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 |
|---|---|---|---|---|---|---|
Let $r$ be a real number, $|r| < 2,$ and let $z$ be a complex number such that
\[z + \frac{1}{z} = r.\]Find $|z|.$ | 1 | 0.9375 | 4,460.9375 | 4,212.2 | 8,192 | |
Rural School USA has 105 students enrolled. There are 60 boys and 45 girls. If $\frac{1}{10}$ of the boys and $\frac{1}{3}$ of the girls are absent on one day, what percent of the total student population is absent? | 20 \% | 1 | 1,620.875 | 1,620.875 | -1 | |
Let $w_1$ and $w_2$ denote the circles $x^2+y^2+10x-24y-87=0$ and $x^2 +y^2-10x-24y+153=0,$ respectively. Let $m$ be the smallest positive value of $a$ for which the line $y=ax$ contains the center of a circle that is externally tangent to $w_2$ and internally tangent to $w_1.$ Given that $m^2=\frac pq,$ where $p$ and ... | 169 | 0.375 | 7,454.3125 | 6,224.833333 | 8,192 | |
The mean of three numbers is $10$ more than the least of the numbers and $15$ less than the greatest. The median of the three numbers is $5$. What is their sum? | 30 | 1. Let the three numbers be $x$, $y$, and $z$. Arrange them in ascending order, so $x \leq y \leq z$. Given that the median of the three numbers is $5$, we have $y = 5$.
2. The mean of the three numbers is given as $10$ more than the least of the numbers and $15$ less than the greatest. This translates to the equation... | 1 | 2,093.5625 | 2,093.5625 | -1 |
A line is parameterized by
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 2 \\ 3 \end{pmatrix} + t \begin{pmatrix} -1 \\ 5 \end{pmatrix}.\]A second line is parameterized by
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 0 \\ 7 \end{pmatrix} + u \begin{pmatrix} -1 \\ 4 \end{pmatrix}.\]Find the point ... | \begin{pmatrix} 6 \\ -17 \end{pmatrix} | 0.6875 | 2,661.0625 | 2,132.272727 | 3,824.4 | |
Chloe chooses a real number uniformly at random from the interval $[0, 3000]$. Independently, Max chooses a real number uniformly at random from the interval $[0, 4500]$. Determine the probability that Max's number is at least twice as large as Chloe's number. | \frac{3}{8} | 0.5625 | 6,893.125 | 5,882.888889 | 8,192 | |
Given that $n$ is an integer and $0 < 4n <30$, what is the sum of all possible integer values of $n$? | 28 | 1 | 1,400.125 | 1,400.125 | -1 | |
Let point O be a point inside triangle ABC with an area of 6, and it satisfies $$\overrightarrow {OA} + \overrightarrow {OB} + 2\overrightarrow {OC} = \overrightarrow {0}$$, then the area of triangle AOC is \_\_\_\_\_\_. | \frac {3}{2} | 0.3125 | 7,580.75 | 7,047.8 | 7,823 | |
How many positive integers less than 1000 are congruent to 6 (mod 11)? | 91 | 1 | 2,962.625 | 2,962.625 | -1 | |
The graph of the line $x+y=b$ intersects the line segment from $(2,5)$ to $(4,9)$ at its midpoint. What is the value of $b$? | 10 | 1 | 2,040.375 | 2,040.375 | -1 | |
Given the numbers 1, 3, 5 and 2, 4, 6, calculate the total number of different three-digit numbers that can be formed when arranging these numbers on three cards. | 48 | 0 | 4,109.6875 | -1 | 4,109.6875 | |
A fair coin is tossed 4 times. What is the probability of at least two consecutive heads? | \frac{5}{8} | 0 | 7,719.9375 | -1 | 7,719.9375 | |
An equilateral hexagon with side length 1 has interior angles $90^{\circ}, 120^{\circ}, 150^{\circ}, 90^{\circ}, 120^{\circ}, 150^{\circ}$ in that order. Find its area. | \frac{3+\sqrt{3}}{2} | The area of this hexagon is the area of a $\frac{3}{2} \times\left(1+\frac{\sqrt{3}}{2}\right)$ rectangle (with the $90^{\circ}$ angles of the hexagon at opposite vertices) minus the area of an equilateral triangle with side length 1. Then this is $$\frac{6+3 \sqrt{3}}{4}-\frac{\sqrt{3}}{4}=\frac{3+\sqrt{3}}{2}$$ | 0 | 7,681.0625 | -1 | 7,681.0625 |
