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 |
|---|---|---|---|---|---|---|
Compute the number of ways to fill each cell in a $8 \times 8$ square grid with one of the letters $H, M$, or $T$ such that every $2 \times 2$ square in the grid contains the letters $H, M, M, T$ in some order. | 1076 | We solve the problem for general $n \times n$ boards where $n$ even. Let the cell in the $i$-th row and $j$-th column be $a_{i, j}$. Claim: In any valid configuration, either the rows (or columns) alternate between ( $\cdots, H, M, H, M, \cdots)$ and $(\cdots, T, M, T, M, \cdots)$ or $(\cdots, M, M, M, M, \cdots)$ and ... | 0 | 8,072.875 | -1 | 8,072.875 |
Determine both the ratio of the volume of the cone to the volume of the cylinder and the ratio of their lateral surface areas. A cone and a cylinder have the same height of 10 cm. However, the cone's base radius is half that of the cylinder's. The radius of the cylinder is 8 cm. | \frac{\sqrt{116}}{40} | 0 | 1,853.0625 | -1 | 1,853.0625 | |
Let n be the number of real values of $p$ for which the roots of
$x^2-px+p=0$
are equal. Then n equals: | 2 | 1. **Identify the condition for equal roots:** For a quadratic equation $Ax^2 + Bx + C = 0$ to have equal roots, the discriminant must be zero. The discriminant is given by $\Delta = B^2 - 4AC$.
2. **Apply the condition to the given equation:** The given quadratic equation is $x^2 - px + p = 0$. Here, $A = 1$, $B = -p... | 1 | 1,319.25 | 1,319.25 | -1 |
What is the smallest base-10 integer that can be represented as $CC_6$ and $DD_8$, where $C$ and $D$ are valid digits in their respective bases? | 63_{10} | 0 | 7,057 | -1 | 7,057 | |
Let \( S = \{1, 2, 3, \ldots, 9, 10\} \). A non-empty subset of \( S \) is considered "Good" if the number of even integers in the subset is more than or equal to the number of odd integers in the same subset. For example, the subsets \( \{4,8\}, \{3,4,7,8\} \) and \( \{1,3,6,8,10\} \) are "Good". How many subsets of \... | 637 | 0.0625 | 8,073 | 6,288 | 8,192 | |
Calculate the volume of the solid formed by rotating around the $O Y$ axis the curvilinear trapezoid which is bounded by the hyperbola $x y=2$ and the lines $y_{1}=1, y_{2}=4$, and $y_{3}=0$. | 3\pi | 0.125 | 7,916.6875 | 5,989.5 | 8,192 | |
What is the number of square meters in the area of a circle with diameter $6$ meters? Express your answer in terms of $\pi$. | 9 \pi | 1 | 1,002.8125 | 1,002.8125 | -1 | |
Let $\clubsuit(x)$ denote the sum of the digits of the positive integer $x$. For example, $\clubsuit(8)=8$ and $\clubsuit(123)=1+2+3=6$. For how many two-digit values of $x$ is $\clubsuit(\clubsuit(x))=3$? | 10 | 1. **Define the function and its application**: Let $\clubsuit(x)$ denote the sum of the digits of the positive integer $x$. For example, $\clubsuit(8) = 8$ and $\clubsuit(123) = 1 + 2 + 3 = 6$.
2. **Determine possible values of $y = \clubsuit(x)$ for two-digit $x$**: Since $x$ is a two-digit number, the maximum value... | 0.9375 | 5,291.1875 | 5,097.8 | 8,192 |
Given that $ \sin \left( \frac{\pi}{3} - \frac{\alpha}{2} \right) = -\frac{\sqrt{3}}{2} $, find the value of $ \cos \left( \frac{\pi}{3} + \alpha \right) $. | \frac{1}{2} | 0.75 | 6,535.875 | 5,983.833333 | 8,192 | |
If $\lceil{\sqrt{x}}\rceil=15$, how many possible integer values of $x$ are there? | 29 | 0.4375 | 3,242.25 | 1,978.142857 | 4,225.444444 | |
In the tetrahedron \(ABCD\), \(CD \perp BC\), \(AB \perp BC\), \(CD = AC\), \(AB = BC = 1\). The dihedral angle between the planes \(BCD\) and \(ABC\) is \(45^\circ\). Find the distance from point \(B\) to the plane \(ACD\). | \frac{\sqrt{3}}{3} | 0 | 6,651.8125 | -1 | 6,651.8125 | |
In acute triangle $ABC$ points $P$ and $Q$ are the feet of the perpendiculars from $C$ to $\overline{AB}$ and from $B$ to $\overline{AC}$, respectively. Line $PQ$ intersects the circumcircle of $\triangle ABC$ in two distinct points, $X$ and $Y$. Suppose $XP=10$, $PQ=25$, and $QY=15$. The value of $AB\cdot AC$ can be w... | 574 | 0 | 8,192 | -1 | 8,192 | |
Mario is once again on a quest to save Princess Peach. Mario enters Peach's castle and finds himself in a room with 4 doors. This room is the first in a sequence of 2 indistinguishable rooms. In each room, 1 door leads to the next room in the sequence (or, for the second room, into Bowser's level), while the other 3 do... | 20 | Let $E_{i}$ be the expected number of doors through which Mario will pass in the future if he is currently in room $i$ for $i=1,2,3$ (we will set $E_{3}=0$). We claim that $E_{i}=1+\frac{3}{4} E_{1}+\frac{1}{4} E_{i+1}$. Indeed, the 1 at the beginning comes from the fact that we need to pass through a door to leave the... | 0 | 7,048.375 | -1 | 7,048.375 |
Let $x$, $y$, and $z$ be nonnegative real numbers such that $x + y + z = 8$. Find the maximum value of
\[
\sqrt{3x + 2} + \sqrt{3y + 2} + \sqrt{3z + 2}.
