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 an ellipse with the equation \\(\\dfrac{x^{2}}{a^{2}}+\\dfrac{y^{2}}{b^{2}}=1(a > b > 0)\\) and an eccentricity of \\(\\dfrac{\\sqrt{3}}{2}\\). A line $l$ is drawn through one of the foci of the ellipse, perpendicular to the $x$-axis, and intersects the ellipse at points $M$ and $N$, with $|MN|=1$. Point $P$ is l... | 26 | 0.1875 | 8,126.8125 | 7,844.333333 | 8,192 | |
Define a new operation: \( x \odot y = 18 + x - a \times y \), where \( a \) is a constant. For example:
\[ 1 \odot 2 = 18 + 1 - a \times 2. \]
If \( 2 \odot 3 = 8 \), then what is \( 3 \odot 5 \) and \( 5 \odot 3 \)? | 11 | 0.5625 | 1,888.875 | 1,721 | 2,104.714286 | |
Find $i + 2i^2 +3i^3 + ... + 2002i^{2002}.$ | -1001 + 1000i | To solve the problem, we need to evaluate the sum $i + 2i^2 + 3i^3 + \cdots + 2002i^{2002}$. We start by recognizing the cyclic nature of the powers of $i$:
- $i^1 = i$
- $i^2 = -1$
- $i^3 = -i$
- $i^4 = 1$
- $i^5 = i$, and so on.
The powers of $i$ repeat every four terms. Therefore, we can group the terms of the sum ... | 0 | 7,335.75 | -1 | 7,335.75 |
Solve for $x$: $0.05x + 0.07(30 + x) = 15.4$. | 110.8333 | 0 | 5,158.4375 | -1 | 5,158.4375 | |
In $\triangle{ABC}$, $\angle ABC=120^\circ,AB=3$ and $BC=4$. If perpendiculars constructed to $\overline{AB}$ at $A$ and to $\overline{BC}$ at $C$ meet at $D$, then $CD=$
$\text{(A) } 3\quad \text{(B) } \frac{8}{\sqrt{3}}\quad \text{(C) } 5\quad \text{(D) } \frac{11}{2}\quad \text{(E) } \frac{10}{\sqrt{3}}$
| \frac{10}{\sqrt{3}} | 0 | 5,762.875 | -1 | 5,762.875 | |
Let $n$ denote the smallest positive integer that is divisible by both $4$ and $9,$ and whose base-$10$ representation consists of only $4$'s and $9$'s, with at least one of each. What are the last four digits of $n?$ | 4944 | 1. **Divisibility by 4**: A number is divisible by 4 if its last two digits form a number that is divisible by 4. Since the number consists only of the digits 4 and 9, the last two digits must be 44 to ensure divisibility by 4, as 44 is divisible by 4.
2. **Divisibility by 9**: A number is divisible by 9 if the sum of... | 0 | 8,176.25 | -1 | 8,176.25 |
The price of Margit néni's favorite chocolate was increased by 30%, and at the same time her pension increased by 15%. By what percentage does Margit néni's chocolate consumption decrease if she can spend only 15% more on chocolate? | 11.54 | 0.875 | 5,803.375 | 5,536.071429 | 7,674.5 | |
Two siblings, Andy and Alexa, bake 24 cookies. Alexa eats some number of cookies that is a positive multiple of the number of cookies eaten by her brother. If the siblings finish all 24 cookies, then what is the maximum number of cookies that the brother, Andy, could have eaten? | 12 | 1 | 3,841.6875 | 3,841.6875 | -1 | |
The product of two positive three-digit palindromes is 436,995. What is their sum? | 1332 | 0.5625 | 6,723.75 | 5,581.777778 | 8,192 | |
Consider an equilateral triangle with a side length of 10 cm, where one side of the triangle is also the diameter of a semicircle. Find the sum of the areas of two small shaded regions formed inside the semicircle but outside the triangle. Express your answer in the form \(a\pi - b\sqrt{c}\) and calculate \(a+b+c\). | 78 | 0 | 8,192 | -1 | 8,192 | |
A torus (donut) having inner radius $2$ and outer radius $4$ sits on a flat table. What is the radius of the largest spherical ball that can be placed on top of the center torus so that the ball still touches the horizontal plane? (If the $xy$-plane is the table, the torus is formed by revolving the circle in the $xz... | \frac{9}{4} | 0.0625 | 7,933.75 | 4,060 | 8,192 | |
Given a set of data 3, 4, 5, a, b with an average of 4 and a median of m, where the probability of selecting the number 4 from the set 3, 4, 5, a, b, m is $\frac{2}{3}$, calculate the variance of the set 3, 4, 5, a, b. | \frac{2}{5} | 0.25 | 6,738.9375 | 4,371.25 | 7,528.166667 | |
Given \(a > 0\), \(b > 0\), \(c > 1\), and \(a + b = 1\). Find the minimum value of \(\left(\frac{2a + b}{ab} - 3\right)c + \frac{\sqrt{2}}{c - 1}\). | 4 + 2\sqrt{2} | 0.375 | 7,995.875 | 7,669 | 8,192 | |
Let $e > 0$ be a given real number. Find the least value of $f(e)$ (in terms of $e$ only) such that the inequality $a^{3}+ b^{3}+ c^{3}+ d^{3} \leq e^{2}(a^{2}+b^{2}+c^{2}+d^{2}) + f(e)(a^{4}+b^{4}+c^{4}+d^{4})$ holds for all real numbers $a, b, c, d$ . | \frac{1}{4e^2} | 0.5 | 7,193.5 | 6,195 | 8,192 | |
Anton, Vasya, Sasha, and Dima were driving from city A to city B, each taking turns at the wheel. The entire journey was made at a constant speed.
