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 |
|---|---|---|---|---|---|---|
Solve the equations
​
(1) $x^{2}-4x+3=0$
​
(2) $(x+1)(x-2)=4$
​
(3) $3x(x-1)=2-2x$
​
(4) $2x^{2}-4x-1=0$ | \frac{2- \sqrt{6}}{2} | 0 | 3,052.375 | -1 | 3,052.375 | |
Find the number of square units in the area of the shaded region. Consider a rectangular plot with vertices at points (0,0), (40,0), (40,20), and (0,20). Within this rectangle, a shaded polygon is formed with vertices at (0,0), (20,0), (40,10), (40,20), and (10,20). Determine the area of this shaded region. | 600 | 0.625 | 6,382.125 | 6,024.9 | 6,977.5 | |
Find the real solution \( x, y, z \) to the equations \( x + y + z = 5 \) and \( xy + yz + zx = 3 \) such that \( z \) is the largest possible value. | \frac{13}{3} | 0.5 | 3,955.0625 | 3,468.25 | 4,441.875 | |
What is the smallest five-digit positive integer which is congruent to 7 (mod 13)? | 10,004 | 0 | 2,529.125 | -1 | 2,529.125 | |
The adult human body has 206 bones. Each foot has 26 bones. Approximately what fraction of the number of bones in the human body is found in one foot? | $\frac{1}{8}$ | 0 | 408.75 | -1 | 408.75 | |
Keisha's basketball team must decide on a new uniform. The seventh-graders will pick the color of the shorts (black, gold, or red) and the eighth-graders will pick the color of the jersey (black, white, gold, or blue), and each group will not confer with the other. Additionally, the ninth-graders will choose whether to... | \frac{5}{6} | 0.4375 | 4,633.6875 | 4,077.857143 | 5,066 | |
The sphere with radius 1 and center $(0,0,1)$ rests on the $xy$-plane. A light source is at $P = (0,-1,2).$ Then the boundary of the shadow of the sphere can be expressed in the form $y = f(x),$ for some function $f(x).$ Find the function $f(x).$ | \frac{x^2}{4} - 1 | 0.0625 | 8,097.125 | 7,087 | 8,164.466667 | |
Find all \( a_{0} \in \mathbb{R} \) such that the sequence defined by
\[ a_{n+1} = 2^{n} - 3a_{n}, \quad n = 0, 1, 2, \cdots \]
is increasing. | \frac{1}{5} | 0.125 | 7,952.875 | 7,816.5 | 7,972.357143 | |
In the unit cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, points $E, F, G$ are the midpoints of edges $A A_{1}, C_{1} D_{1}$, and $D_{1} A_{1}$, respectively. Find the distance from point $B_{1}$ to the plane $E F G$. | \frac{\sqrt{3}}{2} | 0 | 5,262.4375 | -1 | 5,262.4375 | |
At a math competition, a team of $8$ students has $2$ hours to solve $30$ problems. If each problem needs to be solved by $2$ students, on average how many minutes can a student spend on a problem? | 16 | 0.0625 | 691.0625 | 523 | 702.266667 | |
The average of the five numbers in a list is 54. The average of the first two numbers is 48. What is the average of the last three numbers? | 58 | 1 | 1,059.9375 | 1,059.9375 | -1 | |
Given the numbers $1, 2, \cdots, 20$, calculate the probability that three randomly selected numbers form an arithmetic sequence. | \frac{1}{38} | 0 | 6,915.75 | -1 | 6,915.75 | |
Given that the sequence $\{a_n\}$ is an arithmetic sequence, and $a_2=-1$, the sequence $\{b_n\}$ satisfies $b_n-b_{n-1}=a_n$ ($n\geqslant 2, n\in \mathbb{N}$), and $b_1=b_3=1$
(I) Find the value of $a_1$;
(II) Find the general formula for the sequence $\{b_n\}$. | -3 | 0 | 4,120.5 | -1 | 4,120.5 | |