Let \( D \) be a point inside the acute triangle \( \triangle ABC \). Given that \( \angle ADB = \angle ACB + 90^\circ \) and \( AC \cdot BD = AD \cdot BC \), find the value of \( \frac{AB \cdot CD}{AC \cdot BD} \). | \sqrt{2} | 0 | 8,192 | -1 | 8,192 | |
Given a function $f$ for which \[f(x) = f(398 - x) = f(2158 - x) = f(3214 - x)\] holds for all real $x,$ what is the largest number of different values that can appear in the list $f(0),f(1),f(2),\ldots,f(999)?$ | 177 | \begin{align*}f(2158 - x) = f(x) &= f(3214 - (2158 - x)) &= f(1056 + x)\\ f(398 - x) = f(x) &= f(2158 - (398 - x)) &= f(1760 + x)\end{align*}
Since $\mathrm{gcd}(1056, 1760) = 352$ we can conclude that (by the Euclidean algorithm)
\[f(x) = f(352 + x)\]
So we need only to consider one period $f(0), f(1), ... f(351)$, ... | 0 | 8,192 | -1 | 8,192 |
Find the minimum value of $ \int_0^1 |e^{ \minus{} x} \minus{} a|dx\ ( \minus{} \infty < a < \infty)$ . | 1 - 2e^{-1} | 0 | 6,660.8125 | -1 | 6,660.8125 | |
Let $ABC$ be a triangle with $\angle A = 90^o, \angle B = 60^o$ and $BC = 1$ cm. Draw outside of $\vartriangle ABC$ three equilateral triangles $ABD,ACE$ and $BCF$ . Determine the area of $\vartriangle DEF$ . | \frac{9\sqrt{3}}{16} | 0 | 8,119.8125 | -1 | 8,119.8125 | |
Find the maximum real number \( M \) such that for all real numbers \( x \) and \( y \) satisfying \( x + y \geqslant 0 \), the following inequality holds:
$$
\left(x^{2}+y^{2}\right)^{3} \geqslant M\left(x^{3}+y^{3}\right)(xy - x - y). | 32 | 0 | 8,192 | -1 | 8,192 | |
For $y=\frac{1-x}{2x+3}$ and $x\neq-\frac{3}{2}$, what is the value of $y$ that is not attainable? Express your answer as a common fraction. | -\frac12 | 1 | 2,633.0625 | 2,633.0625 | -1 | |
A rectangular field is half as wide as it is long, and it is completely enclosed by 54 meters of fencing. What is the number of square meters in the area of the field? | 162 | 1 | 1,208.875 | 1,208.875 | -1 | |
What is the area enclosed by the geoboard quadrilateral below?
[asy] unitsize(3mm); defaultpen(linewidth(.8pt)); dotfactor=2; for(int a=0; a<=10; ++a) for(int b=0; b<=10; ++b) { dot((a,b)); }; draw((4,0)--(0,5)--(3,4)--(10,10)--cycle); [/asy] | 22\frac{1}{2} | 1. **Identify the vertices of the quadrilateral**: Given the coordinates of the vertices as $(4,0)$, $(0,5)$, $(3,4)$, and $(10,10)$.
2. **Apply the Shoelace Theorem**: The Shoelace Theorem provides a formula to calculate the area of a polygon when the vertices are known. The formula for the area $A$ of a polygon whos... | 0 | 6,670.6875 | -1 | 6,670.6875 |
Let \( M = \{1, 2, 3, \ldots, 1995\} \). Subset \( A \) of \( M \) satisfies the condition: If \( x \in A \), then \( 15x \notin A \). What is the maximum number of elements in \( A \)? | 1870 | 0.125 | 8,126.875 | 7,671 | 8,192 | |
If the two roots of the quadratic $5x^2 + 4x + k$ are $\frac{-4 \pm i \sqrt{379}}{10}$, what is $k$? | 19.75 | 0 | 2,321.9375 | -1 | 2,321.9375 | |
A bag of popping corn contains $\frac{2}{3}$ white kernels and $\frac{1}{3}$ yellow kernels. Only $\frac{1}{2}$ of the white kernels will pop, whereas $\frac{2}{3}$ of the yellow ones will pop. A kernel is selected at random from the bag, and pops when placed in the popper. What is the probability that the kernel selec... | \frac{3}{5} | To solve this problem, we use the concept of conditional probability and the law of total probability. We need to find the probability that a kernel is white given that it popped, denoted as $P(\text{white} \mid \text{popped})$.