Anton drove the car for half the time Vasya did, and Sasha drove for as long as Anton and Dima together. Dima was at the wheel for only one-tenth of the distance. What frac... | 0.4 | 0.625 | 5,212.3125 | 3,693 | 7,744.5 | |
Given \(\lg 2 = 0.30103\), calculate the number of digits in \( M = 1 + 10^4 + \frac{10^4 (10^4 - 1)}{1 \cdot 2} + \frac{10^4 (10^4-1)(10^4-2)}{1 \cdot 2 \cdot 3} + \cdots + \frac{10^4 (10^4 - 1)}{1 \cdot 2} + 10^4 + 1\). | 3011 | 0.1875 | 7,515.0625 | 4,725 | 8,158.923077 | |
There are $2018$ players sitting around a round table. At the beginning of the game we arbitrarily deal all the cards from a deck of $K$ cards to the players (some players may receive no cards). In each turn we choose a player who draws one card from each of the two neighbors. It is only allowed to choose a player whos... | 2017 |
Consider \(2018\) players sitting around a round table, and a deck of \(K\) cards distributed among them. The rules of the game allow a player to draw one card from each of their two neighbors, provided both neighbors have at least one card. The game ends when no player can make such a move. We need to determine the m... | 0.125 | 7,983.3125 | 7,498 | 8,052.642857 |
According to the table below, how many dollars are in the median value of the 59 salaries paid to this company's employees?
\begin{tabular}{|c|c|c|}
\hline
\textbf{Position Title}&\textbf{\# with Title}&\textbf{Salary}\\\hline
President&1&$\$130{,}000$\\\hline
Vice-President&5&$\$90{,}000$\\\hline
Director&10&$\$75{,}... | \$23{,}000 | 0 | 663.25 | -1 | 663.25 | |
What is the sum of all integer solutions to $|n| < |n-3| < 9$? | -14 | 0.875 | 3,118.6875 | 3,106.428571 | 3,204.5 | |
What percent of square $EFGH$ is shaded? All angles in the diagram are right angles. [asy]
import graph;
defaultpen(linewidth(0.7));
xaxis(0,8,Ticks(1.0,NoZero));
yaxis(0,8,Ticks(1.0,NoZero));
fill((0,0)--(2,0)--(2,2)--(0,2)--cycle);
fill((3,0)--(5,0)--(5,5)--(0,5)--(0,3)--(3,3)--cycle);
fill((6,0)--(7,0)--(7,7)--(0,... | 67\% | 0 | 8,086.375 | -1 | 8,086.375 | |
Find the distance between the foci of the ellipse
\[\frac{x^2}{36} + \frac{y^2}{9} = 5.\] | 2\sqrt{5.4} | 0 | 2,480.125 | -1 | 2,480.125 | |
Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7. Play continues until either player wins or indefinitely. If Nathaniel goes f... | 5/11 | 0.0625 | 8,090.5625 | 6,569 | 8,192 | |
The roots of the equation $ax^2 + bx + c = 0$ will be reciprocal if: | c = a | 1. **Start by normalizing the quadratic equation**: Given the quadratic equation $ax^2 + bx + c = 0$, we can divide each term by $a$ (assuming $a \neq 0$) to simplify the equation:
\[
x^2 + \frac{b}{a}x + \frac{c}{a} = 0
\]
2. **Identify the roots**: Let the roots of the equation be $r$ and $s$. According to ... | 0.875 | 4,069.25 | 4,094.214286 | 3,894.5 |
The symbol $|a|$ means $+a$ if $a$ is greater than or equal to zero, and $-a$ if a is less than or equal to zero; the symbol $<$ means "less than";
the symbol $>$ means "greater than."
