problem stringlengths 10 5.15k | answer stringlengths 0 1.22k | solution stringlengths 0 11.1k | reward float64 0 1 | length float64 172 8.19k | correct_length float64 -1 8.19k | incorrect_length float64 -1 8.19k |
|---|---|---|---|---|---|---|
Given the random variable $X \sim N(1, \sigma^{2})$, if $P(0 < x < 3)=0.5$, $P(0 < X < 1)=0.2$, then $P(X < 3)=$\_\_\_\_\_\_\_\_\_\_\_ | 0.8 | 0 | 8,192 | -1 | 8,192 | |
Xiao Xiao did an addition problem, but he mistook the second addend 420 for 240, and the result he got was 390. The correct result is ______. | 570 | 0.875 | 397.9375 | 391.642857 | 442 | |
A rectangular piece of paper $P Q R S$ has $P Q=20$ and $Q R=15$. The piece of paper is glued flat on the surface of a large cube so that $Q$ and $S$ are at vertices of the cube. What is the shortest distance from $P$ to $R$, as measured through the cube? | 18.4 | Since $P Q R S$ is rectangular, then $\angle S R Q=\angle S P Q=90^{\circ}$. Also, $S R=P Q=20$ and $S P=Q R=15$. By the Pythagorean Theorem in $\triangle S P Q$, since $Q S>0$, we have $Q S=\sqrt{S P^{2}+P Q^{2}}=\sqrt{15^{2}+20^{2}}=\sqrt{225+400}=\sqrt{625}=25$. Draw perpendiculars from $P$ and $R$ to $X$ and $Y$, r... | 0 | 8,192 | -1 | 8,192 |
In triangle $\triangle ABC$, the sides opposite angles A, B, and C are denoted as $a$, $b$, and $c$ respectively. Given that $C = \frac{2\pi}{3}$ and $a = 6$:
(Ⅰ) If $c = 14$, find the value of $\sin A$;
(Ⅱ) If the area of $\triangle ABC$ is $3\sqrt{3}$, find the value of $c$. | 2\sqrt{13} | 0.8125 | 5,089.375 | 4,373.384615 | 8,192 | |
Let $k$ be a natural number. For which value of $k$ is $A_k = \frac{19^k + 66^k}{k!}$ maximized? | 65 | 0 | 8,192 | -1 | 8,192 | |
Jack and Jill are going swimming at a pool that is one mile from their house. They leave home simultaneously. Jill rides her bicycle to the pool at a constant speed of $10$ miles per hour. Jack walks to the pool at a constant speed of $4$ miles per hour. How many minutes before Jack does Jill arrive? | 9 | 1. **Calculate Jill's travel time:**
- We use the formula for distance, \(d = rt\), where \(d\) is the distance, \(r\) is the rate, and \(t\) is the time.
- For Jill, \(d = 1\) mile, \(r = 10\) miles per hour. Plugging in the values, we get:
\[
1 = 10t \implies t = \frac{1}{10} \text{ hours}
\]
... | 1 | 1,370.3125 | 1,370.3125 | -1 |
Two mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9 a.m. and 10 a.m., and stay for exactly $m$ minutes. The probability that either one arrives while the other is in the cafeteria is $40 \%,$ and $m = a - b\sqrt {c},$ where $a, b,$ and $c$ are p... | 87 | Case 1:
Case 2:
We draw a number line representing the time interval. If mathematician $M_1$ comes in at the center of the time period, then the two mathematicions will meet if $M_2$ comes in somewhere between $m$ minutes before and after $M_1$ comes (a total range of $2m$ minutes). However, if $M_1$ comes into the c... | 0.9375 | 4,286.25 | 4,025.866667 | 8,192 |
Four pairs of socks in different colors are randomly selected from a wardrobe, and it is known that two of them are from the same pair. Calculate the probability that the other two are not from the same pair. | \frac{8}{9} | 0 | 7,755.0625 | -1 | 7,755.0625 | |
In the interval $[0,\pi]$, a number $x$ is randomly selected. The probability that $\sin x$ falls between $0$ and $\frac{1}{2}$ is ______. | \frac{1}{3} | 1 | 3,623 | 3,623 | -1 | |
A TV station broadcasts 5 advertisements in a row, among which there are 3 different commercial advertisements and 2 different World Expo promotional advertisements. The last advertisement broadcasted is a World Expo promotional advertisement, and the methods in which the 2 World Expo promotional advertisements are not... | 36 | 0.8125 | 7,451.8125 | 7,281 | 8,192 | |
If a worker receives a $20$% cut in wages, he may regain his original pay exactly by obtaining a raise of: | 25\% | 1. **Understanding the wage cut**: Let the original wage of the worker be $W$. A $20\%$ cut in wages means the worker now earns $80\%$ of $W$. Mathematically, this can be expressed as:
\[
\text{New Wage} = 0.8W = \frac{4}{5}W
\]
2. **Calculating the required raise**: To regain the original wage $W$, the worke... | 1 | 1,765.625 | 1,765.625 | -1 |
Find the quadratic polynomial $p(x)$ such that $p(-7) = 0,$ $p(4) = 0,$ and $p(5) = -36.$ | -3x^2 - 9x + 84 | 1 | 1,961.9375 | 1,961.9375 | -1 | |