1. **Calculate $P(\text{white, popped})$:**
- The probability that a kernel is white, $... | 1 | 2,157.8125 | 2,157.8125 | -1 |
The Dingoberry Farm is a 10 mile by 10 mile square, broken up into 1 mile by 1 mile patches. Each patch is farmed either by Farmer Keith or by Farmer Ann. Whenever Ann farms a patch, she also farms all the patches due west of it and all the patches due south of it. Ann puts up a scarecrow on each of her patches that is... | 7 | Whenever Ann farms a patch $P$, she also farms all the patches due west of $P$ and due south of $P$. So, the only way she can put a scarecrow on $P$ is if Keith farms the patch immediately north of $P$ and the patch immediately east of $P$, in which case Ann cannot farm any of the patches due north of $P$ or due east o... | 0 | 8,192 | -1 | 8,192 |
Find the product of the roots and the sum of the roots of the equation $24x^2 + 60x - 600 = 0$. | -2.5 | 0 | 2,809.875 | -1 | 2,809.875 | |
$A$ is located 60 kilometers west of $B$. Individuals A and B depart from location $A$ while individuals C and D depart from location $B$ at the same time. A, B, and D all travel east, while C travels west. It is known that the speeds of A, B, C, and D form an arithmetic sequence, with A having the highest speed. After... | 30 | 0.0625 | 7,860.125 | 8,192 | 7,838 | |
If $a, b, c$ are real numbers such that $a^2 + 2b = 7$, $b^2 + 4c = -7$, and $c^2 + 6a = -14$, find $a^2 + b^2 + c^2$. | 14 | 1. **Given Equations**: We start with the given equations:
- \(a^2 + 2b = 7\)
- \(b^2 + 4c = -7\)
- \(c^2 + 6a = -14\)
2. **Adding Equations**: Add all three equations together:
\[
(a^2 + 2b) + (b^2 + 4c) + (c^2 + 6a) = 7 - 7 - 14
\]
Simplifying, we get:
\[
a^2 + b^2 + c^2 + 6a + 2b + 4c = -... | 0.75 | 4,093.875 | 2,727.833333 | 8,192 |
A trapezoid inscribed in a circle with a radius of $13 \mathrm{~cm}$ has its diagonals located $5 \mathrm{~cm}$ away from the center of the circle. What is the maximum possible area of the trapezoid? | 288 | 0 | 8,192 | -1 | 8,192 | |
How many different positive three-digit integers can be formed using only the digits in the set $\{1, 2, 2, 3, 4, 4, 4\}$ if no digit may be used more times than it appears in the given set of available digits? | 43 | 0.1875 | 7,916.9375 | 6,746.666667 | 8,187 | |
A pair of dice is rolled twice. What is the probability that the sum of the numbers facing up is 5?
A) $\frac{1}{9}$
B) $\frac{1}{4}$
C) $\frac{1}{36}$
D) 97 | \frac{1}{9} | 0 | 7,086.8125 | -1 | 7,086.8125 | |
Calculate the definite integral:
$$
\int_{\pi / 4}^{\arccos (1 / \sqrt{26})} \frac{36 \, dx}{(6 - \tan x) \sin 2x}
$$ | 6 \ln 5 | 0.6875 | 4,542.75 | 4,004.545455 | 5,726.8 | |
Let $x_1,$ $x_2,$ $x_3$ be positive real numbers such that $x_1 + 2x_2 + 3x_3 = 60.$ Find the smallest possible value of
\[x_1^2 + x_2^2 + x_3^2.\] | \frac{1800}{7} | 1 | 4,030.8125 | 4,030.8125 | -1 | |
If $\sin{2x}\sin{3x}=\cos{2x}\cos{3x}$, then one value for $x$ is | 18 | 1. **Start with the given equation:**
\[
\sin{2x}\sin{3x} = \cos{2x}\cos{3x}
\]
2. **Divide both sides by $\cos{2x}\cos{3x}$:**
\[
\frac{\sin{2x}\sin{3x}}{\cos{2x}\cos{3x}} = 1
\]
Assuming $\cos{2x} \neq 0$ and $\cos{3x} \neq 0$, we can simplify this to:
\[
\tan{2x}\tan{3x} = 1
\]
3. **R... | 0 | 3,811.3125 | -1 | 3,811.3125 |
Given non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2|\overrightarrow{b}|$, and $(\overrightarrow{a}-\overrightarrow{b})\bot \overrightarrow{b}$, calculate the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{\pi}{3} | 0.0625 | 2,144.25 | 1,950 | 2,157.2 | |
Three cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a 4, the second card is a $\clubsuit$, and the third card is a 2? | \frac{1}{663} | 0 | 8,032.3125 | -1 | 8,032.3125 | |
Let $S$ be the set $\{1,2,3,...,19\}$. For $a,b \in S$, define $a \succ b$ to mean that either $0 < a - b \le 9$ or $b - a > 9$. How many ordered triples $(x,y,z)$ of elements of $S$ have the property that $x \succ y$, $y \succ z$, and $z \succ x$? | 855 | 1. **Understanding the Relation $\succ$:**
Define $a \succ b$ to mean that either $0 < a - b \le 9$ or $b - a > 9$. This relation can be visualized on a circle where 19 numbers are placed evenly. If $a \succ b$, then $b$ is one of the 9 numbers to the left of $a$ on this circle.
