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 |
|---|---|---|---|---|---|---|
A team consisting of Petya, Vasya, and a single-seat scooter is participating in a race. The distance is divided into 42 equal-length segments, with a checkpoint at the beginning of each segment. Petya completes a segment in 9 minutes, Vasya in 11 minutes, and either of them can cover a segment on the scooter in 3 minu... | 18 | 0.125 | 7,955.3125 | 6,298.5 | 8,192 | |
Dr. Green gives bonus points on a test for students who score above the class average. In a class of 150 students, what is the maximum number of students who can score above the average if their scores are integers? | 149 | 0.1875 | 7,776.375 | 6,213.666667 | 8,137 | |
Let \\(f(x)\\) be defined on \\((-∞,+∞)\\) and satisfy \\(f(2-x)=f(2+x)\\) and \\(f(7-x)=f(7+x)\\). If in the closed interval \\([0,7]\\), only \\(f(1)=f(3)=0\\), then the number of roots of the equation \\(f(x)=0\\) in the closed interval \\([-2005,2005]\\) is . | 802 | 0.125 | 7,749.9375 | 5,511 | 8,069.785714 | |
Estimate the range of the submissions for this problem. Your answer must be between $[0, 1000]$ . An estimate $E$ earns $\frac{2}{1+0.05|A-E|}$ points, where $A$ is the actual answer.
*2022 CCA Math Bonanza Lightning Round 5.2* | 500 | 0.375 | 7,008.5 | 5,807.833333 | 7,728.9 | |
Define $L(x) = x - \frac{x^2}{2}$ for every real number $x$. If $n$ is a positive integer, define $a_n$ by
\[
a_n = L \Bigl( L \Bigl( L \Bigl( \cdots L \Bigl( \frac{17}{n} \Bigr) \cdots \Bigr) \Bigr) \Bigr),
\]where there are $n$ iterations of $L$. For example,
\[
a_4 = L \Bigl( L \Bigl( L \Bigl( L \Bigl( \frac{1... | \frac{34}{19} | 0.375 | 7,378.875 | 6,023.666667 | 8,192 | |
Ed has five identical green marbles, and a large supply of identical red marbles. He arranges the green marbles and some of the red ones in a row and finds that the number of marbles whose right hand neighbor is the same color as themselves is equal to the number of marbles whose right hand neighbor is the other color.... | 3 | We are limited by the number of marbles whose right hand neighbor is not the same color as the marble. By surrounding every green marble with red marbles - RGRGRGRGRGR. That's 10 "not the same colors" and 0 "same colors." Now, for every red marble we add, we will add one "same color" pair and keep all 10 "not the same ... | 0 | 8,192 | -1 | 8,192 |
Ang, Ben, and Jasmin each have $5$ blocks, colored red, blue, yellow, white, and green; and there are $5$ empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives $3$ blocks all of the same color is $\frac... | 471 | 1. **Understanding the Problem:**
- Ang, Ben, and Jasmin each have 5 blocks of different colors and there are 5 boxes.
- They independently place one block in each box.
- We need to find the probability that at least one box ends up with 3 blocks of the same color.
2. **Setting Up the Problem:**
- Each per... | 0 | 7,830.1875 | -1 | 7,830.1875 |
We have created a convex polyhedron using pentagons and hexagons where three faces meet at each vertex. Each pentagon shares its edges with 5 hexagons, and each hexagon shares its edges with 3 pentagons. How many faces does the polyhedron have? | 32 | 0.375 | 5,519.8125 | 3,865.5 | 6,512.4 | |
In an acute-angled $\triangle ABC$, find the minimum value of $3 \tan B \tan C + 2 \tan A \tan C + \tan A \tan B$. | 6 + 2\sqrt{3} + 2\sqrt{2} + 2\sqrt{6} | 0 | 7,999.25 | -1 | 7,999.25 | |
As shown in the figure below, point $E$ lies on the opposite half-plane determined by line $CD$ from point $A$ so that $\angle CDE = 110^\circ$. Point $F$ lies on $\overline{AD}$ so that $DE=DF$, and $ABCD$ is a square. What is the degree measure of $\angle AFE$? | 170 | 1. **Identify the given information and setup**: We know that $ABCD$ is a square, and $E$ lies on the opposite half-plane determined by line $CD$ from point $A$. The angle $\angle CDE = 110^\circ$. Point $F$ lies on $\overline{AD}$ such that $DE = DF$.