The set of values $x$ satisfying the inequality $|3-x|<4$ consists of all $x$ such that: | $-1<x<7$ | The problem involves solving the inequality $|3-x| < 4$. The absolute value inequality $|a| < b$ where $a$ is an expression and $b$ is a positive number, can be rewritten as $-b < a < b$. Applying this to our inequality:
1. Rewrite the inequality:
\[
|3-x| < 4 \implies -4 < 3-x < 4
\]
2. Solve the left part ... | 0 | 547.8125 | -1 | 547.8125 |
Find the minimum value of
\[\frac{\sin^6 x + \cos^6 x + 1}{\sin^4 x + \cos^4 x + 1}\]over all real values $x.$ | \frac{5}{6} | 0.75 | 6,010.8125 | 5,283.75 | 8,192 | |
Given the function $f(x) = \cos(2x - \frac{\pi}{3}) + 2\sin^2x$,
(I) Find the period of the function $f(x)$ and the intervals where it is monotonically increasing;
(II) When $x \in [0, \frac{\pi}{2}]$, find the maximum and minimum values of the function $f(x)$. | \frac{1}{2} | 0.8125 | 6,473.625 | 6,077.076923 | 8,192 | |
Find all values of $x$ so that $\arccos x > \arcsin x.$ | \left[ -1, \frac{1}{\sqrt{2}} \right) | 0 | 4,756.125 | -1 | 4,756.125 | |
Among the first 1500 positive integers, there are n whose hexadecimal representation contains only numeric digits. What is the sum of the digits of n? | 23 | 0.25 | 7,642.625 | 6,491.5 | 8,026.333333 | |
Last summer $100$ students attended basketball camp. Of those attending, $52$ were boys and $48$ were girls. Also, $40$ students were from Jonas Middle School and $60$ were from Clay Middle School. Twenty of the girls were from Jonas Middle School. How many of the boys were from Clay Middle School? | 32 | 1. **Organize the given data into a table**: We start by setting up a table to organize the information provided in the problem. We know the total number of students, the breakdown by gender, the breakdown by school, and the number of girls from Jonas Middle School.
\[
\begin{array}{c|c|c|c}
& \text{Jonas} & ... | 0.9375 | 1,527.875 | 1,540.733333 | 1,335 |
Find the smallest prime which is not the difference (in some order) of a power of $2$ and a power of $3$ . | 41 | 0.1875 | 8,006.75 | 7,204 | 8,192 | |
If the real number sequence: -1, $a_1$, $a_2$, $a_3$, -81 forms a geometric sequence, determine the eccentricity of the conic section $x^2+ \frac{y^2}{a_2}=1$. | \sqrt{10} | 0.875 | 3,767.3125 | 3,527.571429 | 5,445.5 | |
For which natural number \( K \) does the expression \(\frac{n^{2}}{1.001^{n}}\) reach its maximum value? | 2001 | 0.375 | 7,714.1875 | 6,917.833333 | 8,192 | |
A parabola with equation $y=x^2+bx+c$ passes through the points $(-1,-11)$ and $(3,17)$. What is $c$? | -7 | 1 | 1,793.25 | 1,793.25 | -1 | |
Triangle \(ABC\) is isosceles with \(AB = AC\) and \(BC = 65 \, \text{cm}\). \(P\) is a point on \(BC\) such that the perpendicular distances from \(P\) to \(AB\) and \(AC\) are \(24 \, \text{cm}\) and \(36 \, \text{cm}\), respectively. Find the area of \(\triangle ABC\). | 2535 | 0.6875 | 7,125.0625 | 6,640.090909 | 8,192 | |
The Fibonacci sequence $1,1,2,3,5,8,13,21,\ldots$ starts with two 1s, and each term afterwards is the sum of its two predecessors. Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci sequence? | 6 | To solve this problem, we need to determine the first occurrence of each digit from 0 to 9 in the units position of the Fibonacci sequence. We will compute the Fibonacci sequence modulo 10, which will give us the last digit of each Fibonacci number.
