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 |
|---|---|---|---|---|---|---|
Find the number of terms $n \leq 2012$ such that $a_{n}=\frac{3^{n+1}-1}{2}$ is divisible by 7. | 335 | To demonstrate that $7 \mid a_{n}=\frac{3^{n+1}-1}{2}$ is equivalent to showing that $3^{n+1}-1 \equiv 0 \bmod 7$. However $3^{n+1}-1=0 \bmod 7 \Longrightarrow n+1 \equiv 0 \bmod 6$, hence the values of $n \leq 2012$ are $5,11,17, \ldots, 2009$, which is in total $$\frac{2009+1}{6}=335$$ possible terms. | 1 | 3,270.25 | 3,270.25 | -1 |
Given that the random variable $X$ follows a normal distribution $N(2, \sigma^2)$, and $P(X \leq 4) = 0.84$, determine the value of $P(X < 0)$. | 0.16 | 0.3125 | 5,044.25 | 4,995 | 5,066.636364 | |
Mayuki walks once around a track shaped with straight sides and semicircular ends at a constant speed daily. The track has a width of \(4\) meters, and it takes her \(24\) seconds longer to walk around the outside edge than the inside edge. Determine Mayuki's speed in meters per second. | \frac{\pi}{3} | 1 | 2,589.9375 | 2,589.9375 | -1 | |
A radio program has a quiz consisting of $3$ multiple-choice questions, each with $3$ choices. A contestant wins if he or she gets $2$ or more of the questions right. The contestant answers randomly to each question. What is the probability of winning? | \frac{7}{27} | To solve this problem, we need to calculate the probability that the contestant guesses at least 2 out of 3 questions correctly. We will consider the following cases:
1. **Case 1: The contestant guesses all three questions correctly.**
2. **Case 2: The contestant guesses exactly two questions correctly.**
#### Case 1... | 0.9375 | 3,464.9375 | 3,149.8 | 8,192 |
Given $f(x)= \sqrt{2}\sin \left( 2x+ \frac{π}{4} \right)$.
(1) Find the equation of the axis of symmetry of the graph of the function $f(x)$;
(2) Find the interval(s) where $f(x)$ is monotonically increasing;
(3) Find the maximum and minimum values of the function $f(x)$ when $x\in \left[ \frac{π}{4}, \frac{3π}{4} \... | - \sqrt{2} | 0.625 | 6,738.125 | 5,885 | 8,160 | |
Points A and B lie on a circle centered at O, and ∠AOB = 60°. A second circle is internally tangent to the first and tangent to both OA and OB. What is the ratio of the area of the smaller circle to that of the larger circle? | \frac{1}{9} | 0.8125 | 5,377.5625 | 4,728.076923 | 8,192 | |
Evaluate the limit as \( n \) approaches infinity:
$$
\lim _{n \rightarrow \infty} \frac{(1+2n)^{3} - 8n^{5}}{(1+2n)^{2} + 4n^{2}}
$$ | -1 | 0 | 6,305.75 | -1 | 6,305.75 | |
How many of the numbers from the set $\{1, 2, 3, \ldots, 100\}$ have a perfect square factor other than one? | 42 | 0 | 7,851 | -1 | 7,851 | |
A function $f$ is defined by $f(z) = (4 + i) z^2 + \alpha z + \gamma$ for all complex numbers $z$, where $\alpha$ and $\gamma$ are complex numbers and $i^2 = - 1$. Suppose that $f(1)$ and $f(i)$ are both real. What is the smallest possible value of $| \alpha | + |\gamma |$? | \sqrt{2} | 0.6875 | 7,228.375 | 6,790.363636 | 8,192 | |
Three different numbers from the list $2, 3, 4, 6$ have a sum of 11. What is the product of these numbers? | 36 | The sum of 2, 3 and 6 is $2 + 3 + 6 = 11$. Their product is $2 \cdot 3 \cdot 6 = 36$. | 1 | 1,256.4375 | 1,256.4375 | -1 |
Find the sum of all integral values of $c$ with $c\le 25$ for which the equation $y=x^2-7x-c$ has two rational roots. | -2 | 0.625 | 6,358.9375 | 5,259.1 | 8,192 | |
In $\triangle ABC$, $\angle ABC=45^\circ$. Point $D$ is on $\overline{BC}$ so that $2 \cdot BD=CD$ and $\angle DAB=15^\circ$. Find $\angle ACB.$ | 75^\circ | 1. **Given Information and Setup:**
- In $\triangle ABC$, $\angle ABC = 45^\circ$.
- Point $D$ is on $\overline{BC}$ such that $2 \cdot BD = CD$.
- $\angle DAB = 15^\circ$.
- Let $\angle ACB = \theta$.