2. **Extend $\overline{AD}$ to point $G$**: By extending $\overlin... | 0 | 8,144.5 | -1 | 8,144.5 |
Given that $f(x)$ is an odd function defined on $\mathbb{R}$ with a minimal positive period of $3$, and $f(x)=\log_2(1-x)$ when $x \in \left(-\frac{3}{2}, 0\right)$, find the value of $f(2014) + f(2016)$. | -1 | 0.8125 | 4,690.5625 | 4,276.461538 | 6,485 | |
On $5\times 5$ squares, we cover the area with several S-Tetrominos (=Z-Tetrominos) along the square so that in every square, there are two or fewer tiles covering that (tiles can be overlap). Find the maximum possible number of squares covered by at least one tile. | 24 | 0.125 | 7,417.4375 | 7,352 | 7,426.785714 | |
The sum of the areas of all triangles whose vertices are also vertices of a $1$ by $1$ by $1$ cube is $m + \sqrt{n} + \sqrt{p},$ where $m, n,$ and $p$ are integers. Find $m + n + p.$ | 348 | [asy] size(120); import three; unitsize(1cm); draw((0,0,0)--(1,0,0)--(1,1,0)--(0,1,0)--(0,0,0)--(0,0,1)--(0,1,1)--(1,1,1)--(1,0,1)--(0,0,1)); draw((0,1,1)--(0,1,0)); draw((1,1,1)--(1,1,0)); draw((1,0,1)--(1,0,0)); draw((1,1,1)--(1,0,0)--(1,1,0)--(1,1,1), blue); [/asy]
[asy] size(120); import three; unitsize(1cm); draw(... | 0 | 8,192 | -1 | 8,192 |
If set $A=\{-4, 2a-1, a^2\}$, $B=\{a-5, 1-a, 9\}$, and $A \cap B = \{9\}$, then the value of $a$ is. | -3 | 0.625 | 6,989.5 | 7,066.8 | 6,860.666667 | |
Given a geometric series \(\left\{a_{n}\right\}\) with the sum of its first \(n\) terms denoted by \(S_{n}\), and satisfying the equation \(S_{n}=\frac{\left(a_{n}+1\right)^{2}}{4}\), find the value of \(S_{20}\). | 400 | 0 | 4,534.75 | -1 | 4,534.75 | |
A six-digit number is formed by the digits 1, 2, 3, 4, with two pairs of repeating digits, where one pair of repeating digits is not adjacent, and the other pair is adjacent. Calculate the number of such six-digit numbers. | 432 | 0.0625 | 8,192 | 8,192 | 8,192 | |
The graphs of four functions, labelled (2) through (5), are shown below. Note that the domain of function (3) is $$\{-5,-4,-3,-2,-1,0,1,2\}.$$ Find the product of the labels of the functions which are invertible. [asy]
size(8cm);
defaultpen(linewidth(.7pt)+fontsize(8pt));
import graph;
picture pic1,pic2,pic3,pic4;
dr... | 60 | 0.6875 | 3,693.4375 | 3,853.545455 | 3,341.2 | |
Given an ellipse C: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with its right focus F, upper vertex B, the line BF intersects C at another point A, and the projection of point A on the x-axis is A1. O is the origin, and if $\overrightarrow{BO}=2\overrightarrow{A_{1}A}$, determine the eccentricity of C. | \frac{\sqrt{3}}{3} | 0 | 4,055 | -1 | 4,055 | |
A sports team's members have unique numbers taken from the integers 1 to 100. If no member's number is the sum of the numbers of any two other members, nor is it twice the number of any other member, what is the maximum number of people this team can have? | 50 | 0 | 8,192 | -1 | 8,192 | |
How many numbers from the set $\{1, 2, 3, \ldots, 100\}$ have a perfect square factor greater than one? | 40 | 0 | 7,976.4375 | -1 | 7,976.4375 | |
The number of real values of $x$ satisfying the equation $2^{2x^2 - 7x + 5} = 1$ is: | 2 | 1. **Identify the Equation Form**: The given equation is $2^{2x^2 - 7x + 5} = 1$. We know that any number raised to the power of zero equals one, i.e., $a^0 = 1$ for any $a \neq 0$. Therefore, we need $2x^2 - 7x + 5 = 0$.
