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 |
|---|---|---|---|---|---|---|
Chloe wants to buy a hoodie that costs $\$32.75$. She empties her wallet and finds she only has three $\$10$ bills, eight quarters, and a pile of dimes. What is the minimum number of dimes that must be in her pile so she can pay for the hoodie? | 8 | 0.875 | 3,347.125 | 2,655 | 8,192 | |
The rules for a race require that all runners start at $A$, touch any part of the 1500-meter wall, touch any part of the opposite 1500-meter wall, and stop at $B$. What is the minimum distance a participant must run? Assume that $A$ is 400 meters directly south of the first wall, and that $B$ is 600 meters directly nor... | 2915 | 0 | 8,081.6875 | -1 | 8,081.6875 | |
For her zeroth project at Magic School, Emilia needs to grow six perfectly-shaped apple trees. First she plants six tree saplings at the end of Day $0$ . On each day afterwards, Emilia attempts to use her magic to turn each sapling into a perfectly-shaped apple tree, and for each sapling she succeeds in turning it int... | 4910 | 0 | 8,140.125 | -1 | 8,140.125 | |
$f (x) = x + 3$ and $g(x) = x^2 -6$, what is the value of $f (g(2))$? | 1 | 1 | 399.4375 | 399.4375 | -1 | |
Given that the asymptotic line of the hyperbola $\frac{x^2}{a}+y^2=1$ has a slope of $\frac{5π}{6}$, determine the value of $a$. | -3 | 0 | 6,632.6875 | -1 | 6,632.6875 | |
Let $x,$ $y,$ $z$ be nonzero real numbers such that $x + y + z = 0,$ and $xy + xz + yz \neq 0.$ Find all possible values of
\[\frac{x^5 + y^5 + z^5}{xyz (xy + xz + yz)}.\]Enter all possible values, separated by commas. | -5 | 0.875 | 4,136.375 | 3,732.928571 | 6,960.5 | |
There are several pairs of integers $ (a, b) $ satisfying $ a^2 - 4a + b^2 - 8b = 30 $ . Find the sum of the sum of the coordinates of all such points. | 60 | 0 | 5,526.1875 | -1 | 5,526.1875 | |
Given that the terminal side of angle $α$ passes through point $P(\frac{4}{5},-\frac{3}{5})$,
(1) Find the value of $\sin α$;
(2) Find the value of $\frac{\sin (\frac{π}{2}-α)}{\sin (α+π)}-\frac{\tan (α-π)}{\cos (3π-α)}$. | \frac{19}{48} | 0.6875 | 5,306.8125 | 5,423.363636 | 5,050.4 | |
The energy stored by any pair of positive charges is inversely proportional to the distance between them, and directly proportional to their charges. Four identical point charges start at the vertices of a square, and this configuration stores 20 Joules of energy. How much more energy, in Joules, would be stored if one... | 5(3\sqrt{2} - 3) | 0 | 7,919.625 | -1 | 7,919.625 | |
Find the largest natural number whose all digits in its decimal representation are different and which decreases 5 times if you cross out the first digit. | 3750 | 0 | 8,192 | -1 | 8,192 | |
Among all the five-digit numbers formed without repeating any of the digits 0, 1, 2, 3, 4, if they are arranged in ascending order, determine the position of the number 12340. | 10 | 0.0625 | 7,979.375 | 5,495 | 8,145 | |
Given the linear function y=kx+b, where k and b are constants, and the table of function values, determine the incorrect function value. | 12 | 0 | 7,972.3125 | -1 | 7,972.3125 | |
Add $10_7 + 163_7.$ Express your answer in base 7. | 203_7 | 1 | 2,352.3125 | 2,352.3125 | -1 | |
It is known that exactly one of the three (distinguishable) musketeers stole the truffles. Each musketeer makes one statement, in which he either claims that one of the three is guilty, or claims that one of the three is innocent. It is possible for two or more of the musketeers to make the same statement. After hearin... | 99 | We divide into cases, based on the number of distinct people that statements are made about. - The statements are made about 3 distinct people. Then, since exactly one person is guilty, and because exactly one of the three lied, there are either zero statements of guilt or two statements of guilt possible; in either ca... | 0 | 8,192 | -1 | 8,192 |
Fifty students are standing in a line facing the teacher. The teacher first asks everyone to count off from left to right as $1, 2, \cdots, 50$; then asks the students whose numbers are multiples of 3 to turn around, and then asks the students whose numbers are multiples of 7 to turn around. How many students are still... | 31 | 0 | 5,298.3125 | -1 | 5,298.3125 | |
Two reals \( x \) and \( y \) are such that \( x-y=4 \) and \( x^{3}-y^{3}=28 \). Compute \( x y \). | -3 | We have \( 28=x^{3}-y^{3}=(x-y)\left(x^{2}+x y+y^{2}\right)=(x-y)\left((x-y)^{2}+3 x y\right)=4 \cdot(16+3 x y) \), from which \( x y=-3 \). | 1 | 2,134.0625 | 2,134.0625 | -1 |
In how many ways can 4 books be selected from a shelf of 6 books if the order in which the books are selected does not matter? | 15 | 1 | 2,203 | 2,203 | -1 | |
For how many ordered pairs $(b,c)$ of positive integers does neither $x^2+bx+c=0$ nor $x^2+cx+b=0$ have two distinct real solutions? | 6 | To determine the number of ordered pairs $(b,c)$ of positive integers such that neither $x^2+bx+c=0$ nor $x^2+cx+b=0$ have two distinct real solutions, we analyze the discriminants of these quadratic equations.