1. **Compute the Fibonacci sequence modulo 10:**
We start with $F_... | 0.125 | 7,225.125 | 5,440.5 | 7,480.071429 |
Given the ellipse $C$: $\begin{cases}x=2\cos θ \\\\ y=\sqrt{3}\sin θ\end{cases}$, find the value of $\frac{1}{m}+\frac{1}{n}$. | \frac{4}{3} | 0 | 5,756.875 | -1 | 5,756.875 | |
Square $ABCD$ is divided into four rectangles by $EF$ and $GH$ . $EF$ is parallel to $AB$ and $GH$ parallel to $BC$ . $\angle BAF = 18^\circ$ . $EF$ and $GH$ meet at point $P$ . The area of rectangle $PFCH$ is twice that of rectangle $AGPE$ . Given that the value of $\angle FAH$ in degrees is $... | 45 | 0.0625 | 8,107.5 | 7,790 | 8,128.666667 | |
A right triangle has legs of lengths 126 and 168 units. What is the perimeter of the triangle formed by the points where the angle bisectors intersect the opposite sides? | 230.61 | 0 | 8,192 | -1 | 8,192 | |
What is the largest integer \( k \) whose square \( k^2 \) is a factor of \( 10! \)? | 720 | 0.9375 | 4,427.375 | 4,391.6 | 4,964 | |
On the first day, Barry Sotter used his magic wand to make an object's length increase by $\frac{1}{3}$. Meaning if the length of the object was originally $x$, then after the first day, it is $x + \frac{1}{3} x.$ On the second day, he increased the object's new length from the previous day by $\frac{1}{4}$; on the thi... | 147 | 0.6875 | 3,516.5625 | 2,990.545455 | 4,673.8 | |
Find, as a function of $\, n, \,$ the sum of the digits of \[9 \times 99 \times 9999 \times \cdots \times \left( 10^{2^n} - 1 \right),\] where each factor has twice as many digits as the previous one. | \[ 9 \cdot 2^n \] | The answer is $9 \cdot 2^n$ .
Let us denote the quantity $\prod_{k=0}^n \bigl( 10^{2^k}-1 \bigr)$ as $P_n$ . We wish to find the sum of the digits of $P_n$ .
We first note that \[P_{n-1} < \prod_{k=0}^{n-1} 10^{2^k} = 10^{2^n-1},\] so $P_{n-1}$ is a number of at most $2^n$ digits. We also note that the units digit is... | 0 | 7,836.5 | -1 | 7,836.5 |
Given that the positive real numbers \(a_{1}, a_{2}, a_{3}, a_{4}\) satisfy the conditions \(a_{1} \geqslant a_{2} a_{3}^{2}, a_{2} \geqslant a_{3} a_{4}^{2}, a_{3} \geqslant a_{4} a_{1}^{2}, a_{4} \geqslant a_{1} a_{2}^{2}\), find the maximum value of \(a_{1} a_{2} a_{3} a_{4}\left(a_{1}-a_{2} a_{3}^{2}\right)\left(a_... | 1/256 | 0.0625 | 8,014.8125 | 5,357 | 8,192 | |
Let $b > 0$, and let $Q(x)$ be a polynomial with integer coefficients such that
\[Q(2) = Q(4) = Q(6) = Q(8) = b\]and
\[Q(1) = Q(3) = Q(5) = Q(7) = -b.\]
What is the smallest possible value of $b$? | 315 | 0.0625 | 8,053.5 | 5,976 | 8,192 | |
A small class of nine boys are to change their seating arrangement by drawing their new seat numbers from a box. After the seat change, what is the probability that there is only one pair of boys who have switched seats with each other and only three boys who have unchanged seats? | 1/32 | 0 | 7,181.4375 | -1 | 7,181.4375 | |
A transformation of the first quadrant of the coordinate plane maps each point $(x,y)$ to the point $(\sqrt{x},\sqrt{y}).$ The vertices of quadrilateral $ABCD$ are $A=(900,300), B=(1800,600), C=(600,1800),$ and $D=(300,900).$ Let $k_{}$ be the area of the region enclosed by the image of quadrilateral $ABCD.$ Find the g... | 314 | \begin{eqnarray*}A' = & (\sqrt {900}, \sqrt {300})\\ B' = & (\sqrt {1800}, \sqrt {600})\\ C' = & (\sqrt {600}, \sqrt {1800})\\ D' = & (\sqrt {300}, \sqrt {900}) \end{eqnarray*}
First we see that lines passing through $AB$ and $CD$ have equations $y = \frac {1}{3}x$ and $y = 3x$, respectively. Looking at the points abo... | 0 | 8,019.6875 | -1 | 8,019.6875 |
Compute the number of ways a non-self-intersecting concave quadrilateral can be drawn in the plane such that two of its vertices are $(0,0)$ and $(1,0)$, and the other two vertices are two distinct lattice points $(a, b),(c, d)$ with $0 \leq a, c \leq 59$ and $1 \leq b, d \leq 5$. | 366 | We instead choose points $(0,0),(1,0),(a, b),(c, d)$ with $0 \leq a, c \leq 59$ and $0 \leq b, d \leq 5$ with $(c, d)$ in the interior of the triangle formed by the other three points. Any selection of these four points may be connected to form a concave quadrilateral in precisely three ways. Apply Pick's theorem to th... | 0 | 8,192 | -1 | 8,192 |
Given a function $f(x)$ defined on $\mathbb{R}$ that is an odd function, and the period of the function $f(2x+1)$ is 5, if $f(1) = 5$, calculate the value of $f(2009) + f(2010)$. | -5 | 0.9375 | 3,644.375 | 3,341.2 | 8,192 | |
In the diagram, $ABCD$ and $EFGD$ are squares each of area 16. If $H$ is the midpoint of both $BC$ and $EF$, find the total area of polygon $ABHFGD$.
