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 |
|---|---|---|---|---|---|---|
The square $A B C D$ is enlarged from vertex $A$ resulting in the square $A B^{\prime} C^{\prime} D^{\prime}$. The intersection point of the diagonals of the enlarged square is $M$. It is given that $M C = B B^{\prime}$. What is the scale factor of the enlargement? | \sqrt{2} | 0.625 | 5,271.25 | 4,420.8 | 6,688.666667 | |
In a trapezoid $ABCD$ with $AB$ parallel to $CD$, the diagonals $AC$ and $BD$ intersect at $E$. If the area of triangle $ABE$ is 50 square units, and the area of triangle $ADE$ is 20 square units, what is the area of trapezoid $ABCD$? | 98 | 0.375 | 6,899.5 | 5,790.5 | 7,564.9 | |
Given the polar coordinate system with the origin $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis. Point $A(4, \frac{5\pi}{4})$ is known. The polar coordinate equation of curve $E$ is $ρ=ρcos^2θ+\sqrt{2}acosθ (a > 0)$. A perpendicular line $l$ is drawn through point $A$ intersecting curve $... | a = 1 + \sqrt{5} | 0.125 | 7,927.25 | 6,201 | 8,173.857143 | |
What is the greatest common divisor of 1407 and 903? | 21 | 1 | 2,121.9375 | 2,121.9375 | -1 | |
The calculator's keyboard has digits from 0 to 9 and symbols of two operations. Initially, the display shows the number 0. Any keys can be pressed. The calculator performs operations in the sequence of key presses. If an operation symbol is pressed several times in a row, the calculator will remember only the last pres... | 1/3 | 0 | 7,737.9375 | -1 | 7,737.9375 | |
Given $A=\{x|x^{3}+3x^{2}+2x > 0\}$, $B=\{x|x^{2}+ax+b\leqslant 0\}$ and $A\cap B=\{x|0 < x\leqslant 2\}$, $A\cup B=\{x|x > -2\}$, then $a+b=$ ______. | -3 | 0.25 | 6,556.6875 | 4,409.5 | 7,272.416667 | |
Read the following text and answer the questions:<br/>$\because \sqrt{1}＜\sqrt{2}＜\sqrt{4}$, which means $1＜\sqrt{2}＜2$,<br/>$\therefore$ The integer part of $\sqrt{2}$ is $1$, and the decimal part is $\sqrt{2}-1$.<br/>Please answer:<br/>$(1)$ The integer part of $\sqrt{33}$ is ______, and the decimal part is ______;<b... | \sqrt{5} - 8 | 0.9375 | 5,581.9375 | 5,407.933333 | 8,192 | |
What is the median number of moons per planet? (Include Pluto, although arguments rage on about Pluto's status...) \begin{tabular}{c|c}
Planet & $\#$ of Moons\\
\hline
Mercury&0\\
Venus &0\\
Earth &1\\
Mars &2\\
Jupiter&16\\
Saturn&23\\
Uranus&15\\
Neptune&2\\
Pluto&5\\
\end{tabular} | 2 | 0.9375 | 2,099.4375 | 2,123.133333 | 1,744 | |
A plane intersects a right circular cylinder of radius $2$ forming an ellipse. If the major axis of the ellipse is $60\%$ longer than the minor axis, find the length of the major axis. | 6.4 | 0.4375 | 3,797.5 | 3,763.714286 | 3,823.777778 | |
Everyone in a class of 30 students takes math and history. Seven students received an A in history and 13 received an A in math, including four that received an A in both courses. How many students did not receive an A in any of these two courses? | 14 | 0.9375 | 1,936.25 | 1,519.2 | 8,192 | |
Determine the number of ways to arrange the letters of the word "PERCEPTION". | 453,600 | 0 | 1,802.8125 | -1 | 1,802.8125 | |
Determine the total number of distinct, natural-number factors for the number $4^5 \cdot 5^2 \cdot 6^3 \cdot 7!$. | 864 | 0.625 | 3,079.875 | 2,707.9 | 3,699.833333 | |
Let $a, b, c$ be nonzero real numbers such that $a+b+c=0$ and $a^{3}+b^{3}+c^{3}=a^{5}+b^{5}+c^{5}$. Find the value of $a^{2}+b^{2}+c^{2}$. | \frac{6}{5} | Let $\sigma_{1}=a+b+c, \sigma_{2}=ab+bc+ca$ and $\sigma_{3}=abc$ be the three elementary symmetric polynomials. Since $a^{3}+b^{3}+c^{3}$ is a symmetric polynomial, it can be written as a polynomial in $\sigma_{1}, \sigma_{2}$ and $\sigma_{3}$. Now, observe that $\sigma_{1}=0$, and so we only need to worry about the te... | 0.8125 | 4,818 | 4,223.230769 | 7,395.333333 |
Find all quadruples of positive integers $(p, q, a, b)$, where $p$ and $q$ are prime numbers and $a > 1$, such that $$p^a = 1 + 5q^b.$$ | (2, 3, 4, 1) \text{ and } (3, 2, 4, 4) |
We are tasked with finding all quadruples of positive integers \((p, q, a, b)\), where \(p\) and \(q\) are prime numbers, \(a > 1\), and they satisfy the equation:
\[
p^a = 1 + 5q^b.