2. **Solve the Quadratic Equation**: The quadratic equation is $2x^2 - 7x + 5 = 0$. We can attemp... | 1 | 1,550.875 | 1,550.875 | -1 |
Emma writes down the smallest positive multiple of 30 that is a perfect square, the smallest positive multiple of 30 that is a perfect cube, and all the multiples of 30 between them. How many integers are in Emma's list? | 871 | 0.9375 | 5,092.375 | 4,885.733333 | 8,192 | |
Last summer, Pat worked at a summer camp. For each day that he worked, he earned \$100 and he was not charged for food. For each day that he did not work, he was not paid and he was charged \$20 for food. After 70 days, the money that he earned minus his food costs equalled \$5440. On how many of these 70 days did Pat ... | 57 | Let \( x \) be the number of days on which Pat worked. On each of these days, he earned \$100 and had no food costs, so he earned a total of \( 100x \) dollars. Since Pat worked for \( x \) of the 70 days, then he did not work on \( 70-x \) days. On each of these days, he earned no money and was charged \$20 for food, ... | 1 | 1,489.625 | 1,489.625 | -1 |
In triangle \( \triangle ABC \), the sides opposite to angles \( A \), \( B \), and \( C \) are \( a \), \( b \), and \( c \) respectively. Given that \( a^2 - (b - c)^2 = (2 - \sqrt{3})bc \) and \( \sin A \sin B = \cos^2 \frac{C}{2} \), and the length of the median \( AM \) from \( A \) to side \( BC \) is \( \sqrt{7}... | \sqrt{3} | 0.6875 | 6,197.8125 | 5,291.363636 | 8,192 | |
Approximate the increase in the volume of a cylinder with a height of \( H = 40 \) cm and a base radius of \( R = 30 \) cm when the radius is increased by \( 0.5 \) cm. | 1200\pi | 0.4375 | 5,228 | 4,970.285714 | 5,428.444444 | |
8 singers are participating in a festival and are scheduled to perform \( m \) times, with 4 singers performing in each show. Design a plan to minimize the number of performances \( m \) so that any two singers perform together the same number of times. | 14 | 1 | 4,165 | 4,165 | -1 | |
What is the least possible sum of two positive integers $a$ and $b$ where $a \cdot b = 10! ?$ | 3960 | 0 | 8,192 | -1 | 8,192 | |
Let $a_n=6^{n}+8^{n}$. Determine the remainder upon dividing $a_ {83}$ by $49$.
| 35 | 0.875 | 5,091.75 | 4,648.857143 | 8,192 | |
As shown in the diagram, there is a sequence of curves \(P_{0}, P_{1}, P_{2}, \cdots\). It is given that \(P_{0}\) is an equilateral triangle with an area of 1. Each \(P_{k+1}\) is obtained from \(P_{k}\) by performing the following operations: each side of \(P_{k}\) is divided into three equal parts, an equilateral tr... | \frac{8}{5} | 0.0625 | 7,728.8125 | 7,786 | 7,725 | |
1. Given $$\cos\left(\alpha+ \frac {\pi}{6}\right)-\sin\alpha= \frac {3 \sqrt {3}}{5}$$, find the value of $$\sin\left(\alpha+ \frac {5\pi}{6}\right)$$;
2. Given $$\sin\alpha+\sin\beta= \frac {1}{2}$$ and $$\cos\alpha+\cos\beta= \frac {\sqrt {2}}{2}$$, find the value of $$\cos(\alpha-\beta)$$. | -\frac {5}{8} | 0.875 | 5,508.875 | 5,125.571429 | 8,192 | |
A class of 54 students in the fifth grade took a group photo. The fixed price is 24.5 yuan for 4 photos. Additional prints cost 2.3 yuan each. If every student in the class wants one photo, how much money in total needs to be paid? | 139.5 | 0.375 | 523.375 | 453.5 | 565.3 | |
Given positive integers \( a, b, \) and \( c \) such that \( a < b < c \). If the product of any two numbers minus 1 is divisible by the third number, what is \( a^{2} + b^{2} + c^{2} \)? | 38 | 0.5625 | 7,250.375 | 6,518 | 8,192 | |
Two positive integers are written on the blackboard, one being 2002 and the other being a number less than 2002. If the arithmetic mean of the two numbers is an integer $m$, then the following operation can be performed: one of the numbers is erased and replaced by $m$. What is the maximum number of times this operatio... | 10 | 0.0625 | 8,031 | 6,920 | 8,105.066667 | |
a) Calculate the number of triangles whose three vertices are vertices of the cube.
