task_type stringclasses 4
values | problem stringlengths 14 5.23k | solution stringlengths 1 8.29k | problem_tokens int64 9 1.02k | solution_tokens int64 1 1.98k |
|---|---|---|---|---|
math | Every second, the computer displays a number equal to the sum of the digits of the previous number multiplied by 31. On the first second, the number 2020 was displayed. What number will be displayed on the screen on the 2020th second? | 310 | 58 | 3 |
math | Sally's Sandwich Stop offers sandwiches with the following toppings: mayonnaise, mustard, lettuce, tomatoes, pickles, onions, and olives. Additionally, a customer can opt for one of three types of bread (white, wheat, or rye). A customer can choose to have one or two layers of sandwich filling (chicken, turkey, or beef... | 4608 \text{ different kinds of sandwiches can be ordered.} | 85 | 16 |
math | Let \( n \) be a positive integer. Consider the set
\[
S = \{ (x, y, z) : x, y, z \in \{0, 1, \cdots, n\}, x + y + z > 0 \}
\]
This is a set of points in three-dimensional space with \((n+1)^3 - 1\) elements. What is the minimum number of planes required such that their union covers \( S \) but does not contain the poi... | 3n | 117 | 2 |
math | To ensure that the line $\left(3a-1\right)x+\left(2-a\right)y-1=0$ does not pass through the second quadrant, the range of values for $a$ is ______. | \left[2,+\infty \right) | 47 | 11 |
math | Calculate the value of the following expressions:
1. If $a + a^{-1} = 4$, find the value of $a^{\frac{1}{2}} + a^{-\frac{1}{2}}$.
2. Given $2\log_{10} \frac{x-y}{2} = \log_{10} x + \log_{10} y$, find the value of $\log_{(3-2\sqrt{2})} \frac{x}{y}$. | \log_{(3-2\sqrt{2})} \frac{x}{y} = -1 | 107 | 22 |
math | Given the sides $a, b, c, d$ of a trapezoid, how long is the line segment that joins the midpoints of the parallel sides? | \frac{a + b}{2} | 35 | 9 |
math | Let the set \( \mathrm{S} = \{1, 2, 3, \ldots, 10\} \). The subset \( \mathrm{A} \) of \( \mathrm{S} \) satisfies \( \mathrm{A} \cap \{1, 2, 3\} \neq \emptyset \) and \( \mathrm{A} \cup \{4, 5, 6\} \neq \mathrm{S} \). Find the number of such subsets \( \mathrm{A} \). | 888 | 122 | 3 |
math | Please write a monomial containing the letters $m$ and $n$ with a coefficient of $-2$ and a degree of $3$. The monomial is ______. | -2mn^2 | 36 | 5 |
math | Find the distance between the points $(2, 3)$ and $(-6, 7)$. Also, calculate the coordinates of the midpoint between these two points. | (-2, 5) | 34 | 6 |
math | In triangle $XYZ$, $XY = 30$ and $\angle Y = 45^\circ$. Let the perpendicular bisector of $XY$ intersect $XY$ and $YZ$ at point $M$ and point $N$, respectively. Find the length of $MN$. | MN = 15\sqrt{2} | 59 | 10 |
math | The sum of the first $n$ terms of a sequence $\{a_{n}\}$ is denoted as $S_{n}$, satisfying ${S}_{n}={a}_{n+1}-1$, where $n \in \mathbb{N}^{*}$, and $a_{1}=1$.