[asy]
unitsize(3 cm);
pair A, B, C, D, E, F, G, H;
F = (0,0);
G = (1,0);
D = (1,1);
E = (0,1);
H = (E + F)/2;
A = reflect(D,H)*(G);
B = reflect(D,H)*(F);
C = reflect(... | 24 | 0.0625 | 8,086.8125 | 6,956 | 8,162.2 | |
Given the sets \( M = \{1, 2, 3\} \) and \( N = \{1, 2, 3, 4, 5\} \), define the function \( f: M \rightarrow N \). Let the points \( A(1, f(1)), B(2, f(2)), C(3, f(3)) \) form a triangle \( \triangle ABC \). The circumcenter of \( \triangle ABC \) is \( D \), and it is given that \( \mu DA + DC = \lambda DB (\lambda \... | 20 | 0 | 7,628.5625 | -1 | 7,628.5625 | |
If $f^{-1}(g(x))=x^3-1$ and $g$ has an inverse, find $g^{-1}(f(7))$. | 2 | 1 | 1,627.0625 | 1,627.0625 | -1 | |
In the expansion of the polynomial $$(x+ \frac {1}{ \sqrt {x}})^{6}( \sqrt {x}-1)^{10}$$, the constant term is \_\_\_\_\_\_. | -495 | 0.375 | 7,159.8125 | 5,439.5 | 8,192 | |
How many distinct, positive factors does $1100$ have? | 18 | 1 | 3,067.375 | 3,067.375 | -1 | |
What is the sum of the 2009 fractions of the form $\frac{2}{n(n+2)}$ if the values of $n$ are the positive integers from 1 through 2009? Express your answer as a decimal to the nearest thousandth. | 1.499 | 0.5 | 7,686.4375 | 7,180.875 | 8,192 | |
Given the real numbers \( x \) and \( y \) that satisfy \( xy + 6 = x + 9y \) and \( y \in (-\infty, 1) \), find the maximum value of \((x+3)(y+1)\). | 27 - 12\sqrt{2} | 0.75 | 6,641.5625 | 6,466.166667 | 7,167.75 | |
Convex quadrilateral $ABCD$ has $AB = 10$ and $CD = 15$. Diagonals $AC$ and $BD$ intersect at $E$, where $AC = 18$, and $\triangle AED$ and $\triangle BEC$ have equal perimeters. Calculate the length of $AE$.
**A)** $6$
**B)** $7.2$
**C)** $7.5$
**D)** $9$
**E)** $10$ | 7.2 | 0 | 5,815.125 | -1 | 5,815.125 | |
A regular octagon is inscribed in a circle and another regular octagon is circumscribed about the same circle. What is the ratio of the area of the larger octagon to the area of the smaller octagon? Express your answer as a common fraction. | 4 - 2\sqrt{2} | 0.4375 | 7,230.5 | 6,480.714286 | 7,813.666667 | |
Point P is located on side AB of triangle ABC. What is the probability that the area of triangle PBC is less than or equal to 1/3 of the area of triangle ABC. | \frac{1}{3} | 0.875 | 4,705.75 | 4,207.714286 | 8,192 | |
Given two non-zero vectors $\overrightarrow{m}$ and $\overrightarrow{n}$ with an angle of $\frac{\pi}{3}$ between them, and the magnitude of $\overrightarrow{n}$ is a positive scalar multiple of the magnitude of $\overrightarrow{m}$, i.e., $|\overrightarrow{n}| = λ|\overrightarrow{m}| (λ > 0)$. The vector group $\overr... | \frac{8}{3} | 0.3125 | 7,151.8125 | 5,982.4 | 7,683.363636 | |
How many ordered pairs of integers $(x, y)$ satisfy the equation $x^{2020} + y^2 = 2y$? | 4 |
We start by analyzing the given equation:
\[ x^{2020} + y^2 = 2y. \]
#### Step 1: Rearrange the equation
We can rearrange the equation to isolate terms involving \( y \):
\[ y^2 - 2y + x^{2020} = 0. \]
#### Step 2: Complete the square
Completing the square for \( y \) gives:
\[ (y-1)^2 - 1 + x^{2020} = 0 \]
\[ (y-1)... | 1 | 2,745.75 | 2,745.75 | -1 |
Assume that $a$, $b$, $c$, and $d$ are positive integers such that $a^5 = b^4$, $c^3 = d^2$, and $c - a = 19$. Determine $d - b$.
