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 |
|---|---|---|---|---|---|---|
Determine the average daily high temperature in Addington from September 15th, 2008, through September 21st, 2008, inclusive. The high temperatures for the days are as follows: 51, 64, 60, 59, 48, 55, 57 degrees Fahrenheit. Express your answer as a decimal to the nearest tenth. | 56.3 | 0.9375 | 2,418.8125 | 2,547.266667 | 492 | |
Harold, Tanya, and Ulysses paint a very long picket fence.
Harold starts with the first picket and paints every $h$ th picket;
Tanya starts with the second picket and paints every $t$ th picket; and
Ulysses starts with the third picket and paints every $u$ th picket.
Call the positive integer $100h+10t+u$ paintable whe... | 757 | 0 | 8,186.9375 | -1 | 8,186.9375 | |
What is the value of \( \frac{5-2}{2+1} \)? | 1 | Simplifying, \( \frac{5-2}{2+1}=\frac{3}{3}=1 \). | 1 | 1,031.125 | 1,031.125 | -1 |
Given the function $f(x)=\cos^2x-\sin^2x+\frac{1}{2}, x \in (0,\pi)$.
$(1)$ Find the interval of monotonic increase for $f(x)$;
$(2)$ Suppose $\triangle ABC$ is an acute triangle, with the side opposite to angle $A$ being $a=\sqrt{19}$, and the side opposite to angle $B$ being $b=5$. If $f(A)=0$, find the area of $\t... | \frac{15\sqrt{3}}{4} | 0 | 7,028.6875 | -1 | 7,028.6875 | |
How many real solutions are there to the equation $|||| x|-2|-2|-2|=|||| x|-3|-3|-3|$? | 6 | 6. The graphs of the two sides of the equation can be graphed on the same plot to reveal six intersection points. | 0 | 8,057.5 | -1 | 8,057.5 |
What is the base ten equivalent of the base three number $102012_3$? | 302 | 0.5625 | 6,008.5 | 4,460.222222 | 7,999.142857 | |
Given in $\bigtriangleup ABC$, $AB = 75$, and $AC = 120$. A circle with center $A$ and radius $AB$ intersects $\overline{BC}$ at points $B$ and $X$. Moreover, $\overline{BX}$ and $\overline{CX}$ have integer lengths. Find the length of $BC$. | 117 | 0.125 | 7,801.75 | 5,070 | 8,192 | |
Let $N$ be the sum of the divisors of $200$. What is the largest prime factor of $N$? | 31 | 1 | 1,929.6875 | 1,929.6875 | -1 | |
In triangle $\triangle ABC$, where $A B = 16$, $B C = 5 \sqrt{5}$, and $C A = 9$. What is the area of the plane region covered by the set of points outside $\triangle ABC$ such that the distance to points $B$ and $C$ is less than 6? | 54\pi + \frac{5\sqrt{95}}{4} | 0 | 8,192 | -1 | 8,192 | |
A cube has eight vertices (corners) and twelve edges. A segment, such as $x$, which joins two vertices not joined by an edge is called a diagonal. Segment $y$ is also a diagonal. How many diagonals does a cube have? [asy]
/* AMC8 1998 #17 Problem */
pair A=(0,48), B=(0,0), C=(48,0), D=(48,48);
pair E=(24,72), F=(24,24)... | 16 | 1 | 2,768.25 | 2,768.25 | -1 | |
Given that there are 5 balls of each of the three colors: black, white, and red, each marked with the numbers 1, 2, 3, 4, 5, calculate the number of different ways to draw 5 balls such that their numbers are all different and all three colors are present. | 150 | 0.3125 | 7,349 | 6,118 | 7,908.545455 | |
Find the real solutions of $(2 x+1)(3 x+1)(5 x+1)(30 x+1)=10$. | \frac{-4 \pm \sqrt{31}}{15} | $(2 x+1)(3 x+1)(5 x+1)(30 x+1)=[(2 x+1)(30 x+1)][(3 x+1)(5 x+1)]=\left(60 x^{2}+32 x+1\right)\left(15 x^{2}+8 x+1\right)=(4 y+1)(y+1)=10$, where $y=15 x^{2}+8 x$. The quadratic equation in $y$ yields $y=1$ and $y=-\frac{9}{4}$. For $y=1$, we have $15 x^{2}+8 x-1=0$, so $x=\frac{-4 \pm \sqrt{31}}{15}$. For $y=-\frac{9}{... | 0 | 6,994.6875 | -1 | 6,994.6875 |
Calculate $5.47 + 2.58 + 1.95$ and write the sum as a decimal. | 10.00 | 0.8125 | 2,467.3125 | 2,126.923077 | 3,942.333333 | |
What is the greatest common factor of 32 and 48? | 16 | 1 | 1,538.75 | 1,538.75 | -1 | |
A frog makes $3$ jumps, each exactly $1$ meter long. The directions of the jumps are chosen independently at random. What is the probability that the frog's final position is no more than $1$ meter from its starting position? | \frac{1}{4} | To solve this problem, we consider the frog's position in a coordinate system after each jump. Each jump is of length 1 meter, and the direction is chosen randomly. We need to calculate the probability that after 3 jumps, the frog is no more than 1 meter away from its starting point.