2. **Visualizing the Problem:**
C... | 0 | 8,192 | -1 | 8,192 |
Find the last three digits of the product of the positive roots of $\sqrt{1995}x^{\log_{1995}x}=x^2$. | 25 | Taking the $\log_{1995}$ (logarithm) of both sides and then moving to one side yields the quadratic equation $2(\log_{1995}x)^2 - 4(\log_{1995}x) + 1 = 0$. Applying the quadratic formula yields that $\log_{1995}x = 1 \pm \frac{\sqrt{2}}{2}$. Thus, the product of the two roots (both of which are positive) is $1995^{1+\s... | 0 | 5,158.5 | -1 | 5,158.5 |
Two points are chosen inside the square $\{(x, y) \mid 0 \leq x, y \leq 1\}$ uniformly at random, and a unit square is drawn centered at each point with edges parallel to the coordinate axes. The expected area of the union of the two squares can be expressed as $\frac{a}{b}$, where $a, b$ are relatively prime positive ... | 1409 | Let $\Delta x$ and $\Delta y$ be the positive differences between the $x$ coordinates and $y$ coordinates of the centers of the squares, respectively. Then, the length of the intersection of the squares along the $x$ dimension is $1-\Delta x$, and likewise the length along the $y$ dimension is $1-\Delta y$. In order to... | 0.125 | 7,747 | 6,396.5 | 7,939.928571 |
Divide the sequence successively into groups with the first parenthesis containing one number, the second parenthesis two numbers, the third parenthesis three numbers, the fourth parenthesis four numbers, the fifth parenthesis one number, and so on in a cycle: $(3)$, $(5,7)$, $(9,11,13)$, $(15,17,19,21)$, $(23)$, $(25,... | 2072 | 0 | 7,999.3125 | -1 | 7,999.3125 | |
If $n = 2^{10} \cdot 3^{14} \cdot 5^{8}$, how many of the natural-number factors of $n$ are multiples of 150? | 980 | 0.875 | 3,427.8125 | 2,837.357143 | 7,561 | |
If $\log_{25}(x-4)=\frac{1}{2}$, find $\frac{1}{\log_{x}3}$. | 2 | 1 | 1,709.5 | 1,709.5 | -1 | |
Given that $\frac{\pi}{4} < \alpha < \frac{3\pi}{4}$, $0 < \beta < \frac{\pi}{4}$, $\sin(\alpha + \frac{\pi}{4}) = \frac{3}{5}$, and $\cos(\frac{\pi}{4} + \beta) = \frac{5}{13}$, find the value of $\sin(\alpha + \beta)$. | \frac{56}{65} | 0.5 | 6,547.1875 | 5,140.75 | 7,953.625 | |
When $(3 + 2x + x^{2})(1 + mx + m^{2}x^{2})$ is expanded and fully simplified, the coefficient of $x^{2}$ is equal to 1. What is the sum of all possible values of $m$? | -\frac{2}{3} | When $(3 + 2x + x^{2})(1 + mx + m^{2}x^{2})$ is expanded, the terms that include an $x^{2}$ will come from multiplying a constant with a term that includes $x^{2}$ or multiplying two terms that includes $x$. In other words, the term that includes $x^{2}$ will be $3 \cdot m^{2} x^{2} + 2x \cdot mx + x^{2} \cdot 1 = (3m^... | 0.9375 | 3,426.625 | 3,349.533333 | 4,583 |
In triangle $ABC,$ points $D$ and $E$ are on $\overline{AB}$ and $\overline{AC},$ respectively, and angle bisector $\overline{AT}$ intersects $\overline{DE}$ at $F.$ If $AD = 1,$ $DB = 3,$ $AE = 2,$ and $EC = 4,$ compute $\frac{AF}{AT}.$
[asy]
unitsize(1 cm);
pair A, B, C, D, E, F, T;
B = (0,0);
C = (5,0);
A = inte... | \frac{5}{18} | 0.5 | 6,307.625 | 5,241.25 | 7,374 | |
Let $ ABC$ be an isosceles triangle with $ \left|AB\right| \equal{} \left|AC\right| \equal{} 10$ and $ \left|BC\right| \equal{} 12$ . $ P$ and $ R$ are points on $ \left[BC\right]$ such that $ \left|BP\right| \equal{} \left|RC\right| \equal{} 3$ . $ S$ and $ T$ are midpoints of $ \left[AB\right]$ and ... | $ \frac {10\sqrt {13} }{13} $ | 0 | 4,648 | -1 | 4,648 | |
When \(0 < x < \frac{\pi}{2}\), the value of the function \(y = \tan 3x \cdot \cot^3 x\) cannot take numbers within the open interval \((a, b)\). Find the value of \(a + b\). | 34 | 0 | 8,192 | -1 | 8,192 | |
Given a positive integer $n\,$, let $p(n)\,$ be the product of the non-zero digits of $n\,$. (If $n\,$ has only one digits, then $p(n)\,$ is equal to that digit.) Let
$S=p(1)+p(2)+p(3)+\cdots+p(999)$
.
