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 |
|---|---|---|---|---|---|---|
When the set of natural numbers is listed in ascending order, what is the smallest prime number that occurs after a sequence of four consecutive positive integers all of which are nonprime? | 29 | 0.25 | 6,696 | 4,905.75 | 7,292.75 | |
Let $A,B,C,D$ , be four different points on a line $\ell$ , so that $AB=BC=CD$ . In one of the semiplanes determined by the line $\ell$ , the points $P$ and $Q$ are chosen in such a way that the triangle $CPQ$ is equilateral with its vertices named clockwise. Let $M$ and $N$ be two points of the plane be ... | 60 | 0 | 8,192 | -1 | 8,192 | |
Given the decomposition rate $v$ of a certain type of garbage approximately satisfies the relationship $v=a\cdot b^{t}$, where $a$ and $b$ are non-zero constants, and $v=5\%$ after $6$ months and $v=10\%$ after $12$ months, determine the time needed for this type of garbage to completely decompose. | 32 | 0.1875 | 7,439.375 | 5,489.666667 | 7,889.307692 | |
How many times do the graphs $r = 4 \cos \theta$ and $r = 8 \sin \theta$ intersect? | 2 | 0.375 | 7,448.4375 | 6,209.166667 | 8,192 | |
Given that a recipe calls for \( 4 \frac{1}{2} \) cups of flour, calculate the amount of flour needed if only half of the recipe is made. | 2 \frac{1}{4} | 0.875 | 408.1875 | 410.642857 | 391 | |
Create a cube $C_{1}$ with edge length 1. Take the centers of the faces and connect them to form an octahedron $O_{1}$. Take the centers of the octahedron's faces and connect them to form a new cube $C_{2}$. Continue this process infinitely. Find the sum of all the surface areas of the cubes and octahedrons. | \frac{54+9 \sqrt{3}}{8} | The lengths of the second cube are one-third of the lengths of the first cube, so the surface area decreases by a factor of one-ninth. Since the first cube has surface area 6 and the first octahedron has surface area $\sqrt{3}$, the total area is $(6+\sqrt{3}) \cdot\left(1+\frac{1}{9}+\frac{1}{9^{2}}+\cdots\right)=\fra... | 0 | 8,034.0625 | -1 | 8,034.0625 |
Ms.Osborne asks each student in her class to draw a rectangle with integer side lengths and a perimeter of $50$ units. All of her students calculate the area of the rectangle they draw. What is the difference between the largest and smallest possible areas of the rectangles? | 128 | 1. **Understanding the Problem:**
- We are given a rectangle with a perimeter of 50 units.
- The side lengths of the rectangle are integers.
- We need to find the difference between the largest and smallest possible areas of such rectangles.
2. **Expressing the Perimeter in Terms of Side Lengths:**
- Let t... | 0 | 2,291.4375 | -1 | 2,291.4375 |
In a certain sequence of numbers, the first number is $1$, and, for all $n\ge 2$, the product of the first $n$ numbers in the sequence is $n^2$. The sum of the third and the fifth numbers in the sequence is | \frac{61}{16} | 1. **Identify the pattern in the sequence**: Given that the product of the first $n$ numbers in the sequence is $n^2$, we can express this as:
\[
a_1 \cdot a_2 \cdot \ldots \cdot a_n = n^2
\]
where $a_i$ represents the $i$-th number in the sequence.
2. **Determine the formula for each term**: To find the $... | 1 | 2,668.625 | 2,668.625 | -1 |
Sam and Lee run at equal and constant rates. They also cycle and skateboard at equal and constant rates. Sam covers $120$ kilometers after running for $4$ hours, cycling for $5$ hours, and skateboarding for $3$ hours while Lee covers $138$ kilometers after running for $5$ hours, skateboarding for $4$ hours, and cycling... | 436 | 0 | 8,192 | -1 | 8,192 | |
Given that tetrahedron PQRS has edge lengths PQ = 3, PR = 4, PS = 5, QR = 5, QS = √34, and RS = √41, calculate the volume of tetrahedron PQRS. | 10 | 0.1875 | 7,508.9375 | 4,549 | 8,192 | |
Given the function $$f(x)=\sin(x+ \frac {\pi}{6})+2\sin^{2} \frac {x}{2}$$.
(1) Find the equation of the axis of symmetry and the coordinates of the center of symmetry for the function $f(x)$.
(2) Determine the intervals of monotonicity for the function $f(x)$.