1. **Analyze the discriminants:**
- For $x^2 + bx + c = 0$, the discriminant must be non-positive for the... | 0.75 | 5,841.625 | 5,058.166667 | 8,192 |
Winnie-the-Pooh eats 3 cans of condensed milk and a jar of honey in 25 minutes, while Piglet eats them in 55 minutes. Pooh eats one can of condensed milk and 3 jars of honey in 35 minutes, while Piglet eats them in 1 hour 25 minutes. How long will it take for them to eat 6 cans of condensed milk together? | 20 | 0.1875 | 7,348.25 | 3,869 | 8,151.153846 | |
The sum of the first n terms of the sequence $-1, 4, -7, 10, \ldots, (-1)^{n}(3n-2)$ is given by $S_{n}$. Calculate $S_{11}+S_{20}$. | 14 | 0.4375 | 7,164.3125 | 5,843 | 8,192 | |
Point $A$ lies on the line $y=\frac{8}{15} x-6$, and point $B$ on the parabola $y=x^{2}$. What is the minimum length of the segment $AB$? | 1334/255 | 0 | 8,192 | -1 | 8,192 | |
What is the area enclosed by the region defined by the equation $x^2+y^2+12x+16y=0$? | 100\pi | 1 | 1,556.3125 | 1,556.3125 | -1 | |
Find the sum of all positive integers $n$ such that $1.5n - 6.3 < 7.5$. | 45 | 1 | 1,699.5 | 1,699.5 | -1 | |
What is the largest $2$-digit prime factor of the integer $n = {300\choose 150}$? | 97 | 0.8125 | 5,212.8125 | 4,525.307692 | 8,192 | |
Find maximum value of number $a$ such that for any arrangement of numbers $1,2,\ldots ,10$ on a circle, we can find three consecutive numbers such their sum bigger or equal than $a$ . | 18 | 0 | 6,660.75 | -1 | 6,660.75 | |
Find the polynomial $p(x),$ with real coefficients, such that $p(2) = 5$ and
\[p(x) p(y) = p(x) + p(y) + p(xy) - 2\]for all real numbers $x$ and $y.$ | x^2 + 1 | 0.625 | 6,784.4375 | 5,939.9 | 8,192 | |
Given that $\alpha$ and $\beta$ are acute angles, $\tan\alpha= \frac {1}{7}$, $\sin\beta= \frac { \sqrt {10}}{10}$, find $\alpha+2\beta$. | \frac {\pi}{4} | 0.3125 | 4,854.5625 | 4,647 | 4,948.909091 | |
Lines $l$ and $m$ are parallel to each other. $m\angle A = 120^\circ$, and $m\angle B = 150^\circ$. What is the number of degrees in $m\angle C$?