#### Step 1: Understanding the pro... | 0 | 8,192 | -1 | 8,192 |
Let $A B C D E F$ be a convex hexagon with the following properties. (a) $\overline{A C}$ and $\overline{A E}$ trisect $\angle B A F$. (b) $\overline{B E} \| \overline{C D}$ and $\overline{C F} \| \overline{D E}$. (c) $A B=2 A C=4 A E=8 A F$. Suppose that quadrilaterals $A C D E$ and $A D E F$ have area 2014 and 1400, ... | 7295 | From conditions (a) and (c), we know that triangles $A F E, A E C$ and $A C B$ are similar to one another, each being twice as large as the preceding one in each dimension. Let $\overline{A E} \cap \overline{F C}=P$ and $\overline{A C} \cap \overline{E B}=Q$. Then, since the quadrilaterals $A F E C$ and $A E C B$ are s... | 0 | 8,192 | -1 | 8,192 |
The 2007 AMC 10 will be scored by awarding $6$ points for each correct response, $0$ points for each incorrect response, and $1.5$ points for each problem left unanswered. After looking over the $25$ problems, Sarah has decided to attempt the first $22$ and leave only the last $3$ unanswered.
How many of the first $22... | 16 | 1 | 772.8125 | 772.8125 | -1 | |
What is the area of the region defined by the inequality $|3x-18|+|2y+7| \le 3$? | 3 | 1. **Understanding the given inequality**: The inequality provided is $|3x-18|+|2y+7|\le3$. This inequality represents the sum of the absolute values of two linear expressions and is less than or equal to 3.
2. **Translation of coordinates**: To simplify the inequality, we translate the coordinates. Let's define new v... | 0.6875 | 6,343.8125 | 5,812.363636 | 7,513 |
In triangle $XYZ$, which is equilateral with a side length $s$, lines $\overline{LM}$, $\overline{NO}$, and $\overline{PQ}$ are parallel to $\overline{YZ}$, and $XL = LN = NP = QY$. Determine the ratio of the area of trapezoid $PQYZ$ to the area of triangle $XYZ$. | \frac{7}{16} | 0.0625 | 7,979.5 | 4,792 | 8,192 | |
What is the minimum number of cells required to mark on a chessboard so that each cell of the board (marked or unmarked) is adjacent by side to at least one marked cell? | 20 | 0 | 7,269.5625 | -1 | 7,269.5625 | |
Let
\[f(x) = \left\{
\begin{array}{cl}
x + 3 & \text{if $x < 20$}, \\
2x - 2 & \text{if $x \ge 20$}.
\end{array}
\right.\]Find $f^{-1}(7) + f^{-1}(46).$ | 28 | 1 | 1,875.625 | 1,875.625 | -1 | |
Given that a class has 5 students participating in the duty roster from Monday to Friday, with one student arranged each day, student A can only be arranged on Monday or Tuesday, and student B cannot be arranged on Friday, calculate the number of different duty arrangements for them. | 36 | 0.625 | 6,237.5 | 5,064.8 | 8,192 | |
For a real number $x$ let $\lfloor x\rfloor$ be the greatest integer less than or equal to $x$, and define $\{x\} = x - \lfloor x \rfloor$ to be the fractional part of $x$. For example, $\{3\} = 0$ and $\{4.56\} = 0.56$. Define $f(x)=x\{x\}$, and let $N$ be the number of real-valued solutions to the equation $f(f(f(x))... | 10 | To solve $f(f(f(x)))=17$, we need to solve $f(x) = y$ where $f(f(y))=17$, and to solve that we need to solve $f(y) = z$ where $f(z) = 17$.