What is the largest prime factor of $S\,$? | 103 | Note that $p(1)=p(11), p(2)=p(12), p(3)=p(13), \cdots p(19)=p(9)$, and $p(37)=3p(7)$. So $p(10)+p(11)+p(12)+\cdots +p(19)=46$, $p(10)+p(11)+\cdots +p(99)=46*45=2070$. We add $p(1)+p(2)+p(3)+\cdots +p(10)=45$ to get 2115. When we add a digit we multiply the sum by that digit. Thus $2115\cdot (1+1+2+3+4+5+6+7+8+9)=2115\c... | 0.1875 | 6,751.5625 | 6,637 | 6,778 |
Two fair octahedral dice, each with the numbers 1 through 8 on their faces, are rolled. Let $N$ be the remainder when the product of the numbers showing on the two dice is divided by 8. Find the expected value of $N$. | \frac{11}{4} | If the first die is odd, which has $\frac{1}{2}$ probability, then $N$ can be any of $0,1,2,3,4,5,6,7$ with equal probability, because multiplying each element of $\{0, \ldots, 7\}$ with an odd number and taking modulo 8 results in the same numbers, as all odd numbers are relatively prime to 8. The expected value in th... | 0 | 8,161.125 | -1 | 8,161.125 |
Find the sum of the digits of all counting numbers less than 1000. | 13500 | 0.3125 | 6,753.625 | 5,085.8 | 7,511.727273 | |
A beam of light is emitted from point $P(1,2,3)$, reflected by the $Oxy$ plane, and then absorbed at point $Q(4,4,4)$. The distance traveled by the light beam is ______. | \sqrt{62} | 0.8125 | 4,174.8125 | 3,247.769231 | 8,192 | |
Suppose that $y^3$ varies inversely with $\sqrt[3]{z}$. If $y=2$ when $z=1$, find the value of $z$ when $y=4$. Express your answer in simplest fractional form. | \frac1{512} | 1 | 2,996.6875 | 2,996.6875 | -1 | |
Let \( f(x) = x^2 + px + q \) and \( g(x) = x^2 + rx + s \) be two distinct quadratic polynomials where the \( x \)-coordinate of the vertex of \( f \) is a root of \( g \), and the \( x \)-coordinate of the vertex of \( g \) is a root of \( f \), also both \( f \) and \( g \) have the same minimum value. If the graphs... | -200 | 0.75 | 6,681.375 | 6,275.083333 | 7,900.25 | |
Find a polynomial $ p\left(x\right)$ with real coefficients such that
$ \left(x\plus{}10\right)p\left(2x\right)\equal{}\left(8x\minus{}32\right)p\left(x\plus{}6\right)$
for all real $ x$ and $ p\left(1\right)\equal{}210$. | 2(x + 4)(x - 4)(x - 8) |
To find the polynomial \( p(x) \) with real coefficients satisfying the functional equation:
\[
(x + 10)p(2x) = (8x - 32)p(x + 6)
\]
for all real \( x \), and given the condition \( p(1) = 210 \), we start by analyzing the equation.
### Step 1: Analyze and Simplify
Rewrite the given equation:
\[
(x + 10)p(2x) = 8... | 0.0625 | 7,771.875 | 4,988 | 7,957.466667 |
Simplify first, then evaluate: $\dfrac{x^{2}-4x+4}{2x}\div \dfrac{x^{2}-2x}{x^{2}}+1$. Choose a suitable number from $0$, $1$, $2$, substitute it in and evaluate. | \dfrac{1}{2} | 0.9375 | 2,745.5 | 2,717 | 3,173 | |
Let \( f(x) = x^{9} + x^{8} + x^{7} + x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1 \). When \( f(x^{10}) \) is divided by \( f(x) \), the remainder is \( b \). Find the value of \( b \). | 10 | 0.8125 | 4,390.3125 | 3,513 | 8,192 | |
Given the definition: $min\{a,b\} = \begin{cases} a,\quad a\leqslant b \\ b,\quad a > b \end{cases}$. If a point $P(x,y)$ is randomly selected within the region defined by $\begin{cases} 0\leqslant x\leqslant 2 \\ 0\leqslant y\leqslant 6 \end{cases}$, find the probability that $x$ and $y$ satisfy $min\{3x-2y+6,x-y+4\}=... | \frac{2}{3} | 0.875 | 6,171.75 | 5,883.142857 | 8,192 | |
(Experimental Class Question) Given that $\cos \alpha = \frac{1}{7}$ and $\cos (\alpha - \beta) = \frac{13}{14}$, with $0 < \beta < \alpha < \pi$.
1. Find the value of $\sin (2\alpha - \frac{\pi}{6})$;
2. Find the value of $\beta$. | \frac{\pi}{3} | 0.8125 | 6,300.375 | 5,863.846154 | 8,192 | |
Let $p$, $q$, and $r$ be the roots of $x^3 - 2x^2 - x + 3 = 0$. Find $\frac{1}{p-2} + \frac{1}{q-2} + \frac{1}{r-2}$. | -3 | 0.75 | 5,625.5 | 5,148.25 | 7,057.25 | |
Find the dimensions of the cone that can be formed from a $300^{\circ}$ sector of a circle with a radius of 12 by aligning the two straight sides. | 12 | 0.1875 | 1,510.75 | 1,500 | 1,513.230769 | |
Let \( f(x) \) be a monotonic function defined on the interval \( (0, +\infty) \). For any \( x > 0 \), it holds that