b) How many of these triangles are not contained in a face of the cube? | 32 | 0.75 | 4,713.875 | 3,554.5 | 8,192 | |
Find the number of complex solutions to
\[\frac{z^3 - 1}{z^2 + z - 2} = 0.\] | 2 | 1 | 3,105.1875 | 3,105.1875 | -1 | |
The graph of $y^2 + 2xy + 40|x|= 400$ partitions the plane into several regions. What is the area of the bounded region? | 800 | The equation given can be rewritten as:
$40|x| = - y^2 - 2xy + 400$
We can split the equation into a piecewise equation by breaking up the absolute value:
$40x = -y^2 - 2xy + 400\quad\quad x\ge 0$
$40x = y^2 + 2xy - 400 \quad \quad x < 0$
Factoring the first one: (alternatively, it is also possible to complete the sq... | 0.375 | 7,640.75 | 7,509.5 | 7,719.5 |
A coordinate paper is folded once such that the point \((0,2)\) overlaps with the point \((4,0)\). If the point \((7,3)\) overlaps with the point \((m, n)\), what is the value of \(m+n\)? | 6.8 | 0 | 4,061.25 | -1 | 4,061.25 | |
What is the area enclosed by the graph of \( |x| + |3y| = 9 \)? | 54 | 0.9375 | 3,535.0625 | 3,224.6 | 8,192 | |
Given the function $f(x) = \frac{bx}{\ln x} - ax$, where $e$ is the base of the natural logarithm.
(1) If the equation of the tangent line to the graph of the function $f(x)$ at the point $({e}^{2}, f({e}^{2}))$ is $3x + 4y - e^{2} = 0$, find the values of the real numbers $a$ and $b$.
(2) When $b = 1$, if there exis... | \frac{1}{2} - \frac{1}{4e^{2}} | 0 | 7,910.5 | -1 | 7,910.5 | |
In the Cartesian coordinate system, there are points $P_{1}, P_{2}, \ldots, P_{n-1}, P_{n}, \ldots (n \in \mathbb{N}^{*})$. Let the coordinates of point $P_{n}$ be $(n, a_{n})$, where $a_{n}= \frac {2}{n} (n \in \mathbb{N}^{*})$. The line passing through points $P_{n}$ and $P_{n+1}$ forms a triangle with the coordinate... | \frac {125}{6} | 0.9375 | 5,153.125 | 4,950.533333 | 8,192 | |
The positive integers \(a\), \(b\) are such that \(15a + 16b\) and \(16a - 15b\) are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares? | 481 | 0 | 8,192 | -1 | 8,192 | |
When a spring is stretched by 5 cm, 29.43 J of work is done. How far will the spring stretch if 9.81 J of work is done? | 0.029 | 0 | 5,585.6875 | -1 | 5,585.6875 | |
A $k \times k$ array contains each of the numbers $1, 2, \dots, m$ exactly once, with the remaining entries all zero. Suppose that all the row sums and column sums are equal. What is the smallest possible value of $m$ if $k = 3^n$ ($n \in \mathbb{N}^+$)? | 3^{n+1} - 1 |
Consider a \( k \times k \) array, where \( k = 3^n \) for a positive integer \( n \). The array contains each of the integers \( 1, 2, \ldots, m \) exactly once, and the remaining entries are all zeros. We are tasked with finding the smallest possible value of \( m \) such that all row sums and column sums are equal.... | 0 | 7,624.125 | -1 | 7,624.125 |
Given that $z$ is a complex number such that $z+\frac 1z=2\cos 3^\circ$, find the least integer that is greater than $z^{2000}+\frac 1{z^{2000}}$. | 0 | Let $z=re^{i\theta}$. Notice that we have $2\cos(3^{\circ})=e^{i\frac{\pi}{60}}+e^{-i\frac{\pi}{60}}=re^{i\theta}+\frac{1}{r}e^{-i\theta}.$
$r$ must be $1$ (or else if you take the magnitude would not be the same). Therefore, $z=e^{i\frac{\pi}{\theta}}$ and plugging into the desired expression, we get $e^{i\frac{100\pi... | 0.875 | 4,339.5 | 3,789.142857 | 8,192 |
On the $x O y$ coordinate plane, there is a Chinese chess "knight" at the origin $(0,0)$. The "knight" needs to be moved to the point $P(1991,1991)$ using the movement rules of the chess piece. Calculate the minimum number of moves required. | 1328 | 0 | 8,192 | -1 | 8,192 | |
Three cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is an Ace, the second card is a diamond, and the third card is a Jack? | \frac{1}{650} | 0.25 | 7,815.4375 | 6,685.75 | 8,192 | |
If the function
$$
f(x) = |a \sin x + b \cos x - 1| + |b \sin x - a \cos x| \quad (a, b \in \mathbf{R})
$$
attains a maximum value of 11, then $a^{2} + b^{2} = \, \, \, $ . | 50 | 0.125 | 7,185.3125 | 6,468.5 | 7,287.714286 | |
What is the correct order of the fractions $\frac{15}{11}, \frac{19}{15},$ and $\frac{17}{13},$ from least to greatest? | \frac{19}{15}<\frac{17}{13}<\frac{15}{11} | To find the correct order of the fractions $\frac{15}{11}, \frac{19}{15},$ and $\frac{17}{13}$ from least to greatest, we can use the method of comparing each pair of fractions by cross-multiplying, which avoids the need for finding a common denominator.