$(1)$ Find $a_{n}$;
$(2)$ Let ${b}_{n}=(-1)^{n}({a}_{n}-1)$, find the sum $S_{2n}$ of the first $2n$ terms of the sequence $\{b_{n}\}$. | \frac{4^{n} - 1}{3} | 130 | 13 |
math | It is known that a two-digit number, when divided by 3, leaves a remainder of 1, and when divided by 5, leaves a remainder of 3. Find all such numbers. | \{13, 28, 43, 58, 73, 88\} | 41 | 26 |
math | Given that the approximate ratio of the three cases drawn, DD, Dd, dd, is 1:2:1, calculate the probability of drawing dd when two students who have drawn cards are selected, and one card is drawn from each of these two students. | \frac{1}{4} | 54 | 7 |
math | Let \( a \) and \( b \) be non-negative integers such that \( 2^{2a} + 2^b + 5 \) is a perfect square. Find all possible values of \( a + b \). | 4 \text{ or } 5 | 49 | 8 |
math | Given a sequence $\left\{x_{n}\right\}$ that satisfies $x_{n+1}=x_{n}-x_{n-1}$ for $n \geqslant 2$, with $x_{1}=a$ and $x_{2}=b$. Let $S_{n}= x_{1}+x_{2}+\cdots+x_{n}$. Determine $S_{100}$ and $x_{100}$. | -a, \quad 2b - a | 100 | 9 |
math | Ten teams play a tournament in which every team plays every other team exactly once. No ties occur, and each team has a $50\%$ chance of winning any game it plays. The probability that no two teams win the same number of games is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $\log_2 n.$ | 37 | 81 | 2 |
math | Chubby designs another nonstandard checkerboard, but this time it has 33 squares on each side. The board still follows the pattern of having a black square in every corner and alternates red and black squares along every row and column. Determine how many black squares are there on this checkerboard. | 545 | 61 | 3 |
math | At Newton High School, 70% of the students enjoy playing soccer, and the remaining students don't. Out of the ones who enjoy soccer, 75% openly say they enjoy it, while the remaining 25% say they don’t. From the students who don't enjoy soccer, 85% admit that they don't enjoy it, and 15% claim that they do. Determine w... | \frac{35}{86} | 167 | 9 |
math | Your friend sitting to your left (or right?) is unable to solve any of the eight problems on his or her Combinatorics $B$ test, and decides to guess random answers to each of them. To your astonishment, your friend manages to get two of the answers correct. Assuming your friend has equal probability of guessing each ... | 9 | 104 | 1 |
math | The line $x-2y-3=0$ intersects the circle $(x-2)^{2}+(y+3)^{2}=9$ at points $E$ and $F$, calculate the area of $\triangle EOF$ ($O$ is the origin). | \frac{6 \sqrt{5}}{5} | 57 | 12 |
math | Given that a recipe making 8 servings of a smoothie requires 3 bananas, 2 cups of strawberries, 1 liter of yogurt, and 4 tablespoons of honey, determine the number of servings of the smoothie Emily can make using 9 bananas, 8 cups of strawberries, 3 liters of yogurt, and an unspecified amount of honey. | 24 | 72 | 2 |
math | Find all triples $(a,b,c)$ of positive integers such that
(i) $a \leq b \leq c$ ;
(ii) $\text{gcd}(a,b,c)=1$ ; and
(iii) $a^3+b^3+c^3$ is divisible by each of the numbers $a^2b, b^2c, c^2a$ . | (1, 1, 1) | 89 | 9 |
math | Calculate the angle $\theta$ for the complex number sum
\[ e^{11\pi i/120} + e^{31\pi i/120} + e^{51 \pi i/120} + e^{71\pi i /120} + e^{91 \pi i /120} \]
expressed in the form $r e^{i \theta}$, where $0 \le \theta < 2\pi$. | \frac{17\pi}{40} | 106 | 11 |
math | In each cell of a $10 \times 10$ board, there is one rabbit. Between rabbits in adjacent cells, there are partitions that can be removed. What is the minimum number of partitions that need to be removed so that any rabbit can visit any other rabbit, passing through no more than 17 cells on the way (not counting the sta... | 100 | 78 | 3 |
