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 |
|---|---|---|---|---|---|---|
How many solutions does the equation
$$
\{x\}^{2}=\left\{x^{2}\right\}
$$
have in the interval $[1, 100]$? ($\{u\}$ denotes the fractional part of $u$, which is the difference between $u$ and the largest integer not greater than $u$.) | 9901 | 0.125 | 8,132.0625 | 7,755.5 | 8,185.857143 | |
Find $(100110_2 + 1001_2) \times 110_2 \div 11_2$. Express your answer in base 2. | 1011110_2 | 0.1875 | 7,249.625 | 3,166 | 8,192 | |
Let the function \( f(x) \) satisfy the following conditions:
(i) If \( x > y \), and \( f(x) + x \geq w \geq f(y) + y \), then there exists a real number \( z \in [y, x] \), such that \( f(z) = w - z \);
(ii) The equation \( f(x) = 0 \) has at least one solution, and among the solutions of this equation, there exists ... | 2004 | 0.5625 | 6,020.5 | 4,331.555556 | 8,192 | |
Given that a floor is tiled in a similar pattern with a $4 \times 4$ unit repeated pattern and each of the four corners looks like the scaled down version of the original, determine the fraction of the tiled floor made up of darker tiles, assuming symmetry and pattern are preserved. | \frac{1}{2} | 0 | 7,199.375 | -1 | 7,199.375 | |
If $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ are vectors such that $\mathbf{a} \cdot \mathbf{b} = -3,$ $\mathbf{a} \cdot \mathbf{c} = 4,$ and $\mathbf{b} \cdot \mathbf{c} = 6,$ then find
\[\mathbf{b} \cdot (7 \mathbf{c} - 2 \mathbf{a}).\] | 48 | 1 | 1,253.4375 | 1,253.4375 | -1 | |
Define
\[ A' = \frac{1}{1^2} + \frac{1}{7^2} - \frac{1}{11^2} - \frac{1}{13^2} + \frac{1}{19^2} + \frac{1}{23^2} - \dotsb, \]
which omits all terms of the form $\frac{1}{n^2}$ where $n$ is an odd multiple of 5, and
\[ B' = \frac{1}{5^2} - \frac{1}{25^2} + \frac{1}{35^2} - \frac{1}{55^2} + \frac{1}{65^2} - \frac{1}{85^2... | 26 | 0 | 7,961 | -1 | 7,961 | |
Calculate the limit of the function:
$$\lim_{x \rightarrow \frac{1}{3}} \frac{\sqrt[3]{\frac{x}{9}}-\frac{1}{3}}{\sqrt{\frac{1}{3}+x}-\sqrt{2x}}$$ | -\frac{2 \sqrt{2}}{3 \sqrt{3}} | 0 | 7,924.875 | -1 | 7,924.875 | |
The set of $x$-values satisfying the equation $x^{\log_{10} x} = \frac{x^3}{100}$ consists of: | 10 \text{ or } 100 | 1. **Rewrite the given equation**: We start with the equation \(x^{\log_{10} x} = \frac{x^3}{100}\).
2. **Take logarithm on both sides**: To simplify, we take the logarithm with base \(x\) on both sides:
\[
\log_x \left(x^{\log_{10} x}\right) = \log_x \left(\frac{x^3}{100}\right)
\]
Using the logarithmic i... | 0 | 4,669.375 | -1 | 4,669.375 |
What percent of square $EFGH$ is shaded? All angles in the diagram are right angles. [asy]
import graph;
defaultpen(linewidth(0.7));
xaxis(0,6,Ticks(1.0,NoZero));
yaxis(0,6,Ticks(1.0,NoZero));
fill((0,0)--(2,0)--(2,2)--(0,2)--cycle);
fill((3,0)--(4,0)--(4,4)--(0,4)--(0,3)--(3,3)--cycle);
fill((5,0)--(6,0)--(6,6)--(0,... | 61.11\% | 0.125 | 7,569.5625 | 7,601 | 7,565.071429 | |
Given that the vertex of angle $\alpha$ coincides with the origin $O$, its initial side coincides with the non-negative semi-axis of the $x$-axis, and its terminal side passes through point $P(-\frac{3}{5}, -\frac{4}{5})$.
(1) Find the value of $\sin(\alpha + \pi)$;
(2) If angle $\beta$ satisfies $\sin(\alpha + \beta... | \frac{16}{65} | 0.0625 | 7,440.9375 | 7,066 | 7,465.933333 | |
Let $f_0=f_1=1$ and $f_{i+2}=f_{i+1}+f_i$ for all $n\ge 0$. Find all real solutions to the equation
\[x^{2010}=f_{2009}\cdot x+f_{2008}\] | \frac{1 + \sqrt{5}}{2} \text{ and } \frac{1 - \sqrt{5}}{2} |
We begin with the recurrence relation given by \( f_0 = f_1 = 1 \) and \( f_{i+2} = f_{i+1} + f_i \) for all \( i \geq 0 \). This sequence is known as the Fibonacci sequence, where each term is the sum of the two preceding terms.