[asy]
size(100); real h = 1.2; currentpen = fontsize(10pt);
draw(Label("$l$",Relative(1)),(0,0)--(1,0),E);
draw(Label("$m$",Relative(1)),(0,-h)--(1,-h),E);
draw((0,-h)--h/2*... | 90^\circ | 0.625 | 6,413 | 6,256.6 | 6,673.666667 | |
The difference between two positive integers is 6 and their product is 112. What is the sum of the integers? | 22 | 1 | 1,860.8125 | 1,860.8125 | -1 | |
Let $n$ be a positive integer and $a$ be an integer such that $a$ is its own inverse modulo $n$. What is the remainder when $a^2$ is divided by $n$? | 1 | 0.9375 | 4,971.75 | 4,757.066667 | 8,192 | |
Given two fuses, each of which burns for exactly one minute if lit from one end (but may burn non-uniformly), how can you measure 45 seconds using these fuses? (You can light the fuse from either of its two ends.) | 45 | 0.6875 | 4,824 | 3,293.090909 | 8,192 | |
Segments \(AA', BB'\), and \(CC'\), each of length 2, all intersect at a point \(O\). If \(\angle AOC'=\angle BOA'=\angle COB'=60^{\circ}\), find the maximum possible value of the sum of the areas of triangles \(AOC', BOA'\), and \(COB'\). | \sqrt{3} | Extend \(OA\) to \(D\) and \(OC'\) to \(E\) such that \(AD=OA'\) and \(C'E=OC\). Since \(OD=OE=2\) and \(\angle DOE=60^{\circ}\), we have \(ODE\) is an equilateral triangle. Let \(F\) be the point on \(DE\) such that \(DF=OB\) and \(EF=OB'\). Clearly we have \(\triangle DFA \cong \triangle OBA'\) and \(\triangle EFC' \... | 0 | 8,028.125 | -1 | 8,028.125 |
The circle centered at $(2,-1)$ and with radius $4$ intersects the circle centered at $(2,5)$ and with radius $\sqrt{10}$ at two points $A$ and $B$. Find $(AB)^2$. | 15 | 1 | 3,654.0625 | 3,654.0625 | -1 | |
Find the smallest positive integer \( n \) such that the mean of the squares of the first \( n \) natural numbers (\( n > 1 \)) is an integer.
(Note: The mean of the squares of \( n \) numbers \( a_1, a_2, \cdots, a_n \) is given by \( \sqrt{\frac{a_{1}^2 + a_{2}^2 + \cdots + a_{n}^2}{n}} \).)
(Note: Fifteenth Americ... | 337 | 0.0625 | 8,185.3125 | 8,085 | 8,192 | |
Given that $a > 2b$ ($a, b \in \mathbb{R}$), the range of the function $f(x) = ax^2 + x + 2b$ is $[0, +\infty)$. Determine the minimum value of $$\frac{a^2 + 4b^2}{a - 2b}$$. | \sqrt{2} | 0.3125 | 7,259 | 5,206.4 | 8,192 | |
If $a$, $b$, $c$, and $d$ are the solutions of the equation $x^4 - bx - 3 = 0$, then an equation whose solutions are $\frac{a + b + c}{d^2}$, $\frac{a + b + d}{c^2}$, $\frac{a + c + d}{b^2}$, $\frac{b + c + d}{a^2}$ is | 3x^4 - bx^3 - 1 = 0 | 1. **Identify the roots of the given polynomial**: Given the polynomial $x^4 - bx - 3 = 0$, we know that its roots are $a$, $b$, $c$, and $d$.
2. **Use Vieta's formulas**: Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. For the polynomial $x^4 - bx - 3 = 0$, the sum of the r... | 0.1875 | 8,013.5625 | 7,240.333333 | 8,192 |
Let $S=\{(x, y) \mid x>0, y>0, x+y<200$, and $x, y \in \mathbb{Z}\}$. Find the number of parabolas $\mathcal{P}$ with vertex $V$ that satisfy the following conditions: - $\mathcal{P}$ goes through both $(100,100)$ and at least one point in $S$, - $V$ has integer coordinates, and - $\mathcal{P}$ is tangent to the line $... | 264 | We perform the linear transformation $(x, y) \rightarrow(x-y, x+y)$, which has the reverse transformation $(a, b) \rightarrow\left(\frac{a+b}{2}, \frac{b-a}{2}\right)$. Then the equivalent problem has a parabola has a vertical axis of symmetry, goes through $A=(0,200)$, a point $B=(u, v)$ in $S^{\prime}=\{(x, y) \mid x... | 0 | 7,961 | -1 | 7,961 |
We define a function $g(x)$ such that $g(12)=37$, and if there exists an integer $a$ such that $g(a)=b$, then $g(b)$ is defined and follows these rules:
1. $g(b)=3b+1$ if $b$ is odd
2. $g(b)=\frac{b}{2}$ if $b$ is even.