In a WeChat group, five people, namely A, B, C, D, and E, are playing a game of grabbing red packets. There are $4$ red packets in total, each person can grab at most one, and all red packets will be grabbed. Among the $4$ red packets, there are two $2$ yuan packets, one $3$ yuan packet, and one $4$ yuan packet. (Assum... | 36 | 0.0625 | 8,007.375 | 8,164 | 7,996.933333 | |
On a sunny day, 3000 people, including children, boarded a cruise ship. Two-fifths of the people were women, and a third were men. If 25% of the women and 15% of the men were wearing sunglasses, and there were also 180 children on board with 10% wearing sunglasses, how many people in total were wearing sunglasses? | 530 | 0.3125 | 1,066.0625 | 634.2 | 1,262.363636 | |
When a polynomial is divided by $-3x^5 + 10x - 11,$ what are the possible degrees of the remainder? Enter all the possible values, separated by commas. | 0,1,2,3,4 | 0.1875 | 1,616.5 | 2,101.666667 | 1,504.538462 | |
Let $\Gamma$ denote the circumcircle of triangle $A B C$. Point $D$ is on $\overline{A B}$ such that $\overline{C D}$ bisects $\angle A C B$. Points $P$ and $Q$ are on $\Gamma$ such that $\overline{P Q}$ passes through $D$ and is perpendicular to $\overline{C D}$. Compute $P Q$, given that $B C=20, C A=80, A B=65$. | 4 \sqrt{745} | Suppose that $P$ lies between $A$ and $B$ and $Q$ lies between $A$ and $C$, and let line $P Q$ intersect lines $A C$ and $B C$ at $E$ and $F$ respectively. As usual, we write $a, b, c$ for the lengths of $B C, C A, A B$. By the angle bisector theorem, $A D / D B=A C / C B$ so that $A D=\frac{b c}{a+b}$ and $B D=\frac{a... | 0 | 8,192 | -1 | 8,192 |
A paper equilateral triangle $ABC$ has side length 12. The paper triangle is folded so that vertex $A$ touches a point on side $\overline{BC}$ a distance 9 from point $B$. Find the square of the length of the line segment along which the triangle is folded.
[asy]
import cse5;
size(12cm);
pen tpen = defaultpen + 1.337;... | \frac{59319}{1225} | 0 | 8,192 | -1 | 8,192 | |
Given $\alpha \in (0, \pi)$, if $\sin \alpha + \cos \alpha = \frac{\sqrt{3}}{3}$, calculate the value of $\cos^2 \alpha - \sin^2 \alpha$. | -\frac{\sqrt{5}}{3} | 0 | 6,236.875 | -1 | 6,236.875 | |
A 12-hour digital clock has a glitch such that whenever it is supposed to display a 5, it mistakenly displays a 7. For example, when it is 5:15 PM the clock incorrectly shows 7:77 PM. What fraction of the day will the clock show the correct time? | \frac{33}{40} | 0.0625 | 6,339.5 | 6,017 | 6,361 | |
Calculate:<br/>$(1)(1\frac{3}{4}-\frac{3}{8}+\frac{5}{6})÷(-\frac{1}{24})$;<br/>$(2)-2^2+(-4)÷2×\frac{1}{2}+|-3|$. | -2 | 1 | 1,150.75 | 1,150.75 | -1 | |
In a regular tetrahedron the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to that of the larger is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
| 28 | 0.375 | 7,848.75 | 7,276.666667 | 8,192 | |
Given an ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, there is a point $M(2,1)$ inside it. Two lines $l_1$ and $l_2$ passing through $M$ intersect the ellipse $E$ at points $A$, $C$ and $B$, $D$ respectively, and satisfy $\overrightarrow{AM}=\lambda \overrightarrow{MC}, \overrightarrow{BM}=\lambda \o... | \frac{\sqrt{3}}{2} | 0 | 8,164.125 | -1 | 8,164.125 | |
The integer solution for the inequality $|2x-m|\leq1$ with respect to $x$ is uniquely $3$ ($m$ is an integer).