The given equation is:
\[
x^{2010} = f_{2009} \cdot x + f_{2008}
\]
We need to find the... | 0 | 8,108.9375 | -1 | 8,108.9375 |
A point is chosen at random on the number line between 0 and 1, and the point is colored green. Then, another point is chosen at random on the number line between 0 and 1, and this point is colored purple. What is the probability that the number of the purple point is greater than the number of the green point, but les... | \frac{1}{4} | 0.75 | 5,772.1875 | 5,097.166667 | 7,797.25 | |
Rectangular prism P Q R S W T U V has a square base P Q R S. Point X is on the face T U V W so that P X = 12, Q X = 10, and R X = 8. Determine the maximum possible area of rectangle P Q U T. | 67.82 | 0 | 7,408.0625 | -1 | 7,408.0625 | |
Consider the polynomials $P(x) = x^{6} - x^{5} - x^{3} - x^{2} - x$ and $Q(x) = x^{4} - x^{3} - x^{2} - 1.$ Given that $z_{1},z_{2},z_{3},$ and $z_{4}$ are the roots of $Q(x) = 0,$ find $P(z_{1}) + P(z_{2}) + P(z_{3}) + P(z_{4}).$ | 6 | Let $S_k=z_1^k+z_2^k+z_3^k+z_4^k$ then by Vieta's Formula we have \[S_{-1}=\frac{z_1z_2z_3+z_1z_3z_4+z_1z_2z_4+z_1z_2z_3}{z_1z_2z_3z_4}=0\] \[S_0=4\] \[S_1=1\] \[S_2=3\] By Newton's Sums we have \[a_4S_k+a_3S_{k-1}+a_2S_{k-2}+a_1S_{k-1}+a_0S_{k-4}=0\]
Applying the formula couples of times yields $P(z_1)+P(z_2)+P(z_3)+... | 0.8125 | 4,782.5 | 3,995.692308 | 8,192 |
Given the parametric equation of curve \\(C_{1}\\) as \\(\begin{cases}x=3\cos \alpha \\ y=\sin \alpha\end{cases} (\alpha\\) is the parameter\\()\\), and taking the origin \\(O\\) of the Cartesian coordinate system \\(xOy\\) as the pole and the positive half-axis of \\(x\\) as the polar axis to establish a polar coordin... | \dfrac{6 \sqrt{3}}{5} | 0 | 6,535.875 | -1 | 6,535.875 | |
Robert read a book for 10 days. He read an average of 25 pages per day for the first 5 days and an average of 40 pages per day for the next 4 days, and read 30 more pages on the last day. Calculate the total number of pages in the book. | 315 | 1 | 386.5 | 386.5 | -1 | |
The lines $l_1$ and $l_2$ are two tangents to the circle $x^2+y^2=2$. If the intersection point of $l_1$ and $l_2$ is $(1,3)$, then the tangent of the angle between $l_1$ and $l_2$ equals \_\_\_\_\_\_. | \frac{4}{3} | 0.9375 | 3,678.75 | 3,377.866667 | 8,192 | |
Given the function $f(x)$ defined on the interval $[-2011, 2011]$ and satisfying $f(x_1+x_2) = f(x_1) + f(x_2) - 2011$ for any $x_1, x_2 \in [-2011, 2011]$, and $f(x) > 2011$ when $x > 0$, determine the value of $M+N$. | 4022 | 1 | 5,154.875 | 5,154.875 | -1 | |
Find the remainder when $9 \times 99 \times 999 \times \cdots \times \underbrace{99\cdots9}_{\text{999 9's}}$ is divided by $1000$. | 109 | Note that $999\equiv 9999\equiv\dots \equiv\underbrace{99\cdots9}_{\text{999 9's}}\equiv -1 \pmod{1000}$ (see modular arithmetic). That is a total of $999 - 3 + 1 = 997$ integers, so all those integers multiplied out are congruent to $- 1\pmod{1000}$. Thus, the entire expression is congruent to $- 1\times9\times99 = - ... | 0.5625 | 6,554.6875 | 5,647.666667 | 7,720.857143 |
A digital watch displays time in a 24-hour format showing only hours and minutes. Find the largest possible sum of the digits in the display. | 24 | 0.3125 | 7,839.125 | 7,062.8 | 8,192 | |
Find all positive integers $n>1$ for which $\frac{n^{2}+7 n+136}{n-1}$ is the square of a positive integer. | 5, 37 | Write $\frac{n^{2}+7 n+136}{n-1}=n+\frac{8 n+136}{n-1}=n+8+\frac{144}{n-1}=9+(n-1)+\frac{144}{(n-1)}$. We seek to find $p$ and $q$ such that $p q=144$ and $p+q+9=k^{2}$. The possibilities are seen to be $1+144+9=154,2+72+9=83,3+48+9=60,4+36+9=49,6+24+9=39$, $8+18+9=35,9+16+9=34$, and $12+12+9=33$. Of these, $\{p, q\}=\... | 0 | 6,801.8125 | -1 | 6,801.8125 |
The product of two 2-digit numbers is $4536$. What is the smaller of the two numbers? | 54 | 0.5 | 6,293.3125 | 6,195.375 | 6,391.25 | |
If the purchase price of a product is 8 yuan and it is sold for 10 yuan per piece, 200 pieces can be sold per day. If the selling price of each piece increases by 0.5 yuan, the sales volume will decrease by 10 pieces. What should the selling price be set at to maximize the profit? And calculate this maximum profit. | 720 | 0.875 | 3,938.625 | 3,331 | 8,192 | |
A lame king is a chess piece that can move from a cell to any cell that shares at least one vertex with it, except for the cells in the same column as the current cell. A lame king is placed in the top-left cell of a $7 \times 7$ grid. Compute the maximum number of cells it can visit without visiting the same cell twic... | 43 | Color the columns all-black and all-white, alternating by column. Each move the lame king takes will switch the color it's on. Assuming the king starts on a black cell, there are 28 black and 21 white cells, so it can visit at most $22+21=43$ cells in total, which is easily achievable. | 0 | 8,125.125 | -1 | 8,125.125 |
Find all non-negative solutions to the equation $2013^x+2014^y=2015^z$ | (0,1,1) |
To solve the equation \( 2013^x + 2014^y = 2015^z \) for non-negative integer solutions \((x, y, z)\), we will explore small values manually and check if they satisfy the equation due to the rapid growth of exponential terms.