\]
Given the integers involved, we will first check small values for \(p\), and test if they can satisfy the equation.
### Case 1: \(... | 0 | 8,192 | -1 | 8,192 |
Define the sequence $\left\{a_{n}\right\}$ such that $a_{n}$ is the last digit of the sum $1 + 2 + \cdots + n$. Let $S_{n}$ be the sum of the first $n$ terms of the sequence $\left\{a_{n}\right\}$. Find $S_{2016}$. | 7066 | 0.625 | 6,501.1875 | 6,337.7 | 6,773.666667 | |
For a real number $a$, let $\lfloor a \rfloor$ denote the greatest integer less than or equal to $a$. Let $\mathcal{R}$ denote the region in the coordinate plane consisting of points $(x,y)$ such that $\lfloor x \rfloor ^2 + \lfloor y \rfloor ^2 = 25$. The region $\mathcal{R}$ is completely contained in a disk of radiu... | 69 | 1. **Understanding the Problem:**
The problem involves finding the minimum radius $r$ of a disk that completely contains the region $\mathcal{R}$ defined by $\lfloor x \rfloor^2 + \lfloor y \rfloor^2 = 25$ in the coordinate plane. Here, $\lfloor a \rfloor$ denotes the greatest integer less than or equal to $a$.
2. ... | 0 | 7,866 | -1 | 7,866 |
Determine the largest multiple of 36 that consists of all even and distinct digits. | 8640 | 0.375 | 7,751.4375 | 7,017.166667 | 8,192 | |
Given that positive real numbers a and b satisfy $a^{2}+2ab+4b^{2}=6$, calculate the maximum value of a+2b. | 2\sqrt{2} | 0.9375 | 5,543.625 | 5,367.066667 | 8,192 | |
Given a cone with a base radius of $5$ and a lateral area of $65π$, let the angle between the slant height and the height of the cone be $θ$. Find the value of $\sinθ$. | \frac{5}{13} | 1 | 1,883.9375 | 1,883.9375 | -1 | |
Given the function $f(x)=\cos (2x-φ)- \sqrt {3}\sin (2x-φ)(|φ| < \dfrac {π}{2})$, its graph is shifted to the right by $\dfrac {π}{12}$ units and is symmetric about the $y$-axis. Find the minimum value of $f(x)$ in the interval $\[- \dfrac {π}{2},0\]$. | - \sqrt {3} | 0 | 7,460.5625 | -1 | 7,460.5625 | |
A bag contains 3 tan, 2 pink and 4 violet chips. If the 9 chips are randomly drawn from the bag, one at a time and without replacement, what is the probability that the chips are drawn in such a way that the 3 tan chips are drawn consecutively, the 2 pink chips are drawn consecutively, and the 4 violet chips are drawn ... | \frac{1}{210} | 0.75 | 5,818.4375 | 5,076.833333 | 8,043.25 | |
Using the 0.618 method to select a trial point, if the experimental interval is $[2, 4]$, with $x_1$ being the first trial point and the result at $x_1$ being better than that at $x_2$, then the value of $x_3$ is ____. | 3.236 | 0 | 8,192 | -1 | 8,192 | |
Calculate:<br/>$(1)-6-3+\left(-7\right)-\left(-2\right)$;<br/>$(2)\left(-1\right)^{2023}+5\times \left(-2\right)-12\div \left(-4\right)$. | -8 | 1 | 569 | 569 | -1 | |
Patty has $20$ coins consisting of nickels and dimes. If her nickels were dimes and her dimes were nickels, she would have $70$ cents more. How much are her coins worth? | $1.15 | 1. Let $n$ represent the number of nickels Patty has, and $d$ represent the number of dimes. Since Patty has a total of 20 coins, we can express the number of dimes in terms of nickels:
\[
d = 20 - n
\]
2. Calculate the total value of the coins when nickels and dimes are in their original form. The value of a... | 0 | 2,513.625 | -1 | 2,513.625 |
Given that the function $y=f(x)$ is an odd function defined on $\mathbb{R}$ and satisfies $f(x-1)=f(x+1)$ for all $x \in \mathbb{R}$. When $x \in (0,1]$ and $x_1 \neq x_2$, we have $\frac{f(x_2) - f(x_1)}{x_2 - x_1} < 0$. Determine the correct statement(s) among the following:
(1) $f(1)=0$
(2) $f(x)$ has 5 zeros in $... | (1) (2) (3) | 0 | 6,686.0625 | -1 | 6,686.0625 | |
For a positive real number $a$ , let $C$ be the cube with vertices at $(\pm a, \pm a, \pm a)$ and let $T$ be the tetrahedron with vertices at $(2a,2a,2a),(2a, -2a, -2a),(-2a, 2a, -2a),(-2a, -2a, -2a)$ . If the intersection of $T$ and $C$ has volume $ka^3$ for some $k$ , find $k$ . | 4/3 | 0 | 8,157.1875 | -1 | 8,157.1875 | |
For real numbers $t,$ the point
\[(x,y) = (2^t - 3, 4^t - 5 \cdot 2^t - 1)\]is plotted. All the plotted points lie on what kind of curve?