math | The equation of the line on which the chord lies of the ellipse $\dfrac {x^{2}}{16}+ \dfrac {y^{2}}{8}=1$ that is bisected by the point $(2,1)$. | x+y-3=0 | 52 | 6 |
math | Given the sequence \(\{a_n\}\) with the first term 2, and it satisfies
\[ 6 S_n = 3 a_{n+1} + 4^n - 1. \]
Find the maximum value of \(S_n\). | 35 | 56 | 2 |
math | A sequence of integers $a_1, a_2, a_3, \ldots$ is chosen so that $a_n = a_{n - 1} - a_{n - 2}$ for each $n \ge 3.$ What is the sum of the first $2001$ terms of this sequence if the sum of the first $1492$ terms is $1985,$ and the sum of the first $1985$ terms is $1492$? | 986 | 111 | 3 |
math | Given the function $f(x) = ax^2 + x$ (where $a$ is a constant), determine the point through which the graph of the function $f(x-1)$ always passes. | 1, 0 | 42 | 4 |
math | A convex pentagon has interior angles with measures $y$, $2y+2$, $3y-3$, $4y+4$, and $5y-5$ degrees. What is the measure of the largest angle? | 176 | 48 | 3 |
math | In the polar coordinate system, the polar coordinates of the center of the circle $\rho = -2\cos\theta$ are expressed as ($r$, $\theta$). | (1, \pi) | 35 | 6 |
math | Determine the interval of monotonic increase for the function $y = \frac{1}{2}\sin x + \frac{\sqrt{3}}{2}\cos x$ where $x \in [0, \frac{\pi}{2}]$. | \left[0, \frac{\pi}{6}\right] | 53 | 14 |
math | Let \( n \) be a two-digit number written in the decimal system, and \( s \) the sum of the squares of its digits. What are the smallest and largest values of \( n - s \)? | 25 \text{ and } -63 | 44 | 10 |
math | There are four balls in a bag, each with the same shape and size, and their numbers are \\(1\\), \\(2\\), \\(3\\), and \\(4\\).
\\((1)\\) Draw two balls randomly from the bag. Calculate the probability that the sum of the numbers on the balls drawn is no greater than \\(4\\).
\\((2)\\) First, draw a ball randomly fro... | \dfrac{13}{16} | 139 | 9 |
math | Given that $ω > 0$, the graph of the function $y=sin(ωx+ \frac{π}{3})$ is shifted to the right by $\frac{4π}{5}$ units and is symmetric about the $x$-axis with the original graph. What is the minimum value of $ω$? | \frac{5}{4} | 69 | 7 |
math | In pentagon $PQRST$, sides $QR=RS=ST=1$ unit, $\angle T$ is a right angle, and $m \angle Q = m \angle R = m \angle S = 150^\circ$. The extended lines of $QR$ and $ST$ meet at point $U$. Determine the length of segment $PU$ and find its expression in simplest radical form as $c+\sqrt{d}$ units. What is the value of $c+d... | c+d = 4 | 105 | 5 |
math | Let the function $f(x)=|2x-3|$.
$(1)$ Find the solution set of the inequality $f(x) > 5-|x+2|$;
$(2)$ If $g(x)=f(x+m)+f(x-m)$ has a minimum value of $4$, find the value of the real number $m$. | \pm 1 | 72 | 4 |
math | Three people are playing table tennis. The player who loses a game gives up their spot to the player who did not participate in that game. In the end, it turns out that the first player played 21 games, and the second player played 10 games. How many games did the third player play? | 11 | 64 | 2 |
math | Four positive integers $a$, $b$, $c$, and $d$ have a product of $8!$ and satisfy:
\[\begin{array}{rl} ab + a + b & = 524 \\ bc + b + c & = 146 \\ cd + c + d & = 104 \end{array}\]
What is $a-d$? | 10 | 85 | 2 |
math | Positive integer \(a\) and integers \(b\) and \(c\), in the three-dimensional coordinate system \(O-xyz\), point \(O(0,0,0)\), \(A(a, b, c)\), and \(B\left(x^{2}, x, 1\right)\) satisfy that the angle between \(\overrightarrow{OA}\) and \(\overrightarrow{OB}\) is \(\frac{\pi}{2}\). Given that the real number \(x\) has e... | 5 | 143 | 1 |
math | What is the diameter and circumference in meters of a circle whose area is $225\pi \text{m}^2$? | 30\pi \text{ meters} | 29 | 9 |