\] | 3\sqrt{10} | 0.9375 | 4,537.375 | 4,293.733333 | 8,192 | |
A room is 25 feet long and 15 feet wide. Find the ratio of the length of the room to its perimeter and the ratio of the width of the room to its perimeter. Express both your answers in the form $a:b$. | 3:16 | 1 | 1,455.625 | 1,455.625 | -1 | |
Points $P$, $Q$, $R$, and $S$ lie in the plane of the square $EFGH$ so that $EPF$, $FQG$, $GRH$, and $HSE$ are equilateral triangles inside square $EFGH$. If the side length of square $EFGH$ is 8, find the area of quadrilateral $PQRS$. Assume that the vertices $P$, $Q$, $R$, and $S$ lie on the sides of square $EFGH$, r... | 48 | 0 | 8,154.9375 | -1 | 8,154.9375 | |
What is \(1\tfrac{1}{2}\) divided by \(\tfrac{5}{6}\)? | \tfrac{9}{5} | 0.5625 | 2,788.9375 | 3,083.111111 | 2,410.714286 | |
What is the smallest two-digit integer $n$ such that switching its digits and then adding 3 results in $2n$? | 12 | 1 | 3,518.75 | 3,518.75 | -1 | |
Given the circle with center O, and diameters AB and CD where AB is perpendicular to CD, and chord DF intersects AB at E with DE = 6 and EF = 2, find the area of the circle. | 24\pi | 0.6875 | 5,970.5625 | 4,960.818182 | 8,192 | |
A ball was floating in a lake when the lake froze. The ball was removed, leaving a hole $32$ cm across at the top and $16$ cm deep. What was the radius of the ball (in centimeters)? | 16 | 0.9375 | 4,031.25 | 3,753.866667 | 8,192 | |
An isosceles right triangle with side lengths in the ratio 1:1:\(\sqrt{2}\) is inscribed in a circle with a radius of \(\sqrt{2}\). What is the area of the triangle and the circumference of the circle? | 2\pi\sqrt{2} | 0.125 | 1,730.4375 | 1,705.5 | 1,734 | |
Define a sequence $a_{i, j}$ of integers such that $a_{1, n}=n^{n}$ for $n \geq 1$ and $a_{i, j}=a_{i-1, j}+a_{i-1, j+1}$ for all $i, j \geq 1$. Find the last (decimal) digit of $a_{128,1}$. | 4 | By applying the recursion multiple times, we find that $a_{1,1}=1, a_{2, n}=n^{n}+(n+1)^{n+1}$, and $a_{3, n}=n^{n}+2(n+1)^{n+1}+(n+2)^{n+2}$. At this point, we can conjecture and prove by induction that $a_{m, n}=\sum_{k=0}^{m-1}\binom{m-1}{k}(n+k)^{n+k}=\sum_{k \geq 0}\binom{m-1}{k}(n+k)^{n+k}$. (The second expressio... | 0 | 8,192 | -1 | 8,192 |
The graph of $y=f(x)$ for $-3\le x\le 3$ is shown below. On this interval, what is the range of $f(x)-x$? Express your answer in interval notation.