(A) Line
(B) Circle
(C) Parabola
(D) Ellipse
(E) Hyperbola
Enter the letter of the correct option. | \text{(C)} | 0 | 1,631.125 | -1 | 1,631.125 | |
From the $8$ vertices of a cube, select $4$ vertices. The probability that these $4$ vertices lie in the same plane is ______. | \frac{6}{35} | 0.3125 | 7,502.8125 | 5,986.6 | 8,192 | |
Let $S_n$ and $T_n$ be the respective sums of the first $n$ terms of two arithmetic series. If $S_n:T_n=(7n+1):(4n+27)$ for all $n$, the ratio of the eleventh term of the first series to the eleventh term of the second series is: | 4/3 | 1. **Identify the first terms and common differences:**
Let the first term and common difference of the first arithmetic sequence $S$ be $a$ and $d$, respectively. Similarly, let the first term and common difference of the second arithmetic sequence $T$ be $b$ and $e$, respectively.
2. **Expression for $S_n$ and $T... | 0.5625 | 6,209.1875 | 4,667 | 8,192 |
Let $ABCDEF$ be a regular hexagon with side length 10 inscribed in a circle $\omega$ . $X$ , $Y$ , and $Z$ are points on $\omega$ such that $X$ is on minor arc $AB$ , $Y$ is on minor arc $CD$ , and $Z$ is on minor arc $EF$ , where $X$ may coincide with $A$ or $B$ (and similarly for $Y$ and $Z... | 7500 | 0 | 8,192 | -1 | 8,192 | |
The numbers $1,2, \ldots, 2016$ are grouped into pairs in such a way that the product of the numbers in each pair does not exceed a certain natural number $N$. What is the smallest possible value of $N$ for which this is possible? | 1017072 | 0.0625 | 8,119.5625 | 8,192 | 8,114.733333 | |
Find the length of the shortest path on the surface of a unit cube between its opposite vertices. | \sqrt{5} | 0.5625 | 7,017.4375 | 6,103.888889 | 8,192 | |
The polynomial \( p(x) = x^2 - 3x + 1 \) has zeros \( r \) and \( s \). A quadratic polynomial \( q(x) \) has a leading coefficient of 1 and zeros \( r^3 \) and \( s^3 \). Find \( q(1) \). | -16 | 1 | 3,124 | 3,124 | -1 | |
Determine the value of
\[1002 + \frac{1}{3} \left( 1001 + \frac{1}{3} \left( 1000 + \dots + \frac{1}{3} \left( 3 + \frac{1}{3} \cdot 2 \right) \right) \dotsb \right).\] | 1502.25 | 0 | 7,640.1875 | -1 | 7,640.1875 | |
Let $A$, $B$, $C$ and $D$ be the vertices of a regular tetrahedron, each of whose edges measures $1$ meter. A bug, starting from vertex $A$, observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to th... | 182 | We evaluate $P(7)$ recursively: \begin{alignat*}{6} P(0)&=1, \\ P(1)&=\frac13(1-P(0))&&=0, \\ P(2)&=\frac13(1-P(1))&&=\frac13, \\ P(3)&=\frac13(1-P(2))&&=\frac29, \\ P(4)&=\frac13(1-P(3))&&=\frac{7}{27}, \\ P(5)&=\frac13(1-P(4))&&=\frac{20}{81}, \\ P(6)&=\frac13(1-P(5))&&=\frac{61}{243},\\ P(7)&=\frac13(1-P(6))&&=\frac... | 0.625 | 6,585.0625 | 5,620.9 | 8,192 |
Given an ellipse $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with an eccentricity of $\frac{\sqrt{3}}{2}$ and a length of the minor axis of $4$. <br/>$(1)$ Find the equation of the ellipse; <br/>$(2)$ A chord passing through $P(2,1)$ divides $P$ in half. Find the equation of the line containing this chord and the l... | 2\sqrt{5} | 0.875 | 4,416.9375 | 3,933.357143 | 7,802 | |
Given a regular hexagon \( A B C D E F \) with a side length of 1, calculate \((\overrightarrow{A B}+\overrightarrow{D C}) \cdot(\overrightarrow{A D}+\overrightarrow{B E})\). | -3 | 1 | 3,667.8125 | 3,667.8125 | -1 | |
How many positive integers, not exceeding 200, are multiples of 3 or 5 but not 6? | 73 | 0 | 6,348.875 | -1 | 6,348.875 | |
Estimate the population of Island X in the year 2045, given that the population doubles every 15 years and the population in 2020 was 500. | 1587 | 0.6875 | 1,427.125 | 1,809.727273 | 585.4 | |
Two cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a spade and the second card is a king? | \frac{17}{884} | 0 | 7,106.5 | -1 | 7,106.5 | |
Given a real number $a$ satisfying ${a}^{\frac{1}{2}}\leqslant 3$ and $\log _{a}3\leqslant \frac{1}{2}$.