Find the number of sequences consisting of 100 R's and 2011 S's that satisfy the property that among the first \( k \) letters, the number of S's is strictly more than 20 times the number of R's for all \( 1 \leq k \leq 2111 \). | \frac{11}{2111} \binom{2111}{100} | 0 | 7,848.625 | -1 | 7,848.625 | |
Jason borrowed money from his parents to buy a new surfboard. His parents have agreed to let him work off his debt by babysitting under the following conditions: his first hour of babysitting is worth $\$1$, the second hour worth $\$2$, the third hour $\$3$, the fourth hour $\$4$, the fifth hour $\$5$, the sixth hour $... | \$132 | 0.625 | 4,703.125 | 3,333.5 | 6,985.833333 | |
Determine the product of all positive integer values of \( c \) such that \( 9x^2 + 24x + c = 0 \) has real roots. | 20922789888000 | 0.125 | 3,664.4375 | 2,830 | 3,783.642857 | |
Define a function $g(x),$ for positive integer values of $x,$ by \[g(x) = \left\{\begin{aligned} \log_3 x & \quad \text{ if } \log_3 x \text{ is an integer} \\ 1 + g(x + 1) & \quad \text{ otherwise}. \end{aligned} \right.\]Compute $g(200).$ | 48 | 0.25 | 7,043.0625 | 5,762.25 | 7,470 | |
What is the least positive integer greater than 1 that leaves a remainder of 2 when divided by each of 3, 4, 5, 6, 7, 8, 9, and 11? | 27722 | 1 | 3,230.375 | 3,230.375 | -1 | |
Given the function \( f:\{1,2, \cdots, 10\} \rightarrow\{1,2,3,4,5\} \), and for each \( k=1,2, \cdots, 9 \), it is true that \( |f(k+1)-f(k)| \geq 3 \). Find the number of functions \( f \) that satisfy these conditions. | 288 | 0.0625 | 7,991 | 5,106 | 8,183.333333 | |
Given the functions $f(x)= \frac {\ln x}{x}$, $g(x)=kx(k > 0)$, and the function $F(x)=\max\{f(x),g(x)\}$, where $\max\{a,b\}= \begin{cases} a, & \text{if } a\geqslant b\\ b, & \text{if } a < b \end{cases}$
$(I)$ Find the extreme value of $f(x)$;
$(2)$ Find the maximum value of $F(x)$ on the interval $[1,e]$ ($e$ i... | \frac {1}{e} | 0 | 8,183.625 | -1 | 8,183.625 | |
A sequence of real numbers $ x_0, x_1, x_2, \ldots$ is defined as follows: $ x_0 \equal{} 1989$ and for each $ n \geq 1$
\[ x_n \equal{} \minus{} \frac{1989}{n} \sum^{n\minus{}1}_{k\equal{}0} x_k.\]
Calculate the value of $ \sum^{1989}_{n\equal{}0} 2^n x_n.$ | -1989 | 0.0625 | 7,996.5 | 6,705 | 8,082.6 | |
A right circular cone with a base radius $r$ and height $h$ lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base connects with the table traces a circular arc centered at the vertex of the cone. The cone first returns to its original position ... | 400 | 0.9375 | 2,908.1875 | 2,555.933333 | 8,192 | |
Let $P$ be a point on the hyperbola $\frac{x^{2}}{16} - \frac{y^{2}}{20} = 1$, and let $F_{1}$ and $F_{2}$ be the left and right foci, respectively. If $|PF_{1}| = 9$, then find $|PF_{2}|$. | 17 | 0.6875 | 6,184.3125 | 5,271.727273 | 8,192 | |
Express $0.\overline{1}+0.\overline{02}+0.\overline{003}$ as a common fraction. | \frac{164}{1221} | 0.75 | 6,899.3125 | 6,468.416667 | 8,192 | |
I have a picture with dimensions $x$ and $y$ (in inches), such that $x$ and $y$ are both integers greater than one. I would like to place this picture in an elongated frame of dimensions $(2x + 3)$ and $(y+2)$. If I measured the area of the frame to be $34$ square inches, what is the area of the picture in square inch... | 8 | 1 | 2,862 | 2,862 | -1 | |
In the expansion of \((x+y+z)^{8}\), find the sum of the coefficients for all terms of the form \(x^{2} y^{a} z^{b}\) (where \(a, b \in \mathbf{N}\)). | 1792 | 0.125 | 7,212.25 | 5,738.5 | 7,422.785714 | |
Let $ a $ , $ b $ , $ c $ , $ d $ , $ (a + b + c + 18 + d) $ , $ (a + b + c + 18 - d) $ , $ (b + c) $ , and $ (c + d) $ be distinct prime numbers such that $ a + b + c = 2010 $ , $ a $ , $ b $ , $ c $ , $ d \neq 3 $ , and $ d \le 50 $ . Find the maximum value of the difference between two of thes... | 2067 | 0 | 8,192 | -1 | 8,192 | |
The three sides of a right triangle form a geometric sequence. Determine the ratio of the length of the hypotenuse to the length of the shorter leg. | \frac{1+\sqrt{5}}{2} | Let the shorter leg have length $\ell$, and the common ratio of the geometric sequence be $r>1$. Then the length of the other leg is $\ell r$, and the length of the hypotenuse is $\ell r^{2}$. Hence, $$\ell^{2}+(\ell r)^{2}=\left(\ell r^{2}\right)^{2} \Longrightarrow \ell^{2}\left(r^{2}+1\right)=\ell^{2} r^{4} \Longrig... | 0 | 3,928.5 | -1 | 3,928.5 |
