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 |
|---|---|---|---|---|---|---|
In the diagram, $O$ is the center of a circle with radii $OP=OQ=5$. What is the perimeter of the shaded region?
[asy]
size(100);
import graph;
label("$P$",(-1,0),W); label("$O$",(0,0),NE); label("$Q$",(0,-1),S);
fill(Arc((0,0),1,-90,180)--cycle,mediumgray);
draw(Arc((0,0),1,-90,180));
fill((0,0)--(-1,0)--(0,-1)--cyc... | 10 + \frac{15}{2}\pi | 0.625 | 4,570.375 | 4,407.3 | 4,842.166667 | |
Form a six-digit number using the digits 1, 2, 3, 4, 5, 6 without repetition, where both 5 and 6 are on the same side of 3. How many such six-digit numbers are there? | 480 | 0 | 8,069.1875 | -1 | 8,069.1875 | |
Michael is playing basketball. He makes $10 \%$ of his shots, and gets the ball back after $90 \%$ of his missed shots. If he does not get the ball back he stops playing. What is the probability that Michael eventually makes a shot? | \frac{10}{19} | We find the probability Michael never makes a shot. We do casework on the number of shots Michael takes. He takes only one shot with probability $\frac{9}{10} \cdot \frac{1}{10}$ (he misses with probability $\frac{9}{10}$ and does not get the ball back with probability $\frac{1}{10}$). Similarly, he takes two shots wit... | 0.5 | 6,186 | 4,947 | 7,425 |
The function $f(x)$ satisfies
\[xf(y) = yf(x)\]for all real numbers $x$ and $y.$ If $f(15) = 20,$ find $f(3).$ | 4 | 1 | 1,922.1875 | 1,922.1875 | -1 | |
If the sum of two numbers is $1$ and their product is $1$, then the sum of their cubes is: | -2 | Let the two numbers be $a$ and $b$. Given that the sum of the two numbers is $1$ and their product is $1$, we have:
\[ a + b = 1 \]
\[ ab = 1 \]
We need to find the sum of their cubes, $a^3 + b^3$. We can use the identity for the sum of cubes:
\[ a^3 + b^3 = (a+b)(a^2 - ab + b^2) \]
First, we simplify $a^2 - ab + b^2... | 1 | 2,681.75 | 2,681.75 | -1 |
There are 7 students participating in 5 sports events. Students A and B cannot participate in the same event. Each event must have participants, and each student can only participate in one event. How many different arrangements satisfy these conditions? (Answer in numbers) | 15000 | 0.0625 | 8,129.4375 | 7,191 | 8,192 | |
What is the remainder when $333^{333}$ is divided by $11$? | 5 | 1 | 2,739.375 | 2,739.375 | -1 | |
Call a polygon normal if it can be inscribed in a unit circle. How many non-congruent normal polygons are there such that the square of each side length is a positive integer? | 14 | The side lengths of the polygon can only be from the set $\{1, \sqrt{2}, \sqrt{3}, 2\}$, which take up $60^{\circ}, 90^{\circ}, 120^{\circ}, 180^{\circ}$ of the circle respectively. By working modulo 60 degrees we see that $\sqrt{2}$ must be used an even number of times. We now proceed to casework on the longest side o... | 0 | 7,056.8125 | -1 | 7,056.8125 |
Two right triangles share a side as follows: [asy]
pair pA, pB, pC, pD, pE;
pA = (0, 0);
pB = pA + 6 * dir(0);
pC = pA + 10 * dir(90);
pD = pB + 6 * dir(90);
pE = (6 * pA + 10 * pD) / 16;
draw(pA--pB--pC--pA);
draw(pA--pB--pD--pA);
label("$A$", pA, SW);
label("$B$", pB, SE);
label("$C$", pC, NW);
label("$D$", pD, NE);
... | \frac{75}{4} | 0.875 | 5,859.125 | 5,525.857143 | 8,192 | |
The government decided to privatize civil aviation. For each of the 127 cities in the country, the connecting airline between them is sold to one of the private airlines. Each airline must make all acquired airlines one-way but in such a way as to ensure the possibility of travel from any city to any other city (possib... | 63 | 0.1875 | 7,197.9375 | 6,066 | 7,459.153846 | |
Find all composite positive integers \(m\) such that, whenever the product of two positive integers \(a\) and \(b\) is \(m\), their sum is a power of $2$ .