$(1)$ Find the range of real number $a$;
$(2)$ If $a \gt 1$, $f\left(x\right)=mx^{a}+\ln \left(1+x\right)^{a}-a\ln \left(1-x\right)-2\left(m\in R\right)$, and $f(\frac{1}{2})=a$, find the value of $f(-\frac{1}{2... | -13 | 0.5625 | 5,996.875 | 4,289.555556 | 8,192 | |
Given the universal set $U=\{2,3,5\}$, and $A=\{x|x^2+bx+c=0\}$. If $\complement_U A=\{2\}$, then $b=$ ____, $c=$ ____. | 15 | 1 | 1,493.6875 | 1,493.6875 | -1 | |
Compute the number of sets $S$ such that every element of $S$ is a nonnegative integer less than 16, and if $x \in S$ then $(2 x \bmod 16) \in S$. | 678 | For any nonempty $S$ we must have $0 \in S$. Now if we draw a directed graph of dependencies among the non-zero elements, it creates a balanced binary tree where every leaf has depth 3 . In the diagram, if $a$ is a parent of $b$ it means that if $b \in S$, then $a$ must also be in $S$. We wish to find the number of sub... | 0 | 7,923.75 | -1 | 7,923.75 |
One hundred people were surveyed. Of these, $87$ indicated they liked Mozart and $70$ indicated they liked Bach. What is the minimum number of people surveyed who could have said they liked both Mozart and Bach? | 57 | 1 | 1,607.125 | 1,607.125 | -1 | |
Find the greatest common divisor of $7!$ and $(5!)^2.$ | 720 | 0.9375 | 3,518.3125 | 3,206.733333 | 8,192 | |
How many ways are there of using diagonals to divide a regular 6-sided polygon into triangles such that at least one side of each triangle is a side of the original polygon and that each vertex of each triangle is a vertex of the original polygon? | 12 | The number of ways of triangulating a convex $(n+2)$-sided polygon is $\binom{2 n}{n} \frac{1}{n+1}$, which is 14 in this case. However, there are two triangulations of a hexagon which produce one triangle sharing no sides with the original polygon, so the answer is $14-2=12$. | 0 | 7,587.6875 | -1 | 7,587.6875 |
Simplify $(576)^\frac{1}{4}(216)^\frac{1}{2}$. | 72 | 1 | 3,174.25 | 3,174.25 | -1 | |
Given the sequence $\left\{a_{n}\right\}$ defined by \(a_{1}=\frac{2}{3}\), \(a_{n+1}=a_{n}^{2}+a_{n-1}^{2}+\cdots+a_{1}^{2}\) (where \(n \in \mathbf{N}^{*}\)), find the smallest value of \(M\) such that for any \(n \in \mathbf{N}^{*}\), the inequality \(\frac{1}{a_{1}+1}+\frac{1}{a_{2}+1}+\cdots+\frac{1}{a_{n}+1}<M\) ... | \frac{57}{20} | 0 | 7,994.875 | -1 | 7,994.875 | |
Determine the area of the region of the circle defined by $x^2 + y^2 - 8x + 16 = 0$ that lies below the $x$-axis and to the left of the line $y = x - 4$. | 4\pi | 0 | 6,404.3125 | -1 | 6,404.3125 | |
Calculate the expression $8 \times 10^{5}+4 \times 10^{3}+9 \times 10+5$. | 804095 | First, we write out the powers of 10 in full to obtain $8 \times 100000+4 \times 1000+9 \times 10+5$. Simplifying, we obtain $800000+4000+90+5$ or 804095. | 1 | 392.0625 | 392.0625 | -1 |
What is the value of $3 \times (7 - 5) - 5$? | 1 | 1 | 506.1875 | 506.1875 | -1 | |
The perimeter of triangle $APM$ is $152$, and the angle $PAM$ is a right angle. A circle of radius $19$ with center $O$ on $\overline{AP}$ is drawn so that it is tangent to $\overline{AM}$ and $\overline{PM}$. Given that $OP=m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$. | 98 | Let the circle intersect $\overline{PM}$ at $B$. Then note $\triangle OPB$ and $\triangle MPA$ are similar. Also note that $AM = BM$ by power of a point. Using the fact that the ratio of corresponding sides in similar triangles is equal to the ratio of their perimeters, we have \[\frac{19}{AM} = \frac{152-2AM-19+19}{15... | 0.3125 | 7,438.5625 | 5,781 | 8,192 |
If the sum of the digits of a natural number is the same as the sum of the digits of three times that number, but different from the sum of the digits of twice that number, we call such a number a "wonder number." Find the smallest "wonder number." | 144 | 0 | 7,759.625 | -1 | 7,759.625 | |
A $\textit{composite number}$ is a number that has two or more prime factors. The number 87 can be expressed as the sum of two composite numbers in many ways. What is the minimum positive difference between two such numbers? | 3 | 0.875 | 5,645.5625 | 5,281.785714 | 8,192 | |
Steve wrote the digits $1$, $2$, $3$, $4$, and $5$ in order repeatedly from left to right, forming a list of $10,000$ digits, beginning $123451234512\ldots.$ He then erased every third digit from his list (that is, the $3$rd, $6$th, $9$th, $\ldots$ digits from the left), then erased every fourth digit from the resultin... | 11 | 1. **Initial Setup and First Erasure:**
- Steve starts with a repeating sequence of digits: $12345$.