2. **Using the Angle Bisector Theorem:**
- Since $2 \cdot BD = CD$, $D$ is the midpoint of $BC$.
- ... | 0.375 | 7,510.4375 | 6,394.333333 | 8,180.1 |
How many positive, three-digit integers contain at least one $4$ as a digit but do not contain a $6$ as a digit? | 200 | 0.3125 | 6,715.1875 | 3,705.8 | 8,083.090909 | |
Let $d(n)$ denote the number of positive divisors of $n$. For positive integer $n$ we define $f(n)$ as $$f(n) = d\left(k_1\right) + d\left(k_2\right)+ \cdots + d\left(k_m\right),$$ where $1 = k_1 < k_2 < \cdots < k_m = n$ are all divisors of the number $n$. We call an integer $n > 1$ [i]almost perfect[/i] if $f(n) = n$... | 1, 3, 18, 36 |
To find all almost perfect numbers, we first consider the function \( f(n) \). For a given positive integer \( n \), we define \( f(n) \) as:
\[
f(n) = d(k_1) + d(k_2) + \cdots + d(k_m),
\]
where \( 1 = k_1 < k_2 < \cdots < k_m = n \) are all the divisors of the number \( n \). Here, \( d(k) \) denotes the number of... | 0 | 8,192 | -1 | 8,192 |
PQR Entertainment wishes to divide their popular idol group PRIME, which consists of seven members, into three sub-units - PRIME-P, PRIME-Q, and PRIME-R - with each of these sub-units consisting of either two or three members. In how many different ways can they do this, if each member must belong to exactly one sub-un... | 630 | 0.0625 | 5,142.5625 | 8,001 | 4,952 | |
Suppose there are $160$ pigeons and $n$ holes. The $1$ st pigeon flies to the $1$ st hole, the $2$ nd pigeon flies to the $4$ th hole, and so on, such that the $i$ th pigeon flies to the $(i^2\text{ mod }n)$ th hole, where $k\text{ mod }n$ is the remainder when $k$ is divided by $n$ . What is minimum $n... | 326 | 0 | 8,192 | -1 | 8,192 | |
Given a triangle with integral sides and an isosceles perimeter of 11, calculate the area of the triangle. | \frac{5\sqrt{2.75}}{2} | 0 | 8,038.3125 | -1 | 8,038.3125 | |
It takes Clea 60 seconds to walk down an escalator when it is not moving, and 24 seconds when it is moving. How many seconds would it take Clea to ride the escalator down when she is not walking? | 40 | 1. **Identify the variables and equations:**
Let $x$ be Clea's walking speed in units per second and $y$ be the total distance of the escalator in units. When Clea walks down the stationary escalator, it takes her 60 seconds, so the equation is:
\[
60x = y
\]
When the escalator is moving, and Clea is als... | 1 | 2,347.375 | 2,347.375 | -1 |
In an isosceles right triangle \( \triangle ABC \), \( \angle A = 90^\circ \), \( AB = 1 \). \( D \) is the midpoint of \( BC \), \( E \) and \( F \) are two other points on \( BC \). \( M \) is the other intersection point of the circumcircles of \( \triangle ADE \) and \( \triangle ABF \); \( N \) is the other inters... | \sqrt{2} | 0 | 8,192 | -1 | 8,192 | |
Given that five students from Maplewood school worked for 6 days, six students from Oakdale school worked for 4 days, and eight students from Pinecrest school worked for 7 days, and the total amount paid for the students' work was 1240 dollars, determine the total amount earned by the students from Oakdale school, igno... | 270.55 | 0.1875 | 654.375 | 651 | 655.153846 | |
There are 3 numbers A, B, and C, such that $1001C - 2002A = 4004$, and $1001B + 3003A = 5005$. What is the average of A, B, and C? | 3 | 1. **Isolate $C$ and $B$ in terms of $A$:**
- From the equation $1001C - 2002A = 4004$, we can isolate $C$:
\[
1001C = 2002A + 4004 \implies C = 2A + 4
\]
- From the equation $1001B + 3003A = 5005$, we can isolate $B$:
\[
1001B = 5005 - 3003A \implies B = -3A + 5
\]
2. **Calculate t... | 1 | 1,933.375 | 1,933.375 | -1 |
One integer is selected at random from the following list of 15 integers: $1,2,2,3,3,3,4,4,4,4,5,5,5,5,5$. The probability that the selected integer is equal to $n$ is $\frac{1}{3}$. What is the value of $n$? | 5 | Since the list includes 15 integers, then an integer has a probability of $\frac{1}{3}$ of being selected if it occurs $\frac{1}{3} \cdot 15=5$ times in the list. The integer 5 occurs 5 times in the list and no other integer occurs 5 times, so $n=5$. | 1 | 1,308.6875 | 1,308.6875 | -1 |
In how many ways can I choose 3 captains from a team of 11 people? | 165 | 0.875 | 834.75 | 860.071429 | 657.5 | |
Evaluate: $64^2-36^2$ | 2800 | 1 | 627.25 | 627.25 | -1 | |
Factorize \( n^{5} - 5n^{3} + 4n \). What can be concluded in terms of divisibility? | 120 | 0.4375 | 4,834.125 | 5,001.285714 | 4,704.111111 | |
A regular $2015$ -simplex $\mathcal P$ has $2016$ vertices in $2015$ -dimensional space such that the distances between every pair of vertices are equal. Let $S$ be the set of points contained inside $\mathcal P$ that are closer to its center than any of its vertices. The ratio of the volume of $S$ to the v... | 520 | 0 | 8,160.625 | -1 | 8,160.625 | |
Which of the following multiplication expressions has a product that is a multiple of 54? (Fill in the serial number).