(I) Find the value of the integer $m$;
(II) Given that $a, b, c \in R$, if $4a^4+4b^4+4c^4=m$, find the maximum value of $a^2+b^2+c^2$. | \frac{3\sqrt{2}}{2} | 0 | 4,521.25 | -1 | 4,521.25 | |
Find the probability \( P(B^c | A^c) \) given the probabilities:
\[ P(A \cap B) = 0.72, \quad P(A \cap \bar{B}) = 0.18 \] | 0.90 | 0 | 8,017.75 | -1 | 8,017.75 | |
Evaluate $\left|{-4+\frac{7}{6}i}\right|$. | \frac{25}{6} | 1 | 1,519.125 | 1,519.125 | -1 | |
Express $3.\overline{7}$ as a common fraction. | \frac{34}{9} | 1 | 1,755.4375 | 1,755.4375 | -1 | |
If \\(\alpha \in \left( 0, \frac{\pi}{2} \right)\\), and \\(\cos \left( \frac{\pi}{4} - \alpha \right) = 2 \sqrt{2} \cos 2\alpha\\), then \\(\sin 2\alpha = \)_______. | \frac{15}{16} | 0.5625 | 6,365.0625 | 4,944.111111 | 8,192 | |
The whole numbers from 1 to 2016 inclusive are written on a blackboard. Moritz underlines all the multiples of two in red, all the multiples of three in blue, and all the multiples of four in green. How many numbers does Moritz underline exactly twice? | 504 | 0.625 | 6,445.8125 | 5,398.1 | 8,192 | |
Numbers from 1 to 6 are written on the faces of a gaming die. However, the weight of the die is distributed unevenly, and the probability of landing on number $k$ is directly proportional to $k$. The die is rolled two times in a row. What is the probability that the sum of the rolled numbers will be 7? If necessary, ro... | 0.13 | 0.875 | 4,023.375 | 3,976.928571 | 4,348.5 | |
What is the probability that \(2^{n}\), where \(n\) is a randomly chosen positive integer, ends with the digit 2? What is the probability that \(2^{n}\) ends with the digits 12? | 0.05 | 0 | 6,322.375 | -1 | 6,322.375 | |
Let $a<b<c$ be three integers such that $a,b,c$ is an arithmetic progression and $a,c,b$ is a geometric progression. What is the smallest possible value of $c$? | 2 | 1 | 4,812.875 | 4,812.875 | -1 | |
Given that the interior angles \(A, B, C\) of triangle \(\triangle ABC\) are opposite to the sides \(a, b, c\) respectively, and that \(A - C = \frac{\pi}{2}\), and \(a, b, c\) form an arithmetic sequence, find the value of \(\cos B\). | \frac{3}{4} | 0.3125 | 7,374.9375 | 5,749 | 8,114 | |
At the height \(BH\) of triangle \(ABC\), a certain point \(D\) is marked. Line \(AD\) intersects side \(BC\) at point \(E\), and line \(CD\) intersects side \(AB\) at point \(F\). Points \(G\) and \(J\) are the projections of points \(F\) and \(E\) respectively onto side \(AC\). The area of triangle \(H E J\) is twice... | \sqrt{2}: 1 | 0 | 8,192 | -1 | 8,192 | |
In \(\triangle ABC\), \(AB : AC = 4 : 3\) and \(M\) is the midpoint of \(BC\). \(E\) is a point on \(AB\) and \(F\) is a point on \(AC\) such that \(AE : AF = 2 : 1\). It is also given that \(EF\) and \(AM\) intersect at \(G\) with \(GF = 72 \mathrm{~cm}\) and \(GE = x \mathrm{~cm}\). Find the value of \(x\). | 108 | 0.9375 | 4,752.4375 | 4,523.133333 | 8,192 | |
In triangle $ABC$, $BC = 40$ and $\angle C = 45^\circ$. Let the perpendicular bisector of $BC$ intersect $BC$ at $D$ and extend to meet an extension of $AB$ at $E$. Find the length of $DE$. | 20 | 0 | 7,974.25 | -1 | 7,974.25 | |
A triangular pyramid \( S-ABC \) has a base in the shape of an equilateral triangle with side lengths of 4. It is known that \( AS = BS = \sqrt{19} \) and \( CS = 3 \). Find the surface area of the circumscribed sphere of the triangular pyramid \( S-ABC \). | \frac{268\pi}{11} | 0.0625 | 7,780.5 | 7,989 | 7,766.6 | |
The radian measure of a 60° angle is $$\frac {\pi}{6}$$. | \frac{\pi}{3} | 0.75 | 4,663.4375 | 3,705.166667 | 7,538.25 | |
A boss and two engineers are to meet at a park. Each of them arrives at a random time between 1:00 PM and 3:00 PM. The boss leaves instantly if not both engineers are present upon his arrival. Each engineer will wait for up to 1.5 hours for the other before leaving. What is the probability that the meeting takes place? | \frac{1}{4} | 0 | 7,907.625 | -1 | 7,907.625 | |
Petya and Vasya were collecting mushrooms. It turned out that Petya collected as many mushrooms as the percentage of mushrooms Vasya collected from the total number of mushrooms they collected together. Additionally, Vasya collected an odd number of mushrooms. How many mushrooms did Petya and Vasya collect together? | 25 | 0 | 8,192 | -1 | 8,192 | |
Two chords \( PQ \) and \( PR \) are drawn in a circle with diameter \( PS \). The point \( T \) lies on \( PR \) and \( QT \) is perpendicular to \( PR \). The angle \(\angle QPR = 60^\circ\), \( PQ = 24 \text{ cm} \), and \( RT = 3 \text{ cm} \). What is the length of the chord \( QS \) in cm?