| 757 | 1 | 2,457.9375 | 2,457.9375 | -1 | |
In the regular quadrangular pyramid \(P-ABCD\), \(M\) and \(N\) are the midpoints of \(PA\) and \(PB\) respectively. If the tangent of the dihedral angle between a side face and the base is \(\sqrt{2}\), find the cosine of the angle between skew lines \(DM\) and \(AN\). | 1/6 | 0.875 | 6,169.9375 | 5,881.071429 | 8,192 | |
Compute the nearest integer to $$100 \sum_{n=1}^{\infty} 3^{n} \sin ^{3}\left(\frac{\pi}{3^{n}}\right)$$ | 236 | Note that we have $$\sin 3 x=3 \sin x-4 \sin ^{3} x \Longrightarrow \sin ^{3} x=\frac{1}{4}(3 \sin x-\sin 3 x)$$ which implies that $$\frac{\sin ^{3} x}{3 x}=\frac{1}{4}\left(\frac{\sin x}{x}-\frac{\sin 3 x}{3 x}\right)$$ Substituting $x=\frac{\pi}{3^{n}}$ and simplifying gives us $$3^{n} \sin ^{3} \frac{\pi}{3^{n}}=\f... | 0.375 | 7,347.3125 | 6,248.666667 | 8,006.5 |
Determine the maximum possible value of \[\frac{\left(x^2+5x+12\right)\left(x^2+5x-12\right)\left(x^2-5x+12\right)\left(-x^2+5x+12\right)}{x^4}\] over all non-zero real numbers $x$ .
*2019 CCA Math Bonanza Lightning Round #3.4* | 576 | 0.0625 | 8,192 | 8,192 | 8,192 | |
How many ways can one color the squares of a $6 \times 6$ grid red and blue such that the number of red squares in each row and column is exactly 2? | 67950 | Assume the grid is $n \times n$. Let $f(n)$ denote the number of ways to color exactly two squares in each row and column red. So $f(1)=0$ and $f(2)=1$. We note that coloring two squares red in each row and column partitions the set $1,2, \ldots, n$ into cycles such that $i$ is in the same cycle as, and adjacent to, $j... | 0 | 8,192 | -1 | 8,192 |
Find $\left(\frac{1}{2}\right)^{4}$. | \frac{1}{16} | 0.9375 | 1,455 | 1,005.866667 | 8,192 | |
Let $ p$ be an odd prime number. How many $ p$-element subsets $ A$ of $ \{1,2,\dots,2p\}$ are there, the sum of whose elements is divisible by $ p$? | \boxed{2 + \frac{1}{p} \left(\binom{2p}{p} - 2 \right)} |
Let \( p \) be an odd prime number. We are tasked with finding the number of \( p \)-element subsets \( A \) of the set \(\{1, 2, \dots, 2p\}\) such that the sum of the elements in \( A \) is divisible by \( p \).
### Step 1: Representation of Subsets
The set \(\{1, 2, \dots, 2p\}\) contains \( 2p \) elements. We wa... | 0 | 8,192 | -1 | 8,192 |
Find the number of 0-1 binary sequences formed by six 0's and six 1's such that no three 0's are together. For example, 110010100101 is such a sequence but 101011000101 and 110101100001 are not. | 357 | 0.3125 | 7,321.6875 | 5,407 | 8,192 | |
What percent of the positive integers less than or equal to $150$ have no remainders when divided by $6$? | 16.67\% | 0.5 | 3,661.1875 | 2,894.875 | 4,427.5 | |
Let \[\mathbf{N} = \begin{pmatrix} x & y & z \\ y & z & x \\ z & x & y \end{pmatrix}\] be a matrix with real entries such that $\mathbf{N}^3 = \mathbf{I}.$ If $xyz = -1$, find the possible values of $x^3 + y^3 + z^3.$ | -2 | 0 | 7,077.0625 | -1 | 7,077.0625 | |
In $\triangle ABC$, $A=30^{\circ}$, $2 \overrightarrow{AB}\cdot \overrightarrow{AC}=3 \overrightarrow{BC}^{2}$, find the cosine value of the largest angle in $\triangle ABC$. | -\frac{1}{2} | 0.75 | 6,351.5625 | 6,125.25 | 7,030.5 | |
Determine the area of a triangle with side lengths 7, 7, and 5. | \frac{5\sqrt{42.75}}{2} | 0 | 4,151.75 | -1 | 4,151.75 | |
On an auto trip, the distance read from the instrument panel was $450$ miles. With snow tires on for the return trip over the same route, the reading was $440$ miles. Find, to the nearest hundredth of an inch, the increase in radius of the wheels if the original radius was 15 inches. | .34 | To solve this problem, we need to understand how the radius of the wheels affects the distance measurement on the instrument panel. The odometer measures distance based on the number of rotations of the wheels. If the radius of the wheels increases, each rotation covers more distance, but the odometer, calibrated for t... | 0.4375 | 7,228.4375 | 6,576.285714 | 7,735.666667 |
A tetrahedron is formed using the vertices of a cube. How many such distinct tetrahedrons can be formed? | 58 | 0.5 | 7,338.625 | 6,485.25 | 8,192 | |
Determine the base seven product of the numbers $321_7$ and $13_7$. | 4503_7 | 0.625 | 5,643 | 4,284.4 | 7,907.333333 | |
At Typico High School, $60\%$ of the students like dancing, and the rest dislike it. Of those who like dancing, $80\%$ say that they like it, and the rest say that they dislike it. Of those who dislike dancing, $90\%$ say that they dislike it, and the rest say that they like it. What fraction of students who say they d... | 25\% | 1. **Assumptions and Setup**: Assume there are 100 students at Typico High School for simplicity. We know that 60% of the students like dancing, which translates to 60 students. The remaining 40 students do not like dancing.