Given a permutation $\sigma$ of $\{1,2, \ldots, 2013\}$, let $f(\sigma)$ to be the number of fixed points of $\sigma$ - that is, the number of $k \in\{1,2, \ldots, 2013\}$ such that $\sigma(k)=k$. If $S$ is the set of all possible permutations $\sigma$, compute $$\sum_{\sigma \in S} f(\sigma)^{4}$$ (Here, a permutation... | 15(2013!) | First, note that $$\sum_{\sigma \in S} f(\sigma)^{4}=\sum_{\sigma \in S} \sum_{1 \leq a_{1}, a_{2}, a_{3}, a_{4} \leq 2013} g\left(\sigma, a_{1}, a_{2}, a_{3}, a_{4}\right)$$ where $g\left(\sigma, a_{1}, a_{2}, a_{3}, a_{4}\right)=1$ if all $a_{i}$ are fixed points of $\sigma$ and 0 otherwise. (The $a_{i}$ 's need not ... | 0 | 8,192 | -1 | 8,192 |
A regular octahedron has a sphere inscribed within it and a sphere circumscribed about it. For each of the eight faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. A point $Q$ is selected at random inside the circumscribed sphere. Determine the probability that $Q$ li... | \frac{1}{3} | 0 | 7,909.875 | -1 | 7,909.875 | |
Find the remainder when the value of $m$ is divided by 1000 in the number of increasing sequences of positive integers $a_1 \le a_2 \le a_3 \le \cdots \le a_6 \le 1500$ such that $a_i-i$ is odd for $1\le i \le 6$. The total number of sequences can be expressed as ${m \choose n}$ for some integers $m>n$. | 752 | 0.0625 | 7,738.1875 | 5,306 | 7,900.333333 | |
Pompous Vova has an iPhone XXX, and on that iPhone, he has a calculator with voice commands: "Multiply my number by two and subtract two from the result," "Multiply my number by three and then add four," and lastly, "Add seven to my number!" The iPhone knows that initially, Vova's number was 1. How many four-digit numb... | 9000 | 0 | 8,192 | -1 | 8,192 | |
Baron Munchausen told a story. "There were a whole crowd of us. We reached a crossroads. Then half of our group turned left, a third turned right, and a fifth went straight." "But wait, the Duke remarked, the sum of half, a third, and a fifth isn't equal to one, so you are lying!" The Baron replied, "I'm not lying, I'... | 37 | 0 | 8,192 | -1 | 8,192 | |
Simplify the expression and then evaluate: $(a-2b)(a^2+2ab+4b^2)-a(a-5b)(a+3b)$, where $a=-1$ and $b=1$. | -21 | 0.75 | 3,678.3125 | 3,958.25 | 2,838.5 | |
The probability it will rain on Saturday is $60\%$, and the probability it will rain on Sunday is $25\%$. If the probability of rain on a given day is independent of the weather on any other day, what is the probability it will rain on both days, expressed as a percent? | 15 | 1 | 1,492 | 1,492 | -1 | |
How many different-looking arrangements are possible when four balls are selected at random from six identical red balls and three identical green balls and then arranged in a line? | 15 | Since 4 balls are chosen from 6 red balls and 3 green balls, then the 4 balls could include: - 4 red balls, or - 3 red balls and 1 green ball, or - 2 red balls and 2 green balls, or - 1 red ball and 3 green balls. There is only 1 different-looking way to arrange 4 red balls. There are 4 different-looking ways to arrang... | 0.6875 | 5,748.1875 | 4,908.181818 | 7,596.2 |
Digits are placed in the two boxes of $2 \square \square$, with one digit in each box, to create a three-digit positive integer. In how many ways can this be done so that the three-digit positive integer is larger than 217? | 82 | The question is equivalent to asking how many three-digit positive integers beginning with 2 are larger than 217. These integers are 218 through 299 inclusive. There are $299 - 217 = 82$ such integers. | 0.5 | 2,586.625 | 4,338.75 | 834.5 |
Given point $M(\sqrt{6}, \sqrt{2})$ on the ellipse $G$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with an eccentricity of $\frac{\sqrt{6}}{3}$.
1. Find the equation of ellipse $G$.
2. If the line $l$ with a slope of $1$ intersects ellipse $G$ at points $A$ and $B$, and an isosceles triangle is formed... | \frac{9}{2} | 0.8125 | 6,165.1875 | 5,697.461538 | 8,192 | |
Keiko tosses one penny and Ephraim tosses two pennies. What is the probability that Ephraim gets the same number of heads that Keiko gets? Express your answer as a common fraction. | \frac{3}{8} | 1 | 2,058.4375 | 2,058.4375 | -1 | |
Four congruent rectangles are placed as shown. The area of the outer square is 4 times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?