[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=-0... | (-1,0] | 0.3125 | 6,724.8125 | 7,177 | 6,519.272727 | |
In rectangle $PQRS$, $PS=6$ and $SR=3$. Point $U$ is on $QR$ with $QU=2$. Point $T$ is on $PS$ with $\angle TUR=90^{\circ}$. What is the length of $TR$? | 5 | Since $PQRS$ is a rectangle, then $QR=PS=6$. Therefore, $UR=QR-QU=6-2=4$. Since $PQRS$ is a rectangle and $TU$ is perpendicular to $QR$, then $TU$ is parallel to and equal to $SR$, so $TU=3$. By the Pythagorean Theorem, since $TR>0$, then $TR=\sqrt{TU^{2}+UR^{2}}=\sqrt{3^{2}+4^{2}}=\sqrt{25}=5$. Thus, $TR=5$. | 0.8125 | 3,991.8125 | 3,461.307692 | 6,290.666667 |
Alice and Bob play the following "point guessing game." First, Alice marks an equilateral triangle $A B C$ and a point $D$ on segment $B C$ satisfying $B D=3$ and $C D=5$. Then, Alice chooses a point $P$ on line $A D$ and challenges Bob to mark a point $Q \neq P$ on line $A D$ such that $\frac{B Q}{Q C}=\frac{B P}{P C}... | \frac{\sqrt{3}}{3}, 1, \frac{3 \sqrt{3}}{5} | First, if $P=A$ then clearly Bob cannot choose a $Q$. So we can have $B P: P C=1$. Otherwise, we need $A P$ to be tangent to the Apollonius Circle. The key claim is that $A B=A C=A P$. To see why, simply note that since $B$ and $C$ are inverses with respect to the Apollonius Circle, we get that $\odot(A, A B)$ and the ... | 0 | 8,192 | -1 | 8,192 |
How many natural-number factors does $\textit{N}$ have if $\textit{N} = 2^3 \cdot 3^2 \cdot 5^1$? | 24 | 1 | 2,485.25 | 2,485.25 | -1 | |
Find the positive solution to
\[\sqrt{x + 2 + \sqrt{x + 2 + \dotsb}} = \sqrt{x \sqrt{x \dotsm}}.\] | 1 + \sqrt{3} | 1 | 3,316 | 3,316 | -1 | |
Evaluate the limit of the numerical sequence:
\[
\lim _{n \rightarrow \infty}\left(\frac{n^{2}-6 n+5}{n^{2}-5 n+5}\right)^{3 n+2}
\] | e^{-3} | 0.6875 | 5,064.1875 | 5,237.272727 | 4,683.4 | |
Find the matrix $\mathbf{M}$ such that
\[\mathbf{M} \mathbf{v} = \begin{pmatrix} 5 \\ 2 \\ -8 \end{pmatrix} \times \mathbf{v}\]for all vectors $\mathbf{v}.$ | \begin{pmatrix} 0 & 8 & 2 \\ -8 & 0 & -5 \\ -2 & 5 & 0 \end{pmatrix} | 0.9375 | 2,773.1875 | 2,411.933333 | 8,192 | |
Calculate the limit of the numerical sequence:
$$\lim _{n \rightarrow \infty} \frac{\sqrt[4]{2+n^{5}}-\sqrt{2 n^{3}+3}}{(n+\sin n) \sqrt{7 n}}$$ | -\sqrt{\frac{2}{7}} | 0 | 7,760.125 | -1 | 7,760.125 | |
Given that $\binom{18}{8}=31824$, $\binom{18}{9}=48620$, and $\binom{18}{10}=43758$, calculate $\binom{20}{10}$. | 172822 | 0.0625 | 8,106 | 6,816 | 8,192 | |
If $(x^2 - k)(x + k) = x^3 + k(x^2 - x - 5)$ and $k\neq 0$, what is the value of $k$? | 5 | 1 | 1,464.5 | 1,464.5 | -1 | |
Given that the angle between the unit vectors $\overrightarrow{e_{1}}$ and $\overrightarrow{e_{2}}$ is $60^{\circ}$, and $\overrightarrow{a}=2\overrightarrow{e_{1}}-\overrightarrow{e_{2}}$, find the projection of $\overrightarrow{a}$ in the direction of $\overrightarrow{e_{1}}$. | \dfrac{3}{2} | 0.9375 | 3,407.75 | 3,088.8 | 8,192 | |
Given the parabola $y^{2}=2px(p > 0)$ and the hyperbola $\frac {x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}=1(a > 0,b > 0)$ have the same focus $F$, and point $A$ is an intersection point of the two curves, and $AF$ is perpendicular to the x-axis, calculate the eccentricity of the hyperbola. | \sqrt {2} + 1 | 0 | 5,826.625 | -1 | 5,826.625 | |
Given that \(2 \cdot 50N\) is an integer and its representation in base \(b\) is 777, find the smallest positive integer \(b\) such that \(N\) is a fourth power of an integer. | 18 | 0 | 8,192 | -1 | 8,192 | |
In a certain ellipse, the endpoints of the major axis are $(-11,4)$ and $(9,4).$ Also, the ellipse passes through the point $(7,7).$ Find the area of the ellipse. | 50 \pi | 1 | 1,812.375 | 1,812.375 | -1 | |
