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 |
|---|---|---|---|---|---|---|
Find the number of eight-digit positive integers that are multiples of 9 and have all distinct digits. | 181440 | Note that $0+1+\cdots+9=45$. Consider the two unused digits, which must then add up to 9. If it's 0 and 9, there are $8 \cdot 7!$ ways to finish; otherwise, each of the other four pairs gives $7 \cdot 7!$ ways to finish, since 0 cannot be the first digit. This gives a total of $36 \cdot 7!=181440$. | 0.4375 | 6,777.6875 | 5,141 | 8,050.666667 |
For all non-zero real numbers $x$ and $y$ such that $x-y=xy$, $\frac{1}{x}-\frac{1}{y}$ equals | -1 | 1. **Start with the given equation and manipulate it:**
Given that \(x - y = xy\), we can rearrange this equation to:
\[
x - y - xy = 0
\]
Adding 1 to both sides, we get:
\[
x - y - xy + 1 = 1
\]
Factoring the left-hand side, we have:
\[
(x-1)(y+1) = 1
\]
2. **Analyze the expression... | 1 | 2,244.5625 | 2,244.5625 | -1 |
$ABCD$ is a rectangular sheet of paper. Points $E$ and $F$ are located on edges $AB$ and $CD$, respectively, such that $BE < CF$. The rectangle is folded over line $EF$ so that point $C$ maps to $C'$ on side $AD$ and point $B$ maps to $B'$ on side $AD$ such that $\angle{AB'C'} \cong \angle{B'EA}$ and $\angle{B'C'A} = 9... | 57 | 0 | 8,192 | -1 | 8,192 | |
Let $\alpha$ and $\beta$ be conjugate complex numbers such that $\frac{\alpha}{\beta^3}$ is a real number and $|\alpha - \beta| = 6$. Find $|\alpha|$. | 3\sqrt{2} | 0.25 | 7,608.0625 | 7,108.25 | 7,774.666667 | |
When Scott completes the square on the quadratic $x^2 + 8x - 1 = 0$, he obtains an equation of the form $(x + a)^2 = b$. What is $b$? | 17 | 1 | 1,341.125 | 1,341.125 | -1 | |
Given that $\{a_n\}$ is an arithmetic sequence, if $\frac{a_{11}}{a_{10}} < -1$ and its sum of the first $n$ terms, $S_n$, has a maximum value, find the value of $n$ when $S_n$ takes the minimum positive value. | 19 | 0 | 8,192 | -1 | 8,192 | |
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 is not rented out. The maintenance cost for each rented car is 150 yuan per month, and for each car not rented out, ... | 4050 | 0.75 | 5,165.125 | 4,610.333333 | 6,829.5 | |
What is the largest $2$-digit prime factor of the integer $n = {200\choose 100}$? | 61 | Expanding the binomial coefficient, we get ${200 \choose 100}=\frac{200!}{100!100!}$. Let the required prime be $p$; then $10 \le p < 100$. If $p > 50$, then the factor of $p$ appears twice in the denominator. Thus, we need $p$ to appear as a factor at least three times in the numerator, so $3p<200$. The largest such p... | 0.75 | 6,695.125 | 6,521.666667 | 7,215.5 |
A swimming pool can be filled by any of three hoses A, B or C. Hoses A and B together take 4 hours to fill the pool. Hoses A and C together take 5 hours to fill the pool. Hoses B and C together take 6 hours to fill the pool. How many hours does it take hoses A, B and C working together to fill the pool? Express your an... | 3.24 | 1 | 3,732.1875 | 3,732.1875 | -1 | |
Ella walks to her university library, averaging 80 steps per minute, with each of her steps covering 80 cm. It takes her 20 minutes to get to the library. Her friend Tia, going to the same library by the same route, averages 120 steps per minute, but her steps are only 70 cm long. Calculate the time it takes Tia to rea... | 15.24 | 0.6875 | 720.1875 | 737.818182 | 681.4 | |
There are four identical balls numbered $1$, $2$, $3$, $4$, and four boxes also numbered $1$, $2$, $3$, $4$.
$(1)$ If each box contains one ball, find the number of ways such that exactly one box has the same number as the ball inside it.