$261 \times 345$
$234 \times 345$
$256 \times 345$
$562 \times 345$ | $234 \times 345$ | 0 | 1,162.75 | -1 | 1,162.75 | |
Inside a square, 100 points are marked. The square is divided into triangles in such a way that the vertices of the triangles are only the marked 100 points and the vertices of the square, and for each triangle in the division, each marked point either lies outside this triangle or is a vertex of it (such divisions are... | 202 | 0.8125 | 3,932.3125 | 3,632.538462 | 5,231.333333 | |
What is the value of $49^3 + 3(49^2) + 3(49) + 1$? | 125000 | 0.875 | 3,432.9375 | 2,753.071429 | 8,192 | |
A basketball player made the following number of free throws in 8 successive games: 6, 18, 15, 14, 19, 12, 19, and 15. What is the median number of successful free throws? | 15 | 0.9375 | 2,435.5625 | 2,051.8 | 8,192 | |
Given that $\alpha^{2005}+\beta^{2005}$ can be expressed as a two-variable polynomial in terms of $\alpha + \beta$ and $\alpha \beta$, find the sum of the coefficients of this polynomial. | -1 | 0 | 7,326.25 | -1 | 7,326.25 | |
The average years of experience of three employees, David, Emma, and Fiona, at a company is 12 years. Five years ago, Fiona had the same years of experience as David has now. In 4 years, Emma's experience will be $\frac{3}{4}$ of David's experience at that time. How many years of experience does Fiona have now? | \frac{183}{11} | 0.5 | 5,636.125 | 5,500.75 | 5,771.5 | |
The altitudes of a triangle are $12$, $15$, and $20$. What is the area of this triangle? | 150 |
To find the area of a triangle with altitudes \( 12 \), \( 15 \), and \( 20 \), we begin by letting the sides of the triangle be \( a \), \( b \), and \( c \) opposite to the altitudes \( h_a = 12 \), \( h_b = 15 \), and \( h_c = 20 \), respectively. The area \( A \) of the triangle can be expressed as:
\[
A = \frac{... | 0.9375 | 5,130.5625 | 4,926.466667 | 8,192 |
Find the smallest positive angle $x$ that satisfies $\sin 2x \sin 3x = \cos 2x \cos 3x,$ in degrees. | 18^\circ | 1 | 3,561.8125 | 3,561.8125 | -1 | |
What is the sum of all positive integers $\nu$ for which $\mathop{\text{lcm}}[\nu, 45]=180$? | 292 | 0 | 5,570.5625 | -1 | 5,570.5625 | |
For natural numbers _m_ greater than or equal to 2, the _n_-th power of _m_ can be decomposed as follows:
2<sup>2</sup> = 1 + 3, 3<sup>2</sup> = 1 + 3 + 5, 4<sup>2</sup> = 1 + 3 + 5 + 7…
2<sup>3</sup> = 3 + 5, 3<sup>3</sup> = 7 + 9 + 11…
2<sup>4</sup> = 7 + 9…
According to this pattern, the third number in the ... | 125 | 0.25 | 7,848.375 | 6,817.5 | 8,192 | |
Consider all 1000-element subsets of the set $\{1, 2, 3, \dots , 2015\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. | 2016 |
To solve this problem, we need to find the arithmetic mean of the least elements of all 1000-element subsets of the set $\{1, 2, 3, \ldots, 2015\}$. We will use combinatorial arguments to derive the solution.