(3) In triangle $ABC$, $a$, $b$, and $c$ are the sides opp... | \frac {3}{2} | 0.0625 | 7,991.8125 | 7,733 | 8,009.066667 | |
Find the degree measure of the angle
$$
\delta=\arccos \left(\left(\sin 2541^{\circ}+\sin 2542^{\circ}+\cdots+\sin 6141^{\circ}\right)^{\cos 2520^{\circ}}+\cos 2521^{\circ}+\cdots+\cos 6120^{\circ}\right)
$$ | 69 | 0.625 | 5,319.5625 | 4,650.4 | 6,434.833333 | |
Kelvin the Frog likes numbers whose digits strictly decrease, but numbers that violate this condition in at most one place are good enough. In other words, if $d_{i}$ denotes the $i$ th digit, then $d_{i} \leq d_{i+1}$ for at most one value of $i$. For example, Kelvin likes the numbers 43210, 132, and 3, but not the nu... | 14034 | Suppose first that no digit violates the constraint; i.e. the digits are in strictly decreasing order. There are $\binom{10}{5}$ ways to choose the digits of the number, and each set of digits can be arranged in exactly one way, so there are $\binom{10}{5}$ such numbers. We now perform casework on which digit violates ... | 0 | 8,192 | -1 | 8,192 |
Given two real numbers \( p > 1 \) and \( q > 1 \) such that \( \frac{1}{p} + \frac{1}{q} = 1 \) and \( pq = 9 \), what is \( q \)? | \frac{9 + 3\sqrt{5}}{2} | 0 | 6,962.625 | -1 | 6,962.625 | |
Calculate:<br/>$(1)4.7+\left(-2.5\right)-\left(-5.3\right)-7.5$;<br/>$(2)18+48\div \left(-2\right)^{2}-\left(-4\right)^{2}\times 5$;<br/>$(3)-1^{4}+\left(-2\right)^{2}\div 4\times [5-\left(-3\right)^{2}]$;<br/>$(4)(-19\frac{15}{16})×8$ (Solve using a simple method). | -159\frac{1}{2} | 0.0625 | 3,297.625 | 2,440 | 3,354.8 | |
If parallelogram ABCD has an area of 100 square meters, and E and G are the midpoints of sides AD and CD, respectively, while F is the midpoint of side BC, find the area of quadrilateral DEFG. | 25 | 0 | 6,570.6875 | -1 | 6,570.6875 | |
If the non-negative real numbers $x$ and $y$ satisfy $x^{2}+4y^{2}+4xy+4x^{2}y^{2}=32$, find the minimum value of $x+2y$, and the maximum value of $\sqrt{7}(x+2y)+2xy$. | 4\sqrt{7}+4 | 0 | 7,781.4375 | -1 | 7,781.4375 | |
Evaluate $\left|-1 + \frac{2}{3}i\right|$. | \frac{\sqrt{13}}{3} | 0 | 1,510.625 | -1 | 1,510.625 | |
Simplify the expression \(\left(\frac{2-n}{n-1}+4 \cdot \frac{m-1}{m-2}\right):\left(n^{2} \cdot \frac{m-1}{n-1}+m^{2} \cdot \frac{2-n}{m-2}\right)\) given that \(m=\sqrt[4]{400}\) and \(n=\sqrt{5}\). | \frac{\sqrt{5}}{5} | 0 | 6,394.875 | -1 | 6,394.875 | |
Below is a portion of the graph of a function, $y=E(x)$:
[asy]
import graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.5,xmax=4.5,ymin=-0.99,ymax=6.5;
pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 2"); real gx=1,gy=1;
... | 5 | 1 | 1,525.0625 | 1,525.0625 | -1 | |
Triangle $ABC$ has $BC=20.$ The incircle of the triangle evenly trisects the median $AD.$ If the area of the triangle is $m \sqrt{n}$ where $m$ and $n$ are integers and $n$ is not divisible by the square of a prime, find $m+n.$
| 38 | 0 | 8,079.9375 | -1 | 8,079.9375 | |
Calculate the following product: $12 \times 0.5 \times 3 \times 0.2 =$
A) $\frac{20}{5}$
B) $\frac{22}{5}$
C) $\frac{16}{5}$
D) $\frac{18}{5}$
E) $\frac{14}{5}$ | \frac{18}{5} | 0.4375 | 1,477.375 | 406.714286 | 2,310.111111 | |
Given the function $f\left(x\right)=x^{2}-2bx+3$, where $b\in R$.
$(1)$ Find the solution set of the inequality $f\left(x\right) \lt 4-b^{2}$.
$(2)$ When $x\in \left[-1,2\right]$, the function $y=f\left(x\right)$ has a minimum value of $1$. Find the maximum value of the function $y=f\left(x\right)$ when $x\in \left... | 4 + 2\sqrt{2} | 0.125 | 6,217.75 | 4,110.5 | 6,518.785714 | |
On a trip from the United States to Canada, Isabella took $d$ U.S. dollars. At the border she exchanged them all, receiving $10$ Canadian dollars for every $7$ U.S. dollars. After spending $60$ Canadian dollars, she had $d$ Canadian dollars left. What is the sum of the digits of $d$? | 5 | 1. **Understanding the exchange rate and the amount exchanged**: Isabella exchanges her U.S. dollars to Canadian dollars at a rate where $7$ U.S. dollars yield $10$ Canadian dollars. This means for every $7$ U.S. dollars, she receives $10$ Canadian dollars.