*Proposed by Harun Khan* | 15 | 0 | 8,192 | -1 | 8,192 | |
Find all real values of $x$ for which $$\frac{1}{\sqrt{x}+\sqrt{x-2}}+\frac{1}{\sqrt{x+2}+\sqrt{x}}=\frac{1}{4}$$ | \frac{257}{16} | We note that $$\begin{aligned} \frac{1}{4} & =\frac{1}{\sqrt{x}+\sqrt{x-2}}+\frac{1}{\sqrt{x+2}+\sqrt{x}} \\ & =\frac{\sqrt{x}-\sqrt{x-2}}{(\sqrt{x}+\sqrt{x-2})(\sqrt{x}-\sqrt{x-2})}+\frac{\sqrt{x+2}-\sqrt{x}}{(\sqrt{x+2}+\sqrt{x})(\sqrt{x+2}-\sqrt{x})} \\ & =\frac{\sqrt{x}-\sqrt{x-2}}{2}+\frac{\sqrt{x+2}-\sqrt{x}}{2} ... | 1 | 3,983.625 | 3,983.625 | -1 |
Evaluate $|5 - e|$ where $e$ is the base of the natural logarithm. | 2.28172 | 0 | 4,124.0625 | -1 | 4,124.0625 | |
Juan rolls a fair regular decagonal die marked with numbers from 1 to 10. Then Amal rolls a fair eight-sided die marked with numbers from 1 to 8. What is the probability that the product of the two rolls is a multiple of 4? | \frac{19}{40} | 0.1875 | 7,688.875 | 6,008.333333 | 8,076.692308 | |
In the isosceles triangle \(ABC\) (\(AB = BC\)), a point \(D\) is taken on the side \(BC\) such that \(BD : DC = 1 : 4\). In what ratio does the line \(AD\) divide the altitude \(BE\) of triangle \(ABC\), counting from vertex \(B\)? | 1:2 | 0.6875 | 6,525.6875 | 5,768.272727 | 8,192 | |
Team A and Team B each have $n$ members. It is known that each member of Team A shakes hands with each member of Team B exactly once (members within the same team do not shake hands). From these $n^2$ handshakes, two are randomly selected. Let event A be the event that exactly 3 members are involved in these two handsh... | 20 | 0.625 | 5,242.875 | 4,824.9 | 5,939.5 | |
Using 1 digit of '1', 2 digits of '2', and 2 digits of '3', how many different four-digit numbers can be formed? Fill in the blank with the total number of different four-digit numbers. | 30 | 0.625 | 5,889.75 | 5,411 | 6,687.666667 | |
Danielle picks a positive integer $1 \leq n \leq 2016$ uniformly at random. What is the probability that \operatorname{gcd}(n, 2015)=1? | \frac{1441}{2016} | We split the interval $[1,2016]$ into $[1,2015]$ and 2016. The number of integers in $[1,2015]$ that are relatively prime to 2015 is $\phi(2015)=\frac{4}{5} \cdot \frac{12}{13} \cdot \frac{30}{31} \cdot 2015=1440$. Also, 2016 is relatively prime to 2015, so there are a total of 1441 numbers in $[1,2016]$ that are relat... | 0.8125 | 5,056.875 | 4,791.307692 | 6,207.666667 |
Machine tools A, B, and C each independently process the same type of part. It is known that the probabilities of the parts processed by machine tools A, B, and C being first-class are 0.7, 0.6, and 0.8, respectively. The number of parts processed by machine tools B and C are equal, and the number of parts processed by... | 0.6517 | 0.4375 | 6,540.125 | 4,863.714286 | 7,844 | |
In a speech contest held in a class, there are a total of $5$ contestants, including $3$ females (among which is female A) and $2$ males. If the two males cannot appear consecutively, and female A cannot be the first to appear, what is the number of ways for the appearance order? | 60 | 0.1875 | 7,689.5625 | 5,512.333333 | 8,192 | |
Given $\cos\alpha = \frac{5}{13}$ and $\cos(\alpha - \beta) = \frac{4}{5}$, with $0 < \beta < \alpha < \frac{\pi}{2}$,
$(1)$ Find the value of $\tan 2\alpha$;
$(2)$ Find the value of $\cos\beta$. | \frac{56}{65} | 0.6875 | 6,096.3125 | 5,143.727273 | 8,192 | |
Expand $(2z^2 + 5z - 6)(3z^3 - 2z + 1)$. | 6z^5+15z^4-22z^3-8z^2+17z-6 | 0.9375 | 2,395.1875 | 2,413.8 | 2,116 | |
In a rectangular configuration $ABCD$, there are three squares with non-overlapping interiors. One of them, which is shaded, has an area of 4 square inches. The side length of the larger square is twice the side length of the shaded square. What is the area of rectangle $ABCD$, given that all squares fit exactly within... | 24 | 0.625 | 7,066.5625 | 6,391.3 | 8,192 | |
Let $A B C D$ and $W X Y Z$ be two squares that share the same center such that $W X \| A B$ and $W X<A B$. Lines $C X$ and $A B$ intersect at $P$, and lines $C Z$ and $A D$ intersect at $Q$. If points $P, W$, and $Q$ are collinear, compute the ratio $A B / W X$. | \sqrt{2}+1 | Without loss of generality, let $A B=1$. Let $x=W X$. Then, since $B P W X$ is a parallelogram, we have $B P=x$. Moreover, if $T=X Y \cap A B$, then we have $B T=\frac{1-x}{2}$, so $P T=x-\frac{1-x}{2}=\frac{3 x-1}{2}$. Then, from $\triangle P X T \sim \triangle P B C$, we have $$\begin{aligned} \frac{P T}{X T}=\frac{P... | 0 | 7,680 | -1 | 7,680 |
For how many integer values of $n$ between 1 and 1000 inclusive does the decimal representation of $\frac{n}{2520}$ terminate? | 47 | 0 | 6,718.5625 | -1 | 6,718.5625 | |
The graph of the function $y=f(x)$ is symmetric around the line $y=x$. After moving it left by one unit, the graph is still symmetric around the line $y=x$. If $f(1)=0$, then $f(2011)=$ __(A)__. | -2010 | 0.75 | 5,529.8125 | 4,642.416667 | 8,192 | |
The perimeter of triangle $APM$ is $152$, and the angle $PAM$ is a right angle. A circle of radius $19$ with center $O$ on $\overline{AP}$ is drawn so that it is tangent to $\overline{AM}$ and $\overline{PM}$. Given that $OP=m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.
