problem stringlengths 10 5.15k | answer stringlengths 0 1.22k | solution stringlengths 0 11.1k | reward float64 0 1 | length float64 172 8.19k | correct_length float64 -1 8.19k | incorrect_length float64 -1 8.19k |
|---|---|---|---|---|---|---|
Given the function $f(x)=2\cos x(\sin x-\cos x)$, where $x\in R$.
(I) Find the symmetry center of the graph of the function $f(x)$;
(II) Find the minimum and maximum values of the function $f(x)$ on the interval $[\frac{\pi}{8}, \frac{3\pi}{4}]$. | -2 | 0.0625 | 5,499.8125 | 5,168 | 5,521.933333 | |
What is the maximum number of consecutive positive integers that can be added together before the sum exceeds 400? | 27 | 0.875 | 5,823.0625 | 5,484.642857 | 8,192 | |
A truncated cone has horizontal bases with radii 18 and 2. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere? | 6 | 0.4375 | 6,182.875 | 4,927.714286 | 7,159.111111 | |
Let $T$ be a subset of $\{1,2,3,...,60\}$ such that no pair of distinct elements in $T$ has a sum divisible by $5$. What is the maximum number of elements in $T$? | 25 | 0.1875 | 7,729.9375 | 6,783.333333 | 7,948.384615 | |
Perpendiculars $BE$ and $DF$ dropped from vertices $B$ and $D$ of parallelogram $ABCD$ onto sides $AD$ and $BC$, respectively, divide the parallelogram into three parts of equal area. A segment $DG$, equal to segment $BD$, is laid out on the extension of diagonal $BD$ beyond vertex $D$. Line $BE$ intersects segment $AG... | 1:1 | 0 | 8,192 | -1 | 8,192 | |
The equation $y = -4.9t^2 + 23.8t$ describes the height (in meters) of a projectile launched from the ground at 23.8 meters per second. In how many seconds will the projectile first reach 28 meters in height? | 2 | 0.9375 | 4,600.6875 | 4,643.466667 | 3,959 | |
About 40% of students in a certain school are nearsighted, and about 30% of the students in the school use their phones for more than 2 hours per day, with a nearsighted rate of about 50% among these students. If a student who uses their phone for no more than 2 hours per day is randomly selected from the school, calcu... | \frac{5}{14} | 0.8125 | 3,900.25 | 3,399.923077 | 6,068.333333 | |
Pascal High School organized three different trips. Fifty percent of the students went on the first trip, $80 \%$ went on the second trip, and $90 \%$ went on the third trip. A total of 160 students went on all three trips, and all of the other students went on exactly two trips. How many students are at Pascal High Sc... | 800 | Let $x$ be the total number of students at Pascal H.S. Let $a$ be the total number of students who went on both the first trip and the second trip, but did not go on the third trip. Let $b$ be the total number of students who went on both the first trip and the third trip, but did not go on the second trip. Let $c$ be ... | 0.75 | 4,686.8125 | 3,880.5 | 7,105.75 |
As a prank, Tim decides to steal Nathan's fork at dinner, but so he doesn't get caught, he convinces other people to do it for him. On Monday, he convinces Joe to do it. On Tuesday, he could get either Ambie or John to do it. On Wednesday, he can't convince any of those three people to do it, but there are five other p... | 40 | 1 | 1,465.1875 | 1,465.1875 | -1 | |
Let point $O$ be the origin of a three-dimensional coordinate system, and let points $A,$ $B,$ and $C$ be located on the positive $x,$ $y,$ and $z$ axes, respectively. If $OA = \sqrt[4]{75}$ and $\angle BAC = 30^\circ,$ then compute the area of triangle $ABC.$ | \frac{5}{2} | 0.8125 | 5,862.9375 | 5,325.461538 | 8,192 | |
Find the smallest positive integer \( n \) for which there are exactly 2323 positive integers less than or equal to \( n \) that are divisible by 2 or 23, but not both. | 4644 | 0 | 7,252.375 | -1 | 7,252.375 | |
What is the sum of the digits of the base $7$ representation of $2019_{10}$? | 15 | 0.9375 | 3,155.9375 | 3,195.333333 | 2,565 | |
Calculate the number of zeros at the end of 2015!. | 502 | 1 | 1,967.875 | 1,967.875 | -1 | |
Calculate the definite integral:
$$
\int_{-1}^{0}(x+2)^{3} \cdot \ln ^{2}(x+2) \, dx