#### Step 1: Counting subsets with a fixed least element
Let $i$ be the least element of a 1000-element subse... | 0 | 7,785.5 | -1 | 7,785.5 |
In right triangle $DEF$, we have $\sin D = \frac{5}{13}$ and $\sin E = 1$. Find $\sin F$. | \frac{12}{13} | 1 | 2,130.25 | 2,130.25 | -1 | |
Estimate the probability of a rifle student hitting the target using a random simulation method by analyzing the ratio of the number of sets of three random numbers where exactly one represents a hit to the total number of sets of three random numbers. | \frac{9}{20} | 0 | 5,816.875 | -1 | 5,816.875 | |
In convex quadrilateral $ABCD, \angle A \cong \angle C, AB = CD = 180,$ and $AD \neq BC.$ The perimeter of $ABCD$ is $640$. Find $\lfloor 1000 \cos A \rfloor.$ (The notation $\lfloor x \rfloor$ means the greatest integer that is less than or equal to $x.$)
| 777 | 0.875 | 4,007.8125 | 3,410.071429 | 8,192 | |
Let \( OP \) be the diameter of the circle \( \Omega \), and let \( \omega \) be a circle with its center at point \( P \) and a radius smaller than that of \( \Omega \). The circles \( \Omega \) and \( \omega \) intersect at points \( C \) and \( D \). The chord \( OB \) of circle \( \Omega \) intersects the second ci... | \sqrt{5} | 0 | 7,910.1875 | -1 | 7,910.1875 | |
Given that $cos\left( \frac{\pi}{6}-\alpha\right) = \frac{\sqrt{3}}{2}$, find the value of $cos\left( \frac{5\pi}{6}+\alpha\right) - sin^2\left(-\alpha+\frac{7\pi}{6}\right)$.
If $cos\alpha = \frac{2}{3}$ and $\alpha$ is an angle in the fourth quadrant, find the value of $\frac{sin(\alpha-2\pi)+sin(-\alpha-3\pi)cos(-\a... | \frac{\sqrt{5}}{2} | 0 | 7,182.4375 | -1 | 7,182.4375 | |
Given $f(x) = \frac{x^2 + 33}{x}, (x \in \mathbb{N}^*)$, find the minimum value of $f(x)$ in its domain. | \frac{23}{2} | 0.875 | 4,032.4375 | 3,727.928571 | 6,164 | |
In this square array of 16 dots, four dots are to be chosen at random. What is the probability that the four dots will be collinear? Express your answer as a common fraction.
[asy]
size(59);
for(int i = 0; i<4; ++i)
for(int j = 0; j<4; ++j)
dot((i,j),linewidth(7));
[/asy] | \frac{1}{182} | 0.125 | 8,014.6875 | 6,773.5 | 8,192 | |
Two cards are dealt at random from a standard deck of 52 cards (13 hearts, 13 clubs, 13 spades, and 13 diamonds). What is the probability that the first card is a 6 and the second card is a Queen? | \dfrac{4}{663} | 0.25 | 5,665.9375 | 5,496.5 | 5,722.416667 | |
Given that triangle \( ABC \) has all side lengths as positive integers, \(\angle A = 2 \angle B\), and \(CA = 9\), what is the minimum possible value of \( BC \)? | 12 | 0.5625 | 7,277.875 | 6,566.888889 | 8,192 | |
Given that $θ$ is an angle in the second quadrant, and $\sin θ$ and $\cos θ$ are the two roots of the equation $2x^2 + (\sqrt{3} - 1)x + m = 0$ (where $m \in \mathbb{R}$), find the value of $\sin θ - \cos θ$. | \frac{1 + \sqrt{3}}{2} | 0 | 5,205.75 | -1 | 5,205.75 | |
In how many different ways can four couples sit around a circular table such that no couple sits next to each other? | 1488 | 0.0625 | 8,021.6875 | 5,467 | 8,192 | |
Petya and Vasya are playing the following game. Petya thinks of a natural number \( x \) with a digit sum of 2012. On each turn, Vasya chooses any natural number \( a \) and finds out the digit sum of the number \( |x-a| \) from Petya. What is the minimum number of turns Vasya needs to determine \( x \) with certainty? | 2012 | 0 | 8,118.0625 | -1 | 8,118.0625 | |
Vitya has five math lessons a week, one on each day from Monday to Friday. Vitya knows that with a probability of \( \frac{1}{2} \) the teacher will not check his homework at all during the week, and with a probability of \( \frac{1}{2} \) the teacher will check it exactly once during one of the math lessons, but it is... | 1/6 | 0.0625 | 6,948.5 | 6,942 | 6,948.933333 | |
Calculate the definite integral:
$$
\int_{0}^{\pi} 2^{4} \cdot \sin ^{8} x \, dx
$$ | \frac{35\pi}{8} | 0.4375 | 6,572.0625 | 4,578.571429 | 8,122.555556 | |
Compute the definite integral:
$$
\int_{0}^{\frac{\pi}{2}} \frac{\sin x \, dx}{(1+\cos x+\sin x)^{2}}
$$ | \ln 2 - \frac{1}{2} | 0.5625 | 6,691.75 | 5,524.888889 | 8,192 | |
A pyramid with a triangular base has edges of unit length, and the angles between its edges are \(60^{\circ}, 90^{\circ},\) and \(120^{\circ}\). What is the volume of the pyramid? | \frac{\sqrt{2}}{12} | 0 | 7,870.4375 | -1 | 7,870.4375 | |
Find the value of $x,$ if \[|x-20| + |x-18| = |2x-36|.\] | 19 | 1 | 2,306 | 2,306 | -1 | |
Given $F_{1}$ and $F_{2}$ are the foci of a pair of related curves, and $P$ is the intersection point of the ellipse and the hyperbola in the first quadrant, when $\angle F_{1}PF_{2}=60^{\circ}$, determine the eccentricity of the ellipse in this pair of related curves. | \dfrac{\sqrt{3}}{3} | 0 | 8,177.9375 | -1 | 8,177.9375 | |
Three equilateral triangles $ABC$, $BCD$, and $CDE$ are positioned such that $B$, $C$, and $D$ are collinear, and $C$ is the midpoint of $BD$. Triangle $CDE$ is positioned such that $E$ is on the same side of line $BD$ as $A$. What is the value of $AE \div BC$ when expressed in simplest radical form?
