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 |
|---|---|---|---|---|---|---|
A cube with an edge length of 6 is cut into smaller cubes with integer edge lengths. If the total surface area of these smaller cubes is \(\frac{10}{3}\) times the surface area of the original larger cube before cutting, how many of these smaller cubes have an edge length of 1? | 56 | 0 | 8,083.5625 | -1 | 8,083.5625 | |
Given that \(a, b, c, d\) are within the interval \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\), and \(\sin a + \sin b + \sin c + \sin d = 1\), and \(\cos 2a + \cos 2b + \cos 2c + \cos 2d \geq \frac{10}{3}\), what is the maximum value of \(a\)? \(\quad\). | \frac{\pi}{6} | 0.4375 | 7,880 | 7,478.857143 | 8,192 | |
A cylinder has a radius of 2 inches and a height of 3 inches. What is the radius of a sphere that has the same volume as this cylinder? | \sqrt[3]{9} | 1 | 1,948.1875 | 1,948.1875 | -1 | |
Applied to a bill for $\$10,000$ the difference between a discount of $40\%$ and two successive discounts of $36\%$ and $4\%$, expressed in dollars, is: | 144 | 1. **Calculate the single discount of 40% on $\textdollar{10,000}$:**
- A 40% discount means the remaining amount to be paid is 100% - 40% = 60% of the original price.
- Calculation:
\[
10,000 \times 0.6 = \textdollar{6,000}
\]
2. **Calculate the successive discounts of 36% and 4% on $\textdollar... | 0.9375 | 3,453.875 | 3,138 | 8,192 |
In △ABC, the sides opposite to angles A, B, C are a, b, c respectively. If acosB - bcosA = $$\frac {c}{3}$$, then the minimum value of $$\frac {acosA + bcosB}{acosB}$$ is \_\_\_\_\_\_. | \sqrt {2} | 0 | 6,348.1875 | -1 | 6,348.1875 | |
Calculate:<br/>$(1)\frac{\sqrt{20}+\sqrt{5}}{\sqrt{5}}-2$;<br/>$(2)\sqrt[3]{-8}+5\sqrt{\frac{1}{10}}-\sqrt{10}+\sqrt{4}$;<br/>$(3)(\sqrt{3}-\sqrt{2})^2•(5+2\sqrt{6})$;<br/>$(4)(π-3.14)^0+\frac{1}{2+\sqrt{3}}+(-\frac{1}{3})^{-1}+|1-\sqrt{3}|$. | -1 | 0.8125 | 3,214.25 | 3,244.769231 | 3,082 | |
In square $ABCD$, points $P$ and $Q$ lie on $\overline{AD}$ and $\overline{AB}$, respectively. Segments $\overline{BP}$ and $\overline{CQ}$ intersect at right angles at $R$, with $BR = 6$ and $PR = 7$. What is the area of the square? | 117 | 1. **Identify Similar Triangles**: Notice that $\triangle CRB \sim \triangle BAP$ by AA similarity (both have a right angle and share angle $BRP$).
2. **Set Up Ratio of Sides**: From the similarity, we have the ratio of corresponding sides:
\[
\frac{CB}{CR} = \frac{PB}{AB}
\]
Since $CB = AB = s$ (side leng... | 0.5625 | 7,673.0625 | 7,269.444444 | 8,192 |
In the Cartesian coordinate plane, the area of the region formed by the points \((x, y)\) that satisfy \( |x| + |y| + |x - 2| \leqslant 4 \) is ______. | 12 | 0.1875 | 7,935.875 | 6,826 | 8,192 | |
Triangle $ABC$ is a right isosceles triangle. Points $D$, $E$ and $F$ are the midpoints of the sides of the triangle. Point $G$ is the midpoint of segment $DF$ and point $H$ is the midpoint of segment $FE$. What is the ratio of the shaded area to the non-shaded area in triangle $ABC$? Express your answer as a common fr... | \frac{5}{11} | 0.1875 | 7,616.3125 | 6,737.333333 | 7,819.153846 | |
A string has 150 beads of red, blue, and green colors. It is known that among any six consecutive beads, there is at least one green bead, and among any eleven consecutive beads, there is at least one blue bead. What is the maximum number of red beads that can be on the string? | 112 | 0 | 8,033.625 | -1 | 8,033.625 | |
Find the number of real solutions to the equation
\[
\frac{1}{x - 1} + \frac{2}{x - 2} + \frac{3}{x - 3} + \dots + \frac{50}{x - 50} = x + 5.