$$ | 4 \ln^{2} 2 - 2 \ln 2 + \frac{15}{32} | 0 | 5,200.25 | -1 | 5,200.25 | |
Determine the integer \( x \) for which the following equation holds true:
\[ 1 \cdot 2023 + 2 \cdot 2022 + 3 \cdot 2021 + \dots + 2022 \cdot 2 + 2023 \cdot 1 = 2023 \cdot 1012 \cdot x. \] | 675 | 0.9375 | 5,513.75 | 5,335.2 | 8,192 | |
What is the smallest positive value of $x$ such that $x + 4321$ results in a palindrome and that palindrome is at least 200 more than 4321? | 233 | 0.25 | 7,821.3125 | 6,709.25 | 8,192 | |
How many three-digit numbers are there in which the hundreds digit is greater than the tens digit which is in turn greater than the ones digit? | 120 | 0.75 | 5,829.5 | 5,496 | 6,830 | |
Given the function \( f(x) \):
\[ f(x) = \begin{cases}
\ln x & \text{if } x > 1, \\
\frac{1}{2} x + \frac{1}{2} & \text{if } x \leq 1
\end{cases} \]
If \( m < n \) and \( f(m) = f(n) \), what is the minimum value of \( n - m \)? | 3 - 2\ln 2 | 0.9375 | 5,768 | 5,606.4 | 8,192 | |
In the year 2001, the United States will host the International Mathematical Olympiad. Let $I$, $M$, and $O$ be distinct positive integers such that the product $I\cdot M\cdot O=2001$. What is the largest possible value of the sum $I+M+O$? | 671 | 1 | 4,725.6875 | 4,725.6875 | -1 | |
Vijay chooses three distinct integers \(a, b, c\) from the set \(\{1,2,3,4,5,6,7,8,9,10,11\}\). If \(k\) is the minimum value taken on by the polynomial \(a(x-b)(x-c)\) over all real numbers \(x\), and \(l\) is the minimum value taken on by the polynomial \(a(x-b)(x+c)\) over all real numbers \(x\), compute the maximum... | 990 | Quadratics are minimized at the average of their roots, so \(k=a\left(\frac{b+c}{2}-b\right)\left(\frac{b+c}{2}-c\right)=-\frac{a(b-c)^{2}}{4}\), and \(l=a\left(\frac{b-c}{2}-b\right)\left(\frac{b-c}{2}+c\right)=-\frac{a(b+c)^{2}}{4}\). Therefore, \(k-l=-\frac{a}{4}\left((b-c)^{2}-(b+c)^{2}\right)=abc\). Thus, \(k-l=ab... | 0.5 | 6,960.1875 | 5,728.375 | 8,192 |
Find the intersection of the lines $9x-4y=6$ and $7x+y=17$. Express your answer as an ordered pair $(x,y)$. | (2,3) | 1 | 2,032.125 | 2,032.125 | -1 | |
Square $P Q R S$ has an area of 900. $M$ is the midpoint of $P Q$ and $N$ is the midpoint of $P S$. What is the area of triangle $P M N$? | 112.5 | Since square $P Q R S$ has an area of 900, then its side length is $\sqrt{900}=30$. Thus, $P Q=P S=30$. Since $M$ and $N$ are the midpoints of $P Q$ and $P S$, respectively, then $P N=P M=\frac{1}{2}(30)=15$. Since $P Q R S$ is a square, then the angle at $P$ is $90^{\circ}$, so $\triangle P M N$ is right-angled. There... | 0.75 | 2,743.6875 | 1,972.583333 | 5,057 |
Let \( f(n) = 3n^2 - 3n + 1 \). Find the last four digits of \( f(1) + f(2) + \cdots + f(2010) \). | 1000 | 0.0625 | 8,132.3125 | 7,237 | 8,192 | |
Given $\sqrt{15129}=123$ and $\sqrt{x}=0.123$, calculate the value of $x$. | 0.015129 | 1 | 307.125 | 307.125 | -1 | |
In a race on the same distance, two cars and a motorcycle participated. The second car took 1 minute longer to cover the entire distance than the first car. The first car moved 4 times faster than the motorcycle. What portion of the distance per minute did the second car cover if it covered $\frac{1}{6}$ of the distanc... | 2/3 | 0 | 5,798.8125 | -1 | 5,798.8125 | |
The expression $\cos 2x + \cos 6x + \cos 10x + \cos 14x$ can be written in the equivalent form
\[a \cos bx \cos cx \cos dx\] for some positive integers $a,$ $b,$ $c,$ and $d.$ Find $a + b + c + d.$ | 18 | 0.9375 | 3,318.0625 | 2,993.133333 | 8,192 | |
Simplify first, then evaluate: $[\left(2x-y\right)^{2}-\left(y+2x\right)\left(y-2x\right)]\div ({-\frac{1}{2}x})$, where $x=\left(\pi -3\right)^{0}$ and $y={({-\frac{1}{3}})^{-1}}$. | -40 | 1 | 2,454.75 | 2,454.75 | -1 | |
In $\triangle ABC$, $A=120^{\circ}$, $c=5$, $a=7$, find the value of $\frac{\sin B}{\sin C}$____. | \frac{3}{5} | 1 | 2,739.9375 | 2,739.9375 | -1 | |
Given point P(2, 1) is on the parabola $C_1: x^2 = 2py$ ($p > 0$), and the line $l$ passes through point Q(0, 2) and intersects the parabola $C_1$ at points A and B.