| 98 | 0.1875 | 7,820.8125 | 6,212.333333 | 8,192 | |
Using Green's theorem, evaluate the line integral \( I = \oint_{L} 2(x^{2}+y^{2}) \, dx + (x+y)^{2} \, dy \) along the contour \( L \) of triangle \( ABC \) with vertices \( A(1,1) \), \( B(2,2) \), \( C(1,3) \). | -\frac{4}{3} | 0.4375 | 7,688.5625 | 7,041.285714 | 8,192 | |
Let $q(x)$ be a monic quartic polynomial such that $q(1) = 3$, $q(2) = 6, $ $q(3) = 11$, and $q(4) = 18$. Find $q(5)$. | 51 | 0.75 | 6,252.125 | 5,605.5 | 8,192 | |
The Fibonacci sequence $F_{1}, F_{2}, F_{3}, \ldots$ is defined by $F_{1}=F_{2}=1$ and $F_{n+2}=F_{n+1}+F_{n}$. Find the least positive integer $t$ such that for all $n>0, F_{n}=F_{n+t}$. | 60 | 60. | 0 | 8,192 | -1 | 8,192 |
A necklace consists of 50 blue beads and some quantity of red beads. It is known that on any segment of the necklace containing 8 blue beads, there are at least 4 red beads. What is the minimum number of red beads that can be in this necklace? (The beads in the necklace are arranged cyclically, meaning the last bead is... | 29 | 0 | 7,667.875 | -1 | 7,667.875 | |
A regular pentagon has an area numerically equal to its perimeter, and a regular hexagon also has its area numerically equal to its perimeter. Compare the apothem of the pentagon with the apothem of the hexagon. | 1.06 | 0 | 4,872.3125 | -1 | 4,872.3125 | |
When a swing is stationary, with the footboard one foot off the ground, pushing it forward two steps (in ancient times, one step was considered as five feet) equals 10 feet, making the footboard of the swing the same height as a person who is five feet tall, determine the length of the rope when pulled straight at this... | 14.5 | 0 | 2,944.3125 | -1 | 2,944.3125 | |
Petya and Vasya came up with ten fifth-degree polynomials. Then, Vasya sequentially called out consecutive natural numbers (starting from some number), and Petya substituted each called number into one of the polynomials of his choice, writing the resulting values on the board from left to right. It turned out that the... | 50 | 0 | 7,924.375 | -1 | 7,924.375 | |
Given a positive sequence $\{a_n\}$ with the first term being 1, it satisfies $a_{n+1}^2 + a_n^2 < \frac{5}{2}a_{n+1}a_n$, where $n \in \mathbb{N}^*$, and $S_n$ is the sum of the first $n$ terms of the sequence $\{a_n\}$.