It is clear to see for some integer $a \geq 17$ there is exactly one value of $z$ in the interval $[a, a+1)$ where $f(z) = 17$. To understand this, imagine the graph of $f(z)$ on t... | 0 | 8,192 | -1 | 8,192 |
The area of the shaded region BEDC in parallelogram ABCD is to be found, where BC = 15, ED = 9, and the total area of ABCD is 150. If BE is the height of parallelogram ABCD from base BC and is shared with ABE, both of which overlap over BE, calculate the area of the shaded region BEDC. | 120 | 0.375 | 6,858.25 | 4,704.333333 | 8,150.6 | |
What is the largest \( x \) such that \( x^2 \) divides \( 24 \cdot 35 \cdot 46 \cdot 57 \)? | 12 | 0.9375 | 5,488.4375 | 5,308.2 | 8,192 | |
Given sets $M=\{1, 2, a^2 - 3a - 1 \}$ and $N=\{-1, a, 3\}$, and the intersection of $M$ and $N$ is $M \cap N = \{3\}$, find the set of all possible real values for $a$. | \{4\} | 0 | 5,237.625 | -1 | 5,237.625 | |
Compute the definite integral:
$$
\int_{0}^{1} \frac{x^{4}}{\left(2-x^{2}\right)^{3 / 2}} \, dx
$$ | \frac{5}{2} - \frac{3\pi}{4} | 0 | 7,213.6875 | -1 | 7,213.6875 | |
The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers $x$, $y$, and $r$ with $|x|>|y|$,
\[(x+y)^r=x^r+rx^{r-1}y^1+\frac{r(r-1)}2x^{r-2}y^2+\frac{r(r-1)(r-2)}{3!}x^{r-3}y^3+\cdots\]What are the first three digits to the right of the decimal point in the decimal representation... | 428 | 0.1875 | 8,127.25 | 7,902.666667 | 8,179.076923 | |
The set
$$
A=\{\sqrt[n]{n} \mid n \in \mathbf{N} \text{ and } 1 \leq n \leq 2020\}
$$
has the largest element as $\qquad$ . | \sqrt[3]{3} | 0.875 | 5,726.625 | 5,374.428571 | 8,192 | |
There is an equilateral triangle $ABC$ on the plane. Three straight lines pass through $A$ , $B$ and $C$ , respectively, such that the intersections of these lines form an equilateral triangle inside $ABC$ . On each turn, Ming chooses a two-line intersection inside $ABC$ , and draws the straight line determined... | 45853 | 0 | 8,151.875 | -1 | 8,151.875 | |
A solid wooden rectangular prism measures $3 \times 5 \times 12$. The prism is cut in half by a vertical cut through four vertices, creating two congruent triangular-based prisms. What is the surface area of one of these triangular-based prisms? | 150 | Consider the triangular-based prism on the front of the rectangular prism. This prism has five faces: a rectangle on the front, a rectangle on the left, a triangle on the bottom, a triangle on the top, and a rectangle on the back. The rectangle on the front measures $3 \times 12$ and so has area 36. The rectangle on th... | 0 | 8,192 | -1 | 8,192 |
In $\triangle XYZ$, we have $\angle X = 90^\circ$ and $\tan Y = \frac34$. If $YZ = 30$, then what is $XY$? | 24 | 1 | 1,703.3125 | 1,703.3125 | -1 | |
Given that Mike walks to his college, averaging 70 steps per minute with each step being 80 cm long, and it takes him 20 minutes to get there, determine how long it takes Tom to reach the college, given that he averages 120 steps per minute, but his steps are only 50 cm long. | 18.67 | 0.125 | 1,466.1875 | 695 | 1,576.357143 | |
Each vertex of convex pentagon $ABCDE$ is to be assigned a color. There are $6$ colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible? | 3120 | To solve this problem, we need to consider the constraints given by the diagonals of the pentagon. Each diagonal connects two vertices, and the vertices at the ends of each diagonal must have different colors. We will analyze the possible colorings by considering different cases based on the color assignments of the ve... | 0.625 | 7,253.4375 | 6,690.3 | 8,192 |
In the Cartesian plane, a perfectly reflective semicircular room is bounded by the upper half of the unit circle centered at $(0,0)$ and the line segment from $(-1,0)$ to $(1,0)$. David stands at the point $(-1,0)$ and shines a flashlight into the room at an angle of $46^{\circ}$ above the horizontal. How many times do... | 65 | Note that when the beam reflects off the $x$-axis, we can reflect the entire room across the $x$-axis instead. Therefore, the number of times the beam reflects off a circular wall in our semicircular room is equal to the number of times the beam reflects off a circular wall in a room bounded by the unit circle centered... | 0 | 8,192 | -1 | 8,192 |
A positive integer is said to be "nefelibata" if, upon taking its last digit and placing it as the first digit, keeping the order of all the remaining digits intact (for example, 312 -> 231), the resulting number is exactly double the original number. Find the smallest possible nefelibata number. | 105263157894736842 | 0.0625 | 8,132.6875 | 7,243 | 8,192 | |
Samantha has 10 green marbles and 5 purple marbles in a bag. She removes a marble at random, records the color, puts it back, and then repeats this process until she has withdrawn 7 marbles. What is the probability that exactly four of the marbles that she removes are green? Express your answer as a decimal. | 0.256 | 0.0625 | 8,030.5625 | 5,609 | 8,192 | |
Solve for $x$: $100^3 = 10^x$ | 6 | 1 | 2,146.875 | 2,146.875 | -1 | |
If five geometric means are inserted between $8$ and $5832$, the fifth term in the geometric series: | $648$ | 1. **Identify the sequence and terms**: We are given that five geometric means are inserted between $8$ and $5832$. This means that the sequence starts at $8$ and ends at $5832$, with five terms in between, making a total of seven terms in the geometric sequence.