\] | 51 | 0.25 | 7,460.1875 | 7,371.25 | 7,489.833333 | |
In $\triangle ABC$, $AB = 10$, $BC = 6$, $CA = 8$, and side $AB$ is extended to a point $P$ such that $\triangle PCB$ is similar to $\triangle CAB$. Find the length of $PC$.
[asy]
defaultpen(linewidth(0.7)+fontsize(10));
pair A=origin, P=(1.5,5), B=(10,0), C=P+2.5*dir(P--B);
draw(A--P--C--A--B--C);
label("A", A, W);
la... | 4.8 | 0 | 7,751.4375 | -1 | 7,751.4375 | |
Express the quotient and remainder of $3232_5 \div 21_5$ in base $5$. | 130_5 \, R2_5 | 0 | 6,149.625 | -1 | 6,149.625 | |
On a beach $50$ people are wearing sunglasses and $35$ people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is also wearing sunglasses is $\frac{2}{5}$. If instead, someone wearing sunglasses is selected at ... | \frac{7}{25} | 1. **Identify the number of people wearing both sunglasses and caps**: Given that the probability of a person wearing a cap also wearing sunglasses is $\frac{2}{5}$, and there are $35$ people wearing caps, we calculate the number of people wearing both as follows:
\[
\text{Number of people wearing both} = \frac{2... | 1 | 1,415.75 | 1,415.75 | -1 |
If $\det \mathbf{A} = 5,$ then find $\det (\mathbf{A^3}).$ | 125 | 1 | 1,551.75 | 1,551.75 | -1 | |
Ali Baba and the 40 thieves are dividing their loot. The division is considered fair if any 30 participants receive at least half of the loot in total. What is the maximum share that Ali Baba can receive in a fair division? | \frac{1}{3} | 0 | 8,087.1875 | -1 | 8,087.1875 | |
During the flower exhibition in Zhengzhou Green Expo Garden, 6 volunteers are arranged to provide services in 4 exhibition areas. It is required that areas A and B each have one person, and the remaining two areas each have two people. Among them, Little Li and Little Wang cannot be together. Determine the total number... | 156 | 0.375 | 7,412.1875 | 6,673.5 | 7,855.4 | |
How many positive integers have cube roots that are less than 20? | 7999 | 1 | 2,599.25 | 2,599.25 | -1 | |
Calculate the value of the following expression and find angle $\theta$ if the number can be expressed as $r e^{i \theta}$, where $0 \le \theta < 2\pi$:
\[ e^{11\pi i/60} + e^{21\pi i/60} + e^{31 \pi i/60} + e^{41\pi i /60} + e^{51 \pi i /60} \] | \frac{31\pi}{60} | 0 | 8,192 | -1 | 8,192 | |
Let $k$ be a positive integer. Marco and Vera play a game on an infinite grid of square cells. At the beginning, only one cell is black and the rest are white.
A turn in this game consists of the following. Marco moves first, and for every move he must choose a cell which is black and which has more than two white ... | 203 | 0 | 8,176.5 | -1 | 8,176.5 | |
Select 4 out of the numbers $1, 2, 3, 4, 5$ to form a four-digit number. What is the average value of these four-digit numbers? | 3333 | 0.1875 | 7,439.4375 | 4,178.333333 | 8,192 | |
In triangle \( \triangle ABC \), the sides opposite to the angles \( \angle A \), \( \angle B \), and \( \angle C \) are denoted as \( a \), \( b \), and \( c \) respectively. If \( b^{2}=a^{2}+c^{2}-ac \), and \( c-a \) is equal to the height \( h \) from vertex \( A \) to side \( AC \), then find \( \sin \frac{C-A}{2... | \frac{1}{2} | 0.125 | 7,976.375 | 6,467 | 8,192 | |
In equiangular octagon $CAROLINE$, $CA = RO = LI = NE =$ $\sqrt{2}$ and $AR = OL = IN = EC = 1$. The self-intersecting octagon $CORNELIA$ encloses six non-overlapping triangular regions. Let $K$ be the area enclosed by $CORNELIA$, that is, the total area of the six triangular regions. Then $K =$ $\dfrac{a}{b}$, where $... | 23 | We can draw $CORNELIA$ and introduce some points.
The diagram is essentially a 3x3 grid where each of the 9 squares making up the grid have a side length of 1.
In order to find the area of $CORNELIA$, we need to find 4 times the area of $\bigtriangleup$$ACY$ and 2 times the area of $\bigtriangleup$$YZW$.