2. **Calculating the total Canadian dollars received**: Let's... | 1 | 1,407.3125 | 1,407.3125 | -1 |
Given that Sofia has a $5 \times 7$ index card, if she shortens the length of one side by $2$ inches and the card has an area of $21$ square inches, find the area of the card in square inches if instead she shortens the length of the other side by $1$ inch. | 30 | 0.875 | 3,093.125 | 2,561.428571 | 6,815 | |
A finite arithmetic progression \( a_1, a_2, \ldots, a_n \) with a positive common difference has a sum of \( S \), and \( a_1 > 0 \). It is known that if the common difference of the progression is increased by 3 times while keeping the first term unchanged, the sum \( S \) doubles. By how many times will \( S \) inc... | 5/2 | 0.9375 | 4,084.875 | 3,811.066667 | 8,192 | |
In a local government meeting, leaders from five different companies are present. It is known that two representatives are from Company A, and each of the remaining four companies has one representative attending. If three individuals give a speech at the meeting, how many possible combinations are there where these th... | 16 | 0.5625 | 5,682.9375 | 4,489.666667 | 7,217.142857 | |
The circles $C_1$ and $C_2$ are defined by the equations $x^2 + y^2 = 1$ and $(x - 2)^2 + y^2 = 16,$ respectively. Find the locus of the centers $(a,b)$ of all circles externally tangent to $C_1$ and internally tangent to $C_2.$ Enter your answer in the form
\[Pa^2 + Qb^2 + Ra + Sb + T = 0,\]where all the coefficient... | 84a^2 + 100b^2 - 168a - 441 = 0 | 0.25 | 8,105.5625 | 7,846.25 | 8,192 | |
The sum of an infinite geometric series is 64 times the series that results if the first four terms of the original series are removed. What is the value of the series' common ratio? | \frac{1}{2} | 0 | 6,610.1875 | -1 | 6,610.1875 | |
Given the polar equation of a conic section $\rho=\frac{5}{3-2 \cos \theta}$, find the distance between its focus and the corresponding directrix: | $\frac{5}{2}$ | 0 | 7,175.625 | -1 | 7,175.625 | |
In triangle \( \triangle ABC \), \( AB = AC \), \( AD \) and \( BE \) are the angle bisectors of \( \angle A \) and \( \angle B \) respectively, and \( BE = 2 AD \). What is the measure of \( \angle BAC \)? | 108 | 0.25 | 7,851.75 | 6,831 | 8,192 | |
Let $Q$ be the product of the first $50$ positive even integers. Find the largest integer $j$ such that $Q$ is divisible by $2^j$. | 97 | 0.875 | 3,670.25 | 3,024.285714 | 8,192 | |
Find the sum of the values of $x$ such that $\cos^3 3x+ \cos^3 5x = 8 \cos^3 4x \cos^3 x$, where $x$ is measured in degrees and $100< x< 200.$
| 906 | 0.0625 | 7,881.875 | 5,769 | 8,022.733333 | |
Alex chose positive integers $a, b, c, d, e, f$ and completely multiplied out the polynomial product $(1-x)^{a}(1+x)^{b}\left(1-x+x^{2}\right)^{c}\left(1+x^{2}\right)^{d}\left(1+x+x^{2}\right)^{e}\left(1+x+x^{2}+x^{3}+x^{4}\right)^{f}$. After she simplified her result, she discarded any term involving $x$ to any power ... | 23 | Define $f(x)=(1-x)^{a}(1+x)^{b}\left(1-x+x^{2}\right)^{c}\left(1+x^{2}\right)^{d}\left(1+x+x^{2}\right)^{e}\left(1+x+x^{2}+x^{3}+x^{4}\right)^{f}$. We note several algebraic identities, each of which can be checked by expanding and simplifying: $1-x^{5}=(1-x)\left(1+x+x^{2}+x^{3}+x^{4}\right)$, $1-x^{3}=(1-x)\left(1+x+... | 0 | 8,192 | -1 | 8,192 |