What is the smallest possible number of integers in the domain of $g$? | 23 | 0.375 | 7,254.1875 | 5,691.166667 | 8,192 | |
The value of $\sin 210^\circ$ is equal to $\frac{-\sqrt{3}}{2}$. | -\frac{1}{2} | 0.5625 | 5,865.875 | 5,464.111111 | 6,382.428571 | |
Calculate: $\left(-2\right)^{0}-3\tan 30^{\circ}-|\sqrt{3}-2|$. | -1 | 0.9375 | 644.4375 | 646.133333 | 619 | |
Find the product of the greatest common divisor (gcd) and the least common multiple (lcm) of 225 and 252. | 56700 | 0.8125 | 1,345.125 | 1,493.461538 | 702.333333 | |
Given the function $f(x)=2\ln x+8x$, find the value of $\lim_{n\to\infty} \frac{f(1-2\Delta x)-f(1)}{\Delta x}$ ( ). | -20 | 0.75 | 5,530.875 | 4,707.5 | 8,001 | |
As shown below, there is a $40\times30$ paper with a filled $10\times5$ rectangle inside of it. We want to cut out the filled rectangle from the paper using four straight cuts. Each straight cut is a straight line that divides the paper into two pieces, and we keep the piece containing the filled rectangle. The goal is... | 65 |
We are given a \( 40 \times 30 \) rectangle (the paper) with a filled \( 10 \times 5 \) rectangle inside it. The objective is to cut out the filled rectangle using four straight cuts with the aim of minimizing the total length of the cuts. Each cut divides the remaining paper into two pieces, and we keep the piece con... | 0 | 7,859.0625 | -1 | 7,859.0625 |
After the appearance of purple sand flowerpots in the Ming and Qing dynasties, their development momentum was soaring, gradually becoming the collection target of collectors. With the development of pot-making technology, purple sand flowerpots have been integrated into the daily life of ordinary people. A certain purp... | 39 | 0.0625 | 7,823.6875 | 7,716 | 7,830.866667 | |
If $α∈(0, \dfrac{π}{2})$, $\cos ( \dfrac{π}{4}-α)=2 \sqrt{2}\cos 2α$, then $\sin 2α=$____. | \dfrac{15}{16} | 0.5625 | 6,299.3125 | 4,827.222222 | 8,192 | |
Given that point $(a, b)$ moves on the line $x + 2y + 3 = 0$, find the maximum or minimum value of $2^a + 4^b$. | \frac{\sqrt{2}}{2} | 0 | 4,600.1875 | -1 | 4,600.1875 | |
a) Vanya flips a coin 3 times, and Tanya flips a coin 2 times. What is the probability that Vanya gets more heads than Tanya?
b) Vanya flips a coin $n+1$ times, and Tanya flips a coin $n$ times. What is the probability that Vanya gets more heads than Tanya? | \frac{1}{2} | 0.1875 | 7,688.875 | 5,508.666667 | 8,192 | |
Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if neces... | 13 | To solve this problem, we need to find the number of steps in the staircase such that Dash takes 19 fewer jumps than Cozy. Let $n$ be the total number of steps in the staircase. Cozy jumps 2 steps at a time, and Dash jumps 5 steps at a time. We can express the number of jumps each takes as follows:
- Cozy's jumps: $\le... | 0.3125 | 7,701.625 | 6,622.8 | 8,192 |
A foreign investor plans to invest in 3 different projects among 4 candidate cities, with no more than 2 projects in the same city. How many different investment plans are there? | 60 | 0.8125 | 5,234.6875 | 4,552.230769 | 8,192 | |
In right triangle $DEF$, $DE=15$, $DF=9$, and $EF=12$ units. What is the distance from $F$ to the midpoint of segment $DE$? | 7.5 | 0.6875 | 3,361.375 | 3,277.181818 | 3,546.6 | |