A) \( \sqrt{3} \)
B)... | 2\sqrt{3} | 0 | 8,082.8125 | -1 | 8,082.8125 | |
Find the largest prime divisor of $36^2 + 49^2$. | 3697 | 0.625 | 6,559.875 | 5,580.6 | 8,192 | |
A sealed bottle, which contains water, has been constructed by attaching a cylinder of radius \(1 \text{ cm}\) to a cylinder of radius \(3 \text{ cm}\). When the bottle is right side up, the height of the water inside is \(20 \text{ cm}\). When the bottle is upside down, the height of the liquid is \(28 \text{ cm}\). W... | 29 | 0 | 7,721.9375 | -1 | 7,721.9375 | |
A regular octagon is inscribed in a circle of radius 2 units. What is the area of the octagon? Express your answer in simplest radical form. | 16 \sqrt{2} - 8(2) | 0 | 4,733.4375 | -1 | 4,733.4375 | |
If $x$ is real and positive and grows beyond all bounds, then $\log_3{(6x-5)}-\log_3{(2x+1)}$ approaches: | 1 | 1. **Rewrite the expression using logarithmic properties**:
Given the expression $\log_3{(6x-5)}-\log_3{(2x+1)}$, we can use the logarithmic property that states $\log_b a - \log_b b = \log_b \left(\frac{a}{b}\right)$. Applying this property, we get:
\[
\log_3{(6x-5)}-\log_3{(2x+1)} = \log_3 \left(\frac{6x -... | 1 | 2,490.0625 | 2,490.0625 | -1 |
We say that two natural numbers form a perfect pair when the sum and the product of these two numbers are perfect squares. For example, 5 and 20 form a perfect pair because $5+20=25=5^{2}$ and $5 \times 20=100=10^{2}$. Does 122 form a perfect pair with any other natural number? | 122 \times 121 | 0 | 7,820.5625 | -1 | 7,820.5625 | |
Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$. Let $D$ be the midpoint of $\overline{AB}$, and let $E$ be the midpoint of $\overline{AC}$. The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$, respectively. What is the area of quadrilateral $FDBG$? | 75 | 1. **Calculate the area of $\triangle ADE$:**
Since $D$ and $E$ are midpoints of $AB$ and $AC$ respectively, $AD = \frac{1}{2}AB = 25$ and $AE = \frac{1}{2}AC = 5$. The area of $\triangle ABC$ is given as $120$. Using the formula for the area of a triangle, we have:
\[
[\triangle ABC] = \frac{1}{2} \cdot AB ... | 0.1875 | 8,160.25 | 8,022.666667 | 8,192 |
How many six-digit numbers are there in which only the middle two digits are the same? | 90000 | 0.8125 | 4,353.0625 | 3,661.384615 | 7,350.333333 | |
Given a positive integer $k$, let \|k\| denote the absolute difference between $k$ and the nearest perfect square. For example, \|13\|=3 since the nearest perfect square to 13 is 16. Compute the smallest positive integer $n$ such that $\frac{\|1\|+\|2\|+\cdots+\|n\|}{n}=100$. | 89800 | Note that from $n=m^{2}$ to $n=(m+1)^{2},\|n\|$ increases from 0 to a peak of $m$ (which is repeated twice), and then goes back down to 0. Therefore $\sum_{n=1}^{m^{2}}\|n\|=\sum_{k=1}^{m-1} 2(1+2+\cdots+k)=\sum_{k=1}^{m-1} 2\binom{k+1}{2}=2\binom{m+1}{3}=\frac{m}{3}\left(m^{2}-1\right)$. In particular, if $n=m^{2}-1$,... | 0 | 8,192 | -1 | 8,192 |
Let $a$, $b$, and $c$ be three positive real numbers such that $a(a+b+c)=bc$. Determine the maximum value of $\frac{a}{b+c}$. | \frac{\sqrt{2}-1}{2} | 0 | 6,087.9375 | -1 | 6,087.9375 | |
The area of the enclosed shape formed by the line $y=x-2$ and the curve $y^2=x$ can be calculated. | \frac{9}{2} | 0.0625 | 7,582.9375 | 4,797 | 7,768.666667 | |
Find the greatest common divisor of $8!$ and $(6!)^3.$ | 11520 | 0 | 5,333 | -1 | 5,333 | |
Given that the function \( f(x) \) satisfies the equation \( 2 f(x) + x^{2} f\left(\frac{1}{x}\right) = \frac{3 x^{3} - x^{2} + 4 x + 3}{x + 1} \) and \( g(x) = \frac{5}{x + 1} \), determine the minimum value of \( f(x) + g(x) \). | \frac{15}{4} | 0.1875 | 8,048 | 7,930 | 8,075.230769 | |