1. If $a_2 = \frac{3}{2}$, $a_3 = x$, and $a_4 = 4$, find the range of $x$.
2. Suppose the sequenc... | 16 | 0.0625 | 8,030.4375 | 5,607 | 8,192 | |
If $a$ and $b$ are positive integers that can each be written as a sum of two squares, then $a b$ is also a sum of two squares. Find the smallest positive integer $c$ such that $c=a b$, where $a=x^{3}+y^{3}$ and $b=x^{3}+y^{3}$ each have solutions in integers $(x, y)$, but $c=x^{3}+y^{3}$ does not. | 4 | We can't have $c=1=1^{3}+0^{3}$ or $c=2=1^{3}+1^{3}$, and if $c=3$, then $a$ or $b= \pm 3$ which is not a sum of two cubes (otherwise, flipping signs of $x$ and $y$ if necessary, we would get either a sum of two nonnegative cubes to equal 3, which clearly does not happen, or a difference of two nonnegative cubes to equ... | 0.1875 | 7,878.5 | 6,983 | 8,085.153846 |
In triangle $ABC,$ $M$ is the midpoint of $\overline{BC},$ $AB = 12,$ and $AC = 16.$ Let $E$ be on $\overline{AC},$ and $F$ be on $\overline{AB},$ and let $G$ be the intersection of $\overline{EF}$ and $\overline{AM}.$ If $AE = 2AF,$ then find $\frac{EG}{GF}.$
[asy]
unitsize(0.3 cm);
pair A, B, C, E, F, G, M;
real ... | \frac{3}{2} | 0.125 | 8,170.8125 | 8,022.5 | 8,192 | |
(6?3) + 4 - (2 - 1) = 5. To make this statement true, the question mark between the 6 and the 3 should be replaced by | \div | We start by simplifying the expression on the right side of the equation:
\[ (6?3) + 4 - (2 - 1) = 5. \]
First, simplify the parentheses on the right:
\[ 2 - 1 = 1. \]
Now, substitute this back into the equation:
\[ (6?3) + 4 - 1 = 5. \]
Simplify further:
\[ (6?3) + 3 = 5. \]
To isolate $(6?3)$, subtract 3 from bot... | 0.875 | 651.3125 | 650.714286 | 655.5 |
In a division equation, the dividend is 2016 greater than the divisor, the quotient is 15, and the remainder is 0. What is the dividend? | 2160 | 1 | 576.9375 | 576.9375 | -1 | |
$(1)$ If $A=\frac{a-1}{a+2}•\frac{{a}^{2}-4}{{a}^{2}-2a+1}÷\frac{1}{a-1}$, simplify $A$;<br/>$(2)$ If $a$ satisfies $a^{2}-a=0$, find the value of $A$. | -2 | 0.9375 | 3,640.1875 | 3,336.733333 | 8,192 | |
Two different digits from 1 to 9 are chosen. One digit is placed in each box to complete the two 2-digit numbers shown. The result of subtracting the bottom number from the top number is calculated. How many of the possible results are positive? | 36 | 0.125 | 8,073.3125 | 7,658 | 8,132.642857 | |
Let $\triangle{ABC}$ be a triangle with $AB = 10$ and $AC = 11$ . Let $I$ be the center of the inscribed circle of $\triangle{ABC}$ . If $M$ is the midpoint of $AI$ such that $BM = BC$ and $CM = 7$ , then $BC$ can be expressed in the form $\frac{\sqrt{a}-b}{c}$ where $a$ , $b$ , and $c$ are posit... | 622 | 0 | 8,192 | -1 | 8,192 | |
In a magic square, the sum of the three entries in any row, column, or diagonal is the same value. The figure below shows four of the entries of a magic square. Find $x$.
[asy]
size(2cm);
for (int i=0; i<=3; ++i) draw((i,0)--(i,3)^^(0,i)--(3,i));
label("$x$",(0.5,2.5));label("$19$",(1.5,2.5));
label("$96$",(2.5,2.5));l... | 200 | 0.25 | 7,130.4375 | 5,103.75 | 7,806 | |
In an airspace, there are clouds. It turned out that the space can be divided into parts by ten planes so that each part contains no more than one cloud. Through how many clouds could an airplane fly at most while following a straight course? | 11 | 0.625 | 5,343.0625 | 5,010.3 | 5,897.666667 | |
A number has 6 on both its tens and hundredths places, and 0 on both its ones and tenths places. This number is written as \_\_\_\_\_\_. | 60.06 | 0 | 306.875 | -1 | 306.875 | |
Each of the numbers 1, 2, 3, and 4 is substituted, in some order, for \( p, q, r \), and \( s \). Find the greatest possible value of \( p^q + r^s \). | 83 | 0.5625 | 7,774.0625 | 7,449 | 8,192 | |
Given that the median of the numbers $3, 5, 7, 23,$ and $x$ is equal to the mean of those five numbers, calculate the sum of all real numbers $x$. | -13 | 0.9375 | 5,801.3125 | 5,641.933333 | 8,192 | |
How many functions $f:\{1,2,3,4,5\} \rightarrow\{1,2,3,4,5\}$ satisfy $f(f(x))=f(x)$ for all $x \in\{1,2,3,4,5\}$? | 196 | A fixed point of a function $f$ is an element $a$ such that $f(a)=a$. The condition is equivalent to the property that $f$ maps every number to a fixed point. Counting by the number of fixed points of $f$, the total number of such functions is $$\begin{aligned} \sum_{k=1}^{5}\binom{5}{k} k^{5-k} & =1 \cdot\left(5^{0}\r... | 1 | 5,147.8125 | 5,147.8125 | -1 |
How many distinct arrangements of the letters in the word "example" are there? | 5040 | 0 | 1,505.1875 | -1 | 1,505.1875 | |
A triangle $ABC$ with $AC=20$ is inscribed in a circle $\omega$ . A tangent $t$ to $\omega$ is drawn through $B$ . The distance $t$ from $A$ is $25$ and that from $C$ is $16$ .If $S$ denotes the area of the triangle $ABC$ , find the largest integer not exceeding $\frac{S}{20}$ | 10 | 0 | 8,186.9375 | -1 | 8,186.9375 | |
Find the integer that is closest to $1000\sum_{n=3}^{10000}\frac1{n^2-4}$. | 521 | We know that $\frac{1}{n^2 - 4} = \frac{1}{(n+2)(n-2)}$. We can use the process of fractional decomposition to split this into two fractions: $\frac{1}{(n+2)(n-2)} = \frac{A}{(n+2)} + \frac{B}{(n-2)}$ for some A and B.