From the numbers 1, 2, 3, 4, 5, and 6, two distinct numbers are taken out each time and denoted as $a$ and $b$. The total number of distinct values obtained for $3^{\frac{a}{b}}$ is ______. | 22 | 0 | 8,107.8125 | -1 | 8,107.8125 | |
In the plane rectangular coordinate system $xOy$, it is known that $MN$ is a chord of the circle $C: (x-2)^{2} + (y-4)^{2} = 2$, and satisfies $CM\perp CN$. Point $P$ is the midpoint of $MN$. As the chord $MN$ moves on the circle $C$, there exist two points $A$ and $B$ on the line $2x-y-3=0$, such that $\angle APB \geq... | \frac{6\sqrt{5}}{5} + 2 | 0 | 8,192 | -1 | 8,192 | |
Henry's Hamburger Heaven offers its hamburgers with the following condiments: ketchup, mustard, mayonnaise, tomato, lettuce, pickles, cheese, and onions. A customer can choose one, two, or three meat patties, and any collection of condiments. How many different kinds of hamburgers can be ordered? | 768 | 1 | 2,067.0625 | 2,067.0625 | -1 | |
There are 1991 participants at a sporting event. Each participant knows at least $n$ other participants (the acquaintance is mutual). What is the minimum value of $n$ for which there necessarily exists a group of 6 participants who all know each other? | 1593 | 0.4375 | 6,974.9375 | 5,936.571429 | 7,782.555556 | |
A rectangular board of 12 columns and 12 rows has squares numbered beginning in the upper left corner and moving left to right so row one is numbered 1 through 12, row two is 13 through 24, and so on. Determine which number of the form $n^2$ is the first to ensure that at least one shaded square is in each of the 12 co... | 144 | 0 | 8,165.875 | -1 | 8,165.875 | |
An integer $n$ is said to be [i]good[/i] if $|n|$ is not the square of an integer. Determine all integers $m$ with the following property: $m$ can be represented, in infinitely many ways, as a sum of three distinct good integers whose product is the square of an odd integer.
[i] |
To solve the problem, we need to determine all integers \( m \) such that \( m \) can be represented in infinitely many ways as a sum of three distinct good integers whose product is the square of an odd integer.
First, let's clarify the conditions:
- A number \( n \) is said to be good if \( |n| \) is not a perfect... | 0 | 8,192 | -1 | 8,192 | |
In coordinate space, $A = (6,-7,7),$ $B = (16,-17,12),$ $C = (0,3,-6),$ and $D = (2,-5,10).$ Find the intersection point of lines $AB$ and $CD.$ | \left( \frac{4}{3}, -\frac{7}{3}, \frac{14}{3} \right) | 1 | 2,829.5625 | 2,829.5625 | -1 | |
Consider positive integers $n$ where $D(n)$ denotes the number of pairs of different adjacent digits in the binary (base two) representation of $n$. Determine the number of positive integers less than or equal to $50$ for which $D(n) = 3$. | 11 | 0 | 8,192 | -1 | 8,192 | |
Vasya wrote a note on a piece of paper, folded it into quarters, and labeled the top with "MAME". Then he unfolded the note, wrote something else on it, folded the note along the creases randomly (not necessarily as before), and left it on the table with a random side facing up. Find the probability that the inscriptio... | 1/8 | 0.125 | 5,933.9375 | 6,328.5 | 5,877.571429 | |
Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible? | 3 | We are given the conditions:
1. Ralph passed the orange house before the red house.
2. Ralph passed the blue house before the yellow house.
3. The blue house was not next to the yellow house.
We need to determine the possible orderings of the houses under these constraints.
#### Case Analysis:
We will consider the po... | 0.25 | 7,720.5625 | 6,795.5 | 8,028.916667 |
Find the number of ordered quadruples \((a,b,c,d)\) of nonnegative real numbers such that
\[
a^2 + b^2 + c^2 + d^2 = 9,
\]
\[
(a + b + c + d)(a^3 + b^3 + c^3 + d^3) = 81.