(1) Find the equation of the parabola $C_1$ and the equation for the trajectory $C_2$ of the midpoint M of chord AB;
(2) If lines $l_1$ and $l_2$ are tang... | \sqrt{3} | 0.4375 | 7,140.0625 | 6,225.857143 | 7,851.111111 | |
Given vectors $\overrightarrow{a}=(\sin \theta, 2)$ and $\overrightarrow{b}=(\cos \theta, 1)$, which are collinear, where $\theta \in (0, \frac{\pi}{2})$.
1. Find the value of $\tan (\theta + \frac{\pi}{4})$.
2. If $5\cos (\theta - \phi)=3 \sqrt{5}\cos \phi, 0 < \phi < \frac{\pi}{2}$, find the value of $\phi$. | \frac{\pi}{4} | 1 | 2,823.9375 | 2,823.9375 | -1 | |
Given two arithmetic sequences $\{a\_n\}$ and $\{b\_n\}$ with the sum of their first $n$ terms being $S\_n$ and $T\_n$ respectively. If $\frac{S\_n}{T\_n} = \frac{2n}{3n+1}$, find the value of $\frac{a\_{11}}{b\_{11}}$. | \frac{21}{32} | 0.75 | 4,972.0625 | 4,101.583333 | 7,583.5 | |
Given vectors $\overrightarrow {a}$=（sinx，cosx）, $\overrightarrow {b}$=（1，$\sqrt {3}$）.
(1) If $\overrightarrow {a}$$∥ \overrightarrow {b}$, find the value of tanx;
(2) Let f(x) = $\overrightarrow {a}$$$\cdot \overrightarrow {b}$, stretch the horizontal coordinates of each point on the graph of f(x) to twice their orig... | \frac{\pi}{3} | 0.6875 | 5,886.25 | 4,838.181818 | 8,192 | |
Given that $a$ and $b \in R$, the function $f(x) = \ln(x + 1) - 2$ is tangent to the line $y = ax + b - \ln2$ at $x = -\frac{1}{2}$. Let $g(x) = e^x + bx^2 + a$. If the inequality $m \leqslant g(x) \leqslant m^2 - 2$ holds true in the interval $[1, 2]$, determine the real number $m$. | e + 1 | 0 | 7,894.9375 | -1 | 7,894.9375 | |
A chessboard’s squares are labeled with numbers as follows:
[asy]
unitsize(0.8 cm);
int i, j;
for (i = 0; i <= 8; ++i) {
draw((i,0)--(i,8));
draw((0,i)--(8,i));
}
for (i = 0; i <= 7; ++i) {
for (j = 0; j <= 7; ++j) {
label("$\frac{1}{" + string(9 - i + j) + "}$", (i + 0.5, j + 0.5));
}}
[/asy]
Eight of the s... | \frac{8}{9} | 0 | 8,192 | -1 | 8,192 | |
Let \( n \) be a natural number. Define \( 1 = d_{1} < d_{2} < d_{3} < \cdots < d_{k} = n \) as its divisors. It is noted that \( n = d_{2}^{2} + d_{3}^{3} \). Determine all possible values of \( n \). | 68 | 0 | 8,192 | -1 | 8,192 | |
Given the function $f(x)=\cos (x- \frac {π}{4})-\sin (x- \frac {π}{4}).$
(I) Determine the evenness or oddness of the function $f(x)$ and provide a proof;
(II) If $θ$ is an angle in the first quadrant and $f(θ+ \frac {π}{3})= \frac { \sqrt {2}}{3}$, find the value of $\cos (2θ+ \frac {π}{6})$. | \frac {4 \sqrt {2}}{9} | 0 | 5,930.4375 | -1 | 5,930.4375 | |
Consider a four-digit natural number with the following property: if we swap its first two digits with the second two digits, we get a four-digit number that is 99 less.
How many such numbers are there in total, and how many of them are divisible by 9? | 10 | 0.25 | 7,720.5 | 6,306 | 8,192 | |
The fenced area of a yard is a 15-foot by 12-foot rectangular region with a 3-foot by 3-foot square cut out, as shown. What is the area of the region within the fence, in square feet?