$(2)$ Find the number of ways such that exactly one box is empty. | 144 | 0.3125 | 7,354.3125 | 7,060.8 | 7,487.727273 | |
Circles \(P\), \(Q\), and \(R\) are externally tangent to each other and internally tangent to circle \(S\). Circles \(Q\) and \(R\) are congruent. Circle \(P\) has radius 2 and passes through the center of \(S\). What is the radius of circle \(Q\)? | \frac{16}{9} | 0.5625 | 6,405.375 | 5,015.777778 | 8,192 | |
Given that $\{a_{n}\}$ is a geometric sequence, and $a_{3}$ and $a_{7}$ are two roots of the equation $x^{2}+4x+1=0$, calculate the value of $a_{5}$. | -1 | 0.9375 | 5,660.625 | 5,491.866667 | 8,192 | |
How many times does the digit 9 appear in the list of all integers from 1 to 1000? | 300 | 0.375 | 5,762.125 | 4,170.5 | 6,717.1 | |
A circle inscribed in a square has two chords as shown in a pair. It has radius 2, and $P$ bisects $T U$. The chords' intersection is where? Answer the question by giving the distance of the point of intersection from the center of the circle. | 2\sqrt{2} - 2 | The point lies between $X$ and $Q$. Then $M N X Q$ is a parallelogram. For, $O B \| N M$ by homothety at $C$ and $P M \| N X$ because $M N X P$ is an isoceles trapezoid. It follows that $Q X=M N$. Considering that the center of the circle together with points $M, C$, and $N$ determines a square of side length 2, it fol... | 0 | 7,657.125 | -1 | 7,657.125 |
What is the maximum possible product of three different numbers from the set $\{-9, -7, -2, 0, 4, 6, 8\}$, where the product contains exactly one negative number? | -96 | 0.0625 | 6,065.0625 | 6,557 | 6,032.266667 | |
For how many integers $x$ is the number $x^4-51x^2+50$ negative? | 12 | 1. **Substitute and Simplify**: Let $x^2 = u$. Then the expression $x^4 - 51x^2 + 50$ becomes $u^2 - 51u + 50$.
2. **Factorization**: Factor the quadratic expression:
\[
u^2 - 51u + 50 = (u - 1)(u - 50)
\]
This factorization is possible because the roots of the quadratic equation $u^2 - 51u + 50 = 0$ are $... | 0.875 | 4,766.375 | 4,545.571429 | 6,312 |
Evaluate $\left\lfloor \left\lceil \left(\frac{13}{7}\right)^2\right\rceil+\frac{17}{4}\right\rfloor$. | 8 | 0.625 | 5,785.9375 | 4,342.3 | 8,192 | |
What is $2.43-1.2$? | 1.23 | 1 | 331.375 | 331.375 | -1 | |
In writing the integers from 100 through 199 inclusive, how many times is the digit 7 written? | 20 | 0.375 | 6,987.5 | 5,956 | 7,606.4 | |
The closed curve in the figure is made up of 9 congruent circular arcs each of length $\frac{2\pi}{3}$, where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side 2. What is the area enclosed by the curve? | \pi + 6\sqrt{3} | 1. **Understanding the Geometry**: The problem describes a closed curve made up of 9 congruent circular arcs, each with a length of $\frac{2\pi}{3}$. These arcs are centered at the vertices of a regular hexagon with side length 2.
2. **Calculating the Radius of the Arcs**: The length of each arc is given by the formul... | 0 | 8,087.5625 | -1 | 8,087.5625 |
A square pyramid with base $ABCD$ and vertex $E$ has eight edges of length $4$. A plane passes through the midpoints of $AE$, $BC$, and $CD$. The plane's intersection with the pyramid has an area that can be expressed as $\sqrt{p}$. Find $p$. | 80 | [asy]import three; import math; pointpen = black; pathpen = black+linewidth(0.7); currentprojection = perspective(2.5,-12,4); triple A=(-2,2,0), B=(2,2,0), C=(2,-2,0), D=(-2,-2,0), E=(0,0,2*2^.5), P=(A+E)/2, Q=(B+C)/2, R=(C+D)/2, Y=(-3/2,-3/2,2^.5/2),X=(3/2,3/2,2^.5/2), H=(4,2,0), I=(-2,-4,0); draw(A--B--C--D--A--E--B-... | 0 | 7,720.75 | -1 | 7,720.75 |
Let $f$ be a monic cubic polynomial satisfying $f(x)+f(-x)=0$ for all real numbers $x$. For all real numbers $y$, define $g(y)$ to be the number of distinct real solutions $x$ to the equation $f(f(x))=y$. Suppose that the set of possible values of $g(y)$ over all real numbers $y$ is exactly $\{1,5,9\}$. Compute the sum... | 970 | We claim that we must have $f(x)=x^{3}-3 x$. First, note that the condition $f(x)+f(-x)=0$ implies that $f$ is odd. Combined with $f$ being monic, we know that $f(x)=x^{3}+a x$ for some real number $a$. Note that $a$ must be negative; otherwise $f(x)$ and $f(f(x))$ would both be increasing and 1 would be the only possi... | 0 | 8,192 | -1 | 8,192 |
Consider a matrix $A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$ where $a_{11}, a_{12}, a_{21}, a_{22} \in \{0, 1\}$, and the determinant of $A$ is 0. Determine the number of distinct matrices $A$. | 10 | 0.1875 | 7,874.4375 | 6,498.333333 | 8,192 | |
Express $(5-4i)-2(3+6i)$ as a complex number. | -1-16i | 1 | 2,095.5625 | 2,095.5625 | -1 | |
A secret facility is a rectangle measuring $200 \times 300$ meters. Outside the facility, there is one guard at each of the four corners. An intruder approached the perimeter of the facility from the outside, and all the guards ran towards the intruder using the shortest paths along the external perimeter (the intruder... | 150 | 0.3125 | 8,031.8125 | 7,679.4 | 8,192 | |
Given the Cartesian coordinate system $(xOy)$, with the origin as the pole and the positive semi-axis of $x$ as the polar axis, a curve $C$ has the polar equation $ρ^2 - 4ρ\sinθ + 3 = 0$. Points $A$ and $B$ have polar coordinates $(1,π)$ and $(1,0)$, respectively.