1. **Compare $\frac{15}{11}$ and $\frac{19}{15}$:**
\[
\fr... | 0.1875 | 3,775.25 | 3,229.666667 | 3,901.153846 |
Given that point $P$ is a moving point on the parabola $y=\frac{1}{4}x^2$, determine the minimum value of the sum of the distance from point $P$ to the line $x+2y+4=0$ and the $x$-axis. | \frac{6\sqrt{5}}{5}-1 | 0 | 7,331.9375 | -1 | 7,331.9375 | |
Let \(\mathbf{v}\) be a vector such that
\[
\left\| \mathbf{v} + \begin{pmatrix} 4 \\ 2 \end{pmatrix} \right\| = 10.
\]
Find the smallest possible value of \(\|\mathbf{v}\|\). | 10 - 2\sqrt{5} | 0.8125 | 6,587.25 | 6,216.923077 | 8,192 | |
Given that a square $S_1$ has an area of $25$, the area of the square $S_3$ constructed by bisecting the sides of $S_2$ is formed by the points of bisection of $S_2$. | 6.25 | 0 | 6,265.125 | -1 | 6,265.125 | |
In triangle \(ABC\), angle \(C\) equals \(30^\circ\), and angle \(A\) is acute. A line perpendicular to side \(BC\) is drawn, cutting off triangle \(CNM\) from triangle \(ABC\) (point \(N\) lies between vertices \(B\) and \(C\)). The areas of triangles \(CNM\) and \(ABC\) are in the ratio \(3:16\). Segment \(MN\) is ha... | 1/3 | 0.0625 | 8,020.125 | 7,411 | 8,060.733333 | |
What is the sum and the product of the values of $x$ that satisfy the equation $x^2 - 7x + 12 = 0$? | 12 | 1 | 1,805.0625 | 1,805.0625 | -1 | |
How many distinct terms are in the expansion of \[(a+b+c+d)(e+f+g+h+i)\] assuming that terms involving the product of $a$ and $e$, and $b$ and $f$ are identical and combine into a single term? | 19 | 0.4375 | 7,154.1875 | 5,837.285714 | 8,178.444444 | |
Suppose $x$ and $y$ satisfy the system of inequalities $\begin{cases} & x-y \geqslant 0 \\ & x+y-2 \geqslant 0 \\ & x \leqslant 2 \end{cases}$, calculate the minimum value of $x^2+y^2-2x$. | -\dfrac{1}{2} | 0.6875 | 6,155.9375 | 5,663.090909 | 7,240.2 | |
In triangle \(ABC\), the sides \(AC = 14\) and \(AB = 6\) are known. A circle with center \(O\) is constructed using side \(AC\) as the diameter, intersecting side \(BC\) at point \(K\). It turns out that \(\angle BAK = \angle ACB\). Find the area of triangle \(BOC\). | 21 | 0.0625 | 7,931.5 | 4,024 | 8,192 | |
Find the sum of the absolute values of the roots of $x^{4}-4 x^{3}-4 x^{2}+16 x-8=0$. | 2+2\sqrt{2}+2\sqrt{3} | $$\begin{aligned} x^{4}-4 x^{3}-4 x^{2}+16 x-8 & =\left(x^{4}-4 x^{3}+4 x^{2}\right)-\left(8 x^{2}-16 x+8\right) \\ & =x^{2}(x-2)^{2}-8(x-1)^{2} \\ & =\left(x^{2}-2 x\right)^{2}-(2 \sqrt{2} x-2 \sqrt{2})^{2} \\ & =\left(x^{2}-(2+2 \sqrt{2}) x+2 \sqrt{2}\right)\left(x^{2}-(2-2 \sqrt{2}) x-2 \sqrt{2}\right) \end{aligned}... | 0 | 7,403.3125 | -1 | 7,403.3125 |
The cost price of a certain product is 1360 yuan. When it is sold at 80% of the marked price, the profit margin is 15%. What should be the marked price of the product in yuan? | 1955 | 0.5625 | 1,893.625 | 1,682.555556 | 2,165 | |
The stem-and-leaf plot shows the number of minutes and seconds of one ride on each of the $17$ top-rated roller coasters in the world. In the stem-and-leaf plot, $2 \ 20$ represents $2$ minutes, $20$ seconds, which is the same as $140$ seconds. What is the median of this data set? Express your answer in seconds.