[asy]
draw((0,0)-... | \sqrt{3} | 0 | 8,136.125 | -1 | 8,136.125 | |
A monic polynomial of degree $n$, with real coefficients, has its first two terms after $x^n$ as $a_{n-1}x^{n-1}$ and $a_{n-2}x^{n-2}$. It’s known that $a_{n-1} = 2a_{n-2}$. Determine the absolute value of the lower bound for the sum of the squares of the roots of this polynomial. | \frac{1}{4} | 0.6875 | 6,863.0625 | 6,291.090909 | 8,121.4 | |
On average, for every 4 sports cars sold at the local dealership, 7 sedans are sold. The dealership predicts that it will sell 28 sports cars next month. How many sedans does it expect to sell? | 49 | 1. **Identify the given ratio**: The problem states that for every 4 sports cars sold, 7 sedans are sold. This sets up a ratio of sports cars to sedans as $4:7$.
2. **Set up the proportion**: Given that the dealership predicts selling 28 sports cars next month, we need to find the number of sedans expected to be sold,... | 0.9375 | 1,866.75 | 1,445.066667 | 8,192 |
How many sequences of $0$s and $1$s of length $19$ are there that begin with a $0$, end with a $0$, contain no two consecutive $0$s, and contain no three consecutive $1$s? | 65 | To solve this problem, we need to determine the number of valid sequences of length 19 that satisfy the given conditions. We will use a recursive approach to count these sequences.
#### Step 1: Define the Problem
Let $f(n)$ be the number of valid sequences of length $n$ that start with a $0$, end with a $0$, contain n... | 0 | 8,192 | -1 | 8,192 |
Let triangle $PQR$ be a right triangle in the xy-plane with a right angle at $R$. Given that the length of the hypotenuse $PQ$ is $50$, and that the medians through $P$ and $Q$ lie along the lines $y=x+2$ and $y=3x+5$ respectively, find the area of triangle $PQR$. | \frac{125}{3} | 0 | 8,192 | -1 | 8,192 | |
For all positive numbers $a,b \in \mathbb{R}$ such that $a+b=1$, find the supremum of the expression $-\frac{1}{2a}-\frac{2}{b}$. | -\frac{9}{2} | 0.625 | 6,251.6875 | 5,350.5 | 7,753.666667 | |
Let $A = (1,1)$ be a point on the parabola $y = x^2.$ The normal to the parabola at $A$ is drawn, intersecting the parabola again at $B.$ Find $B.$
[asy]
unitsize(1 cm);
pair A, B;
A = (1,1);
B = (-3/2,9/4);
real parab (real x) {
return(x^2);
}
draw(graph(parab,-2,2));
draw((A + (-1,-2))--(A + (1,2)));
draw((A... | \left( -\frac{3}{2}, \frac{9}{4} \right) | 1 | 1,753 | 1,753 | -1 | |
Find the area of rhombus $ABCD$ given that the radii of the circles circumscribed around triangles $ABD$ and $ACD$ are $12.5$ and $25$, respectively.