- The sequence repeats every $5$ digits.
- First, every third digit is erased. To understand the pattern after this erasure, consider the least common multiple (LCM) of the cycle length ($5$) and the erasure in... | 0.0625 | 8,168.0625 | 7,809 | 8,192 |
Let event $A$ be "The line $ax - by = 0$ intersects the circle $(x - 2\sqrt{2})^2 + y^2 = 6$".
(1) If $a$ and $b$ are the numbers obtained by rolling a dice twice, find the probability of event $A$.
(2) If the real numbers $a$ and $b$ satisfy $(a - \sqrt{3})^2 + (b - 1)^2 \leq 4$, find the probability of event $A$. | \frac{1}{2} | 0.1875 | 6,904.125 | 4,886 | 7,369.846154 | |
There is a point inside an equilateral triangle with side length \( d \) whose distances from the vertices are 3, 4, and 5 units. Find the side length \( d \). | \sqrt{25 + 12 \sqrt{3}} | 0 | 8,129 | -1 | 8,129 | |
How many points on the hyperbola \( y = \frac{2013}{x} \) are there such that the tangent line at those points intersects both coordinate axes at points with integer coordinates? | 48 | 0.25 | 7,700.6875 | 6,383 | 8,139.916667 | |
For any positive integer \( k \), let \( f_{1}(k) \) be the square of the sum of the digits of \( k \) when written in decimal notation. For \( n > 1 \), let \( f_{n}(k) = f_{1}\left(f_{n-1}(k)\right) \). What is \( f_{1992}\left(2^{1991}\right) \)? | 256 | 0.1875 | 8,039.375 | 7,378 | 8,192 | |
Simplify $$(x^3+4x^2-7x+11)+(-4x^4-x^3+x^2+7x+3).$$ Express your answer as a polynomial with the terms in order by decreasing degree. | -4x^4+5x^2+14 | 0.8125 | 1,851.75 | 1,409.230769 | 3,769.333333 | |
The operation $\star$ is defined as $a \star b = a^2 \div b$. For how many negative integer values of $x$ will the value of $12 \star x$ be a positive integer? | 15 | 0 | 3,911.125 | -1 | 3,911.125 | |
Triangle $ABC$ is equilateral with side length $6$. Suppose that $O$ is the center of the inscribed circle of this triangle. What is the area of the circle passing through $A$, $O$, and $C$? | 12\pi | 1. **Identify the Triangle and Circle**: Triangle $ABC$ is equilateral with side length $6$. We need to find the area of the circle passing through points $A$, $O$ (the incenter of $\triangle ABC$), and $C$.
2. **Properties of the Incenter**: In an equilateral triangle, the incenter $O$ is also the centroid and the ci... | 1 | 5,595.5625 | 5,595.5625 | -1 |
Two \(10 \times 24\) rectangles are inscribed in a circle as shown. Find the shaded area. | 169\pi - 380 | 0 | 5,619.75 | -1 | 5,619.75 | |
The ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a > b > 0$) has an eccentricity of $e = \frac{2}{3}$. Points A and B lie on the ellipse and are not symmetrical with respect to the x-axis or the y-axis. The perpendicular bisector of segment AB intersects the x-axis at point P(1, 0). Let the midpoint of AB be... | \frac{9}{4} | 0.25 | 7,180.5 | 5,557 | 7,721.666667 | |
For every $x \ge -\frac{1}{e}\,$ , there is a unique number $W(x) \ge -1$ such that
\[
W(x) e^{W(x)} = x.
\]
The function $W$ is called Lambert's $W$ function. Let $y$ be the unique positive number such that
\[
\frac{y}{\log_{2} y} = - \frac{3}{5} \, .