1. **Initial Consideration**:
Consider small non-negative integers for \(x\), \(y\), a... | 0.0625 | 8,192 | 8,192 | 8,192 |
There is a moving point \( P \) on the \( x \)-axis. Given the fixed points \( A(0,2) \) and \( B(0,4) \), what is the maximum value of \( \sin \angle APB \) when \( P \) moves along the entire \( x \)-axis? | \frac{1}{3} | 0.5 | 6,943.6875 | 5,910.125 | 7,977.25 | |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted by $a$, $b$, and $c$ respectively. Given that angle $A = \frac{\pi}{4}$, $\sin A + \sin (B - C) = 2\sqrt{2}\sin 2C$, and the area of triangle $ABC$ is $1$. Find the length of side $BC$. | \sqrt{5} | 0.3125 | 6,684.875 | 5,148.2 | 7,383.363636 | |
Let $x = \frac{\sum\limits_{n=1}^{30} \cos n^\circ}{\sum\limits_{n=1}^{30} \sin n^\circ}$. What is the smallest integer that does not fall below $100x$? | 360 | 0 | 7,191.125 | -1 | 7,191.125 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively, and $\overrightarrow{m}=(\sqrt{3}b-c,\cos C)$, $\overrightarrow{n}=(a,\cos A)$. Given that $\overrightarrow{m} \parallel \overrightarrow{n}$, determine the value of $\cos A$. | \dfrac{\sqrt{3}}{3} | 0 | 5,169.125 | -1 | 5,169.125 | |
Consider the infinite series \(S\) represented by \(2 - 1 - \frac{1}{3} + \frac{1}{9} - \frac{1}{27} - \frac{1}{81} + \frac{1}{243} - \cdots\). Find the sum \(S\). | \frac{3}{4} | 0 | 7,818.5625 | -1 | 7,818.5625 | |
How many solutions does the system have: $ \{\begin{matrix}&(3x+2y) *(\frac{3}{x}+\frac{1}{y})=2 & x^2+y^2\leq 2012 \end{matrix} $ where $ x,y $ are non-zero integers | 102 | 0.8125 | 5,242.5 | 5,055.230769 | 6,054 | |
Charles has two six-sided die. One of the die is fair, and the other die is biased so that it comes up six with probability $\frac{2}{3}$ and each of the other five sides has probability $\frac{1}{15}$. Charles chooses one of the two dice at random and rolls it three times. Given that the first two rolls are both sixes... | 167 | The probability that he rolls a six twice when using the fair die is $\frac{1}{6}\times \frac{1}{6}=\frac{1}{36}$. The probability that he rolls a six twice using the biased die is $\frac{2}{3}\times \frac{2}{3}=\frac{4}{9}=\frac{16}{36}$. Given that Charles rolled two sixes, we can see that it is $16$ times more likel... | 1 | 3,372.9375 | 3,372.9375 | -1 |
Find the number of sets ${a,b,c}$ of three distinct positive integers with the property that the product of $a,b,$ and $c$ is equal to the product of $11,21,31,41,51,61$. | 728 | Note that the prime factorization of the product is $3^{2}\cdot 7 \cdot 11 \cdot 17 \cdot 31 \cdot 41 \cdot 61$. Ignoring overcounting, by stars and bars there are $6$ ways to choose how to distribute the factors of $3$, and $3$ ways to distribute the factors of the other primes, so we have $3^{6} \cdot 6$ ways. Howeve... | 0.25 | 7,647.5 | 6,448.5 | 8,047.166667 |
Let $A=\{m-1,-3\}$, $B=\{2m-1,m-3\}$. If $A\cap B=\{-3\}$, then determine the value of the real number $m$. | -1 | 0.6875 | 4,606.6875 | 2,977 | 8,192 | |
Two pedestrians departed simultaneously from point A in the same direction. The first pedestrian met a tourist heading towards point A 20 minutes after leaving point A, and the second pedestrian met the tourist 5 minutes after the first pedestrian. The tourist arrived at point A 10 minutes after the second meeting. Fin... | 15/8 | 0.125 | 7,562.8125 | 5,747 | 7,822.214286 | |