2. **Set up the formula for the $n$-th term of a geomet... | 0 | 2,842.875 | -1 | 2,842.875 |
Consider real numbers $A$ , $B$ , \dots, $Z$ such that \[
EVIL = \frac{5}{31}, \;
LOVE = \frac{6}{29}, \text{ and }
IMO = \frac{7}{3}.
\] If $OMO = \tfrac mn$ for relatively prime positive integers $m$ and $n$ , find the value of $m+n$ .
*Proposed by Evan Chen* | 579 | 0.25 | 7,519.125 | 7,285.5 | 7,597 | |
Given \(\tan \theta = \frac{5}{12}\), where \(180^{\circ} \leq \theta \leq 270^{\circ}\). If \(A = \cos \theta + \sin \theta\), find the value of \(A\). | -\frac{17}{13} | 0.9375 | 3,132.5 | 2,795.2 | 8,192 | |
In Mr. Abraham's class, $10$ of the $15$ students received an $A$ on the latest exam. If the same ratio of students received an $A$ on Mrs. Berkeley's latest exam, and if Mrs. Berkeley has $24$ students total, how many students in Mrs. Berkeley's class received an $A$? | 16 | 1 | 1,042.4375 | 1,042.4375 | -1 | |
Given the function $f\left(x\right)=\cos 2x+\sin x$, if $x_{1}$ and $x_{2}$ are the abscissas of the maximum and minimum points of $f\left(x\right)$, then $\cos (x_{1}+x_{2})=$____. | \frac{1}{4} | 0.125 | 7,307.5 | 6,656 | 7,400.571429 | |
Given a right circular cone $(P-ABC)$ with lateral edges $(PA)$, $(PB)$, $(PC)$ being pairwise perpendicular, and base edge $AB = \sqrt{2}$, find the surface area of the circumscribed sphere of the right circular cone $(P-ABC)$. | 3\pi | 0.125 | 8,097.125 | 7,433 | 8,192 | |
In acute triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\frac{\sin B\sin C}{\sin A}=\frac{3\sqrt{7}}{2}$, $b=4a$, and $a+c=5$, find the area of $\triangle ABC$. | \frac{3\sqrt{7}}{4} | 0 | 7,192.5625 | -1 | 7,192.5625 | |
Given that $O$ is the center of the circumcircle of $\triangle ABC$, $D$ is the midpoint of side $BC$, and $BC=4$, and $\overrightarrow{AO} \cdot \overrightarrow{AD} = 6$, find the maximum value of the area of $\triangle ABC$. | 4\sqrt{2} | 0.375 | 7,285.8125 | 6,135.5 | 7,976 | |
In the diagram, $\triangle ABE$, $\triangle BCE$ and $\triangle CDE$ are right-angled, with $\angle AEB=\angle BEC = \angle CED = 60^\circ$, and $AE=24$. [asy]
pair A, B, C, D, E;
A=(0,20.785);
B=(0,0);
C=(9,-5.196);
D=(13.5,-2.598);
E=(12,0);
draw(A--B--C--D--E--A);
draw(B--E);
draw(C--E);
label("A", A, N);
label("B",... | 27+21\sqrt{3} | 0.375 | 7,275 | 7,336.166667 | 7,238.3 | |
If a positive integer has eight positive divisors and the sum of these eight positive divisors is 3240, it is called a "good number." For example, 2006 is a good number because the sum of its positive divisors $1, 2, 17, 34, 59, 118, 1003, 2006$ is 3240. Find the smallest good number. | 1614 | 0 | 8,192 | -1 | 8,192 | |
A semipro baseball league has teams with 21 players each. League rules state that a player must be paid at least $15,000 and that the total of all players' salaries for each team cannot exceed $700,000. What is the maximum possible salary, in dollars, for a single player? | 400,000 | 1. **Identify the constraints**: Each team has 21 players, and each player must be paid at least $15,000. The total salary for all players on a team cannot exceed $700,000.