\[ f(x) > -\frac{4}{x}, \]
and
\[ f\left(f(x) + \frac{4}{x}\right) = 3. \]
Find \( f(8) \). | \frac{7}{2} | 1 | 4,079.5 | 4,079.5 | -1 | |
Two circles, one of radius 5 inches, the other of radius 2 inches, are tangent at point P. Two bugs start crawling at the same time from point P, one crawling along the larger circle at $3\pi$ inches per minute, the other crawling along the smaller circle at $2.5\pi$ inches per minute. How many minutes is it before the... | 40 | 0.9375 | 4,240.1875 | 3,976.733333 | 8,192 | |
A polynomial \( P(x) \) of degree 10 with a leading coefficient of 1 is given. The graph of \( y = P(x) \) lies entirely above the x-axis. The polynomial \( -P(x) \) was factored into irreducible factors (i.e., polynomials that cannot be represented as the product of two non-constant polynomials). It is known that at ... | 243 | 0.4375 | 6,121.9375 | 3,759 | 7,959.777778 | |
Given two points $A(-2, 0)$ and $B(0, 2)$, point $C$ is any point on the circle $x^2 + y^2 - 2x = 0$, the minimum value of the area of $\triangle ABC$ is \_\_\_\_\_\_. | 3 - \sqrt{2} | 0.5 | 7,617.0625 | 7,042.125 | 8,192 | |
If points $P, Q, R$, and $S$ are arranged in order on a line segment with $P Q=1, Q R=2 P Q$, and $R S=3 Q R$, what is the length of $P S$? | 9 | Since $P Q=1$ and $Q R=2 P Q$, then $Q R=2$. Since $Q R=2$ and $R S=3 Q R$, then $R S=3(2)=6$. Therefore, $P S=P Q+Q R+R S=1+2+6=9$. | 0.9375 | 315.3125 | 315.2 | 317 |
Three male students and three female students, a total of six students, stand in a row. If male student A does not stand at either end, and exactly two of the three female students stand next to each other, then the number of different arrangements is ______. | 288 | 0 | 8,192 | -1 | 8,192 | |
Natural numbers \( A \) and \( B \) are divisible by all natural numbers from 1 to 65. What is the smallest natural number that \( A + B \) might not be divisible by? | 67 | 0.375 | 6,770.0625 | 4,718.666667 | 8,000.9 | |
Albert starts to make a list, in increasing order, of the positive integers that have a first digit of 1. He writes $1, 10, 11, 12, \ldots$ but by the 1,000th digit he (finally) realizes that the list would contain an infinite number of elements. Find the three-digit number formed by the last three digits he wrote (the... | 116 | 0.3125 | 7,465.5625 | 6,938 | 7,705.363636 | |
Given that $f(x+5)=4x^3 + 5x^2 + 9x + 6$ and $f(x)=ax^3 + bx^2 + cx + d$, find the value of $a+b+c+d$. | -206 | 0.75 | 5,224.625 | 4,235.5 | 8,192 | |
A square flag features a green cross of uniform width, and a yellow square in the center, against a white background. The cross is symmetric with respect to each of the diagonals of the square. Suppose the entire cross (including the green arms and the yellow center) occupies 49% of the area of the flag. If the yellow ... | 45\% | 0.5625 | 7,690.6875 | 7,300.777778 | 8,192 | |
If the integer $k$ is added to each of the numbers $36$, $300$, and $596$, one obtains the squares of three consecutive terms of an arithmetic series. Find $k$. | 925 | Call the terms of the arithmetic progression $a,\ a + d,\ a + 2d$, making their squares $a^2,\ a^2 + 2ad + d^2,\ a^2 + 4ad + 4d^2$.
We know that $a^2 = 36 + k$ and $(a + d)^2 = 300 + k$, and subtracting these two we get $264 = 2ad + d^2$ (1). Similarly, using $(a + d)^2 = 300 + k$ and $(a + 2d)^2 = 596 + k$, subtracti... | 0.875 | 3,965.9375 | 3,362.214286 | 8,192 |
A cuboid has a diagonal $A A^{\prime}$. In what ratio does the plane passing through the endpoints $B, C, D$ of the edges originating from vertex $A$ divide the $A A^{\prime}$ diagonal? | 1:2 | 0.5 | 6,330.1875 | 4,517.75 | 8,142.625 | |
Given real numbers \( a, b, c \) and a positive number \( \lambda \) such that the polynomial \( f(x) = x^3 + a x^2 + b x + c \) has three real roots \( x_1, x_2, x_3 \), and the conditions \( x_2 - x_1 = \lambda \) and \( x_3 > \frac{1}{2}(x_1 + x_2) \) are satisfied, find the maximum value of \( \frac{2 a^3 + 27 c - ... | \frac{3\sqrt{3}}{2} | 0 | 8,192 | -1 | 8,192 | |
Given that
$\frac 1{2!17!}+\frac 1{3!16!}+\frac 1{4!15!}+\frac 1{5!14!}+\frac 1{6!13!}+\frac 1{7!12!}+\frac 1{8!11!}+\frac 1{9!10!}=\frac N{1!18!}$