Solving for A and B gives $1 = (n-2)A + (n+2)B$ or $1 = n(A+B)+ 2(B-A)$. Since there is no n term on... | 0 | 8,192 | -1 | 8,192 |
A circle is circumscribed about an equilateral triangle with side lengths of $12$ units each. Calculate the area of the circle, and then find the perimeter of the triangle. | 36 | 0.8125 | 1,986.5 | 2,044.923077 | 1,733.333333 | |
A line passing through point $P(-2,2)$ intersects the hyperbola $x^2-2y^2=8$ such that the midpoint of the chord $MN$ is exactly at $P$. Find the length of $|MN|$. | 2 \sqrt{30} | 0.875 | 5,076.125 | 4,631 | 8,192 | |
Quadrilateral $ABCD$ satisfies $\angle ABC = \angle ACD = 90^{\circ}, AC=20,$ and $CD=30.$ Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E,$ and $AE=5.$ What is the area of quadrilateral $ABCD?$ | 360 | 1. **Assign Variables and Use Pythagorean Theorem in $\triangle ABC$**:
Let $AB = x$ and $BC = y$. Since $\angle ABC = 90^\circ$, by the Pythagorean theorem, we have:
\[
x^2 + y^2 = AC^2 = 20^2 = 400.
\]
2. **Calculate $DE$ and Area of $\triangle ACD$**:
Since $\angle ACD = 90^\circ$, $CD = 30$, and $CE... | 0.0625 | 7,812.5 | 7,589 | 7,827.4 |
How many natural numbers between 200 and 400 are divisible by 8? | 25 | 0 | 5,475.125 | -1 | 5,475.125 | |
A triangle has sides of lengths 20 and 19. If the triangle is not acute, how many possible integer lengths can the third side have? | 16 | 0.3125 | 6,649.125 | 5,521 | 7,161.909091 | |
Define a function $A(m, n)$ by \[ A(m,n) = \left\{ \begin{aligned} &n+1& \text{ if } m = 0 \\ &A(m-1, 1) & \text{ if } m > 0 \text{ and } n = 0 \\ &A(m-1, A(m, n-1))&\text{ if } m > 0 \text{ and } n > 0. \end{aligned} \right.\]Compute $A(2, 1).$ | 5 | 1 | 4,334.375 | 4,334.375 | -1 | |
The mean of the set of numbers $\{91, 89, 85, 88, 90, 87, y\}$ is 88. What is the median of the set of seven numbers? | 88 | 1 | 2,563.125 | 2,563.125 | -1 | |
For a nonnegative integer $n$, let $r_7(n)$ represent the remainder when $n$ is divided by $7$. Determine the $15^{\text{th}}$ entry in an ordered list of all nonnegative integers $n$ that satisfy $$r_7(3n)\le 3.$$ | 22 | 0.0625 | 6,402.4375 | 4,869 | 6,504.666667 | |
The line segment connecting the focus F of the parabola $y^2=4x$ and the point M(0,1) intersects the parabola at point A. Let O be the origin, then the area of △OAM is _____. | \frac {3}{2} - \sqrt {2} | 0 | 7,032.125 | -1 | 7,032.125 | |
For what values of $x$ is $2x^2+8x\le-6$? Express your answer in interval notation. | [-3, -1] | 1 | 1,721.875 | 1,721.875 | -1 | |
A $10 \times 10$ table consists of 100 unit cells. A block is a $2 \times 2$ square consisting of 4 unit cells of the table. A set $C$ of $n$ blocks covers the table (i.e. each cell of the table is covered by some block of $C$ ) but no $n-1$ blocks of $C$ cover the table. Find the largest possible value of n. | 39 | Consider an infinite table divided into unit cells. Any $2 \times 2$ square consisting of 4 unit cells of the table we also call a block. Fix arbitrary finite set $M$ of blocks lying on the table. Now we will consider arbitrary finite sets of unit cells of the table covered by $M$. For any such set $\Phi$ denote by $|\... | 0 | 6,704 | -1 | 6,704 |
If $a=\frac{1}{2}$ and $(a+1)(b+1)=2$ then the radian measure of $\arctan a + \arctan b$ equals | \frac{\pi}{4} | 1. **Given Values and Equation Setup:**
- We are given $a = \frac{1}{2}$.
- We also have the equation $(a+1)(b+1) = 2$.