Suppose rectangle $F O L K$ and square $L O R E$ are on the plane such that $R L=12$ and $R K=11$. Compute the product of all possible areas of triangle $R K L$. | 414 | There are two possible configurations. If $R L=12$, the side length of the square is $6 \sqrt{2}$. Now $$121=R K^{2}=R E^{2}+E K^{2}=(6 \sqrt{2})^{2}+E K^{2}$$ so $E K=7$. Then the possible values of $L K$ are $6 \sqrt{2} \pm 7$. Note that the area of $\triangle R L K$ is $$\frac{L K \cdot R E}{2}=L K \cdot 3 \sqrt{2}$... | 0.1875 | 7,281.5625 | 5,513 | 7,689.692308 |
On an algebra quiz, $10\%$ of the students scored $70$ points, $35\%$ scored $80$ points, $30\%$ scored $90$ points, and the rest scored $100$ points. What is the difference between the mean and median score of the students' scores on this quiz? | 3 | 1. **Calculate the percentage of students scoring 100 points**:
Given that $10\%$ scored $70$ points, $35\%$ scored $80$ points, and $30\%$ scored $90$ points, the percentage of students scoring $100$ points is:
\[
100\% - (10\% + 35\% + 30\%) = 100\% - 75\% = 25\%
\]
2. **Determine the median score**:
... | 0.9375 | 4,548.8125 | 4,305.933333 | 8,192 |
Two real numbers $x$ and $y$ satisfy $x-y=4$ and $x^3-y^3=28$. Compute $xy$. | -3 | 1 | 1,975 | 1,975 | -1 | |
Find the degree measure of the least positive angle $\theta$ for which
\[\cos 5^\circ = \sin 25^\circ + \sin \theta.\] | 35^\circ | 1 | 2,631.4375 | 2,631.4375 | -1 | |
Let $z$ be a complex number. In the complex plane, the distance from $z$ to 1 is 2 , and the distance from $z^{2}$ to 1 is 6 . What is the real part of $z$ ? | \frac{5}{4} | Note that we must have $|z-1|=2$ and \left|z^{2}-1\right|=6$, so $|z+1|=\frac{\left|z^{2}-1\right|}{|z-1|}=3$. Thus, the distance from $z$ to 1 in the complex plane is 2 and the distance from $z$ to -1 in the complex plane is 3 . Thus, $z, 1,-1$ form a triangle with side lengths $2,3,3$. The area of a triangle with sid... | 0.875 | 4,682.1875 | 4,180.785714 | 8,192 |
Initially, there are 33 ones written on the board. Every minute, Karlson erases any two numbers and writes their sum on the board, then eats a number of candies equal to the product of the two erased numbers. What is the maximum number of candies he can eat in 33 minutes? | 528 | 0.125 | 7,696.5 | 4,695 | 8,125.285714 | |
Evaluate $\left\lceil\sqrt{2}\,\right\rceil+\left\lceil\sqrt{22}\,\right\rceil+\left\lceil\sqrt{222}\,\right\rceil$. | 22 | 0.9375 | 2,950 | 2,600.533333 | 8,192 | |
A board game spinner is divided into three regions labeled $A$, $B$ and $C$. The probability of the arrow stopping on region $A$ is $\frac{1}{3}$ and on region $B$ is $\frac{1}{2}$. What is the probability of the arrow stopping on region $C$? Express your answer as a common fraction. | \frac{1}{6} | 1 | 1,339.75 | 1,339.75 | -1 | |
If any two adjacent digits of a three-digit number have a difference of at most 1, it is called a "steady number". How many steady numbers are there? | 75 | 0.1875 | 7,743 | 5,797.333333 | 8,192 | |
Let event $A$ be "Point $M(x,y)$ satisfies $x^{2}+y^{2}\leqslant a(a > 0)$", and event $B$ be "Point $M(x,y)$ satisfies $\begin{cases} & x-y+1\geqslant 0 \\ & 5x-2y-4\leqslant 0 \\ & 2x+y+2\geqslant 0 \end{cases}$. If $P(B|A)=1$, then find the maximum value of the real number $a$. | \dfrac{1}{2} | 0.1875 | 6,237.0625 | 7,530 | 5,938.692308 | |
Find the distance from vertex \( A_4 \) to the face \( A_1 A_2 A_3 \).