2. **Calculate the minimum total salary for 20 players**: If each of the 20 players receives the minimum salary of $15,000, the total salary paid ... | 0 | 1,932.8125 | -1 | 1,932.8125 |
Isosceles trapezoid \(ABCD\) with bases \(AB\) and \(CD\) has a point \(P\) on \(AB\) with \(AP=11, BP=27\), \(CD=34\), and \(\angle CPD=90^{\circ}\). Compute the height of isosceles trapezoid \(ABCD\). | 15 | Drop projections of \(A, P, B\) onto \(CD\) to get \(A', P', B'\). Since \(A'B'=38\) and \(CD=34\), we get that \(DA'=CB'=2\). Thus, \(P'D=9\) and \(P'C=25\). Hence, the answer is \(PP'=\sqrt{P'D \cdot P'C}=15\). | 1 | 3,145.8125 | 3,145.8125 | -1 |
A ball travels on a parabolic path in which the height (in feet) is given by the expression $-16t^2+32t+15$, where $t$ is the time after launch. What is the maximum height of the ball, in feet? | 31 | 1 | 2,161.1875 | 2,161.1875 | -1 | |
Find the minimum value of
\[2 \cos \theta + \frac{1}{\sin \theta} + \sqrt{2} \tan \theta\]for $0 < \theta < \frac{\pi}{2}.$ | 3 \sqrt{2} | 0.4375 | 7,494.75 | 6,598.285714 | 8,192 | |
All positive integers whose digits add up to 12 are listed in increasing order: $39, 48, 57, ...$. What is the tenth number in that list? | 147 | 0.375 | 7,144.5 | 5,919.5 | 7,879.5 | |
Noelle needs to follow specific guidelines to earn homework points: For each of the first ten homework points she wants to earn, she needs to do one homework assignment per point. For each homework point from 11 to 15, she needs two assignments; for each point from 16 to 20, she needs three assignments and so on. How m... | 80 | 0.3125 | 3,864.625 | 3,782.4 | 3,902 | |
Find the area of triangle $MNP$ given below:
[asy]
unitsize(1inch);
pair M,N,P;
M = (0,0);
N= (sqrt(3),0);
P = (0,1);
draw (M--N--P--M, linewidth(0.9));
draw(rightanglemark(N,M,P,3));
label("$M$",M,S);
label("$N$",N,S);
label("$P$",P,N);
label("$15$",(N+P)/2,NE);
label("$60^\circ$",(0,0.75),E);
[/asy] | 28.125\sqrt{3} | 0 | 6,212.5 | -1 | 6,212.5 | |
A function $f$ is defined for all real numbers and satisfies $f(2+x)=f(2-x)$ and $f(7+x)=f(7-x)$ for all $x.$ If $f(0) = 0,$ what is the least number of roots $f(x)=0$ must have in the interval $-1000\leq x \leq 1000$? | 401 | 0 | 8,192 | -1 | 8,192 | |
A new train goes $20\%$ farther than an older train in the same amount of time. During the time it takes the older train to go 200 miles, how many miles can the newer train complete? | 240 | 1 | 1,489.8125 | 1,489.8125 | -1 | |
The ages of Jonie's four cousins are distinct single-digit positive integers. Two of the cousins' ages multiplied together give $24$, while the other two multiply to $30$. What is the sum of the ages of Jonie's four cousins? | 22 | 1. **Identify possible pairs for the product 24**: The ages of Jonie's cousins are distinct single-digit positive integers. The possible pairs of ages that multiply to 24, considering only single-digit integers, are:
- $3 \times 8 = 24$
- $4 \times 6 = 24$
These pairs are $(3, 8)$ and $(4, 6)$.
2. **Identify ... | 0.875 | 4,090.9375 | 3,505.071429 | 8,192 |
Given the number $S_{20}$, which denotes an integer whose base-ten representation consists of 20 repetitions of the digit "2", and $S_{2}$, which denotes an integer whose base-ten representation consists of 2 repetitions of the digit "2", determine the number of zeros in the base-ten representation of the quotient $T =... | 18 | 0 | 7,265.875 | -1 | 7,265.875 | |
Consider a square-based pyramid (with base vertices $A, B, C, D$) with equal side edges, and let the apex be $E$. Let $P$ be the point that divides the side edge $A E$ in a ratio of 3:1, such that $E P : P A = 3$, and let $Q$ be the midpoint of the side edge $C E$. In what ratio does the plane passing through points $D... | 4/3 | 0.0625 | 6,440.5 | 4,027 | 6,601.4 | |
If a computer executes the following program:
(1) Initial values are $x=3, S=0$.
(2) $x=x+2$.
(3) $S=S+x$.
(4) If $S \geqslant 10000$, proceed to step 5; otherwise, go back to step 2.
(5) Print $x$.
(6) Stop.