Using simil... | 0 | 8,192 | -1 | 8,192 |
For each positive integer $n$, let $S(n)$ denote the sum of the digits of $n$. For how many values of $n$ is $n+S(n)+S(S(n))=2007$? | 4 | 0.125 | 7,511.625 | 5,376 | 7,816.714286 | |
Using $1 \times 2$ tiles to cover a $2 \times 10$ grid, how many different ways are there to cover the grid? | 89 | 0.5625 | 5,716.1875 | 4,239.777778 | 7,614.428571 | |
Without using a calculator, find the largest prime factor of \( 17^4 + 2 \times 17^2 + 1 - 16^4 \). | 17 | 0.9375 | 3,654.8125 | 3,471.6 | 6,403 | |
Point P_{1} is located 600 miles West of point P_{2}. At 7:00 AM a car departs from P_{1} and drives East at a speed of 50 miles per hour. At 8:00 AM another car departs from P_{2} and drives West at a constant speed of x miles per hour. If the cars meet each other exactly halfway between P_{1} and P_{2}, what is the v... | 60 | Each car meets having traveled 300 miles. Therefore the first car traveled for 300 / 50=6 hours, and so the second car traveled for 5 hours. The second car must have traveled 300 / 5=60 miles per hour. | 1 | 2,493.0625 | 2,493.0625 | -1 |
When arranging the performance order of 6 singers, requiring that both singers B and C are either before or after singer A, calculate the total number of different arrangements. | 480 | 0.0625 | 7,739.875 | 7,757 | 7,738.733333 | |
Find one third of 7.2, expressed as a simplified fraction or a mixed number. | 2 \frac{2}{5} | 0.625 | 449.5625 | 446.6 | 454.5 | |
Write the process of using the Horner's algorithm to find the value of the function $\_(f)\_()=1+\_x+0.5x^2+0.16667x^3+0.04167x^4+0.00833x^5$ at $x=-0.2$. | 0.81873 | 0.125 | 6,441 | 5,028 | 6,642.857143 | |
Given that Sia and Kira count sequentially, where Sia skips every fifth number, find the 45th number said in this modified counting game. | 54 | 0 | 6,930.6875 | -1 | 6,930.6875 | |
Given that the sequence $\{a\_n\}$ satisfies $\frac{1}{a_{n+1}} - \frac{1}{a_n} = d (n \in \mathbb{N}^*, d$ is a constant), the sequence $\{\frac{1}{b_n}\}$ is a harmonic sequence, and $b_1 + b_2 + b_3 + ... + b_9 = 90$, find the value of $b_4 + b_6$. | 20 | 0.625 | 5,297.625 | 4,142.5 | 7,222.833333 | |
Let \( x \) and \( y \) be real numbers with \( x > y \) such that \( x^{2} y^{2} + x^{2} + y^{2} + 2xy = 40 \) and \( xy + x + y = 8 \). Find the value of \( x \). | 3 + \sqrt{7} | 1 | 3,146.25 | 3,146.25 | -1 | |
What is the smallest four-digit positive integer which has four different digits? | 1023 | 1 | 4,353 | 4,353 | -1 | |
If $\frac{1}{(2n-1)(2n+1)}=\frac{a}{2n-1}+\frac{b}{2n+1}$ holds for any natural number $n$, then $a=$______, $b=______. | -\frac{1}{2} | 0.9375 | 2,171.125 | 2,258.266667 | 864 | |
Two cars, Car A and Car B, travel towards each other from cities A and B, which are 330 kilometers apart. Car A starts from city A first. After some time, Car B starts from city B. The speed of Car A is $\frac{5}{6}$ of the speed of Car B. When the two cars meet, Car A has traveled 30 kilometers more than Car B. Determ... | 55 | 0.4375 | 4,991.4375 | 4,045.714286 | 5,727 | |
Given $f(x) = -4x^2 + 4ax - 4a - a^2$ has a maximum value of $-5$ in the interval $[0, 1]$, find the value of $a$. | -5 | 0.0625 | 7,974.625 | 8,192 | 7,960.133333 | |
In a diagram, $\triangle ABC$ and $\triangle BDC$ are right-angled, with $\angle ABC = \angle BDC = 45^\circ$, and $AB = 16$. Determine the length of $BC$. | 8\sqrt{2} | 0.1875 | 7,177.4375 | 7,087.666667 | 7,198.153846 | |
The Hoopers, coached by Coach Loud, have 15 players. George and Alex are the two players who refuse to play together in the same lineup. Additionally, if George plays, another player named Sam refuses to play. How many starting lineups of 6 players can Coach Loud create, provided the lineup does not include both George... | 3795 | 0.1875 | 6,950 | 3,486 | 7,749.384615 | |
