haowu89/deepthinking_deepscale · Datasets at Hugging Face

problem stringlengths 10 2.37k	original_solution stringclasses 890 values	answer stringlengths 0 253	source stringclasses 1 value	index int64 6 40.3k
Daniel worked for 50 hours per week for 10 weeks during the summer, earning \$6000. If he wishes to earn an additional \$8000 during the school year which lasts for 40 weeks, how many fewer hours per week must he work compared to the summer if he receives the same hourly wage?		33.33	deepscale	20,926
In the Cartesian coordinate system $xOy$, with the origin as the pole and the positive half-axis of the $x$-axis as the polar axis, the polar coordinate equation of the curve $C_{1}$ is $\rho \cos \theta = 4$. $(1)$ Let $M$ be a moving point on the curve $C_{1}$, point $P$ lies on the line segment $OM$, and satisfies $\|OP\| \cdot \|OM\| = 16$. Find the rectangular coordinate equation of the locus $C_{2}$ of point $P$. $(2)$ Suppose the polar coordinates of point $A$ are $({2, \frac{π}{3}})$, point $B$ lies on the curve $C_{2}$. Find the maximum value of the area of $\triangle OAB$.		2 + \sqrt{3}	deepscale	22,528
Compute \(104 \times 96\).		9984	deepscale	21,496
Let $a, b \in \mathbb{R}$. If the line $l: ax+y-7=0$ is transformed by the matrix $A= \begin{bmatrix} 3 & 0 \\ -1 & b\end{bmatrix}$, and the resulting line is $l′: 9x+y-91=0$. Find the values of the real numbers $a$ and $b$.		13	deepscale	19,414
Find the number of ordered quadruples $(a, b, c, d)$ where each of $a, b, c,$ and $d$ are (not necessarily distinct) elements of $\{1, 2, 3, 4, 5, 6, 7\}$ and $3abc + 4abd + 5bcd$ is even. For example, $(2, 2, 5, 1)$ and $(3, 1, 4, 6)$ satisfy the conditions.		2017	deepscale	30,281
John recently bought a used car for $\$5000$ for his pizza delivery job. He gets $\$10$ for each pizza he delivers, but he has to spend $\$3$ on gas for each pizza he delivers. What is the minimum whole number of pizzas John must deliver in order to earn back the money he spent on the car he bought?		715	deepscale	38,720
Let $x$, $y$, and $z$ be real numbers greater than $1$, and let $z$ be the geometric mean of $x$ and $y$. The minimum value of $\frac{\log z}{4\log x} + \frac{\log z}{\log y}$ is \_\_\_\_\_\_.		\frac{9}{8}	deepscale	21,786
In a class of 120 students, the teacher recorded the following scores for an exam. Calculate the average score for the class. \begin{tabular}{\|c\|c\|} \multicolumn{2}{c}{}\\\hline \textbf{Score (\%)}&\textbf{Number of Students}\\\hline 95&12\\\hline 85&24\\\hline 75&30\\\hline 65&20\\\hline 55&18\\\hline 45&10\\\hline 35&6\\\hline \end{tabular}		69.83	deepscale	22,579
Given $a$ and $b$ are the roots of the equation $x^2-2cx-5d = 0$, and $c$ and $d$ are the roots of the equation $x^2-2ax-5b=0$, where $a,b,c,d$ are distinct real numbers, find $a+b+c+d$.		30	deepscale	25,928
If $5(\cos a + \cos b) + 4(\cos a \cos b + 1) = 0,$ then find all possible values of \[\tan \frac{a}{2} \tan \frac{b}{2}.\]Enter all the possible values, separated by commas.		3,-3	deepscale	39,697
Determine the following number: \[ \frac{12346 \cdot 24689 \cdot 37033 + 12347 \cdot 37034}{12345^{2}} \]		74072	deepscale	15,979
Triangles $\triangle ABC$ and $\triangle DEC$ share side $BC$. Given that $AB = 7\ \text{cm}$, $AC = 15\ \text{cm}$, $EC = 9\ \text{cm}$, and $BD = 26\ \text{cm}$, what is the least possible integral number of centimeters in $BC$?		17	deepscale	27,301
Determine the number of ways to arrange the letters of the word PERSEVERANCE.		19,958,400	deepscale	26,816
How many four-digit numbers greater than 2999 can be formed such that the product of the middle two digits exceeds 5?		4970	deepscale	34,887
A four-meter gas pipe has rusted in two places. Determine the probability that all three resulting pieces can be used as connections to gas stoves, given that regulations require a stove to be at least 1 meter away from the main gas pipeline.		1/8	deepscale	7,486
Let $S$ be the set of all nonzero real numbers. Let $f : S \to S$ be a function such that \[f(x) + f(y) = f(xyf(x + y))\]for all $x,$ $y \in S$ such that $x + y \neq 0.$ Let $n$ be the number of possible values of $f(4),$ and let $s$ be the sum of all possible values of $f(4).$ Find $n \times s.$		\frac{1}{4}	deepscale	36,346
Given $a\ln a=be^{b}$, where $b > 0$, find the maximum value of $\frac{b}{{{a^2}}}$		\frac{1}{2e}	deepscale	11,322
Define $a \Delta b = a^2 -b $. What is the value of $ (2^{4 \Delta13})\Delta(3^{3\Delta5})$		-17	deepscale	33,944