Given the set $P={x|1≦x≦6,x∈N}$, for its non-empty subset $A$, multiply each element $k$ in $A$ by $(-1)^k$ and then sum them up. (For example, if $A={1,3,6}$, the sum would be $(-1)⋅1+(-1)^3⋅3+(-1)^6⋅6=2$.) The total sum of these sums for all non-empty subsets of $M$ is \_\_\_\_\_\_. | 96 | 0.5 | 6,426.5 | 4,661 | 8,192 | |
For a certain natural number $n$, $n^2$ gives a remainder of 4 when divided by 5, and $n^3$ gives a remainder of 2 when divided by 5. What remainder does $n$ give when divided by 5? | 3 | 1 | 1,704.9375 | 1,704.9375 | -1 | |
The real numbers \( x_{1}, x_{2}, \cdots, x_{2001} \) satisfy \( \sum_{k=1}^{2000}\left|x_{k}-x_{k+1}\right| = 2001 \). Let \( y_{k} = \frac{1}{k} \left( x_{1} + x_{2} + \cdots + x_{k} \right) \) for \( k = 1, 2, \cdots, 2001 \). Find the maximum possible value of \( \sum_{k=1}^{2000} \left| y_{k} - y_{k+1} \right| \). | 2000 | 0 | 8,192 | -1 | 8,192 | |
Alice draws three cards from a standard 52-card deck with replacement. Ace through 10 are worth 1 to 10 points respectively, and the face cards King, Queen, and Jack are each worth 10 points. The probability that the sum of the point values of the cards drawn is a multiple of 10 can be written as $\frac{m}{n}$, where $... | 26597 | The probability that all three cards drawn are face cards is $\left(\frac{3}{13}\right)^{3}=\frac{27}{2197}$. In that case, the sum is 30 and therefore a multiple of 10 . Otherwise, one of the cards is not a face card, so its point value $p$ is drawn uniformly from values from 1 to 10 . The sum of the values of the oth... | 0 | 8,108.3125 | -1 | 8,108.3125 |
Let $ABCD$ be a square and $X$ a point such that $A$ and $X$ are on opposite sides of $CD$ . The lines $AX$ and $BX$ intersect $CD$ in $Y$ and $Z$ respectively. If the area of $ABCD$ is $1$ and the area of $XYZ$ is $\frac{2}{3}$ , determine the length of $YZ$ | \frac{2}{3} | 0.5625 | 6,904.6875 | 6,230.222222 | 7,771.857143 | |
Let $z_1$, $z_2$, $z_3$, $\dots$, $z_{16}$ be the 16 zeros of the polynomial $z^{16} - 16^{4}$. For each $j$, let $w_j$ be one of $z_j$ or $iz_j$. Find the maximum possible value of the real part of
\[\sum_{j = 1}^{16} w_j.\] | 16 | 0 | 8,192 | -1 | 8,192 | |
Given vectors $\overrightarrow{a} = (5\sqrt{3}\cos x, \cos x)$ and $\overrightarrow{b} = (\sin x, 2\cos x)$, and the function $f(x) = \overrightarrow{a} \cdot \overrightarrow{b} + |\overrightarrow{b}|^2 + \frac{3}{2}$.
(I) Find the range of $f(x)$ when $x \in [\frac{\pi}{6}, \frac{\pi}{2}]$.
(II) If $f(x) = 8$ when $x ... | \frac{3\sqrt{3}}{2} + 7 | 0 | 6,987.5 | -1 | 6,987.5 | |
Let $\omega_{1}$ be a circle of radius 5, and let $\omega_{2}$ be a circle of radius 2 whose center lies on $\omega_{1}$. Let the two circles intersect at $A$ and $B$, and let the tangents to $\omega_{2}$ at $A$ and $B$ intersect at $P$. If the area of $\triangle ABP$ can be expressed as $\frac{a \sqrt{b}}{c}$, where $... | 19285 | Let $O_{1}$ and $O_{2}$ be the centers of $\omega_{1}$ and $\omega_{2}$, respectively. Because $$\angle O_{2}AP+\angle O_{2}BP=90^{\circ}+90^{\circ}=180^{\circ}$$ quadrilateral $O_{2}APB$ is cyclic. But $O_{2}, A$, and $B$ lie on $\omega_{1}$, so $P$ lies on $\omega_{1}$ and $O_{2}P$ is a diameter of $\omega_{1}$. From... | 0.625 | 6,457.625 | 6,064.9 | 7,112.166667 |
If $α$ is an acute angle, and $\sin (α - \frac{π}{4})= \frac{3}{5}$, then $\cos 2α$ equals ______. | -\frac{24}{25} | 0.375 | 6,456.3125 | 3,569.166667 | 8,188.6 | |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\frac{1}{2}c\sin B=(c-a\cos B)\sin C$.