What is the product of the solutions of the equation $-35=-x^2-2x?$ | -35 | 1 | 1,645 | 1,645 | -1 | |
What integer $n$ satisfies $0\le n<{101}$ and $$100n\equiv 72\pmod {101}~?$$ | 29 | 1 | 2,577.0625 | 2,577.0625 | -1 | |
A number like 45132 is called a "wave number," which means the tens and thousands digits are both larger than their respective neighboring digits. What is the probability of forming a non-repeating five-digit "wave number" using the digits 1, 2, 3, 4, 5? | \frac{1}{15} | 0 | 8,192 | -1 | 8,192 | |
Flat Albert and his buddy Mike are watching the game on Sunday afternoon. Albert is drinking lemonade from a two-dimensional cup which is an isosceles triangle whose height and base measure 9 cm and 6 cm; the opening of the cup corresponds to the base, which points upwards. Every minute after the game begins, the follo... | 26 | Let $A_{0}=\frac{1}{2}(6)(9)=27$ denote the area of Albert's cup; since area varies as the square of length, at time $n$ Mike adds $$A\left(1-\left(1-\frac{1}{9 n^{2}}\right)^{2}\right)$$ whence in all, he adds $$A_{0} \sum_{n=1}^{\infty}\left(\frac{2}{9 n^{2}}-\frac{1}{81 n^{4}}\right)=\frac{2 A_{0} \zeta(2)}{9}-\frac... | 0.3125 | 5,996.875 | 4,734.8 | 6,570.545455 |
In the geometric sequence $\{a_n\}$, it is given that $a_1 + a_4 + a_7 = 2$, and $a_3 + a_6 + a_9 = 18$. Find the sum of the first 9 terms, $S_9$, of the sequence $\{a_n\}$. | 26 | 0.125 | 7,803.0625 | 5,080.5 | 8,192 | |
In the expression $10 \square 10 \square 10 \square 10 \square 10$, fill in the four spaces with each of the operators "+", "-", "×", and "÷" exactly once. The maximum possible value of the resulting expression is: | 109 | 0 | 8,086.5 | -1 | 8,086.5 | |
For a given positive integer $n > 2^3$, what is the greatest common divisor of $n^3 + 3^2$ and $n + 2$? | 1 | 0.9375 | 3,254 | 2,924.8 | 8,192 | |
Find the maximum number of white dominoes that can be cut from the board shown on the left. A domino is a $1 \times 2$ rectangle. | 16 | 0.3125 | 5,924.6875 | 4,361.2 | 6,635.363636 | |
Given the set $A=\{m+2, 2m^2+m\}$, if $3 \in A$, then the value of $m$ is \_\_\_\_\_\_. | -\frac{3}{2} | 0.375 | 5,635.4375 | 6,783.666667 | 4,946.5 | |
An ant starts at the origin, facing in the positive $x$-direction. Each second, it moves 1 unit forward, then turns counterclockwise by $\sin ^{-1}\left(\frac{3}{5}\right)$ degrees. What is the least upper bound on the distance between the ant and the origin? (The least upper bound is the smallest real number $r$ that ... | \sqrt{10} | We claim that the points the ant visits lie on a circle of radius $\frac{\sqrt{10}}{2}$. We show this by saying that the ant stays a constant distance $\frac{\sqrt{10}}{2}$ from the point $\left(\frac{1}{2}, \frac{3}{2}\right)$. Suppose the ant moves on a plane $P$. Consider a transformation of the plane $P^{\prime}$ s... | 0.6875 | 6,227.1875 | 5,390 | 8,069 |
Find the expected value of the number formed by rolling a fair 6-sided die with faces numbered 1, 2, 3, 5, 7, 9 infinitely many times. | \frac{1}{2} | Let $X_{n}$ be the $n$th number rolled. The number formed, $0 . \overline{X_{1} X_{2}} \cdots$, is simply $\sum_{n=1}^{\infty} \frac{X_{n}}{10^{n}}$. By linearity of expectation, the expected value is $\sum_{n=1}^{\infty} \mathbb{E}\left(\frac{X_{n}}{10^{n}}\right)=\sum_{n=1}^{\infty} \frac{\mathbb{E}\left(X_{n}\right)... | 0.1875 | 5,809.3125 | 5,670.333333 | 5,841.384615 |
John scores 93 on this year's AHSME. Had the old scoring system still been in effect, he would score only 84 for the same answers.