[asy]draw((0,0)--(16,0)--(16,12)--(28,12)--(28,0)--(60,0)--(60,48)--(0,48)--cycle);
label("15'",(30,48),N);
label("12'",(60,24),E);
lab... | 171 | 0.9375 | 1,738.1875 | 1,825.466667 | 429 | |
Determine the total number of real solutions for the equation
\[
\frac{x}{50} = \sin x.
\] | 32 | 0 | 8,192 | -1 | 8,192 | |
For n points \[ P_1;P_2;...;P_n \] in that order on a straight line. We colored each point by 1 in 5 white, red, green, blue, and purple. A coloring is called acceptable if two consecutive points \[ P_i;P_{i+1} (i=1;2;...n-1) \] is the same color or 2 points with at least one of 2 points are colored white. How many way... | \frac{3^{n+1} + (-1)^{n+1}}{2} |
To find the number of acceptable colorings for \( n \) points \( P_1, P_2, \ldots, P_n \) on a straight line, we need to adhere to the following rules:
- Each point is colored with one of five colors: white, red, green, blue, or purple.
- A coloring is acceptable if, for any two consecutive points \( P_i \) and \( P_{... | 0 | 8,192 | -1 | 8,192 |
Let A and B be fixed points in the plane with distance AB = 1. An ant walks on a straight
line from point A to some point C in the plane and notices that the distance from itself to B
always decreases at any time during this walk. Compute the area of the region in the plane
containing all points where point C could po... | \frac{\pi}{4} | 0.375 | 7,468.3125 | 6,389.166667 | 8,115.8 | |
In rectangle $A B C D$ with area 1, point $M$ is selected on $\overline{A B}$ and points $X, Y$ are selected on $\overline{C D}$ such that $A X<A Y$. Suppose that $A M=B M$. Given that the area of triangle $M X Y$ is $\frac{1}{2014}$, compute the area of trapezoid $A X Y B$. | \frac{1}{2}+\frac{1}{2014} \text{ OR } \frac{504}{1007} | Notice that $[A M X]+[B Y M]=\frac{1}{2}[A B C D]=\frac{1}{2}$. Thus, $$[A X Y B]=[A M X]+[B Y M]+[M X Y]=\frac{1}{2}+\frac{1}{2014}=\frac{504}{1007}$$ | 0 | 7,553.375 | -1 | 7,553.375 |
Find the greatest common divisor of $8!$ and $(6!)^2.$ | 5760 | 0.9375 | 3,344.875 | 3,021.733333 | 8,192 | |
In the arithmetic sequence $\left\{ a_n \right\}$, $a_{15}+a_{16}+a_{17}=-45$, $a_{9}=-36$, and $S_n$ is the sum of the first $n$ terms.
(1) Find the minimum value of $S_n$ and the corresponding value of $n$;
(2) Calculate $T_n = \left| a_1 \right| + \left| a_2 \right| + \ldots + \left| a_n \right|$. | -630 | 0.25 | 7,781.9375 | 7,890.25 | 7,745.833333 | |
Let $a_n = -n^2 + 10n + 11$, then find the value of $n$ for which the sum of the sequence $\{a_n\}$ from the first term to the nth term is maximized. | 11 | 0.25 | 8,081.8125 | 8,053 | 8,091.416667 | |
Given that point $M$ represents the number $9$ on the number line.<br/>$(1)$ If point $N$ is first moved $4$ units to the left and then $6$ units to the right to reach point $M$, then the number represented by point $N$ is ______.<br/>$(2)$ If point $M$ is moved $4$ units on the number line, then the number represented... | 13 | 0.625 | 3,400.6875 | 4,615.8 | 1,375.5 | |
Given \( s = 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{10^6}} \), what is the integer part of \( s \)? | 1998 | 0.375 | 7,646 | 6,736 | 8,192 | |
A regular octagon is inscribed in a circle and another regular octagon is circumscribed about the same circle. What is the ratio of the area of the larger octagon to the area of the smaller octagon? Express your answer as a common fraction. | \frac{4 - 2\sqrt{2}}{2} | 0 | 7,563.875 | -1 | 7,563.875 | |
Determine how many regions of space are divided by:
a) The six planes of a cube's faces.