(1) Find the parametric equation of curve $C$;
(2) Tak... | 20 | 0.125 | 4,531.625 | 5,896 | 4,336.714286 | |
The positive integer \( N \) is the smallest one whose digits add to 41. What is the sum of the digits of \( N + 2021 \)?
A) 10
B) 12
C) 16
D) 2021
E) 4042 | 10 | 0 | 7,235.6875 | -1 | 7,235.6875 | |
How many different 4-edge trips are there from $A$ to $B$ in a cube, where the trip can visit one vertex twice (excluding start and end vertices)? | 36 | 0 | 8,106.5625 | -1 | 8,106.5625 | |
The legs of a right triangle are 3 and 4. Find the area of the triangle with vertices at the points of tangency of the incircle with the sides of the triangle. | 6/5 | 0.5625 | 6,457.9375 | 6,129.444444 | 6,880.285714 | |
Compute $e^{\pi}+\pi^e$ . If your answer is $A$ and the correct answer is $C$ , then your score on this problem will be $\frac{4}{\pi}\arctan\left(\frac{1}{\left|C-A\right|}\right)$ (note that the closer you are to the right answer, the higher your score is).
*2017 CCA Math Bonanza Lightning Round #5.2* | 45.5999 | 0 | 7,663.75 | -1 | 7,663.75 | |
Find the largest solution to \[\lfloor x \rfloor = 5 + 100 \{ x \},\]where $\{x\} = x - \lfloor x \rfloor.$ | 104.99 | 0.8125 | 3,641.625 | 3,563.384615 | 3,980.666667 | |
What is the smallest positive integer $n$ such that $3n \equiv 1356 \pmod{22}?$ | 12 | 1 | 3,100.6875 | 3,100.6875 | -1 | |
The scent of blooming lily of the valley bushes spreads within a radius of 20 meters around them. How many blooming lily of the valley bushes need to be planted along a straight 400-meter-long alley so that every point along the alley can smell the lily of the valley? | 10 | 0.25 | 7,943.9375 | 7,241.5 | 8,178.083333 | |
Find all functions $f$ defined on the non-negative reals and taking non-negative real values such that: $f(2)=0,f(x)\ne0$ for $0\le x<2$, and $f(xf(y))f(y)=f(x+y)$ for all $x,y$. | f(x) = \begin{cases}
\frac{2}{2 - x}, & 0 \leq x < 2, \\
0, & x \geq 2.
\end{cases} |
We need to find all functions \( f: [0, \infty) \to [0, \infty) \) that satisfy the following conditions:
1. \( f(2) = 0 \).
2. \( f(x) \neq 0 \) for \( 0 \leq x < 2 \).
3. \( f(xf(y))f(y) = f(x+y) \) for all \( x, y \geq 0 \).