Then what is the value printed in step 5? | 201 | 0.5625 | 6,282.1875 | 4,796.777778 | 8,192 | |
If $x^5 - x^4 + x^3 - px^2 + qx + 4$ is divisible by $(x + 2)(x - 1),$ find the ordered pair $(p,q).$ | (-7,-12) | 0.9375 | 3,144.1875 | 2,807.666667 | 8,192 | |
What is the remainder of $8^6 + 7^7 + 6^8$ divided by 5? | 3 | 1 | 2,709.9375 | 2,709.9375 | -1 | |
Alvin, Bingyi, and Cheska play a two-player game that never ends in a tie. In a recent tournament between the three players, a total of 60 games were played and each pair of players played the same number of games. When Alvin and Bingyi played, Alvin won \(20\%\) of the games. When Bingyi and Cheska played, Bingyi won ... | 28 | Since 60 games are played and each of the 3 pairs plays the same number of games, each pair plays \(60 \div 3 = 20\) games. Alvin wins \(20\%\) of the 20 games that Alvin and Bingyi play, so Alvin wins \(\frac{20}{100} \times 20 = \frac{1}{5} \times 20 = 4\) of these 20 games and Bingyi wins \(20 - 4 = 16\) of these 20... | 0.8125 | 4,101.875 | 3,158 | 8,192 |
Let \\((x^{2}+1)(2x+1)^{9}=a_{0}+a_{1}(x+2)+a_{2}(x+2)^{2}+\ldots+a_{n}(x+2)^{n}\\), then \\(a_{0}+a_{1}+a_{2}+\ldots+a_{n}=\\) \_\_\_\_\_\_. | -2 | 0.3125 | 7,400 | 5,972.2 | 8,049 | |
The front tires of a car wear out after 25,000 km, and the rear tires wear out after 15,000 km. When should the tires be swapped so that they wear out at the same time? | 9375 | 0.125 | 7,726.6875 | 7,068 | 7,820.785714 | |
Find the sine of the angle at the vertex of an isosceles triangle, given that the perimeter of any inscribed rectangle, with two vertices lying on the base, is a constant value. | \frac{4}{5} | 0.4375 | 6,086.125 | 4,040.714286 | 7,677 | |
Given point $M(2,0)$, draw two tangent lines $MA$ and $MB$ from $M$ to the circle $x^{2}+y^{2}=1$, where $A$ and $B$ are the tangent points. Calculate the dot product of vectors $\overrightarrow{MA}$ and $\overrightarrow{MB}$. | \dfrac{3}{2} | 1 | 3,818.3125 | 3,818.3125 | -1 | |
Calculate $3 \cdot 7^{-1} + 9 \cdot 13^{-1} \pmod{60}$.
Express your answer as an integer from $0$ to $59$, inclusive. | 42 | 1 | 3,962.4375 | 3,962.4375 | -1 | |
Find the number of collections of $16$ distinct subsets of $\{1,2,3,4,5\}$ with the property that for any two subsets $X$ and $Y$ in the collection, $X \cap Y \not= \emptyset.$ | 081 | Denote the $A$ as $\{ 1,2,3,4,5 \}$ and the collection of subsets as $S$.
Case 1: There are only sets of size $3$ or higher in $S$: Any two sets in $S$ must have at least one element common to both of them (since $3+3>5$). Since there are $16$ subsets of $A$ that have size $3$ or higher, there is only one possibility f... | 0 | 7,493.5 | -1 | 7,493.5 |
Karl bought five folders from Pay-A-Lot at a cost of $\$ 2.50$ each.
Pay-A-Lot had a 20%-off sale the following day. How much could
Karl have saved on the purchase by waiting a day? | $2.50 | 1. **Calculate the original cost per folder**: Karl bought each folder for $\textdollar 2.50$.
2. **Determine the discount rate**: The folders were on sale the next day with a 20% discount. We can express 20% as a decimal by dividing 20 by 100, which gives $0.20$.
3. **Calculate the discount amount per folder**:
... | 0 | 458.5625 | -1 | 458.5625 |
Given the function $f(x)=\ln{x}-\frac{1}{x}-2x$.
(1) Find the monotonic intervals of $f(x)$;
(2) If the line $l: y=ax+b$ is a tangent line of the function $F(x)=f(x)+2x$ and $a,b\in\mathbb{R}$, find the minimum value of $a+b$. | -1 | 1 | 4,427.8125 | 4,427.8125 | -1 | |
How many kings can be placed on an $8 \times 8$ chessboard without any of them being in check? | 16 | 0.375 | 7,113.8125 | 6,757.833333 | 7,327.4 | |
Angle $ABC$ of $\triangle ABC$ is a right angle. The sides of $\triangle ABC$ are the diameters of semicircles as shown. The area of the semicircle on $\overline{AB}$ equals $8\pi$, and the arc of the semicircle on $\overline{AC}$ has length $8.5\pi$. What is the radius of the semicircle on $\overline{BC}$? | 7.5 | 1. **Identify the given information and the relationship between the elements:**
- $\triangle ABC$ is a right triangle with $\angle ABC = 90^\circ$.
- The sides of $\triangle ABC$ are the diameters of semicircles.
- The area of the semicircle on $\overline{AB}$ is $8\pi$.