b) The four planes of a tetrahedron's faces. | 15 | 0.3125 | 7,167.6875 | 5,601.4 | 7,879.636364 | |
Select any 11 numbers from the 20 integers between 1 and 20. Among these, there must be two numbers whose sum equals ( ). | 21 | 0.75 | 4,876.9375 | 3,771.916667 | 8,192 | |
There is a set of 1000 switches, each of which has four positions, called $A, B, C$, and $D$. When the position of any switch changes, it is only from $A$ to $B$, from $B$ to $C$, from $C$ to $D$, or from $D$ to $A$. Initially each switch is in position $A$. The switches are labeled with the 1000 different integers $(2... | 650 | 0 | 7,870.3125 | -1 | 7,870.3125 | |
The segments connecting the feet of the altitudes of an acute-angled triangle form a right triangle with a hypotenuse of 10. Find the radius of the circumcircle of the original triangle. | 10 | 0.625 | 5,666.5625 | 4,286.5 | 7,966.666667 | |
Two circles of radius 3 and 4 are internally tangent to a larger circle. The larger circle circumscribes both the smaller circles. Find the area of the shaded region surrounding the two smaller circles within the larger circle. Express your answer in terms of \(\pi\). | 24\pi | 0.625 | 5,365.625 | 3,754.2 | 8,051.333333 | |
There are 7 students standing in a row. How many different arrangements are there in the following situations?
(1) A and B must stand together;
(2) A is not at the head of the line, and B is not at the end of the line;
(3) There must be exactly one person between A and B. | 1200 | 0.6875 | 4,385 | 3,672.818182 | 5,951.8 | |
Trapezoid \(ABCD\) is inscribed in the parabola \(y = x^2\) such that \(A = (a, a^2)\), \(B = (b, b^2)\), \(C = (-b, b^2)\), and \(D = (-a, a^2)\) for some positive reals \(a, b\) with \(a > b\). If \(AD + BC = AB + CD\), and \(AB = \frac{3}{4}\), what is \(a\)? | \frac{27}{40} | 1 | 4,308.8125 | 4,308.8125 | -1 | |
Given that $\alpha, \beta, \gamma$ are all acute angles and $\cos^{2} \alpha + \cos^{2} \beta + \cos^{2} \gamma = 1$, find the minimum value of $\tan \alpha \cdot \tan \beta \cdot \tan \gamma$. | 2\sqrt{2} | 0.25 | 7,796.9375 | 6,611.75 | 8,192 | |
Given the expressions $(2401^{\log_7 3456})^{\frac{1}{2}}$, calculate its value. | 3456^2 | 0 | 4,822.875 | -1 | 4,822.875 | |
Let $A$ equal the number of four digit odd numbers. Let $B$ equal the number of four digit multiples of 5. Find $A+B$. | 6300 | 0.875 | 3,870.9375 | 4,059.285714 | 2,552.5 | |
A circle with radius 100 is drawn on squared paper with unit squares. It does not touch any of the grid lines or pass through any of the lattice points. What is the maximum number of squares it can pass through? | 800 | 0 | 7,798.25 | -1 | 7,798.25 | |
Given the function $f(x)=x^{3}+3mx^{2}+nx+m^{2}$ has an extreme value of $0$ at $x=-1$, find the value of $m+n$. | 11 | 0.9375 | 4,689.125 | 4,476.266667 | 7,882 | |
In 1991 the population of a town was a perfect square. Ten years later, after an increase of 150 people, the population was 9 more than a perfect square. Now, in 2011, with an increase of another 150 people, the population is once again a perfect square. What is the percent growth of the town's population during this t... | 62 | 1. **Define Variables:**
Let the population of the town in 1991 be $p^2$. In 2001, after an increase of 150 people, the population is $p^2 + 150$. According to the problem, this new population is 9 more than a perfect square, so we can write it as $q^2 + 9$. Thus, we have:
\[
p^2 + 150 = q^2 + 9
\]
Rearr... | 0.6875 | 6,560.1875 | 6,155.181818 | 7,451.2 |
In convex quadrilateral \(EFGH\), \(\angle E = \angle G\), \(EF = GH = 150\), and \(EH \neq FG\). The perimeter of \(EFGH\) is 580. Find \(\cos E\). | \frac{14}{15} | 0.4375 | 6,278.75 | 3,818.857143 | 8,192 | |
Let $x$ and $y$ be real numbers, $y > x > 0,$ such that
\[\frac{x}{y} + \frac{y}{x} = 4.\]Find the value of \[\frac{x + y}{x - y}.\] | \sqrt{3} | 0.125 | 2,971.875 | 3,711 | 2,866.285714 | |