| 400 | 0.6875 | 6,534.5 | 5,781.090909 | 8,192 | |
Given the sequence $\left\{a_{n}\right\}$ where $a_{1}=1$, $a_{2}=2$, and $a_{n} a_{n+1} a_{n+2}=a_{n}+a_{n+1}+a_{n+2}$ for all $n$, and $a_{n+1} a_{n+2} \neq 1$, find the sum $S_{1999}=\sum_{n=1}^{1999} a_{n}$. | 3997 | 1 | 3,451.1875 | 3,451.1875 | -1 | |
Alex and Katy play a game on an \(8 \times 8\) grid made of 64 unit cells. They take turns to play, with Alex going first. On Alex's turn, he writes 'A' in an empty cell. On Katy's turn, she writes 'K' in two empty cells that share an edge. The game ends when one player cannot move. Katy's score is the number of Ks on ... | 32 | 0.25 | 7,816.875 | 6,691.5 | 8,192 | |
In a sequence of positive integers that starts with 1, certain numbers are sequentially colored red according to the following rules. First, 1 is colored red. Then, the next 2 even numbers, 2 and 4, are colored red. After 4, the next three consecutive odd numbers, 5, 7, and 9, are colored red. Following 9, the next fou... | 3943 | 0.3125 | 7,279.1875 | 6,170 | 7,783.363636 | |
A portion of the graph of $f(x)=ax^2+bx+c$ is shown below. The distance between grid lines on the graph is $1$ unit.
What is the value of $a+b+2c$?
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-... | 11 | 1 | 2,673.25 | 2,673.25 | -1 | |
A certain high school has 1000 students in the first year. Their choices of elective subjects are shown in the table below:
| Subject | Physics | Chemistry | Biology | Politics | History | Geography |
|---------|---------|-----------|---------|----------|---------|-----------|
| Number of Students | 300 | 200 | 100 | ... | \frac{3}{10} | 0 | 2,615.25 | -1 | 2,615.25 | |
Given sets \( A = \{1, 2, 3\} \) and \( B = \{4, 5, 6, 7\} \), with a mapping \( f: A \rightarrow B \), if for \( x \in A \), the expression \( x + f(x) + x \cdot f(x) \) is odd, determine the number of such mappings \( f \). | 32 | 0.8125 | 4,963.1875 | 4,929.461538 | 5,109.333333 | |
Suppose a square piece of paper is folded in half vertically. The folded paper is then cut in half along the dashed line. Three rectangles are formed - a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle? | \frac{3}{4} | 1. **Understanding the Problem:**
- We start with a square piece of paper. Let's denote the side length of the square as $s$.
- The paper is folded in half vertically, meaning the new width of the folded paper is $\frac{s}{2}$, and the height remains $s$.
- The paper is then cut in half along the fold line, re... | 0 | 7,917.5625 | -1 | 7,917.5625 |
A right rectangular prism has edge lengths $\log_{5}x, \log_{8}x,$ and $\log_{10}x.$ Given that the sum of its surface area and volume is twice its volume, find the value of $x$.
A) $1,000,000$
B) $10,000,000$
C) $100,000,000$
D) $1,000,000,000$
E) $10,000,000,000$ | 100,000,000 | 0 | 8,170 | -1 | 8,170 | |
Let $A B C$ be a triangle such that $A B=13, B C=14, C A=15$ and let $E, F$ be the feet of the altitudes from $B$ and $C$, respectively. Let the circumcircle of triangle $A E F$ be $\omega$. We draw three lines, tangent to the circumcircle of triangle $A E F$ at $A, E$, and $F$. Compute the area of the triangle these t... | \frac{462}{5} | Note that $A E F \sim A B C$. Let the vertices of the triangle whose area we wish to compute be $P, Q, R$, opposite $A, E, F$ respectively. Since $H, O$ are isogonal conjugates, line $A H$ passes through the circumcenter of $A E F$, so $Q R \| B C$. Let $M$ be the midpoint of $B C$. We claim that $M=P$. This can be see... | 0 | 8,192 | -1 | 8,192 |
In duck language, only letters $q$ , $a$ , and $k$ are used. There is no word with two consonants after each other, because the ducks cannot pronounce them. However, all other four-letter words are meaningful in duck language. How many such words are there?
In duck language, too, the letter $a$ is a vowel, while... | 21 | 0.375 | 7,216.8125 | 6,110.833333 | 7,880.4 | |
Suppose that $20^{21} = 2^a5^b = 4^c5^d = 8^e5^f$ for positive integers $a,b,c,d,e,$ and $f$ . Find $\frac{100bdf}{ace}$ .
*Proposed by Andrew Wu* | 75 | 0.875 | 3,933.75 | 3,708.857143 | 5,508 | |
Number $a$ is such that $\forall a_1, a_2, a_3, a_4 \in \mathbb{R}$, there are integers $k_1, k_2, k_3, k_4$ such that $\sum_{1 \leq i < j \leq 4} ((a_i - k_i) - (a_j - k_j))^2 \leq a$. Find the minimum of $a$. | 1.25 |
Let \( a \) be such that for all \( a_1, a_2, a_3, a_4 \in \mathbb{R} \), there exist integers \( k_1, k_2, k_3, k_4 \) such that
\[
\sum_{1 \leq i < j \leq 4} ((a_i - k_i) - (a_j - k_j))^2 \leq a.