Let's begin by analyzing these conditions:
1. **Condition \( f(2) = 0 \):** According to ... | 0 | 8,192 | -1 | 8,192 |
Given positive numbers $x$ and $y$ satisfying $2x+y=2$, the minimum value of $\frac{1}{x}-y$ is achieved when $x=$ ______, and the minimum value is ______. | 2\sqrt{2}-2 | 1 | 3,182.1875 | 3,182.1875 | -1 | |
How many ordered triples \((x, y, z)\) satisfy the following conditions:
\[ x^2 + y^2 + z^2 = 9, \]
\[ x^4 + y^4 + z^4 = 33, \]
\[ xyz = -4? \] | 12 | 0.5625 | 6,702.625 | 5,784.444444 | 7,883.142857 | |
Arrange the sequence $\{2n+1\}$ ($n\in\mathbb{N}^*$), sequentially in brackets such that the first bracket contains one number, the second bracket two numbers, the third bracket three numbers, the fourth bracket four numbers, the fifth bracket one number, and so on in a cycle: $(3)$, $(5, 7)$, $(9, 11, 13)$, $(15, 17, ... | 403 | 0 | 8,192 | -1 | 8,192 | |
In how many ways can one arrange the natural numbers from 1 to 9 in a $3 \times 3$ square table so that the sum of the numbers in each row and each column is odd? (Numbers can repeat) | 6 * 4^6 * 5^3 + 9 * 4^4 * 5^5 + 5^9 | 0 | 7,968.625 | -1 | 7,968.625 | |
(1) Given the complex number $z=3+bi$ ($i$ is the imaginary unit, $b$ is a positive real number), and $(z-2)^{2}$ is a pure imaginary number, find the complex number $z$;
(2) Given that the sum of all binomial coefficients in the expansion of $(3x+ \frac{1}{ \sqrt{x}})^{n}$ is $16$, find the coefficient of the $x$ term... | 54 | 0.1875 | 4,410 | 2,744.666667 | 4,794.307692 | |
Given that five boys, A, B, C, D, and E, are randomly assigned to stay in 3 standard rooms (with at most two people per room), calculate the probability that A and B stay in the same standard room. | \frac{1}{5} | 0 | 8,192 | -1 | 8,192 | |
Of the 100 students in the science club, 65 take mathematics, 43 take physics and 10 students take both mathematics and physics. How many science club students take neither mathematics nor physics? | 2 | 1 | 1,178.3125 | 1,178.3125 | -1 | |
In a regular hexagon \(ABCDEF\), points \(M\) and \(K\) are taken on the diagonals \(AC\) and \(CE\) respectively, such that \(AM : AC = CK : CE = n\). Points \(B, M,\) and \(K\) are collinear. Find \(n\). | \frac{\sqrt{3}}{3} | 0 | 6,842.3125 | -1 | 6,842.3125 | |
Two circles lie outside regular hexagon $ABCDEF$. The first is tangent to $\overline{AB}$, and the second is tangent to $\overline{DE}$. Both are tangent to lines $BC$ and $FA$. What is the ratio of the area of the second circle to that of the first circle? | 81 | 1. **Assumption and Setup**: Assume without loss of generality (WLOG) that the regular hexagon $ABCDEF$ has a side length of 1. The first circle is tangent to $\overline{AB}$ and the second circle is tangent to $\overline{DE}$. Both circles are tangent to lines $BC$ and $FA$.
2. **First Circle's Radius**: The first ci... | 0 | 8,192 | -1 | 8,192 |
What is the least positive multiple of 45 for which the product of its digits is also a positive multiple of 45? | 945 | 0.0625 | 7,716.9375 | 7,881 | 7,706 | |
When flipping a fair coin, what is the probability that the first two flips are both heads? Express your answer as a common fraction. | \frac{1}{4} | 1 | 1,183.8125 | 1,183.8125 | -1 | |
The first row of a triangle is given as:
$$
1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots, \frac{1}{1993}
$$
Each element of the following rows is calculated as the difference between two elements that are above it. The 1993rd row contains only one element. Find this element. | \frac{1}{1993} | 0.3125 | 7,417.25 | 5,712.8 | 8,192 | |
The sequence $\{a_n\}$ satisfies $a_1 = 1$ and $5^{a_{n + 1} - a_n} - 1 = \frac {1}{n + \frac {2}{3}}$ for $n \geq 1$. Find the least integer $k$ greater than $1$ for which $a_k$ is an integer. | 41 | 0.9375 | 6,059.6875 | 5,917.533333 | 8,192 | |
Given the sequence $\{a_n\}$ with the sum of its first $n$ terms $S_n = 6n - n^2$, find the sum of the first $20$ terms of the sequence $\left\{ \frac{1}{a_na_{n+1}}\right\}$. | -\frac{4}{35} | 0.0625 | 7,682 | 5,434 | 7,831.866667 | |
Find the largest positive integer $n$ such that the number $(2n)!$ ends with $10$ more zeroes than the number $n!$ .