2. **Distribution of Preferences**:
- Among the 60 students who like dancing, 80% say they... | 0 | 3,454.5 | -1 | 3,454.5 |
Given that point $P$ is any point on the curve $(x-1)^2+(y-2)^2=9$ with $y \geq 2$, find the minimum value of $x+ \sqrt {3}y$. | 2\sqrt{3} - 2 | 0.0625 | 8,036.375 | 6,945 | 8,109.133333 | |
A regular 12-sided polygon is inscribed in a circle of radius 1. How many chords of the circle that join two of the vertices of the 12-gon have lengths whose squares are rational? | 42 | 0.125 | 8,007.9375 | 7,792.5 | 8,038.714286 | |
Suppose $f(x)$ is a rational function such that $3f\left(\dfrac{1}{x}\right)+\dfrac{2f(x)}{x}=x^2$ for $x\neq 0$. Find $f(-2)$. | \frac{67}{20} | 0.75 | 5,204.125 | 4,591.166667 | 7,043 | |
What is the least positive integer $n$ such that $6375$ is a factor of $n!$? | 17 | 1 | 3,129 | 3,129 | -1 | |
Triangle $DEF$ has side lengths $DE = 15$, $EF = 36$, and $FD = 39$. Rectangle $WXYZ$ has vertex $W$ on $\overline{DE}$, vertex $X$ on $\overline{DF}$, and vertices $Y$ and $Z$ on $\overline{EF}$. In terms of the side length $WX = \omega$, the area of $WXYZ$ can be expressed as the quadratic polynomial
\[Area(WXYZ) = \... | 17 | 0.1875 | 8,000.75 | 7,172 | 8,192 | |
Someone says that 7 times their birth year divided by 13 gives a remainder of 11, and 13 times their birth year divided by 11 gives a remainder of 7. How old will this person be in the year 1954? | 86 | 0.375 | 7,479.75 | 6,652.333333 | 7,976.2 | |
A train has 18 identical cars. In some of the cars, half of the seats are free, in others, one third of the seats are free, and in the remaining cars, all the seats are occupied. In the entire train, exactly one ninth of all seats are free. How many cars have all seats occupied? | 13 | 0.1875 | 7,336.625 | 6,203 | 7,598.230769 | |
There are 10 different mathematics books, 9 different Chinese language books, and 8 different English books. In how many different ways can you take two books of different subjects? | 242 | 1 | 2,496.4375 | 2,496.4375 | -1 | |
Twenty tiles are numbered 1 through 20 and are placed into box $C$. Twenty other tiles numbered 15 through 34 are placed into box $D$. One tile is randomly drawn from each box. What is the probability that the tile from box $C$ is less than 18 and the tile from box $D$ is either odd or greater than 30? Express your ans... | \frac{17}{40} | 0 | 2,898.0625 | -1 | 2,898.0625 | |
Given the set $A = \{x | x < 1 \text{ or } x > 5\}$, and $B = \{x | a \leq x \leq b\}$, and $A \cup B = \mathbb{R}$, $A \cap B = \{x | 5 < x \leq 6\}$, find the value of $2a - b$. | -4 | 0.5625 | 6,592.375 | 5,767.555556 | 7,652.857143 | |
When plotted in the standard rectangular coordinate system, trapezoid $ABCD$ has vertices $A(1, -2)$, $B(1, 1)$, $C(5, 7)$ and $D(5, 1)$. What is the area of trapezoid $ABCD$? | 18 | 1 | 3,660.125 | 3,660.125 | -1 | |
Given the function $y=\sin (2x+\frac{π}{3})$, determine the horizontal shift required to obtain this graph from the graph of the function $y=\sin 2x$. | \frac{\pi}{6} | 0.8125 | 4,197.875 | 3,837.846154 | 5,758 | |
Given that the graphs of $y=h(x)$ and $y=j(x)$ intersect at $(3,3),$ $(5,5),$ $(7,7),$ and $(9,9),$ determine whether there is a point where the graphs of $y=h(3x)$ and $y=3j(x)$ intersect, and find the sum of the coordinates of that point if it exists. | 12 | 0.375 | 7,226.0625 | 6,220.5 | 7,829.4 | |