Egor, Nikita, and Innokentiy took turns playing chess with each other (two play, one watches). After each game, the loser gave up their place to the spectator (there were no draws). As a result, Egor participated in 13 games, and Nikita participated in 27 games. How many games did Innokentiy play? | 14 | 0 | 8,192 | -1 | 8,192 | |
From the sequence of natural numbers $1, 2, 3, 4, \ldots$, erase every multiple of 3 and 4, but keep every multiple of 5 (for example, 15 and 20 are not erased). After removing the specified numbers, write the remaining numbers in a sequence: $A_{1}=1, A_{2}=2, A_{3}=5, A_{4}=7, \ldots$. Find the value of $A_{1988}$. | 3314 | 0 | 8,061.8125 | -1 | 8,061.8125 | |
Seven cards numbered $1$ through $7$ are to be lined up in a row. Find the number of arrangements of these seven cards where one of the cards can be removed leaving the remaining six cards in either ascending or descending order. | 26 | 0 | 8,192 | -1 | 8,192 | |
Given that the area of a cross-section of sphere O is $\pi$, and the distance from the center O to this cross-section is 1, then the radius of this sphere is __________, and the volume of this sphere is __________. | \frac{8\sqrt{2}}{3}\pi | 0 | 1,708.125 | -1 | 1,708.125 | |
Each unit square of a 3-by-3 unit-square grid is to be colored either blue or red. For each square, either color is equally likely to be used. The probability of obtaining a grid that does not have a 2-by-2 red square is $\frac {m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
| 929 | 0.0625 | 7,443.8125 | 6,262 | 7,522.6 | |
The integers -5 and 6 are shown on a number line. What is the distance between them? | 11 | The distance between two numbers on the number line is equal to their positive difference. Here, this distance is $6-(-5)=11$. | 1 | 1,146.0625 | 1,146.0625 | -1 |
Try to divide the set $\{1,2,\cdots, 1989\}$ into 117 mutually disjoint subsets $A_{i}, i = 1,2,\cdots, 117$, such that
(1) Each $A_{i}$ contains 17 elements;
(2) The sum of the elements in each $A_{i}$ is the same.
| 16915 | 0 | 8,192 | -1 | 8,192 | |
Find the point in the plane $3x - 4y + 5z = 30$ that is closest to the point $(1,2,3).$ | \left( \frac{11}{5}, \frac{2}{5}, 5 \right) | 1 | 3,540.6875 | 3,540.6875 | -1 | |
In Mr. Smith's science class, there are 3 boys for every 4 girls. If there are 42 students in total in his class, what percent of them are boys? | 42.857\% | 0 | 478.625 | -1 | 478.625 | |
Define $n!!$ to be $n(n-2)(n-4)\cdots 3\cdot 1$ for $n$ odd and $n(n-2)(n-4)\cdots 4\cdot 2$ for $n$ even. When $\sum_{i=1}^{2009} \frac{(2i-1)!!}{(2i)!!}$ is expressed as a fraction in lowest terms, its denominator is $2^ab$ with $b$ odd. Find $\dfrac{ab}{10}$.
| 401 | 0 | 8,192 | -1 | 8,192 | |
Find in explicit form all ordered pairs of positive integers $(m, n)$ such that $mn-1$ divides $m^2 + n^2$. | (2, 1), (3, 1), (1, 2), (1, 3) |
To find all ordered pairs of positive integers \((m, n)\) such that \(mn-1\) divides \(m^2 + n^2\), we start by considering the condition:
\[
\frac{m^2 + n^2}{mn - 1} = c \quad \text{where} \quad c \in \mathbb{Z}.
\]
This implies:
\[
m^2 + n^2 = c(mn - 1).
\]
Rewriting, we get:
\[
m^2 - cmn + n^2 + c = 0.