How many questions does he leave unanswered? | 9 | Let $c$, $w$, and $u$ be the number of correct, wrong, and unanswered questions respectively. We are given three pieces of information:
1. Under the old scoring system, John's score is $84$. The old scoring system awards $30$ points initially, $4$ points for each correct answer, subtracts $1$ point for each wrong answ... | 0.125 | 3,591.6875 | 1,314.5 | 3,917 |
A string of digits is defined to be similar to another string of digits if it can be obtained by reversing some contiguous substring of the original string. For example, the strings 101 and 110 are similar, but the strings 3443 and 4334 are not. (Note that a string is always similar to itself.) Consider the string of d... | 1126 | We first count the number of substrings that one could pick to reverse to yield a new substring. If we insert two dividers into the sequence of 50 digits, each arrangement of 2 dividers among the 52 total objects specifies a substring that is contained between the two dividers, for a total of $\binom{52}{2}$ substrings... | 0 | 8,149.75 | -1 | 8,149.75 |
Find, with proof, the least integer $N$ such that if any $2016$ elements are removed from the set $\{1, 2,...,N\}$ , one can still find $2016$ distinct numbers among the remaining elements with sum $N$ . | \boxed{6097392} | Since any $2016$ elements are removed, suppose we remove the integers from $1$ to $2016$ . Then the smallest possible sum of $2016$ of the remaining elements is \[2017+2018+\cdots + 4032 = 1008 \cdot 6049 = 6097392\] so clearly $N\ge 6097392$ . We will show that $N=6097392$ works.
$\vspace{0.2 in}$
$\{1,2\cdots 609739... | 0 | 8,192 | -1 | 8,192 |
Three real numbers \( x, y, \) and \( z \) are such that \( (x+4) / 2=(y+9) /(z-3)=(x+5) /(z-5) \). Determine the value of \( x / y \). | 1/2 | Because the first and third fractions are equal, adding their numerators and denominators produces another fraction equal to the others: \( ((x+4)+(x+5)) /(2+(z-5))=(2 x+9) /(z-3) \). Then \( y+9=2 x+9 \), etc. | 1 | 4,469.3125 | 4,469.3125 | -1 |
Sterling draws 6 circles on the plane, which divide the plane into regions (including the unbounded region). What is the maximum number of resulting regions? | 22 | 0 | 5,514.375 | -1 | 5,514.375 | |
Given positive numbers $x$ and $y$ satisfying $x^2+y^2=1$, find the maximum value of $\frac {1}{x}+ \frac {1}{y}$. | 2\sqrt{2} | 0.125 | 7,845.375 | 5,419 | 8,192 | |
Given that $\theta$ is an angle in the third quadrant, and $sin^{4}\theta+cos^{4}\theta= \frac {5}{9}$, then $sin2\theta= \_\_\_\_\_\_$. | \frac {2 \sqrt {2}}{3} | 0 | 6,142.8125 | -1 | 6,142.8125 | |
Given triangle $\triangle ABC$, $A=120^{\circ}$, $D$ is a point on side $BC$, $AD\bot AC$, and $AD=2$. Calculate the possible area of $\triangle ABC$. | \frac{8\sqrt{3}}{3} | 0 | 8,192 | -1 | 8,192 | |
How many multiples of 5 are between 100 and 400? | 60 | 0.0625 | 5,090.375 | 563 | 5,392.2 | |
Given lines $l_{1}$: $\rho\sin(\theta-\frac{\pi}{3})=\sqrt{3}$ and $l_{2}$: $\begin{cases} x=-t \\ y=\sqrt{3}t \end{cases}$ (where $t$ is a parameter), find the polar coordinates of the intersection point $P$ of $l_{1}$ and $l_{2}$. Additionally, three points $A$, $B$, and $C$ lie on the ellipse $\frac{x^{2}}{4}+y^{2}=... | \frac{15}{8} | 0.125 | 7,784.375 | 4,931 | 8,192 | |
A square has sides of length 10, and a circle centered at one of its vertices has radius 10. What is the area of the union of the regions enclosed by the square and the circle? Express your answer in terms of $\pi$. | 100+75\pi | 1 | 3,452.4375 | 3,452.4375 | -1 | |
What is the base five product of the numbers $121_{5}$ and $11_{5}$? | 1331 | 0 | 3,741.0625 | -1 | 3,741.0625 | |
Let $a + 1 = b + 2 = c + 3 = d + 4 = a + b + c + d + 5$. What is $a + b + c + d$? | -\frac{10}{3} | 1. Let $x = a + 1 = b + 2 = c + 3 = d + 4 = a + b + c + d + 5$. This equation implies that each of $a+1$, $b+2$, $c+3$, and $d+4$ is equal to $x$, and also the sum $a+b+c+d+5$ equals $x$.