\]
The value of $y$ is of the form $e^{-W(z \l... | 5/3 | 0.4375 | 5,715.4375 | 3,766.714286 | 7,231.111111 | |
Given that $3\sin \alpha - 2\cos \alpha = 0$, find the value of the following expressions:
$$(1)\ \frac{\cos \alpha - \sin \alpha}{\cos \alpha + \sin \alpha} + \frac{\cos \alpha + \sin \alpha}{\cos \alpha - \sin \alpha};$$
$$(2)\ \sin^2\alpha - 2\sin \alpha\cos \alpha + 4\cos^2\alpha.$$ | \frac{28}{13} | 0.75 | 6,237.75 | 5,586.333333 | 8,192 | |
How many of the integers from \(2^{10}\) to \(2^{18}\) inclusive are divisible by \(2^{9}\)? | 511 | 0.5625 | 5,704.75 | 4,092.777778 | 7,777.285714 | |
The variables \(a, b, c, d, e\), and \(f\) represent the numbers 4, 12, 15, 27, 31, and 39 in some order. Suppose that
\[
\begin{aligned}
& a + b = c, \\
& b + c = d, \\
& c + e = f,
\end{aligned}
\]
Determine the value of \(a + c + f\). | 73 | 0.625 | 6,348.25 | 5,242 | 8,192 | |
Given that $$(x+y+z)(xy+xz+yz)=18$$and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=6$$for real numbers $x$, $y$, and $z$, what is the value of $xyz$? | 4 | 1 | 1,966.8125 | 1,966.8125 | -1 | |
In the coordinate plane, the curve $xy = 1$ intersects a circle at four points, three of which are $\left( 2, \frac{1}{2} \right),$ $\left( -5, -\frac{1}{5} \right),$ and $\left( \frac{1}{3}, 3 \right).$ Find the fourth point of intersection. | \left( -\frac{3}{10}, -\frac{10}{3} \right) | 0.5 | 6,939.375 | 5,686.75 | 8,192 | |
Given the equation \\((x^{2}-mx+2)(x^{2}-nx+2)=0\\), the four roots of the equation form a geometric sequence with the first term being \\( \frac {1}{2}\\). Find the absolute value of the difference between m and n, i.e., \\(|m-n|\\). | \frac{3}{2} | 0.6875 | 6,058.75 | 5,089.090909 | 8,192 | |
Let
\[ x^6 - 3x^3 - x^2 - x - 2 = q_1(x) q_2(x) \dotsm q_m(x), \]
where each non-constant polynomial $q_i(x)$ is monic with integer coefficients, and cannot be factored further over the integers. Compute $q_1(3) + q_2(3) + \dots + q_m(3)$. | 634 | 0 | 7,601.3125 | -1 | 7,601.3125 | |
For a natural number $b$ , let $N(b)$ denote the number of natural numbers $a$ for which the equation $x^2 + ax + b = 0$ has integer roots. What is the smallest value of $b$ for which $N(b) = 20$ ? | 240 | 0 | 8,055.8125 | -1 | 8,055.8125 | |
For real numbers $a$ and $b$, define $a \diamond b = \sqrt{a^2 + b^2}$. What is the value of $(5 \diamond 12) \diamond ((-12) \diamond (-5))$? | 13\sqrt{2} | 1. **Calculate $5 \diamond 12$:**
\[
5 \diamond 12 = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13
\]
2. **Calculate $(-12) \diamond (-5)$:**
\[
(-12) \diamond (-5) = \sqrt{(-12)^2 + (-5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13
\]
3. **Apply the operation $\diamond$ to the results from steps ... | 1 | 2,521.6875 | 2,521.6875 | -1 |
We randomly choose a function $f:[n] \rightarrow[n]$, out of the $n^{n}$ possible functions. We also choose an integer $a$ uniformly at random from $[n]$. Find the probability that there exist positive integers $b, c \geq 1$ such that $f^{b}(1)=a$ and $f^{c}(a)=1$. $\left(f^{k}(x)\right.$ denotes the result of applying... | \frac{1}{n} | Given a function $f$, define $N(f)$ to be the number of numbers that are in the same cycle as 1 (including 1 itself), if there is one, and zero if there is no such cycle. The problem is equivalent to finding $\mathbb{E}(N(f)) / n$. Note that $P(N(f)=k)=\frac{n-1}{n} \cdot \frac{n-2}{n} \cdots \cdots \cdot \frac{n-k+1}{... | 0.0625 | 8,108.75 | 6,860 | 8,192 |
Given the vertices of a triangle A(0, 5), B(1, -2), C(-6, m), and the midpoint of BC is D, when the slope of line AD is 1, find the value of m and the length of AD. | \frac{5\sqrt{2}}{2} | 0 | 2,245.25 | -1 | 2,245.25 | |
Let $ABC$ be an equilateral triangle with $AB=1.$ Let $M$ be the midpoint of $BC,$ and let $P$ be on segment $AM$ such that $AM/MP=4.$ Find $BP.$ | \frac{\sqrt{7}}{5} | 0 | 6,944.5 | -1 | 6,944.5 | |
Let $n$ be a positive integer. Find, with proof, the least positive integer $d_{n}$ which cannot be expressed in the form \[\sum_{i=1}^{n}(-1)^{a_{i}}2^{b_{i}},\]
where $a_{i}$ and $b_{i}$ are nonnegative integers for each $i.$ | 2 \left( \frac{4^n - 1}{3} \right) + 1 |
Let \( n \) be a positive integer. We aim to find the least positive integer \( d_n \) which cannot be expressed in the form
\[
\sum_{i=1}^{n}(-1)^{a_{i}}2^{b_{i}},
\]
where \( a_i \) and \( b_i \) are nonnegative integers for each \( i \).