\]
Let \((m... | 0 | 8,192 | -1 | 8,192 |
Find the maximum value of
\[\frac{2x + 3y + 4}{\sqrt{x^2 + 4y^2 + 2}}\]
over all real numbers \( x \) and \( y \). | \sqrt{29} | 0 | 8,192 | -1 | 8,192 | |
Given that real numbers $a$ and $b$ satisfy $\frac{1}{a} + \frac{2}{b} = \sqrt{ab}$, calculate the minimum value of $ab$. | 2\sqrt{2} | 0.8125 | 5,724.125 | 5,154.615385 | 8,192 | |
12 Smurfs are seated around a round table. Each Smurf dislikes the 2 Smurfs next to them, but does not dislike the other 9 Smurfs. Papa Smurf wants to form a team of 5 Smurfs to rescue Smurfette, who was captured by Gargamel. The team must not include any Smurfs who dislike each other. How many ways are there to form s... | 36 | 0.6875 | 6,637.5625 | 5,931 | 8,192 | |
Two opposite sides of a rectangle are each divided into $n$ congruent segments, and the endpoints of one segment are joined to the center to form triangle $A$. The other sides are each divided into $m$ congruent segments, and the endpoints of one of these segments are joined to the center to form triangle $B$. [See fig... | \frac{m}{n} | 1. **Positioning the Rectangle**: Place the rectangle on a coordinate grid with diagonal vertices at $(0,0)$ and $(x,y)$. This sets the rectangle's length along the x-axis as $x$ and its height along the y-axis as $y$.
2. **Dividing the Sides**:
- Each horizontal side of the rectangle is divided into $m$ congruent... | 0.625 | 6,251.5625 | 5,312.7 | 7,816.333333 |
Four people, A, B, C, and D, stand on a staircase with 7 steps. If each step can accommodate up to 3 people, and the positions of people on the same step are not distinguished, then the number of different ways they can stand is (answer in digits). | 2394 | 0.25 | 7,667.6875 | 6,294.75 | 8,125.333333 | |
Find the least real number $K$ such that for all real numbers $x$ and $y$ , we have $(1 + 20 x^2)(1 + 19 y^2) \ge K xy$ . | 8\sqrt{95} | 0.25 | 7,651.875 | 6,031.5 | 8,192 | |
The teacher plans to give children a problem of the following type. He will tell them that he has thought of a polynomial \( P(x) \) of degree 2017 with integer coefficients, whose leading coefficient is 1. Then he will tell them \( k \) integers \( n_{1}, n_{2}, \ldots, n_{k} \), and separately he will provide the val... | 2017 | 0.0625 | 8,192 | 8,192 | 8,192 | |
When $n$ standard 6-sided dice are rolled, find the smallest possible value of $S$ such that the probability of obtaining a sum of 2000 is greater than zero and is the same as the probability of obtaining a sum of $S$. | 338 | 0.875 | 4,960.625 | 4,499 | 8,192 | |
In triangle \( \triangle ABC \), the lengths of sides opposite to angles A, B, C are denoted by a, b, c respectively. If \( c = \sqrt{3} \), \( b = 1 \), and \( B = 30^\circ \),
(1) find angles A and C;
(2) find the area of \( \triangle ABC \). | \frac{\sqrt{3}}{2} | 0 | 7,353.25 | -1 | 7,353.25 | |
For positive integers $n$, let $c_{n}$ be the smallest positive integer for which $n^{c_{n}}-1$ is divisible by 210, if such a positive integer exists, and $c_{n}=0$ otherwise. What is $c_{1}+c_{2}+\cdots+c_{210}$? | 329 | In order for $c_{n} \neq 0$, we must have $\operatorname{gcd}(n, 210)=1$, so we need only consider such $n$. The number $n^{c_{n}}-1$ is divisible by 210 iff it is divisible by each of 2, 3, 5, and 7, and we can consider the order of $n$ modulo each modulus separately; $c_{n}$ will simply be the LCM of these orders. We... | 0 | 8,192 | -1 | 8,192 |
In 2019, our county built 4 million square meters of new housing, of which 2.5 million square meters are mid-to-low-priced houses. It is expected that in the coming years, the average annual increase in the area of new housing in our county will be 8% higher than the previous year. In addition, the area of mid-to-low-p... | 2024 | 0.25 | 6,939.5625 | 5,619.5 | 7,379.583333 | |
What is the smallest positive integer $n$ which cannot be written in any of the following forms? - $n=1+2+\cdots+k$ for a positive integer $k$. - $n=p^{k}$ for a prime number $p$ and integer $k$ - $n=p+1$ for a prime number $p$. - $n=p q$ for some distinct prime numbers $p$ and $q$ | 40 | The first numbers which are neither of the form $p^{k}$ nor $p q$ are 12, 18, 20, 24, 28, 30, 36, 40, ... Of these $12,18,20,24,30$ are of the form $p+1$ and 28,36 are triangular. Hence the answer is 40. | 0 | 7,970.625 | -1 | 7,970.625 |
Triangle $\triangle DEF$ has a right angle at $F$, $\angle D = 60^\circ$, and $DF = 6$. Find the radius of the incircle and the circumference of the circumscribed circle around $\triangle DEF$. | 12\pi | 0.8125 | 2,709.5 | 2,759.461538 | 2,493 | |