2. Express $a$, $b$, $c$, and $d$ in terms of $x$:
- $a = x - 1$
- $b = x - 2$
- $c = x - 3$
- $d = x - 4$
3. Substitu... | 0.9375 | 4,164.875 | 3,896.4 | 8,192 |
In $\triangle ABC$, \(a+c=2b\). Find \(\tan \frac{A}{2} \cdot \tan \frac{C}{2}\). | \frac{1}{3} | 0.75 | 6,134.6875 | 5,448.916667 | 8,192 | |
How many positive integer divisors of $3003^{3003}$ are divisible by exactly 3003 positive integers? | 24 | 0.75 | 6,165.875 | 5,519.916667 | 8,103.75 | |
Square \( ABCD \) has a side length of 12 inches. A segment \( AE \) is drawn where \( E \) is on side \( DC \) and \( DE \) is 5 inches long. The perpendicular bisector of \( AE \) intersects \( AE, AD, \) and \( BC \) at points \( M, P, \) and \( Q \) respectively. The ratio of the segments \( PM \) to \( MQ \) is: | 5:19 | 0 | 5,751.125 | -1 | 5,751.125 | |
Bob is trying to decide between two cell phone plans. Plan A has no flat fee, but the user must pay $10$ cents per minute on the phone. Plan B requires a one-time fee of $\$20$, but only requires a payment of $5$ cents per minute on the phone. What is the minimum whole number of minutes Bob has to use the phone for to ... | 401 | 1 | 1,760.1875 | 1,760.1875 | -1 | |
A box contains $2$ pennies, $4$ nickels, and $6$ dimes. Six coins are drawn without replacement, with each coin having an equal probability of being chosen. What is the probability that the value of coins drawn is at least $50$ cents? | \frac{127}{924} | To solve this problem, we need to calculate the probability that the value of the coins drawn is at least 50 cents. We will first determine the total number of ways to draw 6 coins from the 12 available, and then find the number of successful outcomes where the total value is at least 50 cents.
1. **Total Outcomes**:
... | 0.1875 | 7,523.25 | 7,226.666667 | 7,591.692308 |
Given that the regular price for one backpack is $60, and Maria receives a 20% discount on the second backpack and a 30% discount on the third backpack, calculate the percentage of the $180 regular price she saved. | 16.67\% | 0.625 | 616.9375 | 602 | 641.833333 | |
Paco uses a spinner to select a number from 1 through 5, each with equal probability. Manu uses a different spinner to select a number from 1 through 10, each with equal probability. What is the probability that the product of Manu's number and Paco's number is less than 30? Express your answer as a common fraction. | \frac{41}{50} | 0.875 | 4,226.625 | 3,660.142857 | 8,192 | |
One of the factors of $x^4+4$ is: | $x^2-2x+2$ | To solve the problem, we need to verify if any of the given options is a factor of the polynomial $x^4 + 4$. We will test each option by either direct substitution or polynomial division.
#### Option (A) $x^2 + 2$
We can quickly check if $x^2 + 2$ is a factor of $x^4 + 4$ by attempting to factorize $x^4 + 4$ using $x^... | 0 | 3,633.875 | -1 | 3,633.875 |
Find $1273 + 120 \div 60 - 173$. | 1102 | 1 | 833.625 | 833.625 | -1 | |
In $\triangle ABC$, $\angle B=60^{\circ}$, $AC=2\sqrt{3}$, $BC=4$, then the area of $\triangle ABC$ is $\_\_\_\_\_\_$. | 2\sqrt{3} | 0.875 | 4,499.375 | 3,971.857143 | 8,192 | |
In the trapezoid \(ABCD \) with \( AD \parallel BC \), the angle \( \angle ADB \) is twice the angle \( \angle ACB \). It is known that \( BC = AC = 5 \) and \( AD = 6 \). Find the area of the trapezoid. | 22 | 0 | 8,192 | -1 | 8,192 | |
A city uses a lottery system for assigning car permits, with 300,000 people participating in the lottery and 30,000 permits available each month.