$(1)$ Find angle $A$;
$(2)$ If $D$ is a point on side $AB$ such that $AD=2DB$, $AC=2$, and $BC=\sqrt{7}$, find the area of triangle $\triangle ACD$. | \sqrt{3} | 1 | 4,382.1875 | 4,382.1875 | -1 | |
The diagonal of a particular square is 5 inches. The diameter of a particular circle is also 5 inches. By how many square inches is the area of the circle greater than the area of square? Express your answer as a decimal to the nearest tenth. [asy]
draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);
draw((2,0)--(0,2));
draw(c... | 7.1 | 1 | 1,752.3125 | 1,752.3125 | -1 | |
Find the sum of all four-digit numbers in which the digits $0, 3, 6, 9$ are absent. | 6479352 | 0.75 | 5,446.4375 | 4,569.916667 | 8,076 | |
We say a triple $\left(a_{1}, a_{2}, a_{3}\right)$ of nonnegative reals is better than another triple $\left(b_{1}, b_{2}, b_{3}\right)$ if two out of the three following inequalities $a_{1}>b_{1}, a_{2}>b_{2}, a_{3}>b_{3}$ are satisfied. We call a triple $(x, y, z)$ special if $x, y, z$ are nonnegative and $x+y+z=1$. ... | n \geq 4 | The answer is $n \geqslant 4$. Consider the following set of special triples $$\left(0, \frac{8}{15}, \frac{7}{15}\right), \quad\left(\frac{2}{5}, 0, \frac{3}{5}\right), \quad\left(\frac{3}{5}, \frac{2}{5}, 0\right), \quad\left(\frac{2}{15}, \frac{11}{15}, \frac{2}{15}\right)$$ We will prove that any special triple $(x... | 0 | 8,192 | -1 | 8,192 |
(13 points) A rental company owns 100 cars. When the monthly rent for each car is set at 3000 yuan, all cars can be rented out. For every 50 yuan increase in the monthly rent per car, there will be one more car that remains unrented. The maintenance cost for each rented car is 150 yuan per month, and for each unrented ... | 307050 | 0.5625 | 5,171.6875 | 4,105.555556 | 6,542.428571 | |
Compute: \(103 \times 97\). | 9991 | 1 | 456 | 456 | -1 | |
Eight students from a university are planning to carpool for a trip, with two students from each of the grades one, two, three, and four. How many ways are there to arrange the four students in car A, such that the last two students are from the same grade? | 24 | 0 | 7,552.875 | -1 | 7,552.875 | |
What is the probability of rolling four standard, six-sided dice and getting at least three distinct numbers, with at least one die showing a '6'? Express your answer as a common fraction. | \frac{5}{18} | 0.0625 | 7,245.1875 | 3,243 | 7,512 | |
Let $\theta$ be the smallest acute angle for which $\sin \theta,$ $\sin 2 \theta,$ $\sin 3 \theta$ form an arithmetic progression, in some order. Find $\cos \theta.$ | \frac{3}{4} | 0.3125 | 7,401.375 | 6,485 | 7,817.909091 | |
Rectangle $ABCD$ is the base of pyramid $PABCD$. If $AB = 8$, $BC = 4$, $\overline{PA}\perp \overline{AD}$, $\overline{PA}\perp \overline{AB}$, and $PB = 17$, then what is the volume of $PABCD$? | 160 | 1 | 1,626.3125 | 1,626.3125 | -1 | |
The sequence $3, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, ...$ consists of $3$’s separated by blocks of $2$’s with $n$ $2$’s in the $n^{th}$ block. Calculate the sum of the first $1024$ terms of this sequence.
A) $4166$
B) $4248$
C) $4303$
D) $4401$ | 4248 | 0 | 8,192 | -1 | 8,192 | |
Find all ordered triples $(a, b, c)$ of positive reals that satisfy: $\lfloor a\rfloor b c=3, a\lfloor b\rfloor c=4$, and $a b\lfloor c\rfloor=5$, where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$. | \left(\frac{\sqrt{30}}{3}, \frac{\sqrt{30}}{4}, \frac{2 \sqrt{30}}{5}\right),\left(\frac{\sqrt{30}}{3}, \frac{\sqrt{30}}{2}, \frac{\sqrt{30}}{5}\right) | Write $p=a b c, q=\lfloor a\rfloor\lfloor b\rfloor\lfloor c\rfloor$. Note that $q$ is an integer. Multiplying the three equations gives: $$p=\sqrt{\frac{60}{q}}$$ Substitution into the first equation, $$p=3 \frac{a}{\lfloor a\rfloor}<3 \frac{\lfloor a\rfloor+1}{\lfloor a\rfloor} \leq 6$$ Looking at the last equation: $... | 0 | 8,147.625 | -1 | 8,147.625 |
In an arithmetic sequence $\{a_n\}$, it is known that $a_1 + a_3 = 0$ and $a_2 + a_4 = -2$. Find the sum of the first 10 terms of the sequence $\left\{ \frac{a_n}{2^{n-1}} \right\}$. | \frac{5}{256} | 0.25 | 7,696.5625 | 6,210.25 | 8,192 | |
Given the function $f(x) = \frac{e^x - 1}{e^x + 1}$, let $g(x) = f(x - 1) + 1$. Define the sequence $\{a_n\}$ such that $a_n = g\left(\frac{1}{n}\right) + g\left(\frac{2}{n}\right) + g\left(\frac{3}{n}\right) + \dots + g\left(\frac{2n - 1}{n}\right)$, where $n$ is a positive integer. The sum of the first $n$ terms of s... | k = 18 | 0.0625 | 8,047.0625 | 5,873 | 8,192 | |
We suppose that $AB=1$, and that the oblique segments form an angle of $45^{\circ}$ with respect to $(AB)$. There are $n$ vertices above $(AB)$.