Find the minimum possible value of the largest of $x y, 1-x-y+x y$, and $x+y-2 x y$ if $0 \leq x \leq y \leq 1$. | \frac{4}{9} | I claim the answer is $4 / 9$. Let $s=x+y, p=x y$, so $x$ and $y$ are $\frac{s \pm \sqrt{s^{2}-4 p}}{2}$. Since $x$ and $y$ are real, $s^{2}-4 p \geq 0$. If one of the three quantities is less than or equal to $1 / 9$, then at least one of the others is at least $4 / 9$ by the pigeonhole principle since they add up to ... | 0 | 8,192 | -1 | 8,192 |
Square $ABCD$ has side length $13$, and points $E$ and $F$ are exterior to the square such that $BE=DF=5$ and $AE=CF=12$. Find $EF^{2}$.[asy]unitsize(0.2 cm); pair A, B, C, D, E, F; A = (0,13); B = (13,13); C = (13,0); D = (0,0); E = A + (12*12/13,5*12/13); F = D + (5*5/13,-5*12/13); draw(A--B--C--D--cycle); draw(A-... | 578 | 0.875 | 5,200.75 | 4,773.428571 | 8,192 | |
Given that point $P$ is a moving point on the parabola $y^{2}=4x$, the minimum value of the sum of the distance from point $P$ to line $l$: $2x-y+3=0$ and the $y$-axis is ___. | \sqrt{5}-1 | 0.625 | 6,513.6875 | 5,506.7 | 8,192 | |
If \(\frac{1}{4} + 4\left(\frac{1}{2013} + \frac{1}{x}\right) = \frac{7}{4}\), find the value of \(1872 + 48 \times \left(\frac{2013 x}{x + 2013}\right)\). | 2000 | 0.5 | 6,170 | 4,765.625 | 7,574.375 | |
Two springs with stiffnesses of $6 \, \text{kN} / \text{m}$ and $12 \, \text{kN} / \text{m}$ are connected in series. How much work is required to stretch this system by 10 cm? | 20 | 0.75 | 4,143.125 | 3,177.666667 | 7,039.5 | |
A rectangular cake with dimensions $4$ inches, $3$ inches, and $2$ inches (length, width, height respectively) is iced on the sides, the top, and the bottom. The cake is cut from the top center vertex across to the midpoint of the bottom edge on the opposite side face, creating one triangular piece. If the top center v... | 38 | 0 | 8,192 | -1 | 8,192 | |
Write $\frac{5}{8}$ as a decimal. | 0.625 | 1 | 1,847.75 | 1,847.75 | -1 | |
Two cubical dice each have removable numbers $1$ through $6$. The twelve numbers on the two dice are removed, put into a bag, then drawn one at a time and randomly reattached to the faces of the cubes, one number to each face. The dice are then rolled and the numbers on the two top faces are added. What is the probabil... | \frac{1}{6} | To solve this problem, we need to consider the probability of rolling a sum of 7 with two dice, where the numbers on the dice have been randomly reassigned.
#### Step 1: Understand the possible sums
The possible combinations of two numbers summing to 7 are:
- $(1,6)$
- $(2,5)$
- $(3,4)$
- $(4,3)$
- $(5,2)$
- $(6,1)$
... | 0.1875 | 8,001.4375 | 7,175.666667 | 8,192 |
A ream of paper containing $500$ sheets is $5$ cm thick. Approximately how many sheets of this type of paper would there be in a stack $7.5$ cm high? | 750 | 1. **Identify the thickness per sheet**: Given that $500$ sheets of paper have a total thickness of $5$ cm, we can calculate the thickness of one sheet of paper. This is done by dividing the total thickness by the number of sheets:
\[
\text{Thickness per sheet} = \frac{5 \text{ cm}}{500 \text{ sheets}} = 0.01 \te... | 1 | 1,615.375 | 1,615.375 | -1 |
What would the 25th number be in a numeric system where the base is five? | 100 | 0.1875 | 2,212.3125 | 1,058 | 2,478.692308 | |
In a set of $36$ square blocks arranged into a $6 \times 6$ square, how many different combinations of $4$ blocks can be selected from that set so that no two are in the same row or column? | 5400 | 0.8125 | 5,490.75 | 4,896 | 8,068 | |
Square $ABCD$ is circumscribed around a circle. Another square $IJKL$ is inscribed inside a smaller concentric circle. The side length of square $ABCD$ is $4$ units, and the radius of the smaller circle is half the radius of the larger circle. Find the ratio of the area of square $IJKL$ to the area of square $ABCD$. | \frac{1}{4} | 0 | 3,436.1875 | -1 | 3,436.1875 | |
How many ordered triplets $(a, b, c)$ of positive integers such that $30a + 50b + 70c \leq 343$ . | 30 | 0.3125 | 7,517 | 6,032 | 8,192 | |
Let $x,$ $y,$ $z$ be real numbers such that $x + y + z = 2,$ and $x \ge -\frac{1}{2},$ $y \ge -2,$ and $z \ge -3.$ Find the maximum value of:
\[
\sqrt{4x + 2} + \sqrt{4y + 8} + \sqrt{4z + 12}.