There are four distinct codes $A, B, C, D$ used by an intelligence station, with one code being used each week. Each week, a code is chosen randomly with equal probability from the three codes that were not used the previous week. Given that code $A$ is used in the first week, what is the probability that code $A$ is a... | 61/243 | 0.375 | 7,115.25 | 5,379.5 | 8,156.7 | |
The sum of two natural numbers and their greatest common divisor is equal to their least common multiple. Determine the ratio of the two numbers. | 3 : 2 | 0.0625 | 4,788.6875 | 8,192 | 4,561.8 | |
Given $\tan (\alpha+ \frac {π}{3})=2$, find the value of $\frac {\sin (\alpha+ \frac {4π}{3})+\cos ( \frac {2π}{3}-\alpha)}{\cos ( \frac {π}{6}-\alpha )-\sin (\alpha + \frac {5π}{6})}$. | -3 | 0.5625 | 6,669.3125 | 5,492 | 8,183 | |
Given $x^{2}+y^{2}=4$, find the minimum value of $\sqrt{2-y}+\sqrt{5-2x}$. | \sqrt{5} | 0 | 8,192 | -1 | 8,192 | |
Call a fraction $\frac{a}{b}$, not necessarily in the simplest form, special if $a$ and $b$ are positive integers whose sum is $15$. How many distinct integers can be written as the sum of two, not necessarily different, special fractions? | 11 | 1. **Identify Special Fractions**: A fraction $\frac{a}{b}$ is special if $a+b=15$ and $a, b$ are positive integers. The possible pairs $(a, b)$ are:
- $(1, 14), (2, 13), (3, 12), (4, 11), (5, 10), (6, 9), (7, 8), (8, 7), (9, 6), (10, 5), (11, 4), (12, 3), (13, 2), (14, 1)$.
2. **List Special Fractions and Simplify... | 0 | 8,134.625 | -1 | 8,134.625 |
Any seven points are taken inside or on a square with side length $2$. Determine $b$, the smallest possible number with the property that it is always possible to select one pair of points from these seven such that the distance between them is equal to or less than $b$. | \sqrt{2} | 0 | 8,192 | -1 | 8,192 | |
An object in the plane moves from the origin and takes a ten-step path, where at each step the object may move one unit to the right, one unit to the left, one unit up, or one unit down. How many different points could be the final point? | 221 | 0 | 8,192 | -1 | 8,192 | |
Let $(x_1,y_1),$ $(x_2,y_2),$ $\dots,$ $(x_n,y_n)$ be the solutions to
\begin{align*}
|x - 3| &= |y - 9|, \\
|x - 9| &= 2|y - 3|.
\end{align*}Find $x_1 + y_1 + x_2 + y_2 + \dots + x_n + y_n.$ | -4 | 0.8125 | 5,815.8125 | 5,267.461538 | 8,192 | |
Pat wrote a strange example on the board:
$$
550+460+359+340=2012 .
$$
Mat wanted to correct it, so he searched for an unknown number to add to each of the five numbers listed, so that the example would be numerically correct. What was that number?
Hint: How many numbers does Mat add to the left side and how many to... | 75.75 | 0 | 7,687.75 | -1 | 7,687.75 | |
How many distinct four-digit positive integers have a digit product equal to 18? | 48 | 0 | 8,192 | -1 | 8,192 | |
The teacher gave each of her $37$ students $36$ pencils in different colors. It turned out that each pair of students received exactly one pencil of the same color. Determine the smallest possible number of different colors of pencils distributed. | 666 | 0.3125 | 6,797.6875 | 4,693 | 7,754.363636 | |
Alex the Kat has written $61$ problems for a math contest, and there are a total of $187$ problems submitted. How many more problems does he need to write (and submit) before he has written half of the total problems? | 65 | 0.625 | 1,722.625 | 1,938.1 | 1,363.5 | |
Given two lines $l_{1}$: $mx+2y-2=0$ and $l_{2}$: $5x+(m+3)y-5=0$, if $l_{1}$ is parallel to $l_{2}$, calculate the value of $m$. | -5 | 0.3125 | 6,486.125 | 6,681 | 6,397.545455 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$. It is known that $c\sin A= \sqrt {3}a\cos C$.