\] | 15 | 0 | 8,192 | -1 | 8,192 | |
What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 1980 = 0$ has integral solutions? | 290 | 0 | 8,192 | -1 | 8,192 | |
A ship sails eastward at a speed of 15 km/h. At point A, the ship observes a lighthouse B at an angle of 60° northeast. After sailing for 4 hours, the ship reaches point C, where it observes the lighthouse at an angle of 30° northeast. At this time, the distance between the ship and the lighthouse is ______ km. | 60 | 0.125 | 6,654.75 | 6,865 | 6,624.714286 | |
Find all positive integers \( n > 1 \) such that any of its positive divisors greater than 1 has the form \( a^r + 1 \), where \( a \) is a positive integer and \( r \) is a positive integer greater than 1. | 10 | 0 | 8,192 | -1 | 8,192 | |
Five volunteers and two elderly people are taking a photo, and they need to be arranged in a row. The two elderly people must stand next to each other but cannot be at either end of the row. How many different arrangements are possible? | 960 | 0.4375 | 6,452.6875 | 4,216.428571 | 8,192 | |
A person rides a bicycle from place A to place B. If they increase their speed by 6 km/h, they can arrive 5 minutes earlier; if they decrease their speed by 5 km/h, they will be 6 minutes late. What is the distance between place A and place B in kilometers? | 15 | 1 | 3,136.3125 | 3,136.3125 | -1 | |
Given a pyramid \( S-ABCD \) with a square base where each side measures 2, and \( SD \perp \) plane \( ABCD \) and \( SD = AB \). Determine the surface area of the circumscribed sphere of the pyramid \( S-ABCD \). | 12\pi | 0.9375 | 3,902.5625 | 3,616.6 | 8,192 | |
In the center of a circular field stands a geologists' house. From it, 8 straight roads extend, dividing the field into 8 equal sectors. Two geologists embark on a journey from their house at a speed of 5 km/h, each choosing a road at random. Determine the probability that the distance between them will be more than 8 ... | 0.375 | 0 | 7,937.375 | -1 | 7,937.375 | |
Simplify the following expression in $x$: \[2x+8x^2+9-(4-2x-8x^2).\] Express your answer in the form $ax^2 +bx+c$, where $a$, $b$, and $c$ are numbers. | 16x^2+4x+5 | 1 | 1,577.25 | 1,577.25 | -1 | |
Given $|\overrightarrow {a}|=4$, $|\overrightarrow {b}|=2$, and the angle between $\overrightarrow {a}$ and $\overrightarrow {b}$ is $120^{\circ}$, find:
1. $\left(\overrightarrow {a}-2\overrightarrow {b}\right)\cdot \left(\overrightarrow {a}+\overrightarrow {b}\right)$;
2. The projection of $\overrightarrow {a}$ onto... | \dfrac{\pi}{6} | 0 | 4,821 | -1 | 4,821 | |
Amelia and Blaine are playing a modified game where they toss their respective coins. Amelia's coin lands on heads with a probability of $\frac{3}{7}$, and Blaine's lands on heads with a probability of $\frac{1}{3}$. They begin their game only after observing at least one head in a simultaneous toss of both coins. Once... | \frac{21609}{64328} | 0 | 7,962.625 | -1 | 7,962.625 | |
A rectangular tank with a horizontal cross-sectional area of \(S = 6 \ \text{m}^2\) is filled with water up to a height of \(H = 5 \ \text{m}\). Determine the time it takes for all the water to flow out of the tank through a small hole at the bottom with an area of \(s = 0.01 \ \text{m}^2\), assuming that the outflow s... | 1010 | 0.1875 | 7,221.8125 | 5,751 | 7,561.230769 | |
Let the two foci of the conic section \\(\Gamma\\) be \\(F_1\\) and \\(F_2\\), respectively. If there exists a point \\(P\\) on the curve \\(\Gamma\\) such that \\(|PF_1|:|F_1F_2|:|PF_2|=4:3:2\\), then the eccentricity of the curve \\(\Gamma\\) is \_\_\_\_\_\_\_\_ | \dfrac{3}{2} | 0.125 | 6,282.875 | 5,531 | 6,390.285714 | |
ABCD is a trapezium inscribed in a circle centered at O. It is given that AB is parallel to CD, angle COD is three times angle AOB, and the ratio of AB to CD is 2:5. Calculate the ratio of the area of triangle BOC to the area of triangle AOB. | \frac{3}{2} | 0.0625 | 7,365.0625 | 5,509 | 7,488.8 | |
Let \( P \) be a regular 2006-sided polygon. A diagonal of \( P \) is called a "good edge" if its endpoints divide the polygon into two parts, each containing an odd number of the polygon's sides. Each side of \( P \) is also considered a "good edge".
Given that 2003 non-intersecting diagonals divide \( P \) into seve... | 1003 | 0 | 8,192 | -1 | 8,192 | |
Compute: \(93 \times 107\). | 9951 | 0.875 | 366.6875 | 353.071429 | 462 | |
Given three real numbers \(p, q,\) and \(r\) such that
\[ p+q+r=5 \quad \text{and} \quad \frac{1}{p+q}+\frac{1}{q+r}+\frac{1}{p+r}=9 \]
What is the value of the expression
\[ \frac{r}{p+q}+\frac{p}{q+r}+\frac{q}{p+r} ? \] | 42 | 0.9375 | 4,483.875 | 4,236.666667 | 8,192 | |
New definition: Given that $y$ is a function of $x$, if there exists a point $P(a, a+2)$ on the graph of the function, then point $P$ is called a "real point" on the graph of the function. For example, the "real point" on the line $y=2x+1$ is $P(1,3)$.