\( A_1(2, 1, 4) \)
\( A_2(-1, 5, -2) \)
\( A_3(-7, -3, 2) \)
\( A_4(-6, -3, 6) \) | 5 \sqrt{\frac{2}{11}} | 0 | 4,452.9375 | -1 | 4,452.9375 | |
Each of the variables $a, b, c, d$, and $e$ represents a positive integer with the properties that $b+d>a+d$, $c+e>b+e$, $b+d=c$, $a+c=b+e$. Which of the variables has the greatest value? | c | Since $b+d>a+d$, then $b>a$. This means that $a$ does not have the greatest value. Since $c+e>b+e$, then $c>b$. This means that $b$ does not have the greatest value. Since $b+d=c$ and each of $b, c, d$ is positive, then $d<c$, which means that $d$ does not have the greatest value. Consider the last equation $a+c=b+e$ a... | 0.1875 | 5,162.75 | 2,779.666667 | 5,712.692308 |
Derek fills a square $10$ by $10$ grid with $50$ $1$ s and $50$ $2$ s. He takes the product of the numbers in each of the $10$ rows. He takes the product of the numbers in each of the $10$ columns. He then sums these $20$ products up to get an integer $N.$ Find the minimum possible value of $N.$ | 640 | 0.5 | 7,053.4375 | 5,914.875 | 8,192 | |
Given a complex number $z = (a^2 - 4) + (a + 2)i$ where $a \in \mathbb{R}$:
(Ⅰ) If $z$ is a pure imaginary number, find the value of the real number $a$;
(Ⅱ) If the point corresponding to $z$ in the complex plane lies on the line $x + 2y + 1 = 0$, find the value of the real number $a$. | a = -1 | 1 | 2,659.5 | 2,659.5 | -1 | |
Xiaoying goes home at noon to cook noodles by herself, which involves the following steps: ① Wash the pot and fill it with water, taking 2 minutes; ② Wash the vegetables, taking 3 minutes; ③ Prepare the noodles and seasonings, taking 2 minutes; ④ Boil the water in the pot, taking 7 minutes; ⑤ Use the boiling water to c... | 12 | 0.0625 | 6,783.0625 | 7,117 | 6,760.8 | |
Assume we have a calendrical system in which leap years happen every four years, no matter what. In a 150-year period, what is the maximum possible number of leap years? | 38 | 0.3125 | 7,145.875 | 5,420.4 | 7,930.181818 | |
Three cones are placed on a table on their bases, touching each other. The radii of their bases are 1, 12, and 12, and the apex angles of the cones are $-4 \operatorname{arctg} \frac{1}{3}$, $4 \operatorname{arctg} \frac{2}{3}$, and $4 \operatorname{arctg} \frac{2}{3}$ respectively (the apex angle of a cone is the angl... | 40/21 | 0 | 8,121.5 | -1 | 8,121.5 | |
Given an arithmetic sequence $\{a_n\}$ with a non-zero common difference, the sum of the first $n$ terms is $S_n$, satisfying $S_3 = a_4 + 4$, and $a_2$, $a_6$, $a_{18}$ form a geometric sequence.
1. Find the general formula for the sequence $\{a_n\}$.
2. Let $b_n = \frac{a_n}{2^n}$, find the sum of the first $n$ terms... | \frac{1}{4} | 0.3125 | 7,537.625 | 6,098 | 8,192 | |
If the square roots of a positive number are $x+1$ and $4-2x$, then the positive number is ______. | 36 | 0.8125 | 2,220.0625 | 1,767 | 4,183.333333 | |
Right triangle $ABC$ has side lengths $BC=6$, $AC=8$, and $AB=10$. A circle centered at $O$ is tangent to line $BC$ at $B$ and passes through $A$. A circle centered at $P$ is tangent to line $AC$ at $A$ and passes through $B$. What is $OP$? | \frac{35}{12} | 1. **Identify the Geometry of the Problem:**
- Triangle $ABC$ is a right triangle with $AB$ as the hypotenuse, so $AB = 10$, $BC = 6$, and $AC = 8$.
- Circle centered at $O$ is tangent to $BC$ at $B$ and passes through $A$.
- Circle centered at $P$ is tangent to $AC$ at $A$ and passes through $B$.
2. **Proper... | 0.875 | 3,659 | 3,742.642857 | 3,073.5 |
The National High School Mathematics Competition is set up as follows: the competition is divided into the first round and the second round. The first round includes 8 fill-in-the-blank questions (each worth 8 points) and 3 problem-solving questions (worth 16, 20, and 20 points respectively), with a total score of 120 ... | \frac{1}{2} | 0.125 | 8,119.75 | 8,192 | 8,109.428571 | |
Natascha cycles 3 times as fast as she runs. She spends 4 hours cycling and 1 hour running. What is the ratio of the distance that she cycles to the distance that she runs? | 12:1 | Suppose that Natascha runs at $r \mathrm{~km} / \mathrm{h}$. Since she cycles 3 times as fast as she runs, she cycles at $3 r \mathrm{~km} / \mathrm{h}$. In 1 hour of running, Natascha runs $(1 \mathrm{~h}) \cdot(r \mathrm{~km} / \mathrm{h})=r \mathrm{~km}$. In 4 hours of cycling, Natascha cycles $(4 \mathrm{~h}) \cdot... | 0.375 | 1,961.625 | 3,333.5 | 1,138.5 |
Tetrahedron $PQRS$ is such that $PQ=6$, $PR=5$, $PS=4\sqrt{2}$, $QR=3\sqrt{2}$, $QS=5$, and $RS=4$. Calculate the volume of tetrahedron $PQRS$.
**A)** $\frac{130}{9}$
**B)** $\frac{135}{9}$
**C)** $\frac{140}{9}$
**D)** $\frac{145}{9}$ | \frac{140}{9} | 0 | 8,192 | -1 | 8,192 | |
Given the set \( A = \{x \mid (x-2)(x-6) \geqslant 3, x \in \mathbf{Z}, 0 \leqslant x \leq 7\} \), find the number of non-empty subsets of \( A \). | 63 | 0 | 3,540.6875 | -1 | 3,540.6875 | |
The difference of the roots of $x^2-7x-9=0$ is: | \sqrt{85} | 1. **Identify the coefficients of the quadratic equation**: Given the quadratic equation $x^2 - 7x - 9 = 0$, we can identify the coefficients as $a = 1$, $b = -7$, and $c = -9$.
