4$, $a+c=2b$, and the largest angle is $120^\circ$. Find the perimeter of this triangle. | 30 | 0.9375 | 3,851.375 | 3,562 | 8,192 | |
How many natural numbers greater than 10 but less than 100 are relatively prime to 21? | 51 | 0 | 6,314.625 | -1 | 6,314.625 | |
From a plywood circle with a diameter of 30 cm, two smaller circles with diameters of 20 cm and 10 cm are cut out. What is the diameter of the largest circle that can be cut from the remaining piece of plywood? | 20 | 0 | 8,080.1875 | -1 | 8,080.1875 | |
For every subset $T$ of $U = \{ 1,2,3,\ldots,18 \}$, let $s(T)$ be the sum of the elements of $T$, with $s(\emptyset)$ defined to be $0$. If $T$ is chosen at random among all subsets of $U$, the probability that $s(T)$ is divisible by $3$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find ... | 683 | Consider the elements of $U$ modulo $3.$
Ignore the $0$'s because we're gonna multiply $\binom{6}{0}+..+\binom{6}{6}=2^6$ at the end. Let $a$ be the $1's$ and $b$ be the $2's$. The key here is that $2 \equiv -1 \pmod{3}$ so the difference between the number of $a$ and $b$ is a multiple of $3$.
1. Counted twice becaus... | 0.5 | 6,654.125 | 6,320.875 | 6,987.375 |
Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30,$ inclusive. Exactly $300$ points in $S$ lie on or below a line with equation $y=mx.$ The possible values of $m$ lie in an interval of length $\frac ab,$ where $a$ and $b$ are relatively prime positi... | 85 | 1. **Understanding the Problem:**
The problem asks us to find the possible values of the slope $m$ of a line such that exactly $300$ lattice points from the set $S$ (where $S$ consists of points $(x,y)$ with $1 \leq x, y \leq 30$) lie on or below the line $y = mx$. The total number of lattice points in $S$ is $30 \t... | 0 | 7,945.9375 | -1 | 7,945.9375 |
Given a triangle $\triangle ABC$ with angles $A$, $B$, $C$ and their corresponding opposite sides $a$, $b$, $c$. It is known that $2a\sin (C+ \frac{\pi}{6})=b+c$.
1. Find the value of angle $A$.
2. If $B= \frac{\pi}{4}$ and $b-a= \sqrt{2}- \sqrt{3}$, find the area of $\triangle ABC$. | \frac{3 + \sqrt{3}}{4} | 0 | 6,706.5625 | -1 | 6,706.5625 | |
$A B C D$ is a rectangle with $A B=20$ and $B C=3$. A circle with radius 5, centered at the midpoint of $D C$, meets the rectangle at four points: $W, X, Y$, and $Z$. Find the area of quadrilateral $W X Y Z$. | 27 | Suppose that $X$ and $Y$ are located on $A B$ with $X$ closer to $A$ than $B$. Let $O$ be the center of the circle, and let $P$ be the midpoint of $A B$. We have $O P \perp A B$ so $O P X$ and $O P Y$ are right triangles with right angles at $P$. Because $O X=O Y=5$ and $O P=3$, we have $X P=P Y=4$ by the Pythagorean t... | 0.875 | 5,590.375 | 5,218.714286 | 8,192 |
Two spheres are inscribed in a dihedral angle such that they touch each other. The radius of one sphere is 4 times that of the other, and the line connecting the centers of the spheres forms an angle of \(60^\circ\) with the edge of the dihedral angle. Find the measure of the dihedral angle. Provide the cosine of this ... | 0.04 | 0 | 7,757.9375 | -1 | 7,757.9375 | |
Given that the four vertices of the triangular pyramid $P-ABC$ lie on the surface of the sphere $O$, and $PA = PB = PC$. The triangle $ABC$ is an equilateral triangle with side length 2. Points $E$ and $F$ are the midpoints of $AC$ and $BC$ respectively, and $\angle EPF = 60^\circ$. Find the surface area of the sphere ... | 6 \pi | 0.5625 | 6,487.125 | 5,161.111111 | 8,192 | |
What is the least positive integer $n$ such that $n^2 - n$ is divisible by some but not all integer values of $k$ when $1\leq k \leq n$? | 5 | 1 | 5,041.375 | 5,041.375 | -1 | |
Given \(\alpha, \beta \in \left(0, \frac{\pi}{2}\right)\), \(\cos \alpha = \frac{4}{5}\), \(\tan (\alpha - \beta) = -\frac{1}{3}\), find \(\cos \beta\). | \frac{9 \sqrt{10}}{50} | 0 | 4,949.625 | -1 | 4,949.625 | |
Convert the binary number $11011001_2$ to base 4. | 3121_4 | 0.5625 | 6,103.125 | 4,478.444444 | 8,192 | |
What is the base five product of the numbers $203_{5}$ and $14_{5}$? | 3402_5 | 0.75 | 5,310.9375 | 4,350.583333 | 8,192 | |
Let $\{a_{n}\}$ be a geometric sequence, and let $S_{n}$ be the sum of the first n terms of $\{a_{n}\}$. Given that $S_{2}=2$ and $S_{6}=4$, calculate the value of $S_{4}$. | 1+\sqrt{5} | 0.5 | 7,382.6875 | 6,573.375 | 8,192 | |
Select 4 out of 6 sprinters to participate in a 4×100 relay race. If neither A nor B runs the first leg, then there are $\boxed{\text{different}}$ possible team compositions. | 240 | 0.4375 | 5,806.375 | 3,101 | 7,910.555556 | |
How many positive four-digit integers are divisible by both 13 and 7? | 99 | 1 | 3,569 | 3,569 | -1 | |
There are 306 different cards with numbers \(3, 19, 3^{2}, 19^{2}, \ldots, 3^{153}, 19^{153}\) (each card has exactly one number, and each number appears exactly once). How many ways can you choose 2 cards such that the product of the numbers on the selected cards is a perfect square? | 17328 | 0.0625 | 7,884.4375 | 7,569 | 7,905.466667 | |
Evaluate the sum
\[
\sum_{k=1}^{50} (-1)^k \cdot \frac{k^3 + k^2 + 1}{(k+1)!}.
\]
Determine the form of the answer as a difference between two terms, where each term is a fraction involving factorials. | \frac{126001}{51!} | 0 | 8,192 | -1 | 8,192 | |
How many four-digit numbers contain one even digit and three odd digits, with no repeated digits? | 1140 | 0.0625 | 7,901.75 | 8,192 | 7,882.4 | |
Solve for $x$: $\sqrt[3]{20x + \sqrt[3]{20x + 13}} = 13.$ | \frac{546}{5} | 1 | 3,790.125 | 3,790.125 | -1 |
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