$(1)$ Determine whether there is a "real point" on the line $y=\fr... | t=-1 | 0.25 | 7,378.5625 | 5,624 | 7,963.416667 | |
Compute $\tan 40^\circ + 4 \sin 40^\circ.$ | \sqrt{3} | 0 | 8,049 | -1 | 8,049 | |
Let the functions $f(\alpha,x)$ and $g(\alpha)$ be defined as \[f(\alpha,x)=\dfrac{(\frac{x}{2})^\alpha}{x-1}\qquad\qquad\qquad g(\alpha)=\,\dfrac{d^4f}{dx^4}|_{x=2}\] Then $g(\alpha)$ is a polynomial is $\alpha$ . Find the leading coefficient of $g(\alpha)$ . | 1/16 | 0.0625 | 8,112.25 | 6,916 | 8,192 | |
The perimeter of the triangle formed by the line $\frac{x}{3} + \frac{y}{4} = 1$ and the two coordinate axes is $14$. | 12 | 0 | 8,192 | -1 | 8,192 | |
If $10 \%$ of $s$ is $t$, what does $s$ equal? | 10t | The percentage $10 \%$ is equivalent to the fraction $\frac{1}{10}$. Therefore, $t=\frac{1}{10} s$, or $s=10 t$. | 1 | 572.0625 | 572.0625 | -1 |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If the area of $\triangle ABC$ equals $8$, $a=5$, and $\tan B= -\frac{4}{3}$, then $\frac{a+b+c}{\sin A+\sin B+\sin C}=$ \_\_\_\_\_\_. | \frac{5\sqrt{65}}{4} | 0 | 5,961.6875 | -1 | 5,961.6875 | |
What is the value of $(-1)^1+(-1)^2+\cdots+(-1)^{2007}$? | -1 | 1 | 2,383.375 | 2,383.375 | -1 | |
Find the value of $a_2+a_4+a_6+a_8+\ldots+a_{98}$ if $a_1$, $a_2$, $a_3\ldots$ is an arithmetic progression with common difference 1, and $a_1+a_2+a_3+\ldots+a_{98}=137$. | 93 | Since we are dealing with an arithmetic sequence, \[a_2+a_4+a_6+a_8+\ldots+a_{98} = 49a_{50}\] We can also figure out that \[a_1+a_2+a_3+\ldots+a_{98} = a_1 + 97a_{50} = 137\] \[a_1 = a_{50}-49 \Rightarrow 98a_{50}-49 = 137\] Thus, $49a_{50} = \frac{137 + 49}{2} = \boxed{093}$
~Hithere22702 | 0.875 | 6,160.8125 | 5,870.642857 | 8,192 |
When \( \frac{1}{2222} \) is expressed as a decimal, what is the sum of the first 50 digits after the decimal point? | 90 | 0.3125 | 5,677.4375 | 4,093.2 | 6,397.545455 | |
Anca and Bruce left Mathville at the same time. They drove along a straight highway towards Staton. Bruce drove at $50 \mathrm{~km} / \mathrm{h}$. Anca drove at $60 \mathrm{~km} / \mathrm{h}$, but stopped along the way to rest. They both arrived at Staton at the same time. For how long did Anca stop to rest? | 40 \text{ minutes} | Since Bruce drove 200 km at a speed of $50 \mathrm{~km} / \mathrm{h}$, this took him $\frac{200}{50}=4$ hours. Anca drove the same 200 km at a speed of $60 \mathrm{~km} / \mathrm{h}$ with a stop somewhere along the way. Since Anca drove 200 km at a speed of $60 \mathrm{~km} / \mathrm{h}$, the time that the driving port... | 0 | 7,257.4375 | -1 | 7,257.4375 |
A sequence is defined recursively as follows: \( t_{1} = 1 \), and for \( n > 1 \):
- If \( n \) is even, \( t_{n} = 1 + t_{\frac{n}{2}} \).
- If \( n \) is odd, \( t_{n} = \frac{1}{t_{n-1}} \).
Given that \( t_{n} = \frac{19}{87} \), find the sum of the digits of \( n \).