(I) Find $C$;
(II) If $c= \sqrt {7}$ and $\sin C+\sin (B-A)=3\sin 2A$, find the area of $\triangle ABC$. | \frac {3 \sqrt {3}}{4} | 0 | 7,875.0625 | -1 | 7,875.0625 | |
Let $ABCD$ be a convex quadrilateral with $AB=AD, m\angle A = 40^{\circ}, m\angle C = 130^{\circ},$ and $m\angle ADC - m\angle ABC = 20^{\circ}.$ Find the measure of the non-reflex angle $\angle CDB$ in degrees. | 35 | 0.4375 | 6,965 | 5,387.428571 | 8,192 | |
The numbers $1,2, \ldots, 10$ are written in a circle. There are four people, and each person randomly selects five consecutive integers (e.g. $1,2,3,4,5$, or $8,9,10,1,2$). If the probability that there exists some number that was not selected by any of the four people is $p$, compute $10000p$. | 3690 | The unselected numbers must be consecutive. Suppose that $\{1,2, \ldots, k\}$ are the unselected numbers for some $k$. In this case, 1 cannot be selected, so there are 5 possible sets of consecutive numbers the people could have chosen. This leads to $5^{4}$ possibilities. Moreover, 10 must be selected, so we must subt... | 0 | 8,192 | -1 | 8,192 |
In pentagon $ABCDE$, $BC=CD=DE=2$ units, $\angle E$ is a right angle and $m \angle B = m \angle C = m \angle D = 135^\circ$. The length of segment $AE$ can be expressed in simplest radical form as $a+2\sqrt{b}$ units. What is the value of $a+b$? | 6 | 0.0625 | 7,969.8125 | 4,637 | 8,192 | |
Suppose there are initially 1001 townspeople and two goons. What is the probability that, when the game ends, there are exactly 1000 people in jail? | \frac{3}{1003} | By considering the parity of the number of people in jail, we see that this situation arises if and only if the goons win after the 500th night. That means that at this point we must have exactly one townsperson and two goons remaining. In other words, this situation arises if and only if no goon is ever sent to jail. ... | 0 | 7,954.75 | -1 | 7,954.75 |
Let $n \geq 5$ be an integer. Find the largest integer $k$ (as a function of $n$ ) such that there exists a convex $n$ -gon $A_{1}A_{2}\dots A_{n}$ for which exactly $k$ of the quadrilaterals $A_{i}A_{i+1}A_{i+2}A_{i+3}$ have an inscribed circle. (Here $A_{n+j} = A_{j}$ .) | \[
k = \left\lfloor \frac{n}{2} \right\rfloor
\] | Lemma: If quadrilaterals $A_iA_{i+1}A_{i+2}A_{i+3}$ and $A_{i+2}A_{i+3}A_{i+4}A_{i+5}$ in an equiangular $n$ -gon are tangential, and $A_iA_{i+3}$ is the longest side quadrilateral $A_iA_{i+1}A_{i+2}A_{i+3}$ for all $i$ , then quadrilateral $A_{i+1}A_{i+2}A_{i+3}A_{i+4}$ is not tangential.
Proof:
[asy] import geometry;... | 0 | 8,075.125 | -1 | 8,075.125 |
A rectangular cow pasture is enclosed on three sides by a fence and the fourth side is part of the side of a barn that is $400$ feet long. The fence costs $\$5$ per foot, and $\$1,\!200$ altogether. To the nearest foot, find the length of the side parallel to the barn that will maximize the area of the pasture. | 120 | 0.9375 | 4,293.3125 | 4,033.4 | 8,192 | |
Given the two-variable function
$$
f(a, b)=\max _{x \in[-1,1]}\left\{\left|x^{2}-a x-b\right|\right\},
$$
find the minimum value of \( f(a, b) \). | \frac{1}{2} | 0.375 | 7,719.125 | 6,931 | 8,192 | |
A polynomial with integer coefficients is of the form
\[8x^4 + b_3 x^3 + b_2 x^2 + b_1 x + 24 = 0.\]Find the number of different possible rational roots for this polynomial. | 28 | 0.0625 | 7,045.3125 | 7,248 | 7,031.8 | |
Let $n$ be a positive integer. Determine the size of the largest subset of $\{ - n, - n + 1, \ldots , n - 1, n\}$ which does not contain three elements $a, b, c$ (not necessarily distinct) satisfying $a + b + c = 0$ . | \[
\left\lceil \frac{n}{2} \right\rceil
\] | Let $S$ be a subset of $\{-n,-n+1,\dots,n-1,n\}$ of largest size satisfying $a+b+c\neq 0$ for all $a,b,c\in S$ . First, observe that $0\notin S$ . Next note that $|S|\geq \lceil n/2\rceil$ , by observing that the set of all the odd numbers in $\{-n,-n+1,\dots,n-1,n\}$ works. To prove that $|S|\leq \lceil n/2\rceil$ , i... | 0 | 8,192 | -1 | 8,192 |