2. **Solving for $b$:**
- Substitute $a = \frac{1}{2}$ into the equation:
\[
\left(\frac{1}{2} + 1\right)(b+1) = 2
\]
- Simplify the left side:
\[
\frac{3}{2}(b+1... | 1 | 3,090.875 | 3,090.875 | -1 |
Let $ABCD$ be a tetrahedron with $AB=41$, $AC=7$, $AD=18$, $BC=36$, $BD=27$, and $CD=13$, as shown in the figure. Let $d$ be the distance between the midpoints of edges $AB$ and $CD$. Find $d^{2}$. [asy] defaultpen(fontsize(10)+0.8); size(175); pair A,B,C,D,M,P,Q; C=origin; B=(8,0); D=IP(CR(C,6.5),CR(B,8)); A=(4,-3); P... | 137 | Call the midpoint of $\overline{AB}$ $M$ and the midpoint of $\overline{CD}$ $N$. $d$ is the median of triangle $\triangle CDM$. The formula for the length of a median is $m=\sqrt{\frac{2a^2+2b^2-c^2}{4}}$, where $a$, $b$, and $c$ are the side lengths of triangle, and $c$ is the side that is bisected by median $m$. The... | 0.125 | 8,120.9375 | 7,623.5 | 8,192 |
Given that four A's, four B's, four C's, and four D's are to be placed in a 4 × 4 grid so that each row and column contains one of each letter, and A is placed in the upper right corner, calculate the number of possible arrangements. | 216 | 0 | 8,127.9375 | -1 | 8,127.9375 | |
Three boys and three girls are lined up for a photo. Boy A is next to boy B, and exactly two girls are next to each other. Calculate the total number of different ways they can be arranged. | 144 | 0 | 7,970.875 | -1 | 7,970.875 | |
Write $-\left(-3\right)-4+\left(-5\right)$ in the form of algebraic sum without parentheses. | 3-4-5 | 0.0625 | 329 | 316 | 329.866667 | |
The degrees of polynomials $P$ and $Q$ with real coefficients do not exceed $n$. These polynomials satisfy the identity
\[ P(x) x^{n + 1} + Q(x) (x+1)^{n + 1} = 1. \]
Determine all possible values of $Q \left( - \frac{1}{2} \right)$. | 2^n |
Given the problem, we have two polynomials \( P(x) \) and \( Q(x) \) with real coefficients such that their degrees do not exceed \( n \). They satisfy the identity:
\[
P(x) x^{n+1} + Q(x)(x+1)^{n+1} = 1.
\]
We want to determine all possible values of \( Q\left(-\frac{1}{2}\right) \).
### Step 1: Analyzing the Degr... | 0.0625 | 8,027.625 | 6,647 | 8,119.666667 |
Two people are selected from each of the two groups. Group A has 5 boys and 3 girls, and Group B has 6 boys and 2 girls. Calculate the number of ways to select 4 people such that exactly 1 girl is included. | 345 | 0.125 | 6,273 | 3,477.5 | 6,672.357143 | |
Two regular tetrahedrons $A$ and $B$ are made with the 8 vertices of a unit cube. (this way is unique)
What's the volume of $A\cup B$ ? | 1/2 | 0.0625 | 8,060.0625 | 8,192 | 8,051.266667 | |
Given: Circle $C$ passes through point $D(0,1)$, $E(-2,1)$, $F(-1,\sqrt{2})$, $P$ is any point on the line $l_{1}: y=x-2$, and the line $l_{2}: y=x+1$ intersects circle $C$ at points $A$ and $B$. <br/>$(Ⅰ)$ Find the equation of circle $C$;<br/>$(Ⅱ)$ Find the minimum value of $|PA|^{2}+|PB|^{2}$. | 13 | 0.75 | 5,589 | 5,038.75 | 7,239.75 | |
The right triangles $\triangle M D C$ and $\triangle A D K$ share a common right angle $\angle D$. Point $K$ is on $C D$ and divides it in the ratio $2: 3$ counting from point $C$. Point $M$ is the midpoint of side $A D$. Find the sum of $\angle A K D$ and $\angle M C D$, if $A D: C D=2: 5$. | 45 | 0.8125 | 4,476.5 | 3,619.076923 | 8,192 | |
Let $P(x) = 3\sqrt[3]{x}$, and $Q(x) = x^3$. Determine $P(Q(P(Q(P(Q(4))))))$. | 108 | 0.3125 | 6,174.25 | 3,489 | 7,394.818182 | |
If a 31-day month is taken at random, find \( c \), the probability that there are 5 Sundays in the month. | 3/7 | 0.5625 | 5,221.625 | 4,194.555556 | 6,542.142857 | |
Find all natural numbers having exactly six divisors, the sum of which equals 3500. | 1996 | 0.75 | 6,600.625 | 6,070.166667 | 8,192 | |
The Fibonacci numbers are defined by $F_{0}=0, F_{1}=1$, and $F_{n}=F_{n-1}+F_{n-2}$ for $n \geq 2$. There exist unique positive integers $n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}$ such that $\sum_{i_{1}=0}^{100} \sum_{i_{2}=0}^{100} \sum_{i_{3}=0}^{100} \sum_{i_{4}=0}^{100} \sum_{i_{5}=0}^{100} F_{i_{1}+i_{2}+i_{3}+i_... | 1545 | We make use of the identity $\sum_{i=0}^{\ell} F_{i}=F_{\ell+2}-1$ (easily proven by induction) which implies $\sum_{i=k}^{\ell} F_{i}=F_{\ell+2}-F_{k+1}$. Applying this several times yields $\sum_{i_{1}=0}^{100} \sum_{i_{2}=0}^{100} \sum_{i_{3}=0}^{100} \sum_{i_{4}=0}^{100} \sum_{i_{5}=0}^{100} F_{i_{1}+i_{2}+i_{3}+i_... | 0 | 8,127 | -1 | 8,127 |
Given $f(\alpha)= \frac {\sin (\pi-\alpha)\cos (2\pi-\alpha)\tan (-\alpha+ \frac {3\pi}{2})}{\cot (-\alpha-\pi)\sin (-\pi-\alpha)}$.