\] | 3\sqrt{10} | 0.625 | 6,683.9375 | 6,066 | 7,713.833333 | |
The quartic equation \( x^{4} + a x^{3} + b x^{2} + a x + 1 = 0 \) has a real root. Find the minimum value of \( a^{2} + b^{2} \). | 4/5 | 0.25 | 7,791.625 | 7,229.75 | 7,978.916667 | |
In triangle $ABC$ , let $P$ and $R$ be the feet of the perpendiculars from $A$ onto the external and internal bisectors of $\angle ABC$ , respectively; and let $Q$ and $S$ be the feet of the perpendiculars from $A$ onto the internal and external bisectors of $\angle ACB$ , respectively. If $PQ = 7, QR =... | 84 | 0 | 8,192 | -1 | 8,192 | |
Determine the number of triples $0 \leq k, m, n \leq 100$ of integers such that $$ 2^{m} n-2^{n} m=2^{k} $$ | 22 | First consider when $n \geq m$, so let $n=m+d$ where $d \geq 0$. Then we have $2^{m}\left(m+d-2^{d} m\right)=$ $2^{m}\left(m\left(1-2^{d}\right)+d\right)$, which is non-positive unless $m=0$. So our first set of solutions is $m=0, n=2^{j}$. Now, we can assume that $m>n$, so let $m=n+d$ where $d>0$. Rewrite $2^{m} n-2^{... | 0 | 8,192 | -1 | 8,192 |
What is $w + 2 - 3w - 4 + 5w + 6 - 7w - 8$? | -4w - 4 | 0.9375 | 2,034.1875 | 1,623.666667 | 8,192 | |
Isabel wants to save 40 files onto disks, each with a capacity of 1.44 MB. 5 of the files take up 0.95 MB each, 15 files take up 0.65 MB each, and the remaining 20 files each take up 0.45 MB. Calculate the smallest number of disks needed to store all 40 files. | 17 | 0.3125 | 5,481.625 | 965.4 | 7,534.454545 | |
Two cards are dealt from a standard deck of 52 cards. What is the probability that the first card dealt is a $\diamondsuit$ and the second card dealt is a $\spadesuit$? | \frac{13}{204} | 0.5625 | 6,353 | 4,922.666667 | 8,192 | |
Let $X \sim B(4, p)$, and $P(X=2)=\frac{8}{27}$, find the probability of success in one trial. | \frac{2}{3} | 0.6875 | 6,789.375 | 6,310.818182 | 7,842.2 | |
On a circle, points \(B\) and \(D\) are located on opposite sides of the diameter \(AC\). It is known that \(AB = \sqrt{6}\), \(CD = 1\), and the area of triangle \(ABC\) is three times the area of triangle \(BCD\). Find the radius of the circle. | 1.5 | 0 | 7,816.6875 | -1 | 7,816.6875 | |
The sides $PQ$ and $PR$ of triangle $PQR$ are respectively of lengths $4$ inches, and $7$ inches. The median $PM$ is $3\frac{1}{2}$ inches. Then $QR$, in inches, is: | 9 | 1. **Using the Median Formula**: The formula for the length of a median in a triangle, which connects a vertex to the midpoint of the opposite side, is given by:
\[
PM = \frac{1}{2}\sqrt{2PQ^2 + 2PR^2 - QR^2}
\]
where $PQ$, $PR$, and $QR$ are the lengths of the sides of the triangle, and $PM$ is the median ... | 1 | 2,421.625 | 2,421.625 | -1 |
A uniform solid semi-circular disk of radius $R$ and negligible thickness rests on its diameter as shown. It is then tipped over by some angle $\gamma$ with respect to the table. At what minimum angle $\gamma$ will the disk lose balance and tumble over? Express your answer in degrees, rounded to the nearest integ... | 23 | 0.0625 | 7,542.75 | 5,129 | 7,703.666667 | |
In a room containing $N$ people, $N > 3$, at least one person has not shaken hands with everyone else in the room.
What is the maximum number of people in the room that could have shaken hands with everyone else? | $N-1$ | To solve this problem, we need to determine the maximum number of people in a room of $N$ people who could have shaken hands with every other person, given that at least one person has not shaken hands with everyone else.
1. **Understanding the Problem:**
- We have $N$ people in a room.