Let $a_{0}, a_{1}, a_{2}, \ldots$ be an infinite sequence where each term is independently and uniformly random in the set $\{1,2,3,4\}$. Define an infinite sequence $b_{0}, b_{1}, b_{2}, \ldots$ recursively by $b_{0}=1$ and $b_{i+1}=a_{i}^{b_{i}}$. Compute the expected value of the smallest positive integer $k$ such t... | \frac{35}{16} | Do casework on what $a_{0}$ is. If $a_{0}=1$ then $k=1$. If $a_{0}=4$ then $k=2$. If $a_{0}=3$ then - if $a_{1}=1$, then $k=2$ - if $a_{1}=2$ or 4 , then $k=3$ - if $a_{1}=3$, then you make no progress. so in expectation it requires $E=(2+3+(E+1)+3) / 4 \Longrightarrow E=3$. If $a_{0}=2$ then - if $a_{1}=1$ or 4 , then... | 0 | 8,128.75 | -1 | 8,128.75 |
Isosceles trapezoid $ABCD$ has side lengths $AB = 6$ and $CD = 12$ , while $AD = BC$ . It is given that $O$ , the circumcenter of $ABCD$ , lies in the interior of the trapezoid. The extensions of lines $AD$ and $BC$ intersect at $T$ . Given that $OT = 18$ , the area of $ABCD$ can be expressed as $a + b... | 84 | 0.8125 | 6,465 | 6,066.461538 | 8,192 | |
The area of a circle is \( 64\pi \) square units. Calculate both the radius and the circumference of the circle. | 16\pi | 1 | 1,022.4375 | 1,022.4375 | -1 | |
Four circles of radius 1 are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle? Express your answer as a common fraction in simplest radical form. | 1 + \sqrt{2} | 0.5 | 7,415 | 6,638 | 8,192 | |
Luke wants to fence a square piece of land of at least 400 square feet. What should the side of the square be if he wants to use the least amount of fencing? | 20 | 1 | 997.6875 | 997.6875 | -1 | |
Find the sum of all the roots of the equation \( 4x^{2} - 58x + 190 = (29 - 4x - \log_{2} x) \cdot \log_{2} x \). | 12 | 0.4375 | 7,500.75 | 6,612 | 8,192 | |
Three people, A, B, and C, visit three tourist spots, with each person visiting only one spot. Let event $A$ be "the three people visit different spots," and event $B$ be "person A visits a spot alone." Then, the probability $P(A|B)=$ ______. | \dfrac{1}{2} | 0.125 | 6,180.9375 | 5,902.5 | 6,220.714286 | |
Let \[f(x) = \left\{
\begin{array}{cl}
\sqrt{x} &\text{ if }x>4, \\
x^2 &\text{ if }x \le 4.
\end{array}
\right.\]Find $f(f(f(2)))$. | 4 | 1 | 1,838.5625 | 1,838.5625 | -1 | |
According to national regulations, only adults aged between 18 and 70 are eligible to apply for a motor vehicle driver's license. A sixth-grade student, Li Ming, says, "My dad has a driver's license. His age equals the product of the month and day of his birth, and that product is 2975." How old is Li Ming's father? | 35 | 0 | 7,252.1875 | -1 | 7,252.1875 | |
Given Professor Chen has ten different language books, including two Arabic, four German, and four Spanish books, arranged on the shelf so that the Arabic books are together, the Spanish books are together, and two of the German books are treated as indistinguishable, calculate the number of ways to arrange the ten boo... | 576 | 0 | 7,278.1875 | -1 | 7,278.1875 | |
At the Lacsap Hospital, Emily is a doctor and Robert is a nurse. Not including Emily, there are five doctors and three nurses at the hospital. Not including Robert, there are $d$ doctors and $n$ nurses at the hospital. What is the product of $d$ and $n$? | 12 | Since Emily is a doctor and there are 5 doctors and 3 nurses aside from Emily at the hospital, then there are 6 doctors and 3 nurses in total. Since Robert is a nurse, then aside from Robert, there are 6 doctors and 2 nurses. Therefore, $d=6$ and $n=2$, so $dn=12$. | 0 | 2,031.125 | -1 | 2,031.125 |
If the graph of the power function $f(x)=x^{\alpha}$ ($\alpha$ is a constant) always passes through point $A$, and the line ${kx}{-}y{+}2k{+}1{+}\sqrt{3}{=}0$ always passes through point $B$, then the angle of inclination of line $AB$ is _____. | \frac{5\pi}{6} | 0.1875 | 5,152.9375 | 4,185.666667 | 5,376.153846 | |
Consider the curve $y=x^{n+1}$ (where $n$ is a positive integer) and its tangent at the point (1,1). Let the x-coordinate of the intersection point between this tangent and the x-axis be $x_n$.