\]
We aim to find the minimum value of \( a \).
Consider the numbers \( a_i = \frac{i}{4} \) for \( i = 1, 2, 3, 4 \). L... | 0 | 8,000.9375 | -1 | 8,000.9375 |
Three volleyballs with a radius of 18 lie on a horizontal floor, each pair touching one another. A tennis ball with a radius of 6 is placed on top of them, touching all three volleyballs. Find the distance from the top of the tennis ball to the floor. (All balls are spherical in shape.) | 36 | 0.75 | 6,045.3125 | 5,870.25 | 6,570.5 | |
For distinct positive integers $a, b<2012$, define $f(a, b)$ to be the number of integers $k$ with $1\le k<2012$ such that the remainder when $ak$ divided by $2012$ is greater than that of $bk$ divided by $2012$. Let $S$ be the minimum value of $f(a, b)$, where $a$ and $b$ range over all pairs of distinct positive inte... | 502 |
To solve for \( S \), the minimum value of \( f(a, b) \), where distinct positive integers \( a, b < 2012 \), we first need to analyze the function \( f(a, b) \). This function represents the number of integers \( k \) with \( 1 \leq k < 2012 \) such that:
\[
ak \mod 2012 > bk \mod 2012
\]
### Steps to find \( S \):... | 0 | 8,037.0625 | -1 | 8,037.0625 |
Compute all values of $b$ for which the following system has a solution $(x,y)$ in real numbers:
\begin{align*}
\sqrt{xy} &= b^b, \\
\log_b (x^{\log_b y}) + \log_b (y^{\log_b x}) &= 4b^4.
\end{align*} | \left( 0, \frac{1}{\sqrt{2}} \right] | 0 | 6,375.125 | -1 | 6,375.125 | |
Henry walked on a flat field 9 meters due north from a tree. He then turned due east and walked 24 feet. He then turned due south and walked 9 meters plus 32 feet. How many feet away from his original starting point is Henry? | 40\text{ feet} | 1 | 3,396.8125 | 3,396.8125 | -1 | |
Points \(O\) and \(I\) are the centers of the circumscribed and inscribed circles of triangle \(ABC\), \(M\) is the midpoint of the arc \(AC\) of the circumscribed circle (which does not contain \(B\)). It is known that \(AB = 15\), \(BC = 7\), and \(MI = MO\). Find \(AC\). | 13 | 0 | 8,192 | -1 | 8,192 | |
What value of $x$ will give the minimum value of $2x^2 - 12x + 3$? | 3 | 1 | 2,246.0625 | 2,246.0625 | -1 | |
The sides opposite to the internal angles $A$, $B$, and $C$ of $\triangle ABC$ are $a$, $b$, and $c$ respectively. It is given that $b\sin C + c\sin B = 4a\sin B\sin C$ and $b^2 + c^2 - a^2 = 8$. The area of $\triangle ABC$ is __________. | \frac{2\sqrt{3}}{3} | 0 | 6,252.75 | -1 | 6,252.75 | |
The length of one side of the square \(ABCD\) is 4 units. A circle is drawn tangent to \(\overline{BC}\) and passing through the vertices \(A\) and \(D\). Find the area of the circle. | \frac{25 \pi}{4} | 0.25 | 2,962.5 | 2,767.75 | 3,027.416667 | |
Let \[f(x) =
\begin{cases}
x/2 &\quad \text{if } x \text{ is even}, \\
3x+1 &\quad \text{if } x \text{ is odd}.
\end{cases}
\]What is $f(f(f(f(1))))$? | 4 | 1 | 1,981.3125 | 1,981.3125 | -1 | |
Given Liam has written one integer three times and another integer four times. The sum of these seven numbers is 131, and one of the numbers is 17, determine the value of the other number. | 21 | 0.4375 | 6,880.6875 | 7,223.142857 | 6,614.333333 | |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=3$, $|\overrightarrow{b}|=2\sqrt{3}$, and $\overrightarrow{a}\perp(\overrightarrow{a}+\overrightarrow{b})$, find the projection of $\overrightarrow{b}$ in the direction of $\overrightarrow{a}$. | -3 | 0.9375 | 4,633.8125 | 4,608.133333 | 5,019 | |
Let $T$ be the set of all positive integer divisors of $144,000$. Calculate the number of numbers that are the product of two distinct elements of $T$. | 451 | 0 | 8,192 | -1 | 8,192 | |
In $\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:
$\textbf{(A)}\ 1:3 \qquad \textbf{(B)}\ 3:4 \qquad \textbf{(C)}\ 4:3 \qquad \textbf{(D)}\ 3:1 \qquad \textbf{(E)}\ 7:1$
| 3:1 | 0 | 5,033.0625 | -1 | 5,033.0625 | |
Four positive integers $p$, $q$, $r$, $s$ satisfy $p \cdot q \cdot r \cdot s = 9!$ and $p < q < r < s$. What is the smallest possible value of $s-p$? | 12 | 0 | 8,192 | -1 | 8,192 | |
When $3z^3-4z^2-14z+3$ is divided by $3z+5$, the quotient is $z^2-3z+\frac{1}{3}$. What is the remainder? | \frac{4}{3} | 1 | 3,029.9375 | 3,029.9375 | -1 | |
An 8-foot by 10-foot floor is tiled with square tiles of size 1 foot by 1 foot. Each tile has a pattern consisting of four white quarter circles of radius 1/2 foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?