1. If those who win the lottery each month exit the lottery, and those who do not win continue in the following month's lottery, with an additional 30,000 new participants ad... | 10 | 0.125 | 7,591.125 | 6,837 | 7,698.857143 | |
Given an arithmetic sequence $\{a_{n}\}$ and $\{b_{n}\}$, where the sums of the first $n$ terms are $S_{n}$ and $T_{n}$, respectively, and $\left(2n+3\right)S_{n}=nT_{n}$, calculate the value of $\frac{{{a_5}}}{{{b_6}}}$. | \frac{9}{25} | 0.8125 | 3,810.6875 | 3,479.307692 | 5,246.666667 | |
Line segments \( AB \) and \( CD \) are situated between two parallel planes \( \alpha \) and \( \beta \). \( AC \subset \alpha \) and \( BD \subset \beta \). Given \( AB \perp \alpha \), \( AC = BD = 5 \), \( AB = 12 \), and \( CD = 13 \). Points \( E \) and \( F \) divide \( AB \) and \( CD \) in the ratio \( 1:2 \) ... | \frac{5}{3} \sqrt{7} | 0 | 7,496.125 | -1 | 7,496.125 | |
The vectors $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ satisfy $\|\mathbf{a}\| = \|\mathbf{b}\| = 1,$ $\|\mathbf{c}\| = 2,$ and
\[\mathbf{a} \times (\mathbf{a} \times \mathbf{c}) + \mathbf{b} = \mathbf{0}.\]If $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{c},$ then find all possible values of $\theta,$ in d... | 150^\circ | 0.5625 | 4,836.3125 | 4,005.333333 | 5,904.714286 | |
For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$? | 19 | To solve this problem, we need to consider the possible number of intersection points formed by four distinct lines in a plane. Each pair of lines can intersect at most once, and the maximum number of intersection points is determined by the number of ways to choose 2 lines from 4, which is given by the binomial coeffi... | 0.5 | 7,295.875 | 7,206.25 | 7,385.5 |
The real sequence \( x_0, x_1, x_2, \ldots \) is defined by \( x_0 = 1 \), \( x_1 = 2 \), and \( n(n+1) x_{n+1} = n(n-1) x_n - (n-2) x_{n-1} \). Find \( \frac{x_0}{x_1} + x_1 x_2 + \cdots + \frac{x_{50}}{x_{51}} \). | 1326 | 0 | 8,192 | -1 | 8,192 | |
If $p, q,$ and $r$ are three non-zero integers such that $p + q + r = 30$ and \[\frac{1}{p} + \frac{1}{q} + \frac{1}{r} + \frac{240}{pqr} = 1,\] compute $pqr$. | 1080 | 0 | 8,192 | -1 | 8,192 | |
Convert the binary number $111011_{(2)}$ to a decimal number. | 1 \times 2^{5} + 1 \times 2^{4} + 1 \times 2^{3} + 0 \times 2^{2} + 1 \times 2^{1} + 1 | 0 | 2,487.625 | -1 | 2,487.625 | |
We call a positive integer $N$ [i]contagious[/i] if there are $1000$ consecutive non-negative integers such that the sum of all their digits is $N$. Find all contagious positive integers. | \{13500, 13501, 13502, \ldots\} |
To determine which positive integers \( N \) are contagious, we consider 1000 consecutive non-negative integers and the sum of all their digits equating to \( N \).
Let the consecutive integers be \( x, x+1, x+2, \ldots, x+999 \). We need to calculate the sum of the digits of these 1000 numbers.
Let's start by consi... | 0 | 8,192 | -1 | 8,192 |
In a certain exam, students' math scores follow a normal distribution $N(100,100)$. It is known that there are 1000 students taking this exam. Then the number of students whose math scores are between 70 and 110 points is approximately ____.