2. **Apply Vieta's formulas**: Vieta's formulas tell us that for a quadratic equation $ax^2 + bx + c = 0$, the sum of the roots $r_1 + r_2 = ... | 1 | 2,012.875 | 2,012.875 | -1 |
Consider the infinite series $1 - \frac{1}{3} - \frac{1}{9} + \frac{1}{27} - \frac{1}{81} - \frac{1}{243} + \frac{1}{729} - \frac{1}{2187} - \cdots$. Evaluate the sum of this series. | \frac{15}{26} | 0.25 | 7,949.125 | 7,220.5 | 8,192 | |
A big $L$ is formed as shown. What is its area? | 22 | 1. **Identify the dimensions of the big rectangle**: The problem states that the shape is a big "L" formed within a rectangle. Assume the dimensions of the big rectangle are $8$ units by $5$ units based on the typical dimensions used in such problems.
2. **Calculate the area of the big rectangle**:
\[
\text{Are... | 0 | 5,385.5 | -1 | 5,385.5 |
A parking lot has 16 spaces in a row. Twelve cars arrive, each of which requires one parking space, and their drivers chose spaces at random from among the available spaces. Auntie Em then arrives in her SUV, which requires 2 adjacent spaces. What is the probability that she is able to park? | \frac{17}{28} | To solve this problem, we need to calculate the probability that there are at least two adjacent empty spaces among the 16 spaces after 12 cars have parked. Each car occupies one space, and they choose their spaces randomly.
#### Step 1: Total ways to park 12 cars in 16 spaces
The total number of ways to park 12 cars ... | 0.9375 | 4,882.4375 | 4,661.8 | 8,192 |
Given that a website publishes pictures of five celebrities along with five photos of the celebrities when they were teenagers, but only three of the teen photos are correctly labeled, and two are unlabeled, determine the probability that a visitor guessing at random will match both celebrities with the correct unlabel... | \frac{1}{20} | 0 | 7,138.6875 | -1 | 7,138.6875 | |
No two students in Mrs. Vale's 26-student mathematics class have the same two initials. Each student's first name and last name begin with the same letter. If the letter ``Y'' is considered a vowel, what is the probability of randomly picking a student whose initials are vowels? Express your answer as a common fraction... | \frac{3}{13} | 0.8125 | 4,323.6875 | 3,665.461538 | 7,176 | |
Find the smallest solution to the equation \[\frac{1}{x-3} + \frac{1}{x-5} = \frac{5}{x-4}.\] | 4 - \frac{\sqrt{15}}{3} | 0 | 4,492.0625 | -1 | 4,492.0625 | |
When the mean, median, and mode of the list
\[10,2,5,2,4,2,x\]
are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of $x$? | 20 | 1. **Calculate the Mean**:
The mean of the list $10, 2, 5, 2, 4, 2, x$ is calculated as follows:
\[
\text{Mean} = \frac{10 + 2 + 5 + 2 + 4 + 2 + x}{7} = \frac{25 + x}{7}.
\]
2. **Determine the Mode**:
The mode is the number that appears most frequently in the list. Here, the number $2$ appears three t... | 0.5 | 6,844.75 | 6,200.25 | 7,489.25 |
Let $n$ be a fixed integer, $n \geqslant 2$.
1. Determine the smallest constant $c$ such that the inequality
$$
\sum_{1 \leqslant i<j \leqslant n} x_i x_j (x_i^2 + x_j^2) \leqslant c \left(\sum_{i=1}^n x_i \right)^4
$$
holds for all non-negative real numbers $x_1, x_2, \cdots, x_n$.