[asy] unitsize(6mm); defaultpen(linewidth(.8pt)); path p=(1,1)--(-2,1)--(-2,2)--(1,2); draw(p); draw(... | 3 | 1. **Identify the dimensions of the squares and rectangles**:
Let the side length of the inner square be $s$. Assume the shorter side of each rectangle is $y$ and the longer side is $x$. The rectangles are congruent and placed around the inner square such that their longer sides and shorter sides together form the ... | 0.5 | 5,945.9375 | 4,982 | 6,909.875 |
Andreas, Boyu, Callista, and Diane each randomly choose an integer from 1 to 9, inclusive. Each of their choices is independent of the other integers chosen and the same integer can be chosen by more than one person. The probability that the sum of their four integers is even is equal to \(\frac{N}{6561}\) for some pos... | 78 | 0.9375 | 4,057.5625 | 3,781.933333 | 8,192 | |
Initially, there are some red balls and some black balls in a box. After adding some black balls, the red balls account for one-fourth of the total number of balls. Then, after adding some red balls, the number of red balls becomes two-thirds the number of black balls. If the number of added black balls and red balls i... | 1:2 | 0 | 2,554.875 | -1 | 2,554.875 | |
A geometric sequence of positive integers has a first term of 3, and the fourth term is 243. Find the sixth term of the sequence. | 729 | 0 | 8,192 | -1 | 8,192 | |
A bicycle costs 389 yuan, and an electric fan costs 189 yuan. Dad wants to buy a bicycle and an electric fan. He will need approximately \_\_\_\_\_\_ yuan. | 600 | 0 | 193 | -1 | 193 | |
There are 700 cards in a box, in six colors: red, orange, yellow, green, blue, and white. The ratio of the number of red, orange, and yellow cards is $1: 3: 4$, and the ratio of the number of green, blue, and white cards is $3:1:6$. Given that there are 50 more yellow cards than blue cards, determine the minimum number... | 312 | 0.5 | 7,277.375 | 6,751.875 | 7,802.875 | |
Calculate the product of $1101_2 \cdot 111_2$. Express your answer in base 2. | 1100111_2 | 0 | 6,968.9375 | -1 | 6,968.9375 | |
Let $A B C D$ be a convex trapezoid such that $\angle B A D=\angle A D C=90^{\circ}, A B=20, A D=21$, and $C D=28$. Point $P \neq A$ is chosen on segment $A C$ such that $\angle B P D=90^{\circ}$. Compute $A P$. | \frac{143}{5} | Construct the rectangle $A B X D$. Note that $$\angle B A D=\angle B P D=\angle B X D=90^{\circ}$$ so $A B X P D$ is cyclic with diameter $B D$. By Power of a Point, we have $C X \cdot C D=C P \cdot C A$. Note that $C X=C D-X D=C D-A B=8$ and $C A=\sqrt{A D^{2}+D C^{2}}=35$. Therefore, $$C P=\frac{C X \cdot C D}{C A}=\... | 0.8125 | 6,384.375 | 5,985.461538 | 8,113 |
On a lengthy, one-way, single-lane highway, cars travel at uniform speeds and maintain a safety distance determined by their speed: the separation distance from the back of one car to the front of another is one car length for each 10 kilometers per hour of speed or fraction thereof. Cars are exceptionally long, each 5... | 20 | 0 | 8,192 | -1 | 8,192 | |
We have a rectangle of dimensions $x - 2$ by $2x + 5$ such that its area is $8x - 6$. What is the value of $x$? | 4 | 1 | 1,848.6875 | 1,848.6875 | -1 | |
Suppose [$a$ $b$] denotes the average of $a$ and $b$, and {$a$ $b$ $c$} denotes the average of $a$, $b$, and $c$. What is {{1 1 0} {0 1} 0}? | \frac{7}{18} | 1. **Calculate $\{1, 1, 0\}$**:
The average of $1$, $1$, and $0$ is calculated as follows:
\[
\{1, 1, 0\} = \frac{1 + 1 + 0}{3} = \frac{2}{3}
\]
2. **Calculate $[0, 1]$**:
The average of $0$ and $1$ is calculated as follows:
\[
[0, 1] = \frac{0 + 1}{2} = \frac{1}{2}
\]
3. **Calculate $\{\f... | 0.8125 | 1,554.8125 | 1,774.307692 | 603.666667 |
Find a four-digit number that is a perfect square, in which its digits can be grouped into two pairs of equal digits. | 7744 | 0.5625 | 6,757.125 | 6,005.888889 | 7,723 | |
Ted is solving the equation by completing the square: $$64x^2+48x-36 = 0.$$ He aims to write the equation in a form: $$(ax + b)^2 = c,$$ with \(a\), \(b\), and \(c\) as integers and \(a > 0\). Determine the value of \(a + b + c\). | 56 | 0.875 | 4,620.8125 | 4,110.642857 | 8,192 | |
Given \( a_{n} = 4^{2n - 1} + 3^{n - 2} \) (for \( n = 1, 2, 3, \cdots \)), where \( p \) is the smallest prime number dividing infinitely many terms of the sequence \( a_{1}, a_{2}, a_{3}, \cdots \), and \( q \) is the smallest prime number dividing every term of the sequence, find the value of \( p \cdot q \). | 5 \times 13 | 0 | 7,786.875 | -1 | 7,786.875 | |
The perimeter of a square with side length $y$ inches is equal to the circumference of a circle with radius 5 centimeters. If 1 inch equals 2.54 centimeters, what is the value of $y$ in inches? Express your answer as a decimal to the nearest hundredth. | 3.09 | 0.75 | 5,554.6875 | 4,998.25 | 7,224 | |
If \( a + b + c = 1 \), what is the maximum value of \( \sqrt{3a+1} + \sqrt{3b+1} + \sqrt{3c+1} \)? | 3\sqrt{2} | 0.9375 | 4,654.25 | 4,418.4 | 8,192 | |
From the five points consisting of the four vertices and the center of a square, any two points are chosen. The probability that the distance between these two points is not less than the side length of the square is ______. | \frac{3}{5} | 1 | 4,016 | 4,016 | -1 | |
How many ways can we arrange 4 math books, 6 English books, and 2 Science books on a shelf if:
1. All books of the same subject must stay together.
2. The Science books can be placed in any order, but cannot be placed next to each other.
(The math, English, and Science books are all different.) | 207360 | 0.1875 | 7,200.5625 | 4,487 | 7,826.769231 | |
Compute the circumradius of cyclic hexagon $A B C D E F$, which has side lengths $A B=B C=$ $2, C D=D E=9$, and $E F=F A=12$. | 8 | Construct point $E^{\prime}$ on the circumcircle of $A B C D E F$ such that $D E^{\prime}=E F=12$ and $E^{\prime} F=D E=9$; then $\overline{B E^{\prime}}$ is a diameter. Let $B E^{\prime}=d$. Then $C E^{\prime}=\sqrt{B E^{\prime 2}-B C^{2}}=\sqrt{d^{2}-4}$ and $B D=\sqrt{B E^{\prime 2}-D E^{\prime 2}}=\sqrt{d^{2}-144}$... | 0.125 | 8,032.875 | 6,919 | 8,192 |
In a positive geometric sequence $\{a_{n}\}$, it is known that $a_{1}a_{2}a_{3}=4$, $a_{4}a_{5}a_{6}=8$, and $a_{n}a_{n+1}a_{n+2}=128$. Find the value of $n$. | 16 | 0.875 | 4,350.6875 | 3,801.928571 | 8,192 | |
In triangle $ABC$, $AB = AC = 15$ and $BC = 14$. Points $D, E, F$ are on sides $\overline{AB}, \overline{BC},$ and $\overline{AC},$ respectively, such that $\overline{DE} \parallel \overline{AC}$ and $\overline{EF} \parallel \overline{AB}$. What is the perimeter of parallelogram $ADEF$? | 30 | 0.5625 | 7,054.25 | 6,169.333333 | 8,192 | |
What percent of the positive integers less than or equal to $120$ have no remainders when divided by $6$? | 16.\overline{6}\% | 0.1875 | 5,249.0625 | 4,396.333333 | 5,445.846154 | |
Given a $9 \times 9$ chess board, we consider all the rectangles whose edges lie along grid lines (the board consists of 81 unit squares, and the grid lines lie on the borders of the unit squares). For each such rectangle, we put a mark in every one of the unit squares inside it. When this process is completed, how man... | 56 | 56. Consider the rectangles which contain the square in the $i$th row and $j$th column. There are $i$ possible positions for the upper edge of such a rectangle, $10-i$ for the lower edge, $j$ for the left edge, and $10-j$ for the right edge; thus we have $i(10-i) j(10-j)$ rectangles altogether, which is odd iff $i, j$ ... | 0.125 | 7,939.0625 | 6,168.5 | 8,192 |
An $n \times n$ complex matrix $A$ is called $t$-normal if $A A^{t}=A^{t} A$ where $A^{t}$ is the transpose of $A$. For each $n$, determine the maximum dimension of a linear space of complex $n \times n$ matrices consisting of t-normal matrices. | \[
\frac{n(n+1)}{2}
\] | Answer: The maximum dimension of such a space is $\frac{n(n+1)}{2}$. The number $\frac{n(n+1)}{2}$ can be achieved, for example the symmetric matrices are obviously t-normal and they form a linear space with dimension $\frac{n(n+1)}{2}$. We shall show that this is the maximal possible dimension. Let $M_{n}$ denote the ... | 0 | 7,825.0625 | -1 | 7,825.0625 |
In triangle \( ABC \), side \( AC \) is the largest. Points \( M \) and \( N \) on side \( AC \) are such that \( AM = AB \) and \( CN = CB \). It is known that angle \( \angle NBM \) is three times smaller than angle \( \angle ABC \). Find \( \angle ABC \). | 108 | 0 | 8,192 | -1 | 8,192 | |
Today our cat gave birth to kittens! It is known that the two lightest kittens together weigh 80 g, the four heaviest kittens together weigh 200 g, and the total weight of all the kittens is 500 g. How many kittens did the cat give birth to? | 11 | 0.125 | 3,534.5 | 4,619.5 | 3,379.5 | |
Given $2\cos(2\alpha)=\sin\left(\alpha-\frac{\pi}{4}\right)$ and $\alpha\in\left(\frac{\pi}{2},\pi\right)$, calculate the value of $\cos 2\alpha$. | \frac{\sqrt{15}}{8} | 0 | 6,931.625 | -1 | 6,931.625 | |
Find the last three digits of $7^{103}.$ | 343 | 0.9375 | 4,533.4375 | 4,289.533333 | 8,192 | |