find the greatest integer that is less than $\frac N{100}$. | 137 | Let $f(x) = (1+x)^{19}.$ Applying the binomial theorem gives us $f(x) = \binom{19}{19} x^{19} + \binom{19}{18} x^{18} + \binom{19}{17} x^{17}+ \cdots + \binom{19}{0}.$ Since $\frac 1{2!17!}+\frac 1{3!16!}+\dots+\frac 1{8!11!}+\frac 1{9!10!} = \frac{\frac{f(1)}{2} - \binom{19}{19} - \binom{19}{18}}{19!},$ $N = \frac{2^{... | 0.5 | 6,571.25 | 5,348.125 | 7,794.375 |
In the Cartesian coordinate system $(xOy)$, the asymptotes of the hyperbola $({C}_{1}: \frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1 (a > 0,b > 0) )$ intersect with the parabola $({C}_{2}:{x}^{2}=2py (p > 0) )$ at points $O, A, B$. If the orthocenter of $\triangle OAB$ is the focus of $({C}_{2})$, find the eccentri... | \frac{3}{2} | 0.8125 | 5,282.5625 | 4,611.153846 | 8,192 | |
Selected Exercise $4-4$: Coordinate Systems and Parametric Equations
In the rectangular coordinate system $xOy$, the parametric equations of the curve $C$ are given by $\begin{cases} & x=\cos \theta \\ & y=\sin \theta \end{cases}$, where $\theta$ is the parameter. In the polar coordinate system with the same unit leng... | \frac{3\sqrt{2}}{2} + 1 | 0 | 6,485.25 | -1 | 6,485.25 | |
Compute $\sum_{i=1}^{\infty} \frac{a i}{a^{i}}$ for $a>1$. | \left(\frac{a}{1-a}\right)^{2} | The sum $S=a+a x+a x^{2}+a x^{3}+\cdots$ for $x<1$ can be determined by realizing that $x S=a x+a x^{2}+a x^{3}+\cdots$ and $(1-x) S=a$, so $S=\frac{a}{1-x}$. Using this, we have $\sum_{i=1}^{\infty} \frac{a i}{a^{i}}=$ $a \sum_{i=1}^{\infty} \frac{i}{a^{i}}=a\left[\frac{1}{a}+\frac{2}{a^{2}}+\frac{3}{a^{3}}+\cdots\rig... | 0 | 4,944.3125 | -1 | 4,944.3125 |
Find all positive integers $n$ for which there exist positive integers $x_1, x_2, \dots, x_n$ such that
$$ \frac{1}{x_1^2}+\frac{2}{x_2^2}+\frac{2^2}{x_3^2}+\cdots +\frac{2^{n-1}}{x_n^2}=1.$$ | n\ge 3 \text{ and } n=1 |
We need to determine all positive integers \( n \) such that there exist positive integers \( x_1, x_2, \ldots, x_n \) satisfying the equation:
\[
\frac{1}{x_1^2} + \frac{2}{x_2^2} + \frac{2^2}{x_3^2} + \cdots + \frac{2^{n-1}}{x_n^2} = 1.
\]
### Case \( n = 1 \)
For \( n = 1 \), the equation simplifies to:
\[
\frac... | 0 | 8,192 | -1 | 8,192 |
For what values of $j$ does the equation $(2x+7)(x-5) = -43 + jx$ have exactly one real solution? Express your answer as a list of numbers, separated by commas. | 5,\,-11 | 0 | 2,386.0625 | -1 | 2,386.0625 | |
(1) Solve the inequality $|x+1| + 2|x-1| < 3x + 5$.
(2) Given $a, b \in [0, 1]$, find the maximum value of $ab + (1-a-b)(a+b)$. | \frac{1}{3} | 0.9375 | 5,630.9375 | 5,460.2 | 8,192 | |
Given a cluster of circles ${C_n}:{({x-n})^2}+{({y-2n})^2}={n^2}(n≠0)$, the line $l:y=kx+b$ is a common tangent to them, then $k+b=$____. | \frac{3}{4} | 0.875 | 4,591.5625 | 4,077.214286 | 8,192 | |
Positive integers $a$ and $b$ are each less than $6$. What is the smallest possible value for $2 \cdot a - a \cdot b$? | -15 | 1. **Expression Simplification**:
Start by simplifying the given expression:
\[
2a - ab = a(2 - b)
\]
This shows that the expression depends on the values of $a$ and $b$ through the product $a(2 - b)$.
2. **Determine the Range of Values**:
Given that $a$ and $b$ are positive integers each less than $... | 0.9375 | 4,485.75 | 4,238.666667 | 8,192 |
Let $f(x)$ be the polynomial $\prod_{k=1}^{50} \bigl( x - (2k-1) \bigr)$ . Let $c$ be the coefficient of $x^{48}$ in $f(x)$ . When $c$ is divided by 101, what is the remainder? (The remainder is an integer between 0 and 100.) | 60 | 0.75 | 6,908.25 | 6,480.333333 | 8,192 | |
Five points on a circle are numbered 1,2,3,4, and 5 in clockwise order. A bug jumps in a clockwise direction from one point to another around the circle; if it is on an odd-numbered point, it moves one point, and if it is on an even-numbered point, it moves two points. If the bug begins on point 5, after 1995 jumps it ... | 4 | 1. **Identify the movement pattern**: The bug starts at point 5. According to the rules, since 5 is odd, it moves one point to point 1.
2. **Continue the pattern**:
- From point 1 (odd), it moves one point to point 2.
- From point 2 (even), it moves two points to point 4.