We claim that the minimal number that is not \( n \)-good is
\[
d_n = 2 \le... | 0 | 8,074.625 | -1 | 8,074.625 |
Eight coins are arranged in a circle heads up. A move consists of flipping over two adjacent coins. How many different sequences of six moves leave the coins alternating heads up and tails up? | 7680 | Imagine we flip over two adjacent coins by pushing a button halfway between them. Then the outcome depends only on the parities of the number of times that each button is pushed. To flip any coin, we must push the two buttons adjacent to that coin a total of an odd number of times. To flip every other coin, the paritie... | 0 | 8,192 | -1 | 8,192 |
Given the function $y=2\sin \left(3x+ \dfrac{\pi}{4}\right)$, determine the shift required to obtain its graph from the graph of the function $y=2\sin 3x$. | \dfrac{\pi}{12} | 0.6875 | 4,274 | 3,404.545455 | 6,186.8 | |
Given a cylinder with height $OO_1 = 12$ and a base radius $r = 5$. There are points $A$ and $B$ on the circumferences of the top and bottom bases respectively, with $AB = 13$. Find the distance between the axis $OO_1$ and line segment $AB$. | \frac{5}{2} \sqrt{3} | 0 | 7,161.25 | -1 | 7,161.25 | |
The integer $n > 9$ is a root of the quadratic equation $x^2 - ax + b=0$. In this equation, the representation of $a$ in the base-$n$ system is $19$. Determine the base-$n$ representation of $b$. | 90_n | 0 | 2,250.4375 | -1 | 2,250.4375 | |
Let $m$ be a real number where $m > 0$. If for any $x \in (1, +\infty)$, the inequality $2e^{2mx} - \frac{ln x}{m} ≥ 0$ always holds, then find the minimum value of the real number $m$. | \frac{1}{2e} | 0.1875 | 8,045.0625 | 7,408.333333 | 8,192 | |
If $a_0 = \sin^2 \left( \frac{\pi}{45} \right)$ and
\[a_{n + 1} = 4a_n (1 - a_n)\]for $n \ge 0,$ find the smallest positive integer $n$ such that $a_n = a_0.$ | 12 | 0.375 | 7,836.0625 | 7,242.833333 | 8,192 | |
In a sequence of coin tosses, one can keep a record of instances in which a tail is immediately followed by a head, a head is immediately followed by a head, and etc. We denote these by TH, HH, and etc. For example, in the sequence TTTHHTHTTTHHTTH of 15 coin tosses we observe that there are two HH, three HT, four TH, a... | 560 | Let's consider each of the sequences of two coin tosses as an operation instead; this operation takes a string and adds the next coin toss on (eg, THHTH + HT = THHTHT). We examine what happens to the last coin toss. Adding HH or TT is simply an identity for the last coin toss, so we will ignore them for now. However, a... | 0 | 8,192 | -1 | 8,192 |
What is the largest three-digit multiple of 8 whose digits' sum is 24? | 888 | 0.5625 | 7,088.875 | 6,230.888889 | 8,192 | |
Let $n$ be a positive integer. At most how many distinct unit vectors can be selected in $\mathbb{R}^{n}$ such that from any three of them, at least two are orthogonal? | 2n | Solution 1. $2 n$ is the maximal number. An example of $2 n$ vectors in the set is given by a basis and its opposite vectors. In the rest of the text we prove that it is impossible to have $2 n+1$ vectors in the set. Consider the Gram matrix $A$ with entries $a_{i j}=e_{i} \cdot e_{j}$. Its rank is at most $n$, its eig... | 0 | 8,192 | -1 | 8,192 |
Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from $(c,0)$ to $(3,3)$, divides the entire region into two regions of equal area. What is $c$? | \frac{2}{3} |
We are given a configuration of five unit squares in the coordinate plane, and a line extending from $(c,0)$ to $(3,3)$ that divides the entire region into two regions of equal area. We need to find the value of $c$.