$(1)$ Simplify $f(\alpha)$;
$(2)$ If $\alpha$ is an angle in the third quadrant, and $\cos (\alpha- \frac {3\pi}{2})= \frac {1}{5}$, find the value of $f(\alpha)$. | \frac {2 \sqrt {6}}{5} | 0 | 3,808.0625 | -1 | 3,808.0625 | |
On an island, there are two tribes: knights and liars. Knights always tell the truth, and liars always lie. One day, 80 people sat at a round table, and each of them declared: "Among the 11 people sitting immediately after me in a clockwise direction, there are at least 9 liars." How many knights are sitting at the ro... | 20 | 0 | 8,192 | -1 | 8,192 | |
In $\triangle RED$, $\measuredangle DRE=75^{\circ}$ and $\measuredangle RED=45^{\circ}$. $RD=1$. Let $M$ be the midpoint of segment $\overline{RD}$. Point $C$ lies on side $\overline{ED}$ such that $\overline{RC}\perp\overline{EM}$. Extend segment $\overline{DE}$ through $E$ to point $A$ such that $CA=AR$. Then $AE=\fr... | 56 | Let $P$ be the foot of the perpendicular from $A$ to $\overline{CR}$, so $\overline{AP}\parallel\overline{EM}$. Since triangle $ARC$ is isosceles, $P$ is the midpoint of $\overline{CR}$, and $\overline{PM}\parallel\overline{CD}$. Thus, $APME$ is a parallelogram and $AE = PM = \frac{CD}{2}$. We can then use coordinates.... | 0 | 8,192 | -1 | 8,192 |
How many integer solutions does the equation \(\sqrt{x} + \sqrt{y} = \sqrt{1960}\) have? | 15 | 0 | 7,586.6875 | -1 | 7,586.6875 | |
An Ultraman is fighting a group of monsters. It is known that Ultraman has one head and two legs. Initially, each monster has two heads and five legs. During the battle, some monsters split, with each splitting monster creating two new monsters, each with one head and six legs (they cannot split again). At a certain mo... | 13 | 0.0625 | 8,095.625 | 6,650 | 8,192 | |
Given a sequence $\{a_n\}$ with the sum of its first n terms $S_n$, if $S_2 = 4$, and $a_{n+1} = 2S_n + 1$ for $n \in N^*$, find the values of $a_1$ and $S_5$. | 121 | 1 | 2,792.375 | 2,792.375 | -1 | |
A child builds towers using identically shaped cubes of different colors. Determine the number of different towers with a height of 6 cubes that can be built with 3 yellow cubes, 3 purple cubes, and 2 orange cubes (Two cubes will be left out). | 350 | 0.375 | 6,360.9375 | 4,457.666667 | 7,502.9 | |
Suppose that the number $a$ satisfies the equation $4 = a + a^{ - 1}.$ What is the value of $a^{4} + a^{ - 4}?$ | 194 | 1 | 2,605.625 | 2,605.625 | -1 | |
Carl is on a vertex of a regular pentagon. Every minute, he randomly selects an adjacent vertex (each with probability $\frac{1}{2}$ ) and walks along the edge to it. What is the probability that after 10 minutes, he ends up where he had started? | \frac{127}{512} | Let A denote a clockwise move and B denote a counterclockwise move. We want to have some combination of 10 A's and B's, with the number of A's and the number of B's differing by a multiple of 5. We have $\binom{10}{0}+\binom{10}{5}+\binom{10}{10}=254$. Hence the answer is $\frac{254}{2^{10}}=\frac{127}{512}$. | 0 | 8,074.1875 | -1 | 8,074.1875 |
A regular 100-sided polygon is placed on a table, with vertices labeled from 1 to 100. The numbers are written down in the order they appear from the front edge of the table. If two vertices are equidistant from the edge, the number to the left is written down first, followed by the one on the right. All possible sets ... | 10100 | 0 | 5,984.8125 | -1 | 5,984.8125 | |
The integer parts of two finite decimals are 7 and 10, respectively. How many possible values are there for the integer part of the product of these two finite decimals? | 18 | 0.4375 | 7,143.125 | 5,794.571429 | 8,192 | |
Given a dart board is a regular hexagon divided into regions, the center of the board is another regular hexagon formed by joining the midpoints of the sides of the larger hexagon, and the dart is equally likely to land anywhere on the board, find the probability that the dart lands within the center hexagon. | \frac{1}{4} | 0.0625 | 7,036.5625 | 6,267 | 7,087.866667 | |
For an upcoming holiday, the weather forecast indicates a probability of $30\%$ chance of rain on Monday and a $60\%$ chance of rain on Tuesday. Moreover, once it starts raining, there is an additional $80\%$ chance that the rain will continue into the next day without interruption. Calculate the probability that it ra... | 72\% | 0.375 | 7,086.125 | 5,945.833333 | 7,770.3 | |
Which of the following is a possible value of $x$ if given two different numbers on a number line, the number to the right is greater than the number to the left, and the positions of $x, x^{3}$ and $x^{2}$ are marked on a number line? | -\frac{2}{5} | From the number line shown, we see that $x<x^{3}<x^{2}$.