(Ⅰ) Let $a_n = \log{x_n}$. Find the value of $a_1 + a_2 + \ldots + a_9$.
(Ⅱ) Define $nf(n) = x_n$. Determine whether there exi... | 11 | 0.4375 | 7,818.625 | 7,483 | 8,079.666667 | |
Given the equations
$$
z^{2}=4+4 \sqrt{15} i \text { and } z^{2}=2+2 \sqrt{3} i,
$$
the roots are the coordinates of the vertices of a parallelogram in the complex plane. If the area $S$ of the parallelogram can be expressed as $p \sqrt{q} - r \sqrt{s}$ (where $p, q, r, s \in \mathbf{Z}_{+}$, and $r$ and $s$ are not pe... | 20 | 0.0625 | 8,051.4375 | 5,943 | 8,192 | |
There are 35 groups of students, each group containing 3 students. Among these groups:
- 10 groups have only 1 boy.
- 19 groups have at least 2 boys.
- The number of groups with 3 boys is twice the number of groups with 3 girls.
How many boys are there? | 60 | 0.625 | 5,876.5625 | 4,487.3 | 8,192 | |
The probability of rain tomorrow is $\frac{3}{10}$. What is the probability that it will not rain tomorrow? Express your answer as a common fraction. | \frac{7}{10} | 1 | 1,210.5 | 1,210.5 | -1 | |
A spinner is divided into three equal sections labeled 0, 1, and 5. If you spin the spinner three times, use the outcomes to form a three-digit number (hundreds, tens, units), what is the probability that the resulting number is divisible by 5? | \frac{2}{3} | 0.625 | 4,886.0625 | 4,405.4 | 5,687.166667 | |
In the xy-plane, what is the length of the shortest path from $(0,0)$ to $(15,20)$ that does not go inside the circle $(x-7)^2 + (y-9)^2 = 36$?
A) $2\sqrt{94} + 3\pi$
B) $2\sqrt{130} + 3\pi$
C) $2\sqrt{94} + 6\pi$
D) $2\sqrt{130} + 6\pi$
E) $94 + 3\pi$ | 2\sqrt{94} + 3\pi | 0 | 8,192 | -1 | 8,192 | |
Find all triples $(a, b, c)$ of real numbers such that
$$ a^2 + ab + c = 0, $$
$$b^2 + bc + a = 0, $$
$$c^2 + ca + b = 0.$$ | (0, 0, 0)\left(-\frac{1}{2}, -\frac{1}{2}, -\frac{1}{2}\right) |
To find all triples \((a, b, c)\) of real numbers satisfying the system of equations:
\[
\begin{align*}
1. \quad & a^2 + ab + c = 0, \\
2. \quad & b^2 + bc + a = 0, \\
3. \quad & c^2 + ca + b = 0,
\end{align*}
\]
we proceed by considering possible symmetric solutions first and also evaluate the possibility of all v... | 0 | 8,192 | -1 | 8,192 |
Point $F$ is taken on the extension of side $AD$ of parallelogram $ABCD$. $BF$ intersects diagonal $AC$ at $E$ and side $DC$ at $G$. If $EF = 40$ and $GF = 30$, find the length of $BE$. | 20 | 0.6875 | 6,376.625 | 5,615.090909 | 8,052 | |
Twelve congruent rectangles are placed together to make a rectangle $PQRS$. What is the ratio $PQ: QR$?
A) $2: 3$
B) $3: 4$
C) $5: 6$
D) $7: 8$
E) $8: 9$ | 8 : 9 | 0 | 5,727.5 | -1 | 5,727.5 | |
A tetrahedron \( P-ABC \) has edge lengths \( PA = BC = \sqrt{6} \), \( PB = AC = \sqrt{8} \), and \( PC = AB = \sqrt{10} \). Find the radius of the circumsphere of this tetrahedron. | \sqrt{3} | 0.0625 | 8,171.75 | 7,868 | 8,192 | |
For how many integers $n$ is it true that $\sqrt{n} \le \sqrt{4n - 6} < \sqrt{2n + 5}$? | 4 | 1 | 3,638.1875 | 3,638.1875 | -1 | |
Given a rectangle divided into a 2x4 grid of equally spaced points, calculate the total number of distinct triangles that can be formed using three of these points as vertices. | 48 | 0.125 | 7,648.1875 | 6,497 | 7,812.642857 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.