*Proposed by Andy Xu* | 42 | 0.0625 | 8,192 | 8,192 | 8,192 | |
Given $\triangle ABC$ with the sides opposite to angles $A$, $B$, $C$ being $a$, $b$, $c$ respectively, and it satisfies $\frac {\sin (2A+B)}{\sin A}=2+2\cos (A+B)$.
(I) Find the value of $\frac {b}{a}$;
(II) If $a=1$ and $c= \sqrt {7}$, find the area of $\triangle ABC$. | \frac { \sqrt {3}}{2} | 0 | 5,090.75 | -1 | 5,090.75 | |
How many positive integers less than $201$ are multiples of either $6$ or $8$, but not both at once? | 42 | 0.6875 | 4,617.6875 | 3,801.545455 | 6,413.2 | |
There is a card game called "Twelve Months" that is played only during the Chinese New Year. The rules are as follows:
Step 1: Take a brand new deck of playing cards, remove the two jokers and the four Kings, leaving 48 cards. Shuffle the remaining cards.
Step 2: Lay out the shuffled cards face down into 12 columns, ... | 1/12 | 0.25 | 7,295.125 | 4,724.75 | 8,151.916667 | |
In a right triangle $ABC$ (right angle at $C$), the bisector $BK$ is drawn. Point $L$ is on side $BC$ such that $\angle C K L = \angle A B C / 2$. Find $KB$ if $AB = 18$ and $BL = 8$. | 12 | 0.125 | 8,073.5 | 7,507.5 | 8,154.357143 | |
Distribute 4 college students to three factories A, B, and C for internship activities. Factory A can only arrange for 1 college student, the other factories must arrange for at least 1 student each, and student A cannot be assigned to factory C. The number of different distribution schemes is ______. | 12 | 0 | 7,445.6875 | -1 | 7,445.6875 | |
How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots? | 100 | 0.1875 | 8,081.125 | 7,600.666667 | 8,192 | |
Twenty percent less than 60 is one-third more than what number? | 36 | 1. **Calculate 20% less than 60**:
To find 20% less than a number, we subtract 20% of the number from the number itself.
\[
60 - 0.20 \times 60 = 60 - 12 = 48
\]
Alternatively, we can calculate this as 80% of 60:
\[
0.80 \times 60 = 48
\]
or using fractions:
\[
\frac{4}{5} \times 60 =... | 1 | 1,537.625 | 1,537.625 | -1 |
Let $ n$ be a natural number. A cube of edge $ n$ may be divided in 1996 cubes whose edges length are also natural numbers. Find the minimum possible value for $ n$ . | 13 | 0.125 | 8,104.75 | 7,494 | 8,192 | |
Find all real numbers $x,y,z\geq 1$ satisfying \[\min(\sqrt{x+xyz},\sqrt{y+xyz},\sqrt{z+xyz})=\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.\] | \[
\boxed{\left(\frac{c^2+c-1}{c^2}, \frac{c}{c-1}, c\right)}
\] | The key Lemma is: \[\sqrt{a-1}+\sqrt{b-1} \le \sqrt{ab}\] for all $a,b \ge 1$ . Equality holds when $(a-1)(b-1)=1$ .
This is proven easily. \[\sqrt{a-1}+\sqrt{b-1} = \sqrt{a-1}\sqrt{1}+\sqrt{1}\sqrt{b-1} \le \sqrt{(a-1+1)(b-1+1)} = \sqrt{ab}\] by Cauchy.