- At least one person has... | 0 | 6,631.875 | -1 | 6,631.875 |
For how many positive integers $n$ less than or equal to 500 is $$(\sin (t+\frac{\pi}{4})+i\cos (t+\frac{\pi}{4}))^n=\sin (nt+\frac{n\pi}{4})+i\cos (nt+\frac{n\pi}{4})$$ true for all real $t$? | 125 | 0.6875 | 6,073.8125 | 5,111 | 8,192 | |
Mr. Earl E. Bird gets up every day at 8:00 AM to go to work. If he drives at an average speed of 40 miles per hour, he will be late by 3 minutes. If he drives at an average speed of 60 miles per hour, he will be early by 3 minutes. How many miles per hour does Mr. Bird need to drive to get to work exactly on time? | 48 |
#### Step 1: Define Variables
Let $d$ be the distance Mr. Bird needs to travel to work and $t$ be the time he needs to arrive exactly on time (in hours).
#### Step 2: Set Up Equations Based on Given Information
1. If Mr. Bird drives at 40 miles per hour and is late by 3 minutes, he takes $t + \frac{3}{60} = t + \frac... | 1 | 2,294.375 | 2,294.375 | -1 |
Let $x=2001^{1002}-2001^{-1002}$ and $y=2001^{1002}+2001^{-1002}$. Find $x^{2}-y^{2}$. | -4 | -4. | 1 | 3,183.4375 | 3,183.4375 | -1 |
Let $k$ be a positive integer. Suppose that the integers $1, 2, 3, \dots, 3k+1$ are written down in random order. What is the probability that at no time during this process, the sum of the integers that have been written up to that time is a positive integer divisible by 3? Your answer should be in closed form, but ma... | \frac{k!(k+1)!}{(3k+1)(2k)!} | Assume that we have an ordering of $1,2,\dots,3k+1$ such that no initial subsequence sums to $0$ mod $3$. If we omit the multiples of $3$ from this ordering, then the remaining sequence mod $3$ must look like $1,1,-1,1,-1,\ldots$ or $-1,-1,1,-1,1,\ldots$.
Since there is one more integer in the ordering congruent to $1$... | 0 | 8,192 | -1 | 8,192 |
Given that the sum of the first 10 terms of a geometric sequence $\{a_n\}$ is 32 and the sum of the first 20 terms is 56, find the sum of the first 30 terms. | 74 | 0.75 | 4,561.125 | 3,350.833333 | 8,192 | |
Reduced to lowest terms, $\frac{a^{2}-b^{2}}{ab} - \frac{ab-b^{2}}{ab-a^{2}}$ is equal to: | \frac{a}{b} | 1. **Factorize and simplify the expressions:**
- The first term is already simplified: \(\frac{a^2-b^2}{ab}\).
- Factorize the numerator and denominator of the second term:
\[
-\frac{ab-b^2}{ab-a^2} = -\frac{b(a-b)}{a(b-a)}.
\]
Since \(b-a = -(a-b)\), this simplifies to:
\[
-\frac{b(... | 1 | 1,833.5 | 1,833.5 | -1 |
A die with faces showing the numbers $0,1,2,3,4,5$ is rolled until the total sum of the rolled numbers exceeds 12. What is the most likely value of this sum? | 13 | 0 | 8,192 | -1 | 8,192 | |
Find $1 - 0.\overline{123}$. | \frac{292}{333} | 0.875 | 4,199.1875 | 3,628.785714 | 8,192 | |
Tim continues the prank into the next week after a successful first week. This time, he starts on Monday with two people willing to do the prank, on Tuesday there are three options, on Wednesday everyone from Monday and Tuesday refuses but there are six new people, on Thursday four of Wednesday's people can't participa... | 432 | 0.0625 | 6,635.625 | 7,449 | 6,581.4 | |
Five students, labeled as A, B, C, D, and E, are standing in a row to participate in a literary performance. If student A does not stand at either end, calculate the number of different arrangements where students C and D are adjacent. | 24 | 0.125 | 7,790.125 | 6,405 | 7,988 | |
Given an ellipse $E:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with a major axis length of $4$, and the point $P(1,\frac{3}{2})$ lies on the ellipse $E$. <br/>$(1)$ Find the equation of the ellipse $E$; <br/>$(2)$ A line $l$ passing through the right focus $F$ of the ellipse $E$ is drawn such that it doe... | 24 | 0 | 7,417.9375 | -1 | 7,417.9375 | |
In a regular tetrahedron \(ABCD\), points \(E\) and \(F\) are on edges \(AB\) and \(AC\) respectively, satisfying \(BE=3\) and \(EF=4\), and \(EF \parallel\) plane \(BCD\). Find the area of \(\triangle DEF\). | 2 \sqrt{33} | 0 | 8,192 | -1 | 8,192 |
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