What is the length of the broken line? | \sqrt{2} | 0.4375 | 7,040.5625 | 5,560.142857 | 8,192 | |
In triangle \( \triangle ABC \), point \( E \) is on side \( AB \) with \( AE = 1 \) and \( EB = 2 \). Suppose points \( D \) and \( F \) are on sides \( AC \) and \( BC \) respectively, and \( DE \parallel BC \) and \( EF \parallel AC \). What is the ratio of the area of quadrilateral \( CDEF \) to the area of triangl... | 4: 9 | 0 | 6,377.4375 | -1 | 6,377.4375 | |
On a $3 \times 3$ board the numbers from $1$ to $9$ are written in some order and without repeating. We say that the arrangement obtained is *Isthmian* if the numbers in any two adjacent squares have different parity. Determine the number of different Isthmian arrangements.
Note: Two arrangements are considered ... | 720 | 0.5625 | 6,929.3125 | 6,006.666667 | 8,115.571429 | |
There are 5 female students and 2 male students in a class. Find the number of different distribution schemes in which they can be divided into two groups, with each group having both female and male students. | 60 | 0.0625 | 8,092 | 8,192 | 8,085.333333 | |
We will call a ticket with a number from 000000 to 999999 excellent if the difference between some two adjacent digits of its number is 5.
Find the number of excellent tickets. | 409510 | 0 | 8,192 | -1 | 8,192 | |
Given that $| \overrightarrow{a}|=2$, $\overrightarrow{e}$ is a unit vector, and the angle between $\overrightarrow{a}$ and $\overrightarrow{e}$ is $\dfrac {\pi}{3}$, find the projection of $\overrightarrow{a}+ \overrightarrow{e}$ on $\overrightarrow{a}- \overrightarrow{e}$. | \sqrt {3} | 0 | 3,834.0625 | -1 | 3,834.0625 | |
If digits $A$ , $B$ , and $C$ (between $0$ and $9$ inclusive) satisfy
\begin{tabular}{c@{\,}c@{\,}c@{\,}c}
& $C$ & $C$ & $A$
+ & $B$ & $2$ & $B$ \hline
& $A$ & $8$ & $8$
\end{tabular}
what is $A \cdot B \cdot C$ ?
*2021 CCA Math Bonanza Individual Round #5* | 42 | 0.9375 | 3,584.875 | 3,277.733333 | 8,192 | |
A coordinate system is established with the origin as the pole and the positive half of the x-axis as the polar axis. Given the curve $C_1: (x-2)^2 + y^2 = 4$, point A has polar coordinates $(3\sqrt{2}, \frac{\pi}{4})$, and the polar coordinate equation of line $l$ is $\rho \cos (\theta - \frac{\pi}{4}) = a$, with poin... | 2\sqrt{2} | 0.3125 | 7,384.9375 | 6,757 | 7,670.363636 | |
Sierpinski's triangle is formed by taking a triangle, and drawing an upside down triangle inside each upright triangle that appears. A snake sees the fractal, but decides that the triangles need circles inside them. Therefore, she draws a circle inscribed in every upside down triangle she sees (assume that the snake ca... | \frac{\pi}{12} | 0.3125 | 6,828.9375 | 5,216.4 | 7,561.909091 | |
Find the number of permutations $x_1, x_2, x_3, x_4, x_5$ of numbers $1, 2, 3, 4, 5$ such that the sum of five products \[x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_1 + x_5x_1x_2\] is divisible by $3$. | 80 | Since $3$ is one of the numbers, a product with a $3$ in it is automatically divisible by $3,$ so WLOG $x_3=3,$ we will multiply by $5$ afterward since any of $x_1, x_2, \ldots, x_5$ would be $3,$ after some cancelation we see that now all we need to find is the number of ways that $x_5x_1(x_4+x_2)$ is divisible by $3,... | 0 | 8,192 | -1 | 8,192 |
The minimum positive period and the minimum value of the function $y=2\sin(2x+\frac{\pi}{6})+1$ are \_\_\_\_\_\_ and \_\_\_\_\_\_, respectively. | -1 | 1 | 2,031.4375 | 2,031.4375 | -1 | |
A farmer builds a rectangular chicken coop that leans against a wall, covering an area of 36m<sup>2</sup>. Due to geographical constraints, the length of the side of the chicken coop, $x$, cannot exceed 7m. The wall height is 2m. The cost of constructing the front of the chicken coop is 40 yuan/m<sup>2</sup>, the cost ... | 2760 | 0.0625 | 7,888.75 | 4,112 | 8,140.533333 | |