Alice's favorite number is between $90$ and $150$. It is a multiple of $13$, but not a multiple of $4$. The sum of its digits should be a multiple of $4$. What is Alice's favorite number? | 143 | 0.1875 | 7,126.25 | 5,924.666667 | 7,403.538462 | |
A transgalactic ship encountered an astonishing meteor stream. Some meteors fly along a straight line at the same speed, equally spaced from each other. Another group of meteors flies in the exact same manner along another straight line, parallel to the first, but in the opposite direction, also equally spaced. The shi... | 4.6 | 0 | 6,166.8125 | -1 | 6,166.8125 | |
In a different factor tree, each value is also the product of the two values below it, unless the value is a prime number. Determine the value of $X$ for this factor tree:
[asy]
draw((-1,-.3)--(0,0)--(1,-.3),linewidth(1));
draw((-2,-1.3)--(-1.5,-.8)--(-1,-1.3),linewidth(1));
draw((1,-1.3)--(1.5,-.8)--(2,-1.3),linewidt... | 4620 | 0.0625 | 7,248.25 | 2,602 | 7,558 | |
Let
\[f(x)=\int_0^1 |t-x|t \, dt\]
for all real $x$ . Sketch the graph of $f(x)$ . What is the minimum value of $f(x)$ ? | \frac{2 - \sqrt{2}}{6} | 0 | 5,896.625 | -1 | 5,896.625 | |
Find all functions $f$ defined on all real numbers and taking real values such that \[f(f(y)) + f(x - y) = f(xf(y) - x),\] for all real numbers $x, y.$ | f(x) = 0 |
To solve the given functional equation, we need to find all functions \( f : \mathbb{R} \rightarrow \mathbb{R} \) such that:
\[
f(f(y)) + f(x - y) = f(xf(y) - x)
\]
holds for all real numbers \( x \) and \( y \).
### Step-by-Step Analysis:
1. **Substituting Particular Values:... | 0 | 7,893.3125 | -1 | 7,893.3125 |
Determine the value of \(a\) if \(a\) and \(b\) are integers such that \(x^3 - x - 1\) is a factor of \(ax^{19} + bx^{18} + 1\). | 2584 | 0 | 8,192 | -1 | 8,192 | |
Calculate the value of
\[\left(\left(\left((3+2)^{-1}+1\right)^{-1}+2\right)^{-1}+1\right)^{-1}+1.\]
A) $\frac{40}{23}$
B) $\frac{17}{23}$
C) $\frac{23}{17}$
D) $\frac{23}{40}$ | \frac{40}{23} | 0 | 4,566.3125 | -1 | 4,566.3125 | |
Let $T$ be a subset of $\{1,2,3,...,100\}$ such that no pair of distinct elements in $T$ has a sum divisible by $5$. What is the maximum number of elements in $T$? | 41 | 0.4375 | 7,480.75 | 6,969.857143 | 7,878.111111 | |
The circle $2x^2 = -2y^2 + 12x - 4y + 20$ is inscribed inside a square which has a pair of sides parallel to the x-axis. What is the area of the square? | 80 | 1 | 2,491.8125 | 2,491.8125 | -1 | |
Given a sequence $\left\{ a_n \right\}$ satisfying $a_1=4$ and $a_1+a_2+\cdots +a_n=a_{n+1}$, and $b_n=\log_{2}a_n$, calculate the value of $\frac{1}{b_1b_2}+\frac{1}{b_2b_3}+\cdots +\frac{1}{b_{2017}b_{2018}}$. | \frac{3025}{4036} | 0.3125 | 7,415.0625 | 7,779.8 | 7,249.272727 | |
In triangle \(ABC\), the angles \(A\) and \(B\) are \(45^{\circ}\) and \(30^{\circ}\) respectively, and \(CM\) is the median. The circles inscribed in triangles \(ACM\) and \(BCM\) touch segment \(CM\) at points \(D\) and \(E\). Find the area of triangle \(ABC\) if the length of segment \(DE\) is \(4(\sqrt{2} - 1)\). | 16(\sqrt{3}+1) | 0.375 | 7,715.4375 | 6,921.166667 | 8,192 | |
Given the function $f(x) = \cos^4x + 2\sin x\cos x - \sin^4x$
(1) Determine the parity, the smallest positive period, and the intervals of monotonic increase for the function $f(x)$.
(2) When $x \in [0, \frac{\pi}{2}]$, find the maximum and minimum values of the function $f(x)$. | -1 | 0.8125 | 5,865.875 | 5,594.230769 | 7,043 | |
In the polar coordinate system, let curve \(C_1\) be defined by \(\rho\sin^2\theta = 4\cos\theta\). Establish a Cartesian coordinate system \(xOy\) with the pole as the origin and the polar axis as the positive \(x\)-axis. The curve \(C_2\) is described by the parametric equations:
\[ \begin{cases}
x = 2 + \frac{1}{2... | \frac{32}{3} | 0.8125 | 6,501.5625 | 6,213.076923 | 7,751.666667 | |
A pentagon is formed by placing an equilateral triangle atop a square. Each side of the square is equal to the height of the equilateral triangle. What percent of the area of the pentagon is the area of the equilateral triangle? | \frac{3(\sqrt{3} - 1)}{6} \times 100\% | 0 | 7,212.5 | -1 | 7,212.5 | |
The population of a small town is $480$. The graph indicates the number of females and males in the town, but the vertical scale-values are omitted. How many males live in the town? | 160 | 1. **Understanding the Problem**: The total population of the town is given as $480$. The graph, which is not shown here, apparently divides the population into three equal parts, each represented by a square. Each square represents the same number of people, denoted as $x$.