(From the 38th American High School Mathema... | 15 | 0 | 8,192 | -1 | 8,192 | |
Given the function $y=a^{2x}+2a^{x}-1 (a > 0$ and $a \neq 1)$, find the value of $a$ when the maximum value of the function is $14$ for the domain $-1 \leq x \leq 1$. | \frac{1}{3} | 0.8125 | 6,636.5 | 7,197.153846 | 4,207 | |
In $\triangle ABC$, if $\sin B= \sqrt {3}\sin A$, $BC= \sqrt {2}$, and $C= \frac {\pi}{6}$, then the height to side $AC$ is ______. | \frac { \sqrt {2}}{2} | 0 | 4,478.875 | -1 | 4,478.875 | |
Find all positive integer solutions $(m, n)$ to the following equation: $$ m^{2}=1!+2!+\cdots+n! $$ | (1,1), (3,3) | A square must end in the digit $0,1,4,5,6$, or 9 . If $n \geq 4$, then $1!+2!+\cdots+n$ ! ends in the digit 3 , so cannot be a square. A simple check for the remaining cases reveals that the only solutions are $(1,1)$ and $(3,3)$. | 0 | 4,588.9375 | -1 | 4,588.9375 |
In a company, some pairs of people are friends (if $A$ is friends with $B$, then $B$ is friends with $A$). It turns out that among every set of 100 people in the company, the number of pairs of friends is odd. Find the largest possible number of people in such a company. | 101 | 0 | 7,876.25 | -1 | 7,876.25 | |
The function
\[f(z) = \frac{(-1 + i \sqrt{3}) z + (-2 \sqrt{3} - 18i)}{2}\]represents a rotation around some complex number $c$. Find $c$. | \sqrt{3} - 5i | 0.875 | 5,370.9375 | 4,967.928571 | 8,192 | |
The sum of the following seven numbers is exactly 19: $a_1 = 2.56$, $a_2 = 2.61$, $a_3 = 2.65$, $a_4 = 2.71$, $a_5 = 2.79$, $a_6 = 2.82$, $a_7 = 2.86$. It is desired to replace each $a_i$ by an integer approximation $A_i$, $1\le i \le 7$, so that the sum of the $A_i$'s is also 19 and so that $M$, the maximum of the "er... | 61 | If any of the approximations $A_i$ is less than 2 or more than 3, the error associated with that term will be larger than 1, so the largest error will be larger than 1. However, if all of the $A_i$ are 2 or 3, the largest error will be less than 1. So in the best case, we write 19 as a sum of 7 numbers, each of which i... | 0 | 8,107.3125 | -1 | 8,107.3125 |
How many six-digit numbers of the form ababab are there, which are the product of six different prime numbers? | 12 | 0.0625 | 8,163.375 | 7,734 | 8,192 | |
A sphere with radius $r$ is inside a cone, the cross section of which is an equilateral triangle inscribed in a circle. Find the ratio of the total surface area of the cone to the surface area of the sphere. | 9:4 | 0 | 7,011.0625 | -1 | 7,011.0625 | |
Given Mindy made four purchases for $2.96, 6.57, 8.49, and 12.38. Each amount needs to be rounded up to the nearest dollar except the amount closest to a whole number, which should be rounded down. Calculate the total rounded amount. | 31 | 0.3125 | 468.125 | 471 | 466.818182 | |
How many two-digit positive integers are congruent to 1 (mod 3)? | 30 | 1 | 2,540.625 | 2,540.625 | -1 | |
Let $\triangle ABC$ have side lengths $AB=13$, $AC=14$, and $BC=15$. There are two circles located inside $\angle BAC$ which are tangent to rays $\overline{AB}$, $\overline{AC}$, and segment $\overline{BC}$. Compute the distance between the centers of these two circles. | 5\sqrt{13} | 0.0625 | 8,192 | 8,192 | 8,192 | |
Nebraska, the home of the AMC, changed its license plate scheme. Each old license plate consisted of a letter followed by four digits. Each new license plate consists of three letters followed by three digits. By how many times has the number of possible license plates increased? | \frac{26^2}{10} | 1. **Calculate the number of old license plates**:
Each old license plate consists of one letter followed by four digits. Since there are 26 possible letters and 10 possible digits for each position, the total number of old license plates is calculated as:
\[
26 \times 10^4
\]
2. **Calculate the number of... | 0 | 5,814.0625 | -1 | 5,814.0625 |
Given that Xiao Ming's elder brother was born in a year that is a multiple of 19, calculate his age in 2013. | 18 | 0.4375 | 1,177.625 | 1,760.142857 | 724.555556 | |
Given a right prism with all vertices on the same sphere, with a height of $4$ and a volume of $32$, the surface area of this sphere is ______. | 32\pi | 0.25 | 7,657.8125 | 6,055.25 | 8,192 | |
The sum of $n$ terms of an arithmetic progression is $153$, and the common difference is $2$. If the first term is an integer, and $n>1$, then the number of possible values for $n$ is: | 5 | 1. **Identify the formula for the sum of an arithmetic progression (AP)**:
The sum $S_n$ of the first $n$ terms of an AP where the first term is $a$ and the common difference is $d$ is given by:
\[
S_n = \frac{n}{2} \left(2a + (n-1)d\right)
\]
Given $d = 2$ and $S_n = 153$, substituting these values give... | 0.875 | 5,274.9375 | 4,858.214286 | 8,192 |
You want to paint some edges of a regular dodecahedron red so that each face has an even number of painted edges (which can be zero). Determine from How many ways this coloration can be done.