If $x>1$, then successive powers of $x$ are increasing (that is, $x<x^{2}<x^{3}$ ).
Since this is not the case, then it is not true that $x>1$.
If $x=0$ or $x=1$, then successive powers of $x$ are equal. This is not the case either.
If $0<x<1$, then succe... | 0 | 7,780.6875 | -1 | 7,780.6875 |
In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, and it is given that $a\sin C= \sqrt {3}c\cos A$.
$(1)$ Find the measure of angle $A$;
$(2)$ If $a= \sqrt {13}$ and $c=3$, find the area of $\triangle ABC$. | 3 \sqrt {3} | 0 | 4,020.625 | -1 | 4,020.625 | |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|=1$, and $\overrightarrow{a}\cdot \overrightarrow{b}=-\sqrt{2}$, calculate the angle between vector $\overrightarrow{a}$ and $\overrightarrow{b}$. | \dfrac{3\pi}{4} | 0.25 | 1,747.375 | 2,111.25 | 1,626.083333 | |
In the $xy$-plane, the segment with endpoints $(-5,0)$ and $(25,0)$ is the diameter of a circle. If the point $(x,15)$ is on the circle, then $x=$ | 10 | 1. **Identify the center and radius of the circle**:
The endpoints of the diameter of the circle are given as $(-5,0)$ and $(25,0)$. The center of the circle, $C$, is the midpoint of the diameter. Using the midpoint formula:
\[
C = \left(\frac{-5 + 25}{2}, \frac{0 + 0}{2}\right) = (10, 0)
\]
The radius,... | 1 | 2,304 | 2,304 | -1 |
Let $D$, $E$, and $F$ be constants such that the equation \[\frac{(x+E)(Dx+36)}{(x+F)(x+9)} = 3\] has infinitely many solutions for $x$. For these values of $D$, $E$, and $F$, it turns out that there are only finitely many values of $x$ which are not solutions to the equation. Find the sum of these values of $x$. | -21 | 1 | 2,973.3125 | 2,973.3125 | -1 | |
How many negative numbers are there among the numbers of the form $\operatorname{tg}\left(\left(15^{n}\right)^{\circ}\right)$, where $\mathrm{n}$ is a natural number from 1 to 2019? | 1009 | 0.5625 | 6,775.5625 | 6,068.333333 | 7,684.857143 | |
A machine-shop cutting tool has the shape of a notched circle, as shown. The radius of the circle is $\sqrt{50}$ cm, the length of $AB$ is $6$ cm and that of $BC$ is $2$ cm. The angle $ABC$ is a right angle. Find the square of the distance (in centimeters) from $B$ to the center of the circle.
[asy] size(150); default... | 26 | 0.25 | 7,557.875 | 6,834.25 | 7,799.083333 | |
Let $\Omega$ be a unit circle and $A$ be a point on $\Omega$ . An angle $0 < \theta < 180^\circ$ is chosen uniformly at random, and $\Omega$ is rotated $\theta$ degrees clockwise about $A$ . What is the expected area swept by this rotation? | 2\pi | 0 | 8,144.4375 | -1 | 8,144.4375 | |
Given the function $f(x)$ is a monotonic function on the real numbers $\mathbb{R}$, and for any real number $x$, it satisfies the equation $f\left[f(x)+ \frac{2}{2^{x}+1}\right]= \frac{1}{3}$, find the value of $f(\log_23)$. | \frac{1}{2} | 0.75 | 4,674.375 | 3,794.5 | 7,314 | |
In the rectangular coordinate system $xOy$, a polar coordinate system is established with the coordinate origin as the pole and the positive semi-axis of the $x$-axis as the polar axis. The polar coordinate equation of circle $C$ is $\rho^2 - 2m\rho\cos\theta + 4\rho\sin\theta = 1 - 2m$.
(1) Find the rectangular coordi... | 2 + \sqrt{3} | 0.8125 | 4,684.625 | 4,741.076923 | 4,440 | |
Simplify the expression:
\[
\frac{1}{\dfrac{3}{\sqrt{5}+2} - \dfrac{4}{\sqrt{7}+2}}.
\] | \frac{3(9\sqrt{5} + 4\sqrt{7} + 10)}{(9\sqrt{5} - 4\sqrt{7} - 10)(9\sqrt{5} + 4\sqrt{7} + 10)} | 0 | 7,062.0625 | -1 | 7,062.0625 |
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