math | Find the maximum real number $\lambda$ such that for the real coefficient polynomial $f(x) = x^3 + ax^2 + bx + c$ with all non-negative real roots, the inequality $f(x) \geqslant \lambda(x - a)^3$ holds for all $x \geq 0$. Also, determine when equality holds in this expression. | 1 | 79 | 1 |
math | Perform the calculations:
72×54+28×54
60×25×8
2790÷(250×12-2910)
(100-1456÷26)×78. | 3432 | 61 | 4 |
math | Determine the number of ways to arrange the letters of the word PERMUTATIONS. | 239,500,800 | 18 | 11 |
math | What is the minimum number of digits to the right of the decimal point needed to express the fraction $\frac{987654321}{2^{28}\cdot 5^3}$ as a decimal? | 28 | 47 | 2 |
math | Find the distance between the highest and lowest points on the graph of $y=\cos(x+1)$. | \sqrt{\pi^2+4} | 22 | 9 |
math | A number is formed using the digits \(1, 2, \ldots, 9\). Any digit can be used more than once, but adjacent digits cannot be the same. Once a pair of adjacent digits has occurred, that pair, in that order, cannot be used again. How many digits are in the largest such number? | 73 | 69 | 2 |
math | How many non-congruent triangles with perimeter 12 have integer side lengths? | 2 | 17 | 1 |
math | We are allowed to remove exactly one integer from the list $$-1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,$$ and then choose two distinct integers at random from the remaining list. What number should we remove if we wish to maximize the probability that the sum of the two chosen numbers is 12? | 6 | 92 | 1 |
math | Let $A$ denote the set of functions whose range is $R$, and $B$ denote the set of functions $\phi(x)$ that have the following property: for the function $\phi(x)$, there exists a positive number $M$ such that the range of $\phi(x)$ is contained in the interval $[-M,M]$. For example, when $\phi_{1}(x)=x^{3}$, $\phi_{2}(... | 1,3,4 | 344 | 5 |
math | Find the eccentricity of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad (a > b > 0)$ given that the inscribed circle of the rhombus formed by its four vertices passes through the focus. | \frac{\sqrt{5} - 1}{2} | 63 | 13 |
math | Given that the Purple Parrot High School chess team consists of three boys and four girls, with two specific girls, Anna and Beth, who must sit next to each other, determine the number of possible arrangements of the team with a boy at each end and the remaining members in the middle. | 72 | 58 | 2 |
math | Find the value of $\dfrac {\sin 330^{\circ} \cdot \tan (-\dfrac {13}{3}\pi)}{\cos (-\dfrac {19}{6}\pi) \cdot \cos 690^{\circ} }$. | - \dfrac {2 \sqrt {3}}{3} | 61 | 14 |
math | If the internal angles A, C, B of $\triangle ABC$ form an arithmetic sequence, and the area of $\triangle ABC$ is $2\sqrt{3}$, then the minimum value of side AB is ______. | 2\sqrt{2} | 46 | 6 |
math | The maximum distance from a point on the ellipse $$\frac {x^{2}}{16}+ \frac {y^{2}}{4}=1$$ to the line $$x+2y- \sqrt {2}=0$$ is \_\_\_\_\_\_. | \sqrt{10} | 58 | 6 |
math | Each of the equations \( a x^{2} - b x + c = 0 \) and \( c x^{2} - a x + b = 0 \) has two distinct real roots. The sum of the roots of the first equation is non-negative, and the product of the roots of the first equation is 9 times the sum of the roots of the second equation. Find the ratio of the sum of the roots of ... | -3 | 104 | 2 |
math | A scanning code consists of a $5 \times 5$ grid of squares, where some squares are black and others are white. Each scanning code must contain at least one square of each color. A code is $\textit{symmetric}$ if it appears the same when rotated by any multiple of $90^{\circ}$ around its center or reflected across any d... | 62 | 94 | 2 |
math | Given the line $l: \rho\sin(\theta - \frac{\pi}{4}) = 4$ and the circle $C: \rho = 2k\cdot\cos(\theta + \frac{\pi}{4})$ ($k \neq 0$), if the minimum distance from a point on line $l$ to a point on circle $C$ equals 2.