2. For this constant $c$, determine ... | \frac{1}{8} | 0 | 8,192 | -1 | 8,192 | |
Roger the ant is traveling on a coordinate plane, starting at $(0,0)$. Every second, he moves from one lattice point to a different lattice point at distance 1, chosen with equal probability. He will continue to move until he reaches some point $P$ for which he could have reached $P$ more quickly had he taken a differe... | 1103 | Roger is guaranteed to be able to take at least one step. Suppose he takes that step in a direction $u$. Let $e_{1}$ be the expectation of the number of additional steps Roger will be able to take after that first move. Notice that Roger is again guaranteed to be able to make a move, and that three types of steps are p... | 0 | 8,192 | -1 | 8,192 |
Given the line \( y = x - 1 \) intersects the ellipse \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{a^{2} - 1} = 1 \) (where \( a > 1 \)) at points \( A \) and \( B \). If the circle with diameter \( AB \) passes through the left focus of the ellipse, find the value of \( a \). | \frac{\sqrt{6} + \sqrt{2}}{2} | 0 | 7,963.9375 | -1 | 7,963.9375 | |
The increasing sequence $1, 3, 4, 9, 10, 12, 13, \dots$ consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Find the $150^{\text{th}}$ term of this sequence. | 2280 | 0.5 | 6,362.875 | 4,533.75 | 8,192 | |
Given a circle $x^2 + (y-1)^2 = 1$ with its tangent line $l$, which intersects the positive x-axis at point A and the positive y-axis at point B. Determine the y-intercept of the tangent line $l$ when the distance AB is minimized. | \frac{3+\sqrt{5}}{2} | 0 | 7,019.4375 | -1 | 7,019.4375 | |
Two congruent squares, $ABCD$ and $JKLM$, each have side lengths of 12 units. Square $JKLM$ is placed such that its center coincides with vertex $C$ of square $ABCD$. Determine the area of the region covered by these two squares in the plane. | 216 | 0.0625 | 7,997.75 | 7,914 | 8,003.333333 | |
A 40-foot by 10-foot rectangular garden is enclosed by a fence. To make the garden larger, while using the same amount of fencing, its shape is changed to a square. How many square feet larger than the old garden is the new garden? | 225 | 1 | 1,329.375 | 1,329.375 | -1 | |
In $\triangle ABC$, point $M$ lies inside the triangle such that $\angle MBA = 30^\circ$ and $\angle MAB = 10^\circ$. Given that $\angle ACB = 80^\circ$ and $AC = BC$, find $\angle AMC$. | 70 | 0.3125 | 7,198 | 5,011.2 | 8,192 | |
Let $ABCDEF$ be a regular hexagon. Let $G$, $H$, $I$, $J$, $K$, and $L$ be the midpoints of sides $AB$, $BC$, $CD$, $DE$, $EF$, and $AF$, respectively. The segments $\overline{AH}$, $\overline{BI}$, $\overline{CJ}$, $\overline{DK}$, $\overline{EL}$, and $\overline{FG}$ bound a smaller regular hexagon. Let the ratio of ... | 11 | 0.125 | 8,146.875 | 7,831 | 8,192 | |
An inverted cone with base radius $12 \mathrm{cm}$ and height $18 \mathrm{cm}$ is full of water. The water is poured into a tall cylinder whose horizontal base has radius of $24 \mathrm{cm}$. What is the height in centimeters of the water in the cylinder? | 1.5 | 1. **Calculate the volume of the water in the cone**:
The formula for the volume $V$ of a cone is given by:
\[
V = \frac{1}{3} \pi r^2 h
\]
where $r$ is the radius of the base and $h$ is the height of the cone. For the given cone, $r = 12 \text{ cm}$ and $h = 18 \text{ cm}$. Plugging in these values, w... | 0.25 | 798.3125 | 880 | 771.083333 |
Find the integer $n,$ $-90 \le n \le 90,$ such that $\sin n^\circ = \cos 456^\circ.$ | -6 | 1 | 2,927 | 2,927 | -1 | |
Given that $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, $a_6$, $a_7$ are distinct positive integers whose sum equals 159, find the maximum value of the smallest number $a_1$. | 19 | 0.375 | 7,343.5625 | 6,835.833333 | 7,648.2 | |
Given numbers \(a, b, c\) satisfy \(a b c+a+c-b\). Then the maximum value of the algebraic expression \(\frac{1}{1+a^{2}}-\frac{1}{1+b^{2}}+\frac{1}{1+c^{2}}\) is | \frac{5}{4} | 0 | 8,192 | -1 | 8,192 | |
How many integer quadruples $a,b,c,d$ are there such that $7$ divides $ab-cd$ where $0\leq a,b,c,d < 7$ ? | 385 | 0.5 | 6,332.6875 | 5,116.5 | 7,548.875 | |
Solve for t: $3 \cdot 3^t + \sqrt{9 \cdot 9^t} = 18$. | 1 | 1 | 2,395 | 2,395 | -1 | |
A line segment starts at $(2, 4)$ and ends at the point $(7, y)$ with $y > 0$. The segment is 6 units long. Find the value of $y$. | 4 + \sqrt{11} | 0 | 3,926.125 | -1 | 3,926.125 | |
Every second, the computer displays a number equal to the sum of the digits of the previous number multiplied by 31. On the first second, the number 2020 was displayed. What number will be displayed on the screen on the 2020th second? | 310 | 0.9375 | 5,282.5625 | 5,088.6 | 8,192 | |
Three years ago, you invested some money at $12\%$ interest. You now have $\$504.32$ in the account. If the interest was compounded yearly, how much did you invest 3 years ago? | 359 | 0 | 7,858.875 | -1 | 7,858.875 | |
Compute
\[
\sum_{n=1}^{\infty} \frac{4n + 1}{(4n - 1)^3 (4n + 3)^3}.