- The arc length of the semicircle ... | 0.125 | 2,580.4375 | 1,781.5 | 2,694.571429 |
For what value of $x$ does $10^{x} \cdot 100^{2x}=1000^{5}$? | 3 | 1. **Rewrite the equation using properties of exponents:**
Given the equation \(10^x \cdot 100^{2x} = 1000^5\), we start by expressing all terms with base 10:
\[
100 = 10^2 \quad \text{and} \quad 1000 = 10^3
\]
Therefore, the equation becomes:
\[
10^x \cdot (10^2)^{2x} = (10^3)^5
\]
2. **Simpli... | 1 | 1,428.5 | 1,428.5 | -1 |
Define a function $g :\mathbb{N} \rightarrow \mathbb{R}$ Such that $g(x)=\sqrt{4^x+\sqrt {4^{x+1}+\sqrt{4^{x+2}+...}}}$ .
Find the last 2 digits in the decimal representation of $g(2021)$ . | 53 | 0.3125 | 7,221.0625 | 6,012.4 | 7,770.454545 | |
Six positive integers are written on the faces of a cube. Each vertex is labeled with the product of the three numbers on the faces adjacent to the vertex. If the sum of the numbers on the vertices is equal to $2310$, then what is the sum of the numbers written on the faces? | 40 | 0.0625 | 8,087.625 | 6,522 | 8,192 | |
Find the coefficient of the $x^2$ term in the expansion of the product $$(2x^2 +3x +4)(5x^2 +6x +7).$$ | 52 | 1 | 2,958.6875 | 2,958.6875 | -1 | |
What is the largest integer less than $\log_2 \frac{3}{2} + \log_2 \frac{6}{3} + \cdots + \log_2 \frac{3030}{3029}$? | 10 | 0 | 8,142.5 | -1 | 8,142.5 | |
In a particular year, the price of a commodity increased by $30\%$ in January, decreased by $10\%$ in February, increased by $20\%$ in March, decreased by $y\%$ in April, and finally increased by $15\%$ in May. Given that the price of the commodity at the end of May was the same as it had been at the beginning of Janua... | 38 | 0 | 8,049.5 | -1 | 8,049.5 | |
Let $P(x)$ be a polynomial of degree $3n$ such that
\begin{align*} P(0) = P(3) = \dots = P(3n) &= 2, \\ P(1) = P(4) = \dots = P(3n+1-2) &= 1, \\ P(2) = P(5) = \dots = P(3n+2-2) &= 0. \end{align*}
Also, $P(3n+1) = 730$. Determine $n$. | 1 | To solve for $n$, we start by analyzing the polynomial $P(x)$ given its values at specific points and its degree. We use Lagrange Interpolation Formula to express $P(x)$, and then evaluate it at $x = 3n+1$ to find $n$.
1. **Constructing the Polynomial Using Lagrange Interpolation:**
The polynomial $P(x)$ is defined... | 0 | 8,192 | -1 | 8,192 |
Calculate the definite integral:
$$
\int_{0}^{\pi}\left(9 x^{2}+9 x+11\right) \cos 3 x \, dx
$$ | -2\pi - 2 | 0.5 | 6,589.375 | 5,762.375 | 7,416.375 | |
Does there exist a point \( M \) on the parabola \( y^{2} = 2px \) such that the ratio of the distance from point \( M \) to the vertex and the distance from point \( M \) to the focus is maximized? If such a point \( M \) exists, find its coordinates and the maximum ratio. If the point \( M \) does not exist, provide ... | \frac{2}{\sqrt{3}} | 0 | 6,209.25 | -1 | 6,209.25 | |
In a square, points \(P\) and \(Q\) are placed such that \(P\) is the midpoint of the bottom side and \(Q\) is the midpoint of the right side of the square. The line segment \(PQ\) divides the square into two regions. Calculate the fraction of the square's area that is not in the triangle formed by the points \(P\), \(... | \frac{7}{8} | 0 | 5,068.9375 | -1 | 5,068.9375 | |
Given that $\underbrace{9999\cdots 99}_{80\text{ nines}}$ is multiplied by $\underbrace{7777\cdots 77}_{80\text{ sevens}}$, calculate the sum of the digits in the resulting product. | 720 | 0.25 | 7,665.875 | 6,087.5 | 8,192 | |
In the permutation \(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\) of \(1, 2, 3, 4, 5\), how many permutations are there that satisfy \(a_{1} < a_{2}, a_{2} > a_{3}, a_{3} < a_{4}, a_{4} > a_{5}\)? | 16 | 0.4375 | 6,888.4375 | 5,212.428571 | 8,192 | |
All sides of the convex pentagon $ABCDE$ are of equal length, and $\angle A = \angle B = 90^\circ$. What is the degree measure of $\angle E$? | 150^\circ | 0.3125 | 6,623.625 | 3,884.4 | 7,868.727273 | |
Fifteen square tiles with side 10 units long are arranged as shown. An ant walks along the edges of the tiles, always keeping a black tile on its left. Find the shortest distance that the ant would walk in going from point \( P \) to point \( Q \). | 80 | 0.1875 | 5,882.625 | 5,769.666667 | 5,908.692308 | |
Given that 3 females and 2 males participate in a performance sequence, and the 2 males cannot appear consecutively, and female A cannot be the first to appear, determine the total number of different performance sequences. | 60 | 0.25 | 7,790.3125 | 6,585.25 | 8,192 | |
There are $2017$ distinct points in the plane. For each pair of these points, construct the midpoint of the segment joining the pair of points. What is the minimum number of distinct midpoints among all possible ways of placing the points? | 2016 | 0 | 8,192 | -1 | 8,192 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $a = 2c \cos A$ and $\sqrt{5} \sin A = 1$, find:
1. $\sin C$
2. $\frac{b}{c}$ | \frac{2\sqrt{5} + 5\sqrt{3}}{5} | 0 | 6,108.9375 | -1 | 6,108.9375 | |
If $S = i^n + i^{-n}$, where $i = \sqrt{-1}$ and $n$ is an integer, then the total number of possible distinct values for $S$ is: | 3 | 1. **Understanding the Powers of $i$:**
Recall that $i = \sqrt{-1}$, and the powers of $i$ cycle every four terms:
- $i^1 = i$
- $i^2 = -1$
- $i^3 = -i$
- $i^4 = 1$
- $i^5 = i$, and so on.