Let $A B C$ be an isosceles triangle with $A B=A C$. Let $D$ and $E$ be the midpoints of segments $A B$ and $A C$, respectively. Suppose that there exists a point $F$ on ray $\overrightarrow{D E}$ outside of $A B C$ such that triangle $B F A$ is similar to triangle $A B C$. Compute $\frac{A B}{B C}$. | \sqrt{2} | Let $\alpha=\angle A B C=\angle A C B, A B=2 x$, and $B C=2 y$, so $A D=D B=A E=E C=x$ and $D E=y$. Since $\triangle B F A \sim \triangle A B C$ and $B A=A C$, we in fact have $\triangle B F A \cong \triangle A B C$, so $B F=B A=2 x, F A=2 y$, and $\angle D A F=\alpha$. But $D E \| B C$ yields $\angle A D F=\angle A B ... | 0.375 | 7,745.375 | 7,001 | 8,192 |
The product of two consecutive integers is $20{,}412$. What is the sum of these two integers? | 287 | 0 | 8,192 | -1 | 8,192 | |
Convert the decimal number 89 to binary. | 1011001 | 0.6875 | 716.25 | 721.272727 | 705.2 | |
The Brookhaven College Soccer Team has 16 players, including 2 as designated goalkeepers. In a training session, each goalkeeper takes a turn in the goal, while every other player on the team gets a chance to shoot a penalty kick. How many penalty kicks occur during the session to allow every player, including the goal... | 30 | 0 | 401.1875 | -1 | 401.1875 | |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, where $b=2$.
$(1)$ If $A+C=120^{\circ}$ and $a=2c$, find the length of side $c$.
$(2)$ If $A-C=15^{\circ}$ and $a=\sqrt{2}c\sin A$, find the area of triangle $\triangle ABC$. | 3 - \sqrt{3} | 0.5 | 7,113.1875 | 6,034.375 | 8,192 | |
What is the value of $102^{4} - 4 \cdot 102^{3} + 6 \cdot 102^2 - 4 \cdot 102 + 1$? | 104060401 | 0.9375 | 4,569.0625 | 4,327.533333 | 8,192 | |
Let us call a ticket with a number from 000000 to 999999 excellent if the difference between some two neighboring digits of its number is 5. Find the number of excellent tickets. | 409510 | 0.0625 | 8,061.9375 | 6,111 | 8,192 | |
Let numbers $x$ and $y$ be chosen independently at random from the intervals $[0, \pi]$ and $[-\frac{\pi}{2}, \frac{\pi}{2}]$, respectively. Define $P(\alpha)$ as the probability that
\[\cos^2{x} + \cos^2{y} < \alpha\]
where $\alpha$ is a constant with $1 < \alpha \leq 2$. Find the maximum value of $P(\alpha)$.
A) $\fr... | \frac{\pi}{2} | 0 | 7,869.4375 | -1 | 7,869.4375 | |
Consider a convex pentagon $FGHIJ$ where $\angle F = \angle G = 100^\circ$. Let $FI = IJ = JG = 3$ and $GH = HF = 5$. Calculate the area of pentagon $FGHIJ$. | \frac{9\sqrt{3}}{4} + 24.62 | 0 | 8,192 | -1 | 8,192 | |
In a class, there are 15 boys and 15 girls. On Women's Day, some boys called some girls to congratulate them (no boy called the same girl more than once). It turned out that the children can be uniquely divided into 15 pairs, such that each pair consists of a boy and a girl whom he called. What is the maximum number of... | 120 | 0 | 7,113.375 | -1 | 7,113.375 | |
When you simplify $\left[ \sqrt [3]{\sqrt [6]{a^9}} \right]^4\left[ \sqrt [6]{\sqrt [3]{a^9}} \right]^4$, the result is: | a^4 | 1. **Simplify each term inside the brackets:**
- For the first term $\sqrt[3]{\sqrt[6]{a^9}}$, we simplify the exponents:
\[
\sqrt[3]{\sqrt[6]{a^9}} = a^{\frac{9}{6} \cdot \frac{1}{3}} = a^{\frac{9}{18}} = a^{\frac{1}{2}}
\]
- For the second term $\sqrt[6]{\sqrt[3]{a^9}}$, we simplify the exponents... | 1 | 3,724 | 3,724 | -1 |
Find the units digit of $n$ given that $mn = 21^6$ and $m$ has a units digit of 7. | 3 | 1 | 1,680 | 1,680 | -1 | |
A trapezoid has side lengths 3, 5, 7, and 11. The sum of all the possible areas of the trapezoid can be written in the form of $r_1\sqrt{n_1}+r_2\sqrt{n_2}+r_3$, where $r_1$, $r_2$, and $r_3$ are rational numbers and $n_1$ and $n_2$ are positive integers not divisible by the square of any prime. What is the greatest in... | 63 | To solve this problem, we need to consider all possible configurations of the trapezoid with sides 3, 5, 7, and 11. We will use Heron's formula to calculate the area of triangles formed by these sides and then determine the area of the trapezoid.