Equality then holds when $a-1 =\frac{1}{b-1} \implies (a-1)(b-1) ... | 0 | 8,192 | -1 | 8,192 |
Consider a right rectangular prism \(B\) with edge lengths \(2,\ 5,\) and \(6\), including its interior. For any real \(r \geq 0\), let \(T(r)\) be the set of points in 3D space within a distance \(r\) from some point in \(B\). The volume of \(T(r)\) is expressed as \(ar^{3} + br^{2} + cr + d\), where \(a,\) \(b,\) \(c... | \frac{8112}{240} | 0 | 7,947.25 | -1 | 7,947.25 | |
Calculate
$$
\operatorname{tg} \frac{\pi}{43} \cdot \operatorname{tg} \frac{2 \pi}{43}+\operatorname{tg} \frac{2 \pi}{43} \cdot \operatorname{tg} \frac{3 \pi}{43}+\ldots+\operatorname{tg} \frac{k \pi}{43} \cdot \operatorname{tg} \frac{(k+1) \pi}{43}+\ldots+\operatorname{tg} \frac{2019 \pi}{43} \cdot \operatorname{tg} ... | -2021 | 0 | 8,192 | -1 | 8,192 | |
In the expansion of \((x + y + z)^8\), determine the sum of the coefficients of all terms of the form \(x^2 y^a z^b\) (\(a, b \in \mathbf{N}\)). | 1792 | 0 | 7,781.0625 | -1 | 7,781.0625 | |
At what value of $a$ do the graphs of $y=ax^2+3x+1$ and $y=-x-1$ intersect at exactly one point? | 2 | 0.9375 | 2,079.8125 | 1,672.333333 | 8,192 | |
The graph of the function y=sin(2x+φ) is shifted to the left by π/6 units along the x-axis, resulting in an even function graph. Determine the value of φ such that the equation 2(x + π/6) + φ = -x + 2πk is satisfied for some integer k. | \frac{\pi}{6} | 0.3125 | 7,208.1875 | 6,339 | 7,603.272727 | |
Daniel writes over a board, from top to down, a list of positive integer numbers less or equal to 10. Next to each number of Daniel's list, Martin writes the number of times exists this number into the Daniel's list making a list with the same length. If we read the Martin's list from down to top, we get the same
list... | 10 | 0.0625 | 8,103.25 | 6,772 | 8,192 | |
Trapezoid $A B C D$ is inscribed in the parabola $y=x^{2}$ such that $A=\left(a, a^{2}\right), B=\left(b, b^{2}\right)$, $C=\left(-b, b^{2}\right)$, and $D=\left(-a, a^{2}\right)$ for some positive reals $a, b$ with $a>b$. If $A D+B C=A B+C D$, and $A B=\frac{3}{4}$, what is $a$? | \frac{27}{40} | Let $t=\frac{3}{4}$, so $2 a+2 b=2 \sqrt{(a-b)^{2}+\left(a^{2}-b^{2}\right)^{2}}=2 t$ gives $a=b=t$ and $t^{2}=(a-b)^{2}\left[1+(a+b)^{2}\right]=(a-b)^{2}\left[1+t^{2}\right]$. Thus $a=\frac{t+\frac{t}{\sqrt{1+t^{2}}}}{2}=\frac{\frac{3}{4}+\frac{3}{5}}{2}=\frac{27}{40}$. | 0.75 | 5,773.375 | 4,967.166667 | 8,192 |
It is known that \( b^{16} - 1 \) has four distinct prime factors. Determine the largest one, denoted by \( c \). | 257 | 0.375 | 7,791.1875 | 7,123.166667 | 8,192 | |
Let $n$ be a positive integer. Initially, a $2n \times 2n$ grid has $k$ black cells and the rest white cells. The following two operations are allowed :
(1) If a $2\times 2$ square has exactly three black cells, the fourth is changed to a black cell;
(2) If there are exactly two black cells in a $2 \times 2$ square, t... | n^2 + n + 1 |
Let \( n \) be a positive integer. Initially, a \( 2n \times 2n \) grid has \( k \) black cells and the rest white cells. The following two operations are allowed:
1. If a \( 2 \times 2 \) square has exactly three black cells, the fourth is changed to a black cell.
2. If there are exactly two black cells in a \( 2 \ti... | 0 | 8,074.25 | -1 | 8,074.25 |
For any real number $x$ , we let $\lfloor x \rfloor$ be the unique integer $n$ such that $n \leq x < n+1$ . For example. $\lfloor 31.415 \rfloor = 31$ . Compute \[2020^{2021} - \left\lfloor\frac{2020^{2021}}{2021} \right \rfloor (2021).\]
*2021 CCA Math Bonanza Team Round #3* | 2020 | 1 | 4,642.5625 | 4,642.5625 | -1 | |
A triangular array of $2016$ coins has $1$ coin in the first row, $2$ coins in the second row, $3$ coins in the third row, and so on up to $N$ coins in the $N$th row. What is the sum of the digits of $N$? | 9 | 1. **Identify the formula for the sum of the first $N$ natural numbers**: The sum of the first $N$ natural numbers is given by the formula:
\[
S = 1 + 2 + 3 + \cdots + N = \frac{N(N+1)}{2}
\]
We need to find $N$ such that this sum equals $2016$.
2. **Set up the equation**:
\[
\frac{N(N+1)}{2} = 2016
... | 1 | 1,528.6875 | 1,528.6875 | -1 |
The points $(2, 9), (12, 14)$, and $(4, m)$, where $m$ is an integer, are vertices of a triangle. What is the sum of the values of $m$ for which the area of the triangle is a minimum? | 20 | 0.6875 | 3,501.375 | 3,396.454545 | 3,732.2 | |
Given $f(\alpha) = \frac{\sin(\frac{\pi}{2} + \alpha) + 3\sin(-\pi - \alpha)}{2\cos(\frac{11\pi}{2} - \alpha) - \cos(5\pi - \alpha)}$.