2. **Setting Up the Equation**: According t... | 0 | 7,269.9375 | -1 | 7,269.9375 |
Given the function $f(x)=\sin(2x- \frac{\pi}{6})$, determine the horizontal shift required to obtain the graph of the function $g(x)=\sin(2x)$. | \frac{\pi}{12} | 0.875 | 4,820.1875 | 4,338.5 | 8,192 | |
Let \( M \) be a set composed of a finite number of positive integers,
\[
M = \bigcup_{i=1}^{20} A_i = \bigcup_{i=1}^{20} B_i, \text{ where}
\]
\[
A_i \neq \varnothing, B_i \neq \varnothing \ (i=1,2, \cdots, 20)
\]
satisfying the following conditions:
1. For any \( 1 \leqslant i < j \leqslant 20 \),
\[
A_i \cap A_j... | 180 | 0 | 8,122.625 | -1 | 8,122.625 | |
Determine with proof the number of positive integers $n$ such that a convex regular polygon with $n$ sides has interior angles whose measures, in degrees, are integers. | 22 | 0.8125 | 4,725.9375 | 3,926.076923 | 8,192 | |
How many solutions does the equation $\tan x = \tan (\tan x + \frac{\pi}{4})$ have in the interval $0 \leq x \leq \tan^{-1} 1884$? | 600 | 0.125 | 7,545.4375 | 7,143 | 7,602.928571 | |
Rectangle $EFGH$ has area $4032.$ An ellipse with area $4032\pi$ passes through points $E$ and $G$ and has its foci at points $F$ and $H$. Determine the perimeter of the rectangle. | 8\sqrt{2016} | 0 | 7,004.3125 | -1 | 7,004.3125 | |
If $\frac{x-y}{x+y}=5$, what is the value of $\frac{2x+3y}{3x-2y}$? | 0 | Since $\frac{x-y}{x+y}=5$, then $x-y=5(x+y)$. This means that $x-y=5x+5y$ and so $0=4x+6y$ or $2x+3y=0$. Therefore, $\frac{2x+3y}{3x-2y}=\frac{0}{3x-2y}=0$. | 1 | 3,486.75 | 3,486.75 | -1 |
Given $\binom{18}{11}=31824$, $\binom{18}{12}=18564$, and $\binom{20}{13}=77520$, find the value of $\binom{19}{13}$. | 27132 | 0.75 | 5,549.875 | 4,840.166667 | 7,679 | |
Any six points are taken inside or on a rectangle with dimensions $2 \times 1$. Let $b$ be the smallest possible number with the property that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than $b$. Determine the value of $b$. | \frac{\sqrt{5}}{2} | 0 | 8,192 | -1 | 8,192 | |
Given points $A$, $B$, $C$ with coordinates $(4,0)$, $(0,4)$, $(3\cos \alpha,3\sin \alpha)$ respectively, and $\alpha\in\left( \frac {\pi}{2}, \frac {3\pi}{4}\right)$. If $\overrightarrow{AC} \perp \overrightarrow{BC}$, find the value of $\frac {2\sin ^{2}\alpha-\sin 2\alpha}{1+\tan \alpha}$. | - \frac {7 \sqrt {23}}{48} | 0 | 8,129.5 | -1 | 8,129.5 | |
Let $S$ be the set of numbers of the form $n^5 - 5n^3 + 4n$ , where $n$ is an integer that is not a multiple of $3$ . What is the largest integer that is a divisor of every number in $S$ ? | 360 | 0.25 | 7,721 | 6,473 | 8,137 | |
Given that point $(x, y)$ moves on the circle $x^{2}+(y-1)^{2}=1$.
(1) Find the maximum and minimum values of $\frac{y-1}{x-2}$;
(2) Find the maximum and minimum values of $2x+y$. | 1 - \sqrt{5} | 0.875 | 4,489.875 | 4,617.214286 | 3,598.5 | |
Find the smallest integer \( n \) such that the expanded form of \( (xy - 7x - 3y + 21)^n \) has 2012 terms. | 44 | 0.375 | 7,111.1875 | 5,367.666667 | 8,157.3 | |
The numbers $-2, 4, 6, 9$ and $12$ are rearranged according to these rules:
1. The largest isn't first, but it is in one of the first three places.
2. The smallest isn't last, but it is in one of the last three places.
3. The median isn't first or last.
What is the average of the first and l... | 6.5 | 1. **Identify the largest and smallest numbers**: The largest number in the set $\{-2, 4, 6, 9, 12\}$ is $12$, and the smallest is $-2$.