#### Step 1: Identify possible configurations
A trapezoid has one pair o... | 0 | 7,775.5625 | -1 | 7,775.5625 |
An six-digit integer is formed by repeating a positive three-digit integer. For example, 123,123 or 456,456 are integers of this form. What is the greatest common divisor of all six-digit integers of this form? | 1001 | 0.75 | 5,369.875 | 4,429.166667 | 8,192 | |
The sizes of circular pizzas are determined by their diameter. If Lana's initial pizza was 14 inches in diameter and she decides to order a larger pizza with a diameter of 18 inches instead, what is the percent increase in the area of her pizza? | 65.31\% | 0.5625 | 5,553 | 4,786.222222 | 6,538.857143 | |
Given the function $f(x) = \frac{\ln{x}}{x + a}$ where $a \in \mathbb{R}$:
(1) If the tangent to the curve $y = f(x)$ at the point $(1, f(1))$ is perpendicular to the line $x + y + 1 = 0$, find the value of $a$.
(2) Discuss the number of real roots of the equation $f(x) = 1$. | a = 0 | 0.5 | 5,501.3125 | 5,372.25 | 5,630.375 | |
In the diagram, there are more than three triangles. If each triangle has the same probability of being selected, what is the probability that a selected triangle has all or part of its interior shaded? Express your answer as a common fraction.
[asy]
draw((0,0)--(1,0)--(0,1)--(0,0)--cycle,linewidth(1));
draw((0,0)--(.... | \frac{3}{5} | 0 | 7,725.625 | -1 | 7,725.625 | |
Let $c>0$ be a given positive real and $\mathbb{R}_{>0}$ be the set of all positive reals. Find all functions $f \colon \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ such that \[f((c+1)x+f(y))=f(x+2y)+2cx \quad \textrm{for all } x,y \in \mathbb{R}_{>0}.\] | f(x) = 2x |
To solve the functional equation
\[
f((c+1)x + f(y)) = f(x + 2y) + 2cx
\]
for all \( x, y \in \mathbb{R}_{>0} \), we aim to find all functions \( f \colon \mathbb{R}_{>0} \to \mathbb{R}_{>0} \) that satisfy this condition.
### Step 1: Analyze the given functional equation
Consider substituting specific values for... | 0 | 7,832.0625 | -1 | 7,832.0625 |
A polynomial $p(x)$ is called self-centered if it has integer coefficients and $p(100) = 100.$ If $p(x)$ is a self-centered polynomial, what is the maximum number of integer solutions $k$ to the equation $p(k) = k^3$? | 10 | 0 | 8,192 | -1 | 8,192 | |
A math test starts at 12:35 PM and lasts for $4 \frac{5}{6}$ hours. At what time does the test end? | 17:25 | 0 | 482.6875 | -1 | 482.6875 | |
How many three-digit positive integers $x$ satisfy $3874x+481\equiv 1205 \pmod{23}$? | 40 | 0.9375 | 3,255.875 | 3,008.466667 | 6,967 | |
Three male students and three female students, a total of six students, stand in a row. If female students do not stand at the end of the row, and female students A and B are not adjacent to female student C, then find the number of different arrangements. | 144 | 0 | 8,059.9375 | -1 | 8,059.9375 | |
Find the volume of the region in space defined by
\[|x + y + z| + |x + y - z| \le 8\]and $x,$ $y,$ $z \ge 0.$ | 32 | 0.5625 | 7,087.75 | 6,228.888889 | 8,192 | |
The line $l_{1}$: $x+my+6=0$ is parallel to the line $l_{2}$: $(m-2)x+3y+2m=0$. Find the value of $m$. | -1 | 0.5625 | 6,849.3125 | 7,116.222222 | 6,506.142857 | |
From a school of 2100 students, a sample of 30 students is randomly selected. The time (in minutes) each student spends on homework outside of class is as follows:
75, 80, 85, 65, 95, 100, 70, 55, 65, 75, 85, 110, 120, 80, 85, 80, 75, 90, 90, 95, 70, 60, 60, 75, 90, 95, 65, 75, 80, 80. The number of students in this ... | 630 | 0.3125 | 3,965.25 | 4,407.4 | 3,764.272727 | |
Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths $3$ and $4$ units. In the corner where those sides meet at a right angle, he leaves a small unplanted square $S$ so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortes... | \frac{145}{147} | 1. **Identify the vertices and setup the problem**: Let $A$, $B$, and $C$ be the vertices of the right triangle field with $AB = 3$ units, $AC = 4$ units, and $\angle BAC = 90^\circ$. Let $S$ be the square in the corner at $A$, and let its vertices be $A$, $M$, $D$, and $N$ with $AM$ and $AN$ along $AB$ and $AC$ respec... | 0.5 | 7,012.1875 | 5,904 | 8,120.375 |
Given $\alpha $, $\beta \in (0, \frac{π}{2})$, $\sin \alpha = \frac{{\sqrt{5}}}{5}$, $\cos \beta = \frac{1}{{\sqrt{10}}}$, find the value of $\alpha - \beta$. | -\frac{\pi}{4} | 0.9375 | 4,711.5 | 4,479.466667 | 8,192 | |
A box contains 5 white balls and 6 black balls. Five balls are drawn out of the box at random. What is the probability that they all are white? | \dfrac{1}{462} | 1 | 2,503.0625 | 2,503.0625 | -1 | |
Find all functions $f: \mathbb R \to \mathbb R$ such that for any $x,y \in \mathbb R$, the multiset $\{(f(xf(y)+1),f(yf(x)-1)\}$ is identical to the multiset $\{xf(f(y))+1,yf(f(x))-1\}$.