(1) Find the Cartesian coordinates of the center of circle $C$;
(2) Find the value of $k$. | -1 | 112 | 2 |
math | How many solutions does the equation
\[
\frac{(x-1)(x-2)(x-3)\dotsm(x-120)}{(x-1^2)(x-2^2)(x-3^2)\dotsm(x-10^2)(x-120)}
\]
have for $x$? | 109 | 74 | 3 |
math | Let $x$ and $y$ be positive real numbers such that $x^3 + y^3 = x + y.$ Simplify
\[\frac{x}{y} + \frac{y}{x} - \frac{2}{xy}.\] | 0 | 55 | 1 |
math | Form a sequence from the consecutive positive integers from 1 to $n$ (where $n > 1$) in such a way that the sum of any two adjacent terms in the sequence is a perfect square. What is the smallest value of the positive integer $n$? | 15 | 57 | 2 |
math | How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5 \times 5$ square array of dots? | 100 | 33 | 3 |
math | Given
$$
(\sqrt{2017} x-\sqrt{2027})^{2017} = a_{1} x^{2017}+a_{2} x^{2016}+\cdots+a_{2017} x+a_{2018}.
$$
Then, evaluate $\left(a_{1}+a_{3}+\cdots+a_{2017}\right)^{2} - \left(a_{2}+a_{4}+\cdots+a_{2018}\right)^{2}$. | -10^{2017} | 128 | 9 |
math | Given the hyperbola C: $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \quad (a > 0, b > 0)$$, whose right vertex is A, construct a circle A with its center at A and a radius of b. Circle A intersects one of the asymptotes of the hyperbola C at points M and N. If ∠MAN=60°, the eccentricity of C is __________. | e = \frac{2}{\sqrt{3}} | 107 | 12 |
math | Point P is a center of symmetry of the graph of the function $f(x) = \cos \omega x$ ($\omega > 0$). If the minimum distance from point P to the axis of symmetry of the graph C is $\pi$, find the smallest positive period of the function $f(x)$. | 2\pi | 65 | 3 |
math | For a finite set \( A \), there exists a function \( f: \mathbb{N} \rightarrow A \) with the following property: If \( |i - j| \) is a prime number, then \( f(i) \neq f(j) \), where \( \mathbb{N} = \{1, 2, \ldots\} \). Find the minimum number of elements in the finite set \( A \). | 4 | 95 | 1 |
math | Find the number of square units in the area of the shaded region. Consider a rectangular plot with vertices at points (0,0), (40,0), (40,20), and (0,20). Within this rectangle, a shaded polygon is formed with vertices at (0,0), (20,0), (40,10), (40,20), and (10,20). Determine the area of this shaded region. | 600 | 100 | 3 |
math | Given $a$, $b$, and $c$ are the three side lengths of a triangle.
$(1)$ Fill in the blanks: $a+b-c \_\_\_\_\_\_0$, $a-b+c \_\_\_\_\_\_0$, $a-b-c \_\_\_\_\_\_0$ (Fill in "$ \gt $", "$ \lt $", or "$=$").
$(2)$ Simplify: $|a+b-c|-|a-b+c|+|a-b-c|$. | -a + 3b - c | 110 | 7 |
math | The lyrics of the popular song "Digging for Mr. Huang" on the internet recently include the line "种什么样的种子开什么样的花" which means "What kind of seed you plant determines what kind of flower will bloom." Suppose the probability of a small seed growing into a small flower is $0.9$, and the probability of a large seed growing ... | 0.62 | 134 | 4 |
math | Several gnomes live at mutually distinct points on the surface of a pentagonal pyramid. They can live inside the faces, on the edges, or at the vertices. It is found that each face (including its vertices and edges) contains a different number of gnomes. What is the minimum number of gnomes that live on the pyramid? | 6 | 71 | 1 |
math | Euler was not only a great mathematician but also a father who was good at educating his children. The following problem is one he created to enlighten his children: "Before his death, the father made a will to distribute his estate in the following manner: the eldest son would receive 100 francs plus 1/10 of the remai... | 900 | 153 | 3 |
math | Compute the sum:
\[
\sum_{k = 1}^\infty \frac{20^k}{(4^k - 3^k)(4^{k+1} - 3^{k+1})}.