(Reference data: $P(\mu -\sigma < X < \mu +\sigma )\approx 0.6827, P(\mu -3\s... | 840 | 0.4375 | 6,896.1875 | 5,883.857143 | 7,683.555556 | |
What is $2\frac{1}{4}$ divided by $\frac{2}{3}$? | \frac{27}{8} | 0.4375 | 2,923.75 | 2,460.857143 | 3,283.777778 | |
Construct a new sequence $\{a_n\}$ by extracting every 2nd, 4th, 8th, ..., $2^n$th, ... term from the sequence $\{3n+\log_2 n\}$ in the original order. Find the general term $a_n$ for the new sequence $\{a_n\}$ and calculate the sum of the first five terms, $S_5$. | 201 | 1 | 3,753.0625 | 3,753.0625 | -1 | |
A quadrilateral has vertices $P(a,b)$, $Q(b,a)$, $R(-a, -b)$, and $S(-b, -a)$, where $a$ and $b$ are integers with $a>b>0$. The area of $PQRS$ is $16$. What is $a+b$? | 4 | 1. **Identify the slopes of the sides to determine the shape of the quadrilateral**:
- The slope of line segment $PQ$ is calculated as:
\[
\text{slope of } PQ = \frac{Q_y - P_y}{Q_x - P_x} = \frac{a - b}{b - a} = -1
\]
- The slope of line segment $PS$ is calculated as:
\[
\text{slope of }... | 0.9375 | 3,207.8125 | 2,875.533333 | 8,192 |
In triangle \( \triangle ABC \), if \( \overrightarrow{AB} \cdot \overrightarrow{AC} = 7 \) and \( |\overrightarrow{AB} - \overrightarrow{AC}| = 6 \), find the maximum possible area of \( \triangle ABC \). | 12 | 0.875 | 4,777.3125 | 4,420.5 | 7,275 | |
The postal department stipulates that for letters weighing up to $100$ grams (including $100$ grams), each $20$ grams requires a postage stamp of $0.8$ yuan. If the weight is less than $20$ grams, it is rounded up to $20$ grams. For weights exceeding $100$ grams, the initial postage is $4$ yuan. For each additional $10... | 5.6 | 0 | 7,985.4375 | -1 | 7,985.4375 | |
Given that the function f(x) (x ∈ R) satisfies f(x + π) = f(x) + sin(x), and f(x) = 0 when 0 ≤ x ≤ π. Find f(23π/6). | \frac{1}{2} | 0.3125 | 7,779.9375 | 6,966.2 | 8,149.818182 | |
Given $\|\mathbf{v}\| = 4,$ find $\|-3 \mathbf{v}\|.$ | 12 | 1 | 1,380.125 | 1,380.125 | -1 | |
Compute $\sum_{n=1}^{\infty} \sum_{k=1}^{n-1} \frac{k}{2^{n+k}}$. | \frac{4}{9} | We change the order of summation: $\sum_{n=1}^{\infty} \sum_{k=1}^{n-1} \frac{k}{2^{n+k}}=\sum_{k=1}^{\infty} \frac{k}{2^{k}} \sum_{n=k+1}^{\infty} \frac{1}{2^{n}}=\sum_{k=1}^{\infty} \frac{k}{4^{k}}=\frac{4}{9}$. (The last two steps involve the summation of an infinite geometric series, and what is sometimes called an... | 0.375 | 6,674.25 | 4,144.666667 | 8,192 |
The circle is divided by points \(A\), \(B\), \(C\), and \(D\) such that \(AB: BC: CD: DA = 3: 2: 13: 7\). Chords \(AD\) and \(BC\) are extended to intersect at point \(M\).
Find the angle \( \angle AMB \). | 72 | 0.3125 | 6,669.25 | 4,380.2 | 7,709.727273 | |
Given a rectangular grid constructed with toothpicks of equal length, with a height of 15 toothpicks and a width of 12 toothpicks, calculate the total number of toothpicks required to build the grid. | 387 | 0.5 | 3,556.25 | 2,755.375 | 4,357.125 | |
Given that the random variable $x$ follows a normal distribution $N(3, \sigma^2)$, and $P(x \leq 4) = 0.84$, find $P(2 < x < 4)$. | 0.68 | 0.375 | 4,743.3125 | 5,756.333333 | 4,135.5 | |
A quadrilateral is inscribed in a circle of radius $250$. Three sides of this quadrilateral have lengths of $250$, $250$, and $100$ respectively. What is the length of the fourth side? | 200 | 0 | 8,177.9375 | -1 | 8,177.9375 | |
In the triangular pyramid $P-ABC$, $PA\bot $ plane $ABC$, $\triangle ABC$ is an isosceles triangle, where $AB=BC=2$, $\angle ABC=120{}^\circ $, and $PA=4$. The surface area of the circumscribed sphere of the triangular pyramid $P-ABC$ is __________. | 32\pi | 0 | 8,192 | -1 | 8,192 | |
Determine the volume of the original cube given that one dimension is increased by $3$, another is decreased by $2$, and the third is left unchanged, and the volume of the resulting rectangular solid is $6$ more than that of the original cube. | (3 + \sqrt{15})^3 | 0 | 5,613.4375 | -1 | 5,613.4375 | |
A list of $2023$ positive integers has a unique mode, which occurs exactly $11$ times. Determine the least number of distinct values that can occur in the list. | 203 | 0.6875 | 6,706.375 | 6,031.090909 | 8,192 | |
A square is constructed on one side of a regular octagon, outward. In the octagon, two diagonals intersect at point \( B \) (see the drawing). Find the measure of angle \( ABC \).
(A polygon is called regular if all its sides are equal and all its angles are equal.) | 22.5 | 0 | 8,001.25 | -1 | 8,001.25 |
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