\begi... | 163 | 0.8125 | 3,812.9375 | 3,284.846154 | 6,101.333333 | |
If the integer part of $\sqrt{10}$ is $a$ and the decimal part is $b$, then $a=$______, $b=\_\_\_\_\_\_$. | \sqrt{10} - 3 | 0.375 | 1,526.9375 | 997 | 1,844.9 | |
Find the area of quadrilateral ABCD given that $\angle A = \angle D = 120^{\circ}$, $AB = 5$, $BC = 7$, $CD = 3$, and $DA = 4$. | \frac{47\sqrt{3}}{4} | 0 | 8,011.4375 | -1 | 8,011.4375 | |
The number of sets of two or more consecutive positive integers whose sum is 100 is | 2 | 1. **Formulate the sum of consecutive integers**: Let's consider a sequence of $n$ consecutive integers starting from $a$. The integers are $a, a+1, a+2, \ldots, a+n-1$. The sum $S$ of these integers can be calculated using the formula for the sum of an arithmetic series:
\[
S = \frac{n}{2} \times (\text{first te... | 0.125 | 7,974 | 6,448 | 8,192 |
On the board, the number 0 is written. Two players take turns appending to the expression on the board: the first player appends a + or - sign, and the second player appends one of the natural numbers from 1 to 1993. The players make 1993 moves each, and the second player uses each of the numbers from 1 to 1993 exactly... | 1993 | 0 | 8,192 | -1 | 8,192 | |
A three-quarter sector of a circle of radius $4$ inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches? | $3 \pi \sqrt{7}$ |
1. **Understanding the Geometry of the Problem**:
The problem states that a three-quarter sector of a circle with radius $4$ inches is rolled up to form a cone. The arc length of this sector becomes the circumference of the base of the cone, and the radius of the sector becomes the slant height of the cone.
2. **C... | 0 | 1,839.6875 | -1 | 1,839.6875 |
There are $10$ people standing equally spaced around a circle. Each person knows exactly $3$ of the other $9$ people: the $2$ people standing next to her or him, as well as the person directly across the circle. How many ways are there for the $10$ people to split up into $5$ pairs so that the members of each pair know... | 12 | To solve this problem, we need to consider the relationships each person has with others in the circle. Each person knows exactly three others: the two adjacent people and the one directly across the circle. We need to find how many ways we can pair up these 10 people such that each pair consists of people who know eac... | 0.125 | 8,042.125 | 6,993 | 8,192 |
Evaluate the sum \[\frac{1}{2^1} + \frac{2}{2^2} + \frac{3}{2^3} + \cdots + \frac{k}{2^k} + \cdots \] | 2 | 1 | 3,995 | 3,995 | -1 | |
$(1)$ Calculate: $\sqrt{3}\tan 45^{\circ}-\left(2023-\pi \right)^{0}+|2\sqrt{3}-2|+(\frac{1}{4})^{-1}-\sqrt{27}$; <br/>$(2)$ Simplify first, then evaluate: $\frac{{{x^2}-x}}{{{x^2}+2x+1}}\div (\frac{2}{{x+1}}-\frac{1}{x})$, simplify it first, then choose an integer you like as the value of $x$ within the range of $-2 \... | \frac{4}{3} | 0.8125 | 4,238.6875 | 4,342.076923 | 3,790.666667 | |
Find the common ratio of the infinite geometric series: $$\frac{4}{7} + \frac{16}{21} + \frac{64}{63} + \dots$$ | \frac{4}{3} | 1 | 2,273.1875 | 2,273.1875 | -1 | |
Find a positive integer that is divisible by 14 and has a square root between 25 and 25.3. | 630 | 1 | 1,844.6875 | 1,844.6875 | -1 | |
In triangle $ABC,$ if median $\overline{AD}$ makes an angle of $45^\circ$ with side $\overline{BC},$ then find the value of $|\cot B - \cot C|.$ | 2 | 0.625 | 6,892 | 6,112 | 8,192 | |
Find the least positive integer of the form <u>a</u> <u>b</u> <u>a</u> <u>a</u> <u>b</u> <u>a</u>, where a and b are distinct digits, such that the integer can be written as a product of six distinct primes | 282282 | 0 | 8,192 | -1 | 8,192 | |
Call a set of positive integers "conspiratorial" if no three of them are pairwise relatively prime. What is the largest number of elements in any "conspiratorial" subset of the integers $1$ to $16$ ? | 11 | 0 | 8,192 | -1 | 8,192 | |
How many two-digit numbers have digits whose sum is a perfect square? | 17 | 0.9375 | 4,712.0625 | 4,480.066667 | 8,192 | |