[asy]
fill((5,5)--(5,-5)--(-... | 80 - 20\pi | 1 | 3,384.5 | 3,384.5 | -1 | |
Given that $\{a_{n}\}$ is an arithmetic progression, $\{b_{n}\}$ is a geometric progression, and $a_{2}+a_{5}=a_{3}+9=8b_{1}=b_{4}=16$.
$(1)$ Find the general formulas for $\{a_{n}\}$ and $\{b_{n}\}$.
$(2)$ Arrange the terms of $\{a_{n}\}$ and $\{b_{n}\}$ in ascending order to form a new sequence $\{c_{n}\}$. Let t... | 2726 | 0 | 8,192 | -1 | 8,192 | |
Find the square root of $\dfrac{10!}{210}$. | 24\sqrt{30} | 0.875 | 5,088.6875 | 4,645.357143 | 8,192 | |
A function $f$ is defined recursively by $f(1)=f(2)=1$ and \[f(n)=f(n-1)-f(n-2)+n\]for all integers $n \geq 3$. What is $f(2018)$? | 2017 | 0.75 | 7,221 | 6,897.333333 | 8,192 | |
A shooter fires 5 shots in succession, hitting the target with scores of: $9.7$, $9.9$, $10.1$, $10.2$, $10.1$. The variance of this set of data is __________. | 0.032 | 0.75 | 5,027 | 4,145.166667 | 7,672.5 | |
How many real numbers $x$ satisfy the equation $3^{2x+2}-3^{x+3}-3^x+3=0$? | 2 | 1. **Substitute and Simplify**: Let $a = 3^x$. Then, we can rewrite the given equation $3^{2x+2} - 3^{x+3} - 3^x + 3 = 0$ using the substitution:
\[
3^{2x+2} = (3^x)^2 \cdot 3^2 = 9a^2, \quad 3^{x+3} = 3^x \cdot 3^3 = 27a, \quad 3^x = a
\]
Substituting these into the equation, we get:
\[
9a^2 - 27a - ... | 1 | 2,252.875 | 2,252.875 | -1 |
A man, standing on a lawn, is wearing a circular sombrero of radius 3 feet. Unfortunately, the hat blocks the sunlight so effectively that the grass directly under it dies instantly. If the man walks in a circle of radius 5 feet, what area of dead grass will result? | 60\pi | 0.6875 | 4,552.9375 | 3,771.636364 | 6,271.8 | |
On sides \( BC \) and \( AC \) of triangle \( ABC \), points \( D \) and \( E \) are chosen respectively such that \( \angle BAD = 50^\circ \) and \( \angle ABE = 30^\circ \). Find \( \angle BED \) if \( \angle ABC = \angle ACB = 50^\circ \). | 40 | 0.125 | 8,175.1875 | 8,057.5 | 8,192 | |
As $t$ takes on all real values, the set of points $(x,y)$ defined by
\begin{align*}
x &= t^2 - 2, \\
y &= t^3 - 9t + 5
\end{align*}forms a curve that crosses itself. Compute the ordered pair $(x,y)$ where this crossing occurs. | (7,5) | 1 | 2,629.0625 | 2,629.0625 | -1 | |
Find all positive integers $n, k_1, \dots, k_n$ such that $k_1 + \cdots + k_n = 5n-4$ and \[ \frac{1}{k_1} + \cdots + \frac{1}{k_n} = 1. \] | n = 1, k_1 = 1; n = 3, (k_1,k_2,k_3) = (2,3,6); n = 4, (k_1,k_2,k_3,k_4) = (4,4,4,4) | By the arithmetic-harmonic mean inequality or the Cauchy-Schwarz inequality, \[ (k_1 + \cdots + k_n)\left(\frac{1}{k_1} + \cdots + \frac{1}{k_n} \right) \geq n^2. \] We must thus have $5n-4 \geq n^2$, so $n \leq 4$. Without loss of generality, we may suppose that $k_1 \leq \cdots \leq k_n$. If $n=1$, we must have $k_1 ... | 0 | 8,112 | -1 | 8,112 |
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