\] | \frac{1}{432} | 0 | 8,192 | -1 | 8,192 | |
Compute $\sqrt{(31)(30)(29)(28)+1}$. | 869 | Note that the four numbers to multiply are symmetric with the center at $29.5$. Multiply the symmetric pairs to get $31\cdot 28=868$ and $30\cdot 29=870$. $\sqrt{868\cdot 870 + 1} = \sqrt{(869-1)(869+1) + 1} = \sqrt{869^2 - 1^2 + 1} = \sqrt{869^2} = \boxed{869}$. | 0.9375 | 2,924.0625 | 2,716 | 6,045 |
What is $1254_6 - 432_6 + 221_6$? Express your answer in base $6$. | 1043_6 | 0.4375 | 6,662.5 | 4,696 | 8,192 | |
Given that the leftmost position can be occupied by student A or B, and the rightmost position cannot be occupied by student A, find the number of different arrangements of the six high-performing students from Class 1, Grade 12. | 216 | 0.375 | 7,256.0625 | 6,222 | 7,876.5 | |
We placed 6 different dominoes in a closed chain on the table. The total number of points on the dominoes is $D$. What is the smallest possible value of $D$? (The number of points on each side of the dominoes ranges from 0 to 6, and the number of points must be the same on touching sides of the dominoes.) | 12 | 0 | 8,182.875 | -1 | 8,182.875 | |
The integer $m$ is the largest positive multiple of $18$ such that every digit of $m$ is either $9$ or $0$. Compute $\frac{m}{18}$. | 555 | 0 | 7,970.625 | -1 | 7,970.625 | |
How many sequences of $0$s and $1$s of length $19$ are there that begin with a $0$, end with a $0$, contain no two consecutive $0$s, and contain no three consecutive $1$s? | 65 | To solve this problem, we need to determine the number of valid sequences of length 19 that satisfy the given conditions. We define $f(n)$ as the number of valid sequences of length $n$ that meet the criteria.
#### Step 1: Understanding the Problem
The sequences must:
- Start and end with a $0$.
- Contain no two conse... | 0 | 8,192 | -1 | 8,192 |
In a class of 50 students, it is decided to use systematic sampling to select 10 students out of these 50. The students are randomly assigned numbers from 1 to 50 and grouped, with the first group being numbers 1 to 5, the second group 6 to 10, and so on, up to the tenth group which is 46 to 50. If a student with the n... | 37 | 0.5625 | 5,861.4375 | 4,834.333333 | 7,182 | |
Given the function $f(x) = 2\sin\omega x \cdot \cos(\omega x) + (\omega > 0)$ has the smallest positive period of $4\pi$.
(1) Find the value of the positive real number $\omega$;
(2) In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively, and it satisfies $2b\cos A = a... | \frac{\sqrt{3}}{2} | 0 | 2,896.875 | -1 | 2,896.875 | |
The difference between two numbers is 9, and the sum of the squares of each number is 153. What is the value of the product of the two numbers? | 36 | 1 | 1,837.9375 | 1,837.9375 | -1 | |
The altitude of an equilateral triangle is $\sqrt6$ units. What is the area of the triangle, in square units? Express your answer in simplest radical form. | 2\sqrt{3} | 1 | 2,022.4375 | 2,022.4375 | -1 | |
Chinese mathematician Hua Luogeng saw a brain teaser in a magazine that the passenger next to him was reading while on a trip abroad: find the cube root of $59319$. Hua Luogeng blurted out the answer, astonishing everyone. They quickly asked about the calculation's mystery. Do you know how he calculated the result quic... | 58 | 0.75 | 5,922.875 | 5,166.5 | 8,192 | |
In a cube with edge length 1, two cross-sections in the form of regular hexagons are made. Find the length of the segment at which these cross-sections intersect. | \sqrt{2} | 0 | 8,074 | -1 | 8,074 | |
Positive integers \( d, e, \) and \( f \) are chosen such that \( d < e < f \), and the system of equations
\[ 2x + y = 2010 \quad \text{and} \quad y = |x-d| + |x-e| + |x-f| \]
has exactly one solution. What is the minimum value of \( f \)? | 1006 | 0.0625 | 8,192 | 8,192 | 8,192 | |
You are in a completely dark room with a drawer containing 10 red, 20 blue, 30 green, and 40 khaki socks. What is the smallest number of socks you must randomly pull out in order to be sure of having at least one of each color? | 91 | 0.375 | 5,243.9375 | 4,361.833333 | 5,773.2 | |
For each positive integer $n$, there is a circle around the origin with radius $n$. Rainbow Dash starts off somewhere on the plane, but not on a circle. She takes off in some direction in a straight path. She moves \frac{\sqrt{5}}{5}$ units before crossing a circle, then \sqrt{5}$ units, then \frac{3 \sqrt{5}}{5}$ unit... | \frac{2 \sqrt{170}-9 \sqrt{5}}{5} | Note that the distance from Rainbow Dash's starting point to the first place in which she hits a circle is irrelevant, except in checking that this distance is small enough that she does not hit another circle beforehand. It will be clear at the end that our configuration does not allow this (by the Triangle Inequality... | 0 | 8,192 | -1 | 8,192 |
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