2. **Apply rule 1**: The largest number, $12$, cannot be first but must be in one of the first three places. Thus, $12$ can be in the second or third position.
3. **Apply rule 2**: ... | 0 | 7,629.625 | -1 | 7,629.625 |
What is the maximum value that the expression \(\frac{1}{a+\frac{2010}{b+\frac{1}{c}}}\) can take, where \(a, b, c\) are distinct non-zero digits? | 1/203 | 0 | 8,191.6875 | -1 | 8,191.6875 | |
Compute \(\arccos(\cos 8.5)\). All functions are in radians. | 2.217 | 0 | 7,282.5625 | -1 | 7,282.5625 | |
Find the largest prime factor of $15^3+10^4-5^5$. | 41 | 1 | 2,043.0625 | 2,043.0625 | -1 | |
Angie, Bridget, Carlos, and Diego are seated at random around a square table, one person to a side. What is the probability that Angie and Carlos are seated opposite each other? | \frac{1}{3} | 1. **Fix Angie's Position:** Assume Angie is seated at one side of the square table. This does not affect the generality of the problem due to the symmetry of the seating arrangement.
2. **Count Total Arrangements:** With Angie's position fixed, there are 3 remaining seats for Bridget, Carlos, and Diego. The number of... | 1 | 4,660.0625 | 4,660.0625 | -1 |
Given that \( f(x) \) is an odd function with a period of 4, and for \( x \in (0,2) \), \( f(x) = x^2 - 16x + 60 \). Find the value of \( f(2 \sqrt{10}) \). | -36 | 0.1875 | 7,882.0625 | 6,539 | 8,192 | |
A rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n+76$ equal squares. Find $n$ . | 324 | 0.3125 | 7,545.0625 | 6,487 | 8,026 | |
Given the function $f(x)={x^3}+\frac{{{{2023}^x}-1}}{{{{2023}^x}+1}}+5$, if real numbers $a$ and $b$ satisfy $f(2a^{2})+f(b^{2}-2)=10$, then the maximum value of $a\sqrt{1+{b^2}}$ is ______. | \frac{3\sqrt{2}}{4} | 0 | 6,523 | -1 | 6,523 | |
Find $A^2$, where $A$ is the sum of the absolute values of all roots of the following equation:
\[x = \sqrt{19} + \frac{91}{{\sqrt{19}+\frac{91}{{\sqrt{19}+\frac{91}{{\sqrt{19}+\frac{91}{{\sqrt{19}+\frac{91}{x}}}}}}}}}.\] | 383 | 0.1875 | 7,453.4375 | 4,253 | 8,192 | |
Given that $A$, $B$, and $C$ are the three interior angles of $\triangle ABC$, $a$, $b$, and $c$ are the three sides, $a=2$, and $\cos C=-\frac{1}{4}$.
$(1)$ If $\sin A=2\sin B$, find $b$ and $c$;
$(2)$ If $\cos (A-\frac{π}{4})=\frac{4}{5}$, find $c$. | \frac{5\sqrt{30}}{2} | 0 | 7,803 | -1 | 7,803 | |
Points $M$ and $N$ are the midpoints of sides $PA$ and $PB$ of $\triangle PAB$. As $P$ moves along a line that is parallel to side $AB$, how many of the four quantities listed below change?
(a) the length of the segment $MN$
(b) the perimeter of $\triangle PAB$
(c) the area of $\triangle PAB$
(d) the area of trapezoid ... | 1 | To determine how many of the four quantities change as $P$ moves along a line parallel to side $AB$, we analyze each quantity individually:
1. **Length of segment $MN$:**
- Since $M$ and $N$ are midpoints of $PA$ and $PB$ respectively, segment $MN$ is parallel to $AB$ and half its length due to the midpoint theorem... | 0.875 | 3,459.4375 | 2,783.357143 | 8,192 |
What is the remainder when $3001 \cdot 3002 \cdot 3003 \cdot 3004 \cdot 3005$ is divided by 17? | 14 | 0 | 6,510.6875 | -1 | 6,510.6875 | |
How many positive integers less than 1998 are relatively prime to 1547 ? (Two integers are relatively prime if they have no common factors besides 1.) | 1487 | The factorization of 1547 is \(7 \cdot 13 \cdot 17\), so we wish to find the number of positive integers less than 1998 that are not divisible by 7, 13, or 17. By the Principle of Inclusion-Exclusion, we first subtract the numbers that are divisible by one of 7, 13, and 17, add back those that are divisible by two of 7... | 0.6875 | 5,524.125 | 5,063.181818 | 6,538.2 |
A portion of the graph of a quadratic function $f(x)$ is shown below.
Let $g(x)=-f(x)$ and $h(x)=f(-x)$. If $a$ is the number of points where the graphs of $y=f(x)$ and $y=g(x)$ intersect, and $b$ is the number of points where the graphs of $y=f(x)$ and $y=h(x)$ intersect, then what is $10a+b$?
[asy]
size(150);
real ... | 21 | 0.9375 | 5,207.875 | 5,008.933333 | 8,192 |
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