\] | 5 | 52 | 1 |
math | Given the parabola $y^2 = 2x$ with focus $F$ and a moving point $P$ on the parabola, and another point $A(3, 2)$, find the coordinates of point $P$ such that the sum of distances $|PA| + |PF|$ is minimized. | (2, 2) | 69 | 6 |
math | Simplify first, then evaluate: $\left(\frac{1}{x+3}-1\right) \div \frac{{x}^{2}-4}{{x}^{2}+6x+9}$, and then choose a suitable number from $-3$, $-2$, $0$, $2$, $3$ to substitute and evaluate. | -6 | 76 | 2 |
math | Solve for \( x \):
\[ \sqrt{9+\sqrt{27+9x}}+ \sqrt{3+\sqrt{3+x}} = 3+3\sqrt{3} \] | 33 | 45 | 2 |
math | From the subset of complex numbers \( \left\{ x + y i \mid 1 \leqslant x < y \leqslant 4, x, y \in \mathbf{N}_{+} \right\} \), determine the number of different differences that can be obtained when two different elements are randomly selected and then subtracted. | 9 | 76 | 1 |
math | Point \( B \) lies on segment \( AC \), with \( AB = 14 \) and \( BC = 28 \). Semicircles are constructed on segments \( AB \), \( BC \), and \( AC \) as diameters in the same half-plane. Find the radius of the circle that is tangent to all three semicircles. | 7 | 75 | 1 |
math | Two numbers are both composite and are coprime. Their least common multiple is 120. The greatest common divisor of these two numbers is \_\_\_\_\_\_, and the smaller composite number is \_\_\_\_\_\_. | 8 | 50 | 1 |
math | If the integer $m$ makes the roots of the equation $x^2 - mx + m + 2006 = 0$ non-zero integers, then the number of such integers $m$ is. | 5 | 45 | 1 |
math | Given the function $f(x)= \frac {3}{2}\sin 2x+ \frac { \sqrt {3}}{2}\cos 2x+ \frac {π}{12}$, its graph is symmetric about the point $(a,b)$. If $a\in(-\frac {π}{2},0)$, then find the value of $a+b$. | \frac {0}{1} | 81 | 7 |
math | Let $a, b, c$ be real numbers. Factor the expression
\[ a^4(b^3 - c^3) + b^4(c^3 - a^3) + c^4(a^3 - b^3) \]
into the form $(a-b)(b-c)(c-a)q(a,b,c)$, where $q(a,b,c)$ is some polynomial in $a, b, c$. Find $q(a,b,c)$. | a^2 + ab + ac + b^2 + bc + c^2 | 99 | 17 |
math | The graph of the equation $10x + 270y = 2700$ is drawn on graph paper where each square represents one unit in each direction. A second line defined by $x + y = 10$ also passes through the graph. How many of the $1$ by $1$ graph paper squares have interiors lying entirely below both graphs and entirely in the first qua... | 50 | 85 | 2 |
math | Given that $\frac{x}{30-x} + \frac{y}{75-y} + \frac{z}{50-z} = 9$, evaluate $\frac{6}{30-x} + \frac{15}{75-y} + \frac{10}{50-z}$. | 2.4 | 67 | 3 |
math | Given the equation \( \sqrt{a(x-a)} + \sqrt{a(y-a)} = \sqrt{x-a} - \sqrt{a-y} \) holds true within the range of real numbers, where \( a, x, y \) are three distinct real numbers. Find the value of \( \frac{3x^2 + xy - y^2}{x^2 - xy + y^2} \). | \frac{1}{3} | 89 | 7 |
math | Factorize:
(1) $x^4-9x^2$
(2) $25x^2y+20xy^2+4y^3$
(3) $x^2(a-1)+y^2(1-a)$ | (a-1)(x+y)(x-y) | 59 | 10 |
math | Given \(f(x)=2\sin 2\omega x\), if its period is \(\pi\), then \(\omega=\) ______, and when \(x \in \left[ \frac{\pi}{6}, \frac{\pi}{3} \right]\), the maximum value of the function \(f(x)\) is ______. | 2 | 73 | 1 |
math | Let \( a < b < c < d < e \) be real numbers. Among the 10 sums of the pairs of these numbers, the least three are 32, 36, and 37, while the largest two are 48 and 51. Find all possible values of \( e \). | 27.5 | 69 | 4 |