Determine the maximal size of a set of positive integers with the following properties: 1. The integers consist of digits from the set {1,2,3,4,5,6}. 2. No digit occurs more than once in the same integer. 3. The digits in each integer are in increasing order. 4. Any two integers have at least one digit in common (possibly at different positions). 5. There is no digit which appears in all the integers.		32	deepscale	13,386
In the trapezoid \(ABCD\), the lengths of the bases \(AD = 24 \text{ cm}\), \(BC = 8 \text{ cm}\), and the diagonals \(AC = 13 \text{ cm}\), \(BD = 5\sqrt{17} \text{ cm}\) are known. Calculate the area of the trapezoid.		80	deepscale	13,548
Construct a square \(A B C D\) with side length \(6 \text{ cm}\). Construct a line \(p\) parallel to the diagonal \(A C\) passing through point \(D\). Construct a rectangle \(A C E F\) such that vertices \(E\) and \(F\) lie on the line \(p\). Using the given information, calculate the area of rectangle \(A C E F\).		36	deepscale	8,076
Let $ a, b \in \mathbb{N}$ with $ 1 \leq a \leq b,$ and $ M \equal{} \left[\frac {a \plus{} b}{2} \right].$ Define a function $ f: \mathbb{Z} \mapsto \mathbb{Z}$ by \[ f(n) \equal{} \begin{cases} n \plus{} a, & \text{if } n \leq M, \\ n \minus{} b, & \text{if } n >M. \end{cases} \] Let $ f^1(n) \equal{} f(n),$ $ f_{i \plus{} 1}(n) \equal{} f(f^i(n)),$ $ i \equal{} 1, 2, \ldots$ Find the smallest natural number $ k$ such that $ f^k(0) \equal{} 0.$	Let \( a, b \in \mathbb{N} \) with \( 1 \leq a \leq b \), and let \( M = \left\lfloor \frac{a + b}{2} \right\rfloor \). The function \( f: \mathbb{Z} \to \mathbb{Z} \) is defined as: \[ f(n) = \begin{cases} n + a, & \text{if } n \leq M, \\ n - b, & \text{if } n > M. \end{cases} \] We are required to find the smallest natural number \( k \) such that \( f^k(0) = 0 \), where \( f^1(n) = f(n) \) and for \( i \geq 1 \), \( f_{i+1}(n) = f(f^i(n)) \). ### Step-by-step Analysis: 1. Initial Application of \( f \): Start with \( n = 0 \). Since \( 0 \leq M \) (as \( M \geq 0 \)), we apply the first case of the function: \[ f(0) = 0 + a = a. \] 2. Subsequent Applications of \( f \): Next, apply \( f \) to \( a \): \[ f(a) = a + a = 2a \quad \text{(since } a \leq M \text{ if } 2a \leq M \text{, which is usually true for small } a). \] Continue applying \( f \) on subsequent results until \( n > M \). As soon as \( n > M \) and since each application adds \( a \) until the threshold is reached, beyond this point, \( n \) will start decreasing by \( b \). 3. Loop Detection: The behavior of \( f \) may create a cycle due to the periodic nature of additions of \( a \) and subtractions of \( b \). To return precisely to zero, a full cycle must have a net effect of zero. This means the total sum of increases and decreases (by adding \( a \) and subtracting \( b \)) must cancel out over some iterations. 4. Formulating the Cycle Condition: We need \( f^k(0) = 0 \), meaning: \[ j \cdot a = l \cdot b \quad \text{for some integers } j, l, \] or equivalently, the number of times \( a \) is added must equal multiples of \( b \). Hence, the smallest \( k \) is connected to multiples that sum to zero. 5. Using \( \gcd(a, b) \): The smallest \( k \) is determined by the period after which multiples of \( a \) transform into some form of \( b \)-cycle to return to zero. For this, the period length is directly linked to the least common multiple cycles determined by: \[ k = \frac{a + b}{\gcd(a, b)}, \] where \( \gcd(a, b) \) ensures the minimal integer multiple cycle to zero. Thus, the smallest \( k \) satisfying \( f^k(0) = 0 \) is: \[ \boxed{\frac{a + b}{\gcd(a, b)}}. \]	\frac {a + b}{\gcd(a,b)}	deepscale	6,356
The year 2009 has a unique property: by rearranging the digits of the number 2009, it is impossible to form a smaller four-digit number (numbers do not start with zero). In which future year will this property first repeat again?		2022	deepscale	10,026
Given the ellipse \( x^{2}+ \frac {y^{2}}{b^{2}+1}=1(b > 0) \) has an eccentricity of \( \frac {\sqrt {10}}{10} \), determine the value of \( b \).		\frac{1}{3}	deepscale	12,252
How many even integers between 4000 and 7000 have four different digits?	The thousands digit is $\in \{4,5,6\}$. Case $1$: Thousands digit is even $4, 6$, two possibilities, then there are only $\frac{10}{2} - 1 = 4$ possibilities for the units digit. This leaves $8$ possible digits for the hundreds and $7$ for the tens places, yielding a total of $2 \cdot 8 \cdot 7 \cdot 4 = 448$. Case