The hyperbola $C:\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ $(a > 0,b > 0)$ has an asymptote perpendicular to the line $x+2y+1=0$. Let $F_1$ and $F_2$ be the foci of $C$, and let $A$ be a point on the hyperbola. If $|F_1A|=2|F_2A|$, then $\cos \angle AF_2F_1=$ __________. | \dfrac{\sqrt{5}}{5} | 0 | 7,258.875 | -1 | 7,258.875 | |
In each cell of a $5 \times 5$ board, there is either an X or an O, and no three Xs are consecutive horizontally, vertically, or diagonally. What is the maximum number of Xs that can be on the board? | 16 | 0 | 8,186.9375 | -1 | 8,186.9375 | |
Javier is excited to visit Disneyland during spring break. He plans on visiting five different attractions, but he is particularly excited about the Space Mountain ride and wants to visit it twice during his tour before lunch. How many different sequences can he arrange his visits to these attractions, considering his ... | 360 | 0.125 | 1,697.625 | 656.5 | 1,846.357143 | |
The values of $y$ which will satisfy the equations $2x^{2}+6x+5y+1=0$, $2x+y+3=0$ may be found by solving: | $y^{2}+10y-7=0$ | 1. **Substitute $x$ from the second equation into the first equation:**
Given the equations:
\[ 2x^2 + 6x + 5y + 1 = 0 \]
\[ 2x + y + 3 = 0 \]
Solve the second equation for $x$:
\[ 2x = -y - 3 \]
\[ x = -\frac{y+3}{2} \]
2. **Substitute $x = -\frac{y+3}{2}$ into the first equation:**
\[ 2\left(-\f... | 0 | 4,255.6875 | -1 | 4,255.6875 |
In a bag, there are 10 red balls, 8 white balls, and 7 black balls. The balls are randomly drawn one by one. To ensure that a red ball is drawn for the first time on the $k$-th draw or before, the minimum value of $k$ is ______. | 16 | 0.625 | 4,386.625 | 2,790.4 | 7,047 | |
Given the scores $93$, $89$, $92$, $95$, $93$, $94$, $93$ of seven referees, determine the average value of the remaining data after removing the highest and lowest scores. | 93 | 0.5625 | 470 | 462 | 480.285714 | |
Each of the letters in "GEOMETRY" is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word "RHYME"? Express your answer as a common fraction. | \frac{1}{2} | 0.75 | 4,304.0625 | 3,840.5 | 5,694.75 | |
Given $\sin (α- \frac {π}{6})= \frac {2}{3}$, $α∈(π, \frac {3π}{2})$, $\cos ( \frac {π}{3}+β)= \frac {5}{13}$, $β∈(0,π)$, find the value of $\cos (β-α)$. | - \frac{10+12 \sqrt{5}}{39} | 0 | 8,160.125 | -1 | 8,160.125 | |
\angle 1 + \angle 2 = 180^\circ
\angle 3 = \angle 4
Find \angle 4. | 35^\circ | 1. **Using the left triangle's angle sum property**: The sum of angles in any triangle is $180^\circ$. Given $\angle 1 + 70^\circ + 40^\circ = 180^\circ$, we can solve for $\angle 1$:
\[
\angle 1 + 110^\circ = 180^\circ
\]
\[
\angle 1 = 180^\circ - 110^\circ = 70^\circ
\]
2. **Using the given relatio... | 0 | 4,906.25 | -1 | 4,906.25 |
In $\triangle ABC$, $D$ is a point on $BC$ such that $\frac{BD}{DC}=\frac{1}{3}$. $E$ is the midpoint of $AC$. $AD$ and $BE$ intersect at $O$, and $CO$ intersects $AB$ at $F$. Find the ratio of the area of quadrilateral $BDOF$ to the area of $\triangle ABC$. | 1/10 | 0.6875 | 7,007.875 | 6,469.636364 | 8,192 | |
How many prime positive integers are divisors of 555? | 3 | 1 | 1,322.375 | 1,322.375 | -1 | |
Complex numbers are often used when dealing with alternating current (AC) circuits. In the equation $V = IZ$, $V$ is voltage, $I$ is current, and $Z$ is a value known as impedance. If $V = 1-i$ and $Z=1+3i$, find $I$. Express your answer as a complex number in the form $a+bi$, where $a$ and $b$ are real numbers. | -\frac{1}{5} - \frac{2}{5}i | 1 | 3,171.125 | 3,171.125 | -1 | |
What is the value of $2^{0^{1^9}} + (2^0)^{1^9}$? | 2 | To solve the expression \[2^{\left(0^{\left(1^9\right)}\right)}+\left(\left(2^0\right)^1\right)^9,\] we need to evaluate each term separately.
1. **Evaluate the first term: $2^{\left(0^{\left(1^9\right)}\right)}$**
- First, simplify the exponent's base: $1^9 = 1$.
- Then, calculate $0^1 = 0$.