math | Let $ABCDEF$ be a regular hexagon, and $J, K, L$ are the midpoints of $AB, CD, EF$ respectively. If the area of $\triangle JKL$ is $100$, what is the area of hexagon $ABCDEF$? | 400 | 60 | 3 |
math | Given that $\log_{10}\sin x + \log_{10}\cos x = -2$ and that $\log_{10}(\sin x + \cos x) = \frac{1}{2}(\log_{10}n + 1)$, find the value of $n$. | 0.102 | 66 | 5 |
math | Given a sequence of positive integers $\left\{a_{n}\right\}$ such that:
$$
a_{0}=m, \quad a_{n+1}=a_{n}^{5}+487 \quad (n \in \mathbf{N}),
$$
find the value of the positive integer $m$ such that the sequence $\left\{a_{n}\right\}$ contains the maximum number of perfect squares. | 9 | 94 | 1 |
math | Find the natural numbers \(k\) of the form \(k=\frac{a^{2}+a b+b^{2}}{a b-1}\) where \(a\) and \(b\) are natural numbers such that \(a b \neq 1\). | 4 \text{ and } 7 | 56 | 8 |
math | A rubber band is 4 inches long. An ant begins at the left end. Every minute, the ant walks one inch along rightwards along the rubber band, but then the band is stretched (uniformly) by one inch. For what value of $n$ will the ant reach the right end during the $n$th minute? | 7 | 69 | 1 |
math | Find the geometric mean of the two numbers \\(\sqrt {2}+1\\) and \\(\sqrt {2}-1\\). | 1 | 28 | 1 |
math | Consider a geometric sequence with terms $a$, $a(a-1)$, $a(a-1)^2$, ..., and let the sum of the first $n$ terms be denoted as $S_n$.
(1) Determine the range of the real number $a$ and the expression for $S_n$;
(2) Does there exist a real number $a$ such that $S_1$, $S_3$, $S_2$ form an arithmetic sequence? If it ex... | \frac{1}{2} | 120 | 7 |
math | There are four weights of different masses. Katya weighs the weights in pairs. As a result, she gets $1700,1870,2110,2330,$ and 2500 grams. How many grams does the sixth pair of weights weigh? | 2090 | 62 | 4 |
math | Call an integer $n$ oddly powerful if there exist positive integers $a$ and $b$, where $b>1$, $b$ is odd, and $a^b = n$. How many oddly powerful integers are less than $2010$? | 16 | 55 | 2 |
math | Ten identical crates each with dimensions $2\mathrm{ft}\times 3\mathrm{ft}\times 5\mathrm{ft}$ are to be stacked one on top of the other. Each crate's orientation is chosen at random, and placed flat on the previous crate. Calculate the probability that the total height of the stack is exactly $38\mathrm{ft}$. Express ... | 980 | 112 | 3 |
math | What is the base ten equivalent of $45327_8$? | 19159 | 17 | 5 |
math | What is the last digit of $1^{1}+2^{2}+3^{3}+\cdots+100^{100}$? | 0 | 34 | 1 |
math | In the $xy$-plane, an L-shaped region is defined by vertices at $(0,0), (0,4), (4,4), (4,2), (6,2)$, and $(6,0)$. Find the slope of the line through the origin that divides the area of this region exactly in half.
A) $\frac{1}{2}$
B) $1$
C) $1.5$
D) $2$
E) $2.5$ | 2 | 104 | 1 |
math | Determine the value of \(n\) for which the quantity
\[\frac{\tan \frac{\pi}{7} + i}{\tan \frac{\pi}{7} - i}\]
is a fourteenth root of unity, i.e., it equals \(\cos \frac{2n \pi}{14} + i \sin \frac{2n \pi}{14}\) for some integer \(n\) between 0 and 13 inclusive. | n = 4 | 99 | 4 |
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