Note: A regular dodecahedron has twelve pentagonal faces and in each vertex concur three edges. The edges of the dodecahedron a... | 2048 | 0.1875 | 6,796.375 | 4,720 | 7,275.538462 | |
Let $z$ be a complex number such that
\[|z - 12| + |z - 5i| = 13.\]Find the smallest possible value of $|z|.$ | \frac{60}{13} | 0.9375 | 5,341.5625 | 5,151.533333 | 8,192 | |
Let $A = \left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$ be a set of numbers, and let the arithmetic mean of all elements in $A$ be denoted by $P(A)\left(P(A)=\frac{a_{1}+a_{2}+\cdots+a_{n}}{n}\right)$. If $B$ is a non-empty subset of $A$ such that $P(B) = P(A)$, then $B$ is called a "balance subset" of $A$. Find the numbe... | 51 | 0.0625 | 7,926.5625 | 6,981 | 7,989.6 | |
From the 2015 natural numbers between 1 and 2015, what is the maximum number of numbers that can be found such that their product multiplied by 240 is a perfect square? | 134 | 0 | 8,148.875 | -1 | 8,148.875 | |
If $f(x) = 5x^2 - 2x - 1$, then $f(x + h) - f(x)$ equals: | h(10x+5h-2) | To find $f(x+h) - f(x)$ for the function $f(x) = 5x^2 - 2x - 1$, we start by substituting $x+h$ into the function and then subtracting $f(x)$.
1. **Substitute $x+h$ into $f$:**
\[
f(x+h) = 5(x+h)^2 - 2(x+h) - 1
\]
Expand $(x+h)^2$:
\[
(x+h)^2 = x^2 + 2xh + h^2
\]
Substitute back into $f(x+h)$:
... | 0 | 4,250.75 | -1 | 4,250.75 |
For any real number $k,$ the graph of
\[y = 7x^2 + kx - 4k\]passes through a fixed point $(a,b).$ Find $(a,b).$ | (4,112) | 1 | 1,444.25 | 1,444.25 | -1 | |
Suppose \( a, b \), and \( c \) are real numbers with \( a < b < 0 < c \). Let \( f(x) \) be the quadratic function \( f(x) = (x-a)(x-c) \) and \( g(x) \) be the cubic function \( g(x) = (x-a)(x-b)(x-c) \). Both \( f(x) \) and \( g(x) \) have the same \( y \)-intercept of -8 and \( g(x) \) passes through the point \( (... | \frac{8}{3} | 0.875 | 4,602.125 | 4,089.285714 | 8,192 | |
What was the range of temperatures on Monday in Fermatville, given that the minimum temperature was $-11^{\circ} \mathrm{C}$ and the maximum temperature was $14^{\circ} \mathrm{C}$? | 25^{\circ} \mathrm{C} | Since the maximum temperature was $14^{\circ} \mathrm{C}$ and the minimum temperature was $-11^{\circ} \mathrm{C}$, then the range of temperatures was $14^{\circ} \mathrm{C} - (-11^{\circ} \mathrm{C}) = 25^{\circ} \mathrm{C}$. | 0.875 | 423.8125 | 356.142857 | 897.5 |
For a finite sequence $B = (b_1, b_2, \dots, b_n)$ of numbers, the Cesaro sum is defined as
\[\frac{T_1 + T_2 + \cdots + T_n}{n},\]
where $T_k = b_1 + b_2 + \cdots + b_k$ for $1 \leq k \leq n$.
If the Cesaro sum of the 100-term sequence $(b_1, b_2, \dots, b_{100})$ is 1200, where $b_1 = 2$, calculate the Cesaro sum o... | 1191 | 0 | 8,069.25 | -1 | 8,069.25 | |
For a natural number \( N \), if at least six of the nine natural numbers from 1 to 9 are factors of \( N \), then \( N \) is called a “six-match number.” Find the smallest "six-match number" greater than 2000. | 2016 | 0.125 | 8,065.125 | 7,177 | 8,192 | |
Given the integers \( a, b, c \) that satisfy \( a + b + c = 2 \), and
\[
S = (2a + bc)(2b + ca)(2c + ab) > 200,
\]
find the minimum value of \( S \). | 256 | 0 | 8,192 | -1 | 8,192 |
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