(I) Simplify $f(\alpha)$;
(II) If $\tan \alpha = 3$, find the value of $f(\alpha)$. | -2 | 0.8125 | 3,811.5 | 4,013.076923 | 2,938 | |
Find all positive values of $c$ so that the inequality $x^2-6x+c<0$ has real solutions for $x$. Express your answer in interval notation. | (0,9) | 1 | 1,459.375 | 1,459.375 | -1 | |
If $a + b + c = 11$ and $ab + ac + bc = 25,$ then find
\[a^3 + b^3 + c^3 - 3abc.\] | 506 | 1 | 2,344.6875 | 2,344.6875 | -1 | |
The rational numbers $x$ and $y$, when written in lowest terms, have denominators 60 and 70 , respectively. What is the smallest possible denominator of $x+y$ ? | 84 | Write $x+y=a / 60+b / 70=(7 a+6 b) / 420$. Since $a$ is relatively prime to 60 and $b$ is relatively prime to 70 , it follows that none of the primes $2,3,7$ can divide $7 a+6 b$, so we won't be able to cancel any of these factors in the denominator. Thus, after reducing to lowest terms, the denominator will still be a... | 0.125 | 7,825.5 | 6,519.5 | 8,012.071429 |
What is the greatest common divisor of $7979$ and $3713$? | 79 | 1 | 2,503.75 | 2,503.75 | -1 | |
There are five positive integers that are common divisors of each number in the list $$36, 72, -24, 120, 96.$$ Find the sum of these five positive integers. | 16 | 0.3125 | 7,127 | 5,761.4 | 7,747.727273 | |
A $\textit{palindrome}$ is a number which reads the same forward as backward, for example 313 or 1001. Ignoring the colon, how many different palindromes are possible on a 12-hour digital clock displaying only the hours and minutes? (Notice a zero may not be inserted before a time with a single-digit hour value. Theref... | 57 | 0 | 8,192 | -1 | 8,192 | |
The function $f(x)$ satisfies
\[f(x) + 2f(1 - x) = 3x^2\]for all real numbers $x.$ Find $f(3).$ | -1 | 1 | 2,902.875 | 2,902.875 | -1 | |
Let $Q(x) = 0$ be the polynomial equation of the least possible degree, with rational coefficients, having $\sqrt[4]{13} + \sqrt[4]{169}$ as a root. Compute the product of all of the roots of $Q(x) = 0.$ | -13 | 0 | 7,868.5625 | -1 | 7,868.5625 | |
On each horizontal line in the figure below, the five large dots indicate the populations of cities $A, B, C, D$ and $E$ in the year indicated.
Which city had the greatest percentage increase in population from $1970$ to $1980$? | C | To find the city with the greatest percentage increase in population from 1970 to 1980, we calculate the percentage increase for each city using the formula:
\[
\text{Percentage Increase} = \left(\frac{\text{Population in 1980} - \text{Population in 1970}}{\text{Population in 1970}}\right) \times 100\%
\]
However, for ... | 0.125 | 7,403.125 | 5,161.5 | 7,723.357143 |
Given a set of seven positive integers with the unique mode being 6 and the median being 4, find the minimum possible sum of these seven integers. | 26 | 0 | 8,189.9375 | -1 | 8,189.9375 | |
Three different numbers are chosen at random from the list \(1, 3, 5, 7, 9, 11, 13, 15, 17, 19\). The probability that one of them is the mean of the other two is \(p\). What is the value of \(\frac{120}{p}\) ? | 720 | 0.1875 | 7,774.5 | 6,437 | 8,083.153846 | |
In a regular pentagon $PQRST$, what is the measure of $\angle PRS$? | 72^{\circ} | Join $P$ to $R$. Since $PQRST$ is a regular pentagon, then $\angle PQR = \angle QRS = 108^{\circ}$. Since $PQ = QR$, then $\triangle PQR$ is isosceles with $\angle QPR = \angle QRP$. Since $\angle PQR = 108^{\circ}$, then $\angle PQR + \angle QPR + \angle QRP = 180^{\circ}$, $108^{\circ} + 2\angle QRP = 180^{\circ}$, $... | 0.25 | 7,878.25 | 7,464.25 | 8,016.25 |
Circle $\omega$ has radius 5 and is centered at $O$. Point $A$ lies outside $\omega$ such that $OA=13$. The two tangents to $\omega$ passing through $A$ are drawn, and points $B$ and $C$ are chosen on them (one on each tangent), such that line $BC$ is tangent to $\omega$ and $\omega$ lies outside triangle $ABC$. Comput... | 17 | 0 | 8,192 | -1 | 8,192 | |
For a real number $x,$ find the maximum value of
\[\frac{x^4}{x^8 + 2x^6 - 4x^4 + 8x^2 + 16}.\] | \frac{1}{12} | 0.625 | 7,182.375 | 6,576.6 | 8,192 | |
Determine how many ordered pairs of positive integers $(x, y)$, where $x < y$, have a harmonic mean of $5^{20}$. | 20 | 0.4375 | 7,171.5625 | 5,859.571429 | 8,192 | |
A hexagon is inscribed in a circle. Five of the sides have length $81$ and the sixth, denoted by $\overline{AB}$, has length $31$. Find the sum of the lengths of the three diagonals that can be drawn from $A$.