#### Step-by-step Analysis:
1. **Total Area of the Squares**: The total area of the five unit square... | 0 | 8,192 | -1 | 8,192 |
Triangle $OAB$ has $O=(0,0)$, $B=(5,0)$, and $A$ in the first quadrant. In addition, $\angle ABO=90^\circ$ and $\angle AOB=30^\circ$. Suppose that $OA$ is rotated $90^\circ$ counterclockwise about $O$. What are the coordinates of the image of $A$? | $\left( - \frac {5}{3}\sqrt {3},5\right)$ | 1. **Identify the Coordinates of Point $A$:**
Given that $\triangle OAB$ has $O = (0,0)$, $B = (5,0)$, and $\angle ABO = 90^\circ$, point $A$ must lie on the line perpendicular to $OB$ at $B$. Since $\angle AOB = 30^\circ$, we can use trigonometric relationships to find the coordinates of $A$.
2. **Using the Pythag... | 0 | 5,393.125 | -1 | 5,393.125 |
According to the standard convention for exponentiation,
\[2^{2^{2^{2}}} = 2^{(2^{(2^2)})} = 2^{16} = 65536.\]
If the order in which the exponentiations are performed is changed, how many other values are possible? | 1 | To solve this problem, we need to evaluate the expression $2^{2^{2^{2}}}$ with different parenthesizations and determine how many distinct values can be obtained. We will use the notation $a \uparrow b$ to denote $a^b$ for clarity.
The possible parenthesizations of $2 \uparrow 2 \uparrow 2 \uparrow 2$ are:
1. $2 \upar... | 1 | 5,444.3125 | 5,444.3125 | -1 |
What is the sum of all two-digit positive integers whose squares end with the digits 25? | 644 | 0 | 3,357.625 | -1 | 3,357.625 | |
Eight spheres of radius 2 are each tangent to the faces of an inner cube centered at the origin, with each sphere located in one of the octants, and the side length of the cube is 6. Find the radius of the smallest sphere, centered at the origin, that encloses these eight spheres. | 3\sqrt{3} + 2 | 0 | 6,419.5 | -1 | 6,419.5 | |
If \( (2^{a})(2^{b})=64 \), what is the mean (average) of \( a \) and \( b \)? | 3 | Since \( (2^{a})(2^{b})=64 \), then \( 2^{a+b}=64 \), using an exponent law. Since \( 64=2^{6} \), then \( 2^{a+b}=2^{6} \) and so \( a+b=6 \). Therefore, the average of \( a \) and \( b \) is \( \frac{1}{2}(a+b)=3 \). | 1 | 1,338.75 | 1,338.75 | -1 |
In how many ways can we fill the cells of a $4\times4$ grid such that each cell contains exactly one positive integer and the product of the numbers in each row and each column is $2020$? | 576 |
To solve the problem, we need to fill the cells of a \(4 \times 4\) grid such that each cell contains exactly one positive integer, and the product of the numbers in each row and each column is 2020. We must determine the number of ways to achieve this configuration.
First, observe that the prime factorization of 202... | 0 | 7,897.75 | -1 | 7,897.75 |
Let \( r(\theta) = \frac{1}{1-2\theta} \). Calculate \( r(r(r(r(r(r(10)))))) \) (where \( r \) is applied 6 times). | 10 | 0 | 6,640.375 | -1 | 6,640.375 | |
Let \( a, b, c, x, y, z \) be nonzero complex numbers such that
\[ a = \frac{b+c}{x-3}, \quad b = \frac{a+c}{y-3}, \quad c = \frac{a+b}{z-3}, \]
and \( xy + xz + yz = 10 \) and \( x + y + z = 6 \), find \( xyz \). | 15 | 0 | 6,523.25 | -1 | 6,523.25 | |
Determine the value of $k$ such that the equation
\[\frac{x + 3}{kx - 2} = x\] has exactly one solution. | -\frac{3}{4} | 0.1875 | 7,862.5 | 6,671.333333 | 8,137.384615 | |
Does there exist a three-digit number whose cube ends in three sevens? | 753 | 0.75 | 5,937.5625 | 5,186.083333 | 8,192 | |
A frustum of a right circular cone is formed by cutting a smaller cone from a larger cone. Suppose the frustum has a lower base radius of 8 inches, an upper base radius of 2 inches, and a height of 5 inches. Calculate the total surface area of the frustum. | 10\pi \sqrt{61} + 68\pi | 0.1875 | 3,748.1875 | 3,597 | 3,783.076923 | |
Given the function $f(x)=2x-\sin x$, if the positive real numbers $a$ and $b$ satisfy $f(a)+f(2b-1)=0$, then the minimum value of $\dfrac {1}{a}+ \dfrac {4}{b}$ is ______. | 9+4 \sqrt {2} | 0 | 7,621 | -1 | 7,621 |
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