Given the hyperbola $\frac{x^{2}}{a-3} + \frac{y^{2}}{2-a} = 1$, with foci on the $y$-axis and a focal distance of $4$, determine the value of $a$. The options are:
A) $\frac{3}{2}$
B) $5$
C) $7$
D) $\frac{1}{2}$ | \frac{1}{2} | 0 | 5,369.8125 | -1 | 5,369.8125 | |
Mitya is 11 years older than Shura. When Mitya was as old as Shura is now, he was twice as old as she was. How old is Mitya? | 27.5 | 0 | 1,784.0625 | -1 | 1,784.0625 | |
Rectangle $ABCD$ has area $4032$. An ellipse with area $4032\pi$ passes through points $A$ and $C$ and has foci at points $B$ and $D$. Determine the perimeter of the rectangle. | 8\sqrt{2016} | 0 | 7,303.125 | -1 | 7,303.125 | |
A triangle has side lengths of 8, 15 and 17 units. What is the area of the triangle, in square units? | 60 | 1 | 2,260.875 | 2,260.875 | -1 | |
If \(\frac{5+6+7+8}{4} = \frac{2014+2015+2016+2017}{N}\), calculate the value of \(N\). | 1240 | 0.125 | 3,066.625 | 524.5 | 3,429.785714 | |
Given four circles with radii $2, 4, 6,$ and $8$ tangent to two perpendicular lines $\ell_1$ and $\ell_2$ intersecting at point $A$, and region $S$ consisting of all the points that lie inside exactly one of these four circles, find the maximum possible area of region $S$. | 120\pi | 0.25 | 7,586.3125 | 5,769.25 | 8,192 | |
A cowboy is initially 6 miles south and 2 miles west of a stream that flows due northeast. His cabin is located 12 miles east and 9 miles south of his initial position. He wants to water his horse at the stream and then return to his cabin. What is the shortest distance he can travel to accomplish this?
A) $\sqrt{289} ... | 8 + \sqrt{545} | 0 | 8,192 | -1 | 8,192 | |
In the expansion of \((x+y+z)^{8}\), what is 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 | 7,712.125 | -1 | 7,712.125 | |
A store sells a batch of football souvenir books, with a cost price of $40$ yuan per book and a selling price of $44$ yuan per book. The store can sell 300 books per day. The store decides to increase the selling price, and after investigation, it is found that for every $1$ yuan increase in price, the daily sales decr... | 2640 | 0.8125 | 5,003.375 | 5,125.846154 | 4,472.666667 | |
Given the function $f(x)=\sqrt{2}\sin(2\omega x-\frac{\pi}{12})+1$ ($\omega > 0$) has exactly $3$ zeros in the interval $\left[0,\pi \right]$, determine the minimum value of $\omega$. | \frac{5}{3} | 0 | 8,192 | -1 | 8,192 | |
Let $a,$ $b,$ and $c$ be nonnegative real numbers such that $a + b + c = 1.$ Find the maximum value of
\[a + \sqrt{ab} + \sqrt[3]{abc}.\] | \frac{4}{3} | 0 | 8,192 | -1 | 8,192 | |
Find $2^{-1} \pmod{185}$, as a residue modulo 185. (Give an answer between 0 and 184, inclusive.) | 93 | 1 | 2,696.375 | 2,696.375 | -1 | |
Two standard six-sided dice are rolled. What is the probability that the product of the two numbers rolled is 12? | \frac{4}{36} | When two standard six-sided dice are rolled, there are $6 \times 6 = 36$ possibilities for the pair of numbers that are rolled. Of these, the pairs $2 \times 6, 3 \times 4, 4 \times 3$, and $6 \times 2$ each give 12. (If one of the numbers rolled is 1 or 5, the product cannot be 12.) Since there are 4 pairs of possible... | 0 | 3,046.125 | -1 | 3,046.125 |
Given a tetrahedron O-ABC, where $\angle BOC=90^\circ$, $OA \perpendicular$ plane BOC, and $AB= \sqrt{10}$, $BC= \sqrt{13}$, $AC= \sqrt{5}$. Points O, A, B, and C are all on the surface of sphere S. Find the surface area of sphere S. | 14\pi | 0.4375 | 6,711.5 | 4,808 | 8,192 | |
A point $ M$ is taken on the perpendicular bisector of the side $ AC$ of an acute-angled triangle $ ABC$ so that $ M$ and $ B$ are on the same side of $ AC$ . If $ \angle BAC\equal{}\angle MCB$ and $ \angle ABC\plus{}\angle MBC\equal{}180^{\circ}$ , find $ \angle BAC.$ | 30 | 0 | 7,971.25 | -1 | 7,971.25 |
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