[i]Note:[/i] The multiset $\{a,b\}$ is identical to the multiset $\{c,d\}$ if and only if $a=c,b=d$ or $a=d,b=c$. | f(x) \equiv x \text{ or } f(x) \equiv -x |
Let \( f: \mathbb{R} \to \mathbb{R} \) be a function such that for any \( x, y \in \mathbb{R} \), the multiset \( \{ f(xf(y) + 1), f(yf(x) - 1) \} \) is identical to the multiset \( \{ xf(f(y)) + 1, yf(f(x)) - 1 \} \).
We aim to find all such functions \( f \).
Let \( P(x, y) \) denote the assertion that \( \{ f(xf(... | 0 | 7,567.5625 | -1 | 7,567.5625 |
Given the hyperbola $\dfrac {x^{2}}{a^{2}}- \dfrac {y^{2}}{b^{2}}=1(a > 0,b > 0)$ with eccentricity $e= \dfrac {2 \sqrt {3}}{3}$, calculate the angle between the two asymptotes. | \frac{\pi}{3} | 0.0625 | 4,136.4375 | 2,361 | 4,254.8 | |
Let $x_1$ , $x_2$ , and $x_3$ be the roots of the polynomial $x^3+3x+1$ . There are relatively prime positive integers $m$ and $n$ such that $\tfrac{m}{n}=\tfrac{x_1^2}{(5x_2+1)(5x_3+1)}+\tfrac{x_2^2}{(5x_1+1)(5x_3+1)}+\tfrac{x_3^2}{(5x_1+1)(5x_2+1)}$ . Find $m+n$ . | 10 | 0 | 8,192 | -1 | 8,192 | |
In triangle $PQR$, the sides $PQ$, $QR$, and $RP$ measure 17, 15, and 8 units, respectively. Let $J$ be the incenter of triangle $PQR$. The incircle of triangle $PQR$ touches side $QR$, $RP$, and $PQ$ at points $K$, $L$, and $M$, respectively. Determine the length of $PJ$. | \sqrt{34} | 0.625 | 5,728 | 4,823.8 | 7,235 | |
The complex numbers \( \alpha_{1}, \alpha_{2}, \alpha_{3}, \) and \( \alpha_{4} \) are the four distinct roots of the equation \( x^{4}+2 x^{3}+2=0 \). Determine the unordered set \( \left\{\alpha_{1} \alpha_{2}+\alpha_{3} \alpha_{4}, \alpha_{1} \alpha_{3}+\alpha_{2} \alpha_{4}, \alpha_{1} \alpha_{4}+\alpha_{2} \alpha_... | \{1 \pm \sqrt{5},-2\} | Employing the elementary symmetric polynomials \( \left(s_{1}=\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4}=\right. -2, s_{2}=\alpha_{1} \alpha_{2}+\alpha_{1} \alpha_{3}+\alpha_{1} \alpha_{4}+\alpha_{2} \alpha_{3}+\alpha_{2} \alpha_{4}+\alpha_{3} \alpha_{4}=0, s_{3}=\alpha_{1} \alpha_{2} \alpha_{3}+\alpha_{2} \alpha_{3} ... | 0 | 8,192 | -1 | 8,192 |
In the diagram below, $ABCD$ is a trapezoid such that $\overline{AB}\parallel \overline{CD}$ and $\overline{AC}\perp\overline{CD}$. If $CD = 20$, $\tan D = 2$, and $\tan B = 2.5$, then what is $BC$?
[asy]
pair A,B,C,D;
C = (0,0);
D = (20,0);
A = (20,40);
B= (30,40);
draw(A--B--C--D--A);
label("$A$",A,N);
label("$B$",B... | 4\sqrt{116} | 0 | 5,753.3125 | -1 | 5,753.3125 |
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