- Finally, compu... | 1 | 1,906.375 | 1,906.375 | -1 |
Calculate the value of the expression $2+3 imes 5+2$. | 19 | Calculating, $2+3 imes 5+2=2+15+2=19$. | 1 | 283.375 | 283.375 | -1 |
Given \(3 \sin^{2} \alpha + 2 \sin^{2} \beta = 1\) and \(3 (\sin \alpha + \cos \alpha)^{2} - 2 (\sin \beta + \cos \beta)^{2} = 1\), find \(\cos 2 (\alpha + \beta) = \quad \) . | -\frac{1}{3} | 0.625 | 6,619.75 | 5,856 | 7,892.666667 | |
Three balls numbered 1, 2, and 3 are placed in a bag. A ball is drawn, the number recorded, and then the ball is returned. This process is repeated three times. Calculate the probability that the sum of the three recorded numbers is less than 8. | \frac{23}{27} | 0.625 | 6,525.75 | 5,526 | 8,192 | |
Given a right triangle \( ABC \) with legs \( AC = 3 \) and \( BC = 4 \). Construct triangle \( A_1 B_1 C_1 \) by successively translating point \( A \) a certain distance parallel to segment \( BC \) to get point \( A_1 \), then translating point \( B \) parallel to segment \( A_1 C \) to get point \( B_1 \), and fina... | 12 | 0 | 8,192 | -1 | 8,192 | |
How many 4-digit positive integers, where each digit is odd, are divisible by 3? | 208 | 0.25 | 7,797.375 | 6,613.5 | 8,192 | |
On the International Space Station, there was an electronic clock displaying time in the HH:MM format. Due to an electromagnetic storm, the device started malfunctioning, causing each digit on the display to either increase by 1 or decrease by 1. What was the actual time when the storm occurred, if immediately after th... | 18:18 | 0.3125 | 7,447.25 | 5,808.8 | 8,192 | |
Given $w$ and $z$ are complex numbers such that $|w+z|=2$ and $|w^2+z^2|=10,$ find the smallest possible value of $|w^4+z^4|$. | 82 | 0 | 8,133.9375 | -1 | 8,133.9375 | |
In a set of 10 programs, there are 6 singing programs and 4 dance programs. The requirement is that there must be at least one singing program between any two dance programs. Determine the number of different ways to arrange these programs. | 604800 | 0.5 | 6,122 | 5,714.875 | 6,529.125 | |
The sum of the non-negative numbers \(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7}\) is 1. Let \(M\) be the maximum of the quantities \(a_{1} + a_{2} + a_{3}, a_{2} + a_{3} + a_{4}, a_{3} + a_{4} + a_{5}, a_{4} + a_{5} + a_{6}, a_{5} + a_{6} + a_{7}\).
How small can \(M\) be? | 1/3 | 0.1875 | 7,473.875 | 6,249.666667 | 7,756.384615 | |
Several points were marked on a line, and then two additional points were placed between each pair of neighboring points. This procedure was repeated once more with the entire set of points. Could there have been 82 points on the line as a result? | 10 | 0 | 5,331.4375 | -1 | 5,331.4375 | |
Which fractions $ \dfrac{p}{q},$ where $p,q$ are positive integers $< 100$, is closest to $\sqrt{2} ?$ Find all digits after the point in decimal representation of that fraction which coincide with digits in decimal representation of $\sqrt{2}$ (without using any table). | \sqrt2\approx1+\frac1{2+\frac1{2+\frac1{2+\frac1{2+\frac12}}}}=\boxed{\frac{99}{70}} |
We are tasked with finding the fraction \(\frac{p}{q}\), where \( p, q \) are positive integers less than 100, that is closest to \(\sqrt{2}\). Additionally, we aim to determine how many digits after the decimal point coincide between this fraction and \(\sqrt{2}\).
### Step 1: Representation of \(\sqrt{2}\) via Cont... | 0 | 8,041.625 | -1 | 8,041.625 |
It is known that the numbers \(x, y, z \) form an arithmetic progression in that order with a common difference \(\alpha = \arcsin \frac{\sqrt{7}}{4}\), and the numbers \(\frac{1}{\sin x}, \frac{4}{\sin y}, \frac{1}{\sin z}\) also form an arithmetic progression in that order. Find \(\sin ^{2} y\). | \frac{7}{13} | 0.1875 | 7,723 | 5,690.666667 | 8,192 | |
If $-2x - 7 = 7x + 2$, what is the value of $x$? | -1 | 1 | 1,234.8125 | 1,234.8125 | -1 | |
What is the largest four-digit number that is divisible by 6? | 9996 | 1 | 1,150.625 | 1,150.625 | -1 |
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