| 384 | 0.125 | 8,025.4375 | 6,859.5 | 8,192 | |
The cities of Coco da Selva and Quixajuba are connected by a bus line. From Coco da Selva, buses leave for Quixajuba every hour starting at midnight. From Quixajuba, buses leave for Coco da Selva every hour starting at half past midnight. The bus journey takes exactly 5 hours.
If a bus leaves Coco da Selva at noon, ho... | 10 | 0 | 7,586.25 | -1 | 7,586.25 | |
The area of the parallelogram generated by the vectors $\mathbf{a}$ and $\mathbf{b}$ is 8. Find the area of the parallelogram generated by the vectors $2 \mathbf{a} + 3 \mathbf{b}$ and $\mathbf{a} - 5 \mathbf{b}.$ | 104 | 1 | 1,993.9375 | 1,993.9375 | -1 | |
A certain intelligence station has four different kinds of passwords $A$, $B$, $C$, and $D$. Each week, one of these passwords is used, and each week a password is chosen uniformly at random from the three passwords that were not used the previous week. Given that password $A$ is used in the first week, what is the pro... | 61/243 | 0.4375 | 6,945.6875 | 5,343.285714 | 8,192 | |
What is the largest 2-digit prime factor of the integer $n = {180 \choose 90}$? | 59 | 0.0625 | 6,213.8125 | 7,023 | 6,159.866667 | |
How many distinct $x$-intercepts does the graph of $y = (x-5)(x^2+5x+6)$ have? | 3 | 1 | 1,262.25 | 1,262.25 | -1 | |
Compute: $55\times1212-15\times1212$ . | 48480 | 0.8125 | 324.125 | 323.692308 | 326 | |
What is the coefficient of $x^8$ in the expansion of $(x-1)^9$? | -9 | 1 | 3,054.6875 | 3,054.6875 | -1 | |
In an isosceles right triangle $ABC$ with $\angle A = 90^{\circ}$ and $AB = AC = 2$, calculate the projection of the vector $\vec{AB}$ in the direction of $\vec{BC}$. | -\sqrt{2} | 0.75 | 5,589.875 | 5,082.416667 | 7,112.25 | |
During a long voyage of a passenger ship, it was observed that at each dock, a quarter of the passenger composition is renewed, that among the passengers leaving the ship, only one out of ten boarded at the previous dock, and finally, that the ship is always fully loaded.
Determine the proportion of passengers at any ... | 21/40 | 0 | 8,066.5 | -1 | 8,066.5 | |
In trapezoid $PQRS$ with $PQ$ parallel to $RS$, the diagonals $PR$ and $QS$ intersect at $T$. If the area of triangle $PQT$ is 75 square units, and the area of triangle $PST$ is 30 square units, calculate the area of trapezoid $PQRS$. | 147 | 0.3125 | 6,780.0625 | 4,803 | 7,678.727273 | |
Borya and Vova play the following game on an initially white $8 \times 8$ board. Borya goes first and, on each of his turns, colors any four white cells black. After each of his turns, Vova colors an entire row or column white. Borya aims to color as many cells black as possible, while Vova tries to hinder him. What is... | 25 | 0 | 7,983.9375 | -1 | 7,983.9375 | |
On the ellipse $\frac {x^{2}}{3}+ \frac {y^{2}}{2}=1$, the distance from a point P to the left focus is $\frac { \sqrt {3}}{2}$. Find the distance from P to the right directrix. | \frac{9}{2} | 0.4375 | 6,218.5 | 5,074.285714 | 7,108.444444 |
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