2. **Expression for $i^{-n}$:**
We know that $i^{-n} = \frac{1}{i^n}$. Using the property of $i$, $\frac{... | 1 | 2,934.875 | 2,934.875 | -1 |
Two distinct integers, $x$ and $y$, are randomly chosen from the set $\{1,2,3,4,5,6,7,8,9,10,11,12\}$. What is the probability that $(x-1)(y-1)$ is odd? | \frac{5}{22} | 1 | 2,631.5625 | 2,631.5625 | -1 | |
The three points A, B, C form a triangle. AB=4, BC=5, AC=6. Let the angle bisector of \angle A intersect side BC at D. Let the foot of the perpendicular from B to the angle bisector of \angle A be E. Let the line through E parallel to AC meet BC at F. Compute DF. | \frac{1}{2} | Since AD bisects \angle A, by the angle bisector theorem \frac{AB}{BD}=\frac{AC}{CD}, so BD=2 and CD=3. Extend BE to hit AC at X. Since AE is the perpendicular bisector of BX, AX=4. Since B, E, X are collinear, applying Menelaus' Theorem to the triangle ADC, we have \frac{AE}{ED} \cdot \frac{DB}{BC} \cdot \frac{CX}{XA}... | 0.8125 | 4,701.5 | 4,142.153846 | 7,125.333333 |
If $x = 2$ and $y = 5$, then what is the value of $\frac{x^4+2y^2}{6}$ ? | 11 | 1 | 1,495.3125 | 1,495.3125 | -1 | |
For the function $f(x)=a- \frac {2}{2^{x}+1}(a\in\mathbb{R})$
$(1)$ Determine the monotonicity of the function $f(x)$ and provide a proof;
$(2)$ If there exists a real number $a$ such that the function $f(x)$ is an odd function, find $a$;
$(3)$ For the $a$ found in $(2)$, if $f(x)\geqslant \frac {m}{2^{x}}$ holds tr... | \frac {12}{5} | 0.875 | 5,359.8125 | 4,955.214286 | 8,192 | |
What is the greatest integer not exceeding the number $\left( 1 + \frac{\sqrt 2 + \sqrt 3 + \sqrt 4}{\sqrt 2 + \sqrt 3 + \sqrt 6 + \sqrt 8 + 4}\right)^{10}$ ? | 32 | 0.3125 | 7,398.5625 | 5,653 | 8,192 | |
Express 826,000,000 in scientific notation. | 8.26 \times 10^{8} | 0.1875 | 387.1875 | 383.333333 | 388.076923 | |
In $\triangle ABC$, $B(-\sqrt{5}, 0)$, $C(\sqrt{5}, 0)$, and the sum of the lengths of the medians on sides $AB$ and $AC$ is $9$.
(Ⅰ) Find the equation of the trajectory of the centroid $G$ of $\triangle ABC$.
(Ⅱ) Let $P$ be any point on the trajectory found in (Ⅰ), find the minimum value of $\cos\angle BPC$. | -\frac{1}{9} | 0 | 8,192 | -1 | 8,192 | |
During the festive season when the moon is full and the country is celebrating together, a supermarket plans to reduce the selling price of grapes that cost $16$ yuan per kilogram. Through statistical analysis, it was found that when the selling price is $26$ yuan per kilogram, $320$ kilograms can be sold per day. If t... | 21 | 0.375 | 6,649.0625 | 5,925.666667 | 7,083.1 |
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