Let's consider two fictional states, Sunland and Moonland, which have different license plate formats. Sunland license plates have the format LLDDLLL (where 'L' stands for a letter and 'D' for a digit), while Moonland license plates have the format LLLDDD. Assuming all 10 digits and all 26 letters are equally likely to... | 1170561600 | 0.3125 | 3,723.1875 | 5,274.2 | 3,018.181818 | |
Given that in the geometric sequence $\{a_n\}$ where all terms are positive, $a_1a_3=16$ and $a_3+a_4=24$, find the value of $a_5$. | 32 | 1 | 2,755.375 | 2,755.375 | -1 | |
Let $f(n)$ be the number of distinct prime divisors of $n$ less than 6. Compute $$\sum_{n=1}^{2020} f(n)^{2}$$ | 3431 | Define $$\mathbf{1}_{a \mid n}= \begin{cases}1 & a \mid n \\ 0 & \text { otherwise }\end{cases}$$ Then $$\begin{aligned} f(n)^{2} & =\left(\mathbf{1}_{2 \mid n}+\mathbf{1}_{3 \mid n}+\mathbf{1}_{5 \mid n}\right)^{2} \\ & =\mathbf{1}_{2 \mid n}+\mathbf{1}_{3 \mid n}+\mathbf{1}_{5 \mid n}+2\left(\mathbf{1}_{2 \mid n} \ma... | 0.375 | 7,474.8125 | 6,693.166667 | 7,943.8 |
In 1960, there were 450,000 cases of measles reported in the U.S. In 1996, there were 500 cases reported. How many cases of measles would have been reported in 1987 if the number of cases reported from 1960 to 1996 decreased linearly? | 112,\!875 | 0 | 5,906.9375 | -1 | 5,906.9375 | |
Let
\[z = \frac{(-11 + 13i)^3 \cdot (24 - 7i)^4}{3 + 4i},\]and let $w = \frac{\overline{z}}{z}.$ Compute $|w|.$ | 1 | 1 | 2,308.8125 | 2,308.8125 | -1 | |
A triangle has altitudes of lengths 15, 21, and 35. Find its area. | 210 | 0 | 6,904.625 | -1 | 6,904.625 | |
The positive integer divisors of 175, except 1, are arranged around a circle so that every pair of adjacent integers has a common factor greater than 1. What is the sum of the two integers adjacent to 7? | 210 | 0.5625 | 6,939.125 | 5,964.666667 | 8,192 | |
On a road of length $A B = 8 \text{ km}$, buses travel in both directions at a speed of $12 \text{ km/h}$. The first bus from each location starts at 6 o'clock, with subsequent buses departing every 10 minutes.
A pedestrian starts walking from $A$ to $B$ at $\frac{81}{4}$ hours; their speed is $4 \text{ km/h}$.
Deter... | 16 | 0 | 8,192 | -1 | 8,192 | |
Igor Gorshkov has all seven books about Harry Potter. In how many ways can Igor arrange these seven volumes on three different shelves, such that each shelf has at least one book? (Arrangements that differ in the order of books on a shelf are considered different). | 75600 | 0.0625 | 7,564.9375 | 6,064 | 7,665 | |
The greatest common divisor (GCD) of 17 and 51 is , and the least common multiple (LCM) is . The GCD of 6 and 8 is , and the LCM of 8 and 9 is . | 72 | 1 | 887.3125 | 887.3125 | -1 | |
How many natural numbers between 200 and 400 are divisible by 8? | 25 | 0 | 5,264.75 | -1 | 5,264.75 | |
Paint both sides of a small wooden board. It takes 1 minute to paint one side, but you must wait 5 minutes for the paint to dry before painting the other side. How many minutes will it take to paint 6 wooden boards in total? | 12 | 0 | 444.5 | -1 | 444.5 |
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