- From point 4 (even), it moves two p... | 0.5 | 6,743.75 | 5,295.5 | 8,192 |
13. Given that $a$, $b$, $c$, are the lengths of the sides opposite to angles $A$, $B$, $C$ in $\triangle ABC$ respectively, with $a=2$, and $(2+b)(\sin A-\sin B)=(c-b)\sin C$, find the maximum area of $\triangle ABC$. | \sqrt{3} | 0.3125 | 7,567.6875 | 6,750.8 | 7,939 | |
How many distinct trees with exactly 7 vertices exist? | 11 | 0.0625 | 3,055.0625 | 2,415 | 3,097.733333 | |
Inside an isosceles triangle \( ABC \), a point \( K \) is marked such that \( CK = AB = BC \) and \(\angle KAC = 30^\circ\). Find the angle \( AKB \). | 150 | 0 | 8,192 | -1 | 8,192 | |
A rectangle has sides of length $a$ and 36. A hinge is installed at each vertex of the rectangle, and at the midpoint of each side of length 36. The sides of length $a$ can be pressed toward each other keeping those two sides parallel so the rectangle becomes a convex hexagon as shown. When the figure is a hexagon with... | 720 | Alternatively, use basic geometry. First, scale everything down by dividing everything by 6. Let $a/6=p$. Then, the dimensions of the central rectangle in the hexagon is p x 4, and the original rectangle is 6 x p. By Pythagorean theorem and splitting the end triangles of the hexagon into two right triangles, the altitu... | 0 | 7,113.4375 | -1 | 7,113.4375 |
When $555_{10}$ is expressed in this base, it has 4 digits, in the form ABAB, where A and B are two different digits. What base is it? | 6 | 0.875 | 4,328.9375 | 3,777.071429 | 8,192 | |
In triangle \( \triangle ABC \), if \( \sin A = 2 \sin C \) and the three sides \( a, b, c \) form a geometric sequence, find the value of \( \cos A \). | -\frac{\sqrt{2}}{4} | 0 | 6,373.3125 | -1 | 6,373.3125 | |
Given that vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=\sqrt{2}$, $|\overrightarrow{b}|=2$, and $\overrightarrow{a}\bot (\overrightarrow{a}-\overrightarrow{b})$, calculate the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{\pi}{4} | 0.5 | 2,638.75 | 2,488.625 | 2,788.875 | |
The red parabola shown is the graph of the equation $x = ay^2 + by + c$. Find $a+b+c$.
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
re... | -\frac{15}{2} | 0.4375 | 6,134.75 | 3,489.714286 | 8,192 | |
A man on his way to dinner shortly after $6:00$ p.m. observes that the hands of his watch form an angle of $110^{\circ}$. Returning before $7:00$ p.m. he notices that again the hands of his watch form an angle of $110^{\circ}$. The number of minutes that he has been away is: | 40 | 1. **Define the angles of the hands**: Let $n$ be the number of minutes after 6:00 PM. The angle of the hour hand from the top (12:00 position) can be calculated as follows:
- In one hour, the hour hand moves $30^\circ$ (since $360^\circ / 12 = 30^\circ$).
- In $n$ minutes, the hour hand moves $\frac{30}{60}n = \... | 0.9375 | 4,990.5625 | 4,777.133333 | 8,192 |
Consider a square where each side measures 1 unit. At each vertex of the square, a quarter circle is drawn outward such that each side of the square serves as the radius for two adjoining quarter circles. Calculate the total perimeter formed by these quarter circles. | 2\pi | 0.3125 | 6,228.25 | 6,053.4 | 6,307.727273 | |
A small town has fewer than 6000 inhabitants. We know that there are $10\%$ more girls than boys among the children, and $15\%$ more men than women among the adults. There are $20\%$ more children than adults in the town.
How many people live in the town? | 3311 | 0.5625 | 6,425.6875 | 5,051.888889 | 8,192 | |
Suppose that a parabola has vertex $\left(\frac{1}{4},-\frac{9}{8}\right)$ and equation $y = ax^2 + bx + c$, where $a > 0$ and $a + b + c$ is an integer. Find the smallest possible value of $a.$ | \frac{2}{9} | 1 | 4,913.9375 | 4,913.9375 | -1 | |
For $1 \le n \le 200$, how many integers are there such that $\frac{n}{n+1}$ is a repeating decimal? | 182 | 0.125 | 6,394.1875 | 5,919.5 | 6,462 | |
Let $U$ be a square with side length 1. Two points are randomly chosen on the sides of $U$. The probability that the distance between these two points is at least $\frac{1}{2}$ is $\frac{a - b \pi}{c}\left(a, b, c \in \mathbf{Z}_{+}, (a, b, c)=1\right)$. Find the value of $a + b + c$. | 59 | 0 | 8,138.3125 | -1 | 8,138.3125 | |
If I have a $5\times 5$ chess board, in how many ways can I place five distinct pawns on the board such that each column and row of the board contains no more than one pawn? | 14400 | 0.875 | 4,642.125 | 4,135 | 8,192 | |
Let $f(x) = 3x^2-2$ and $g(f(x)) = x^2 + x +1$. Find the sum of all possible values of $g(25)$. | 20 | 1 | 2,325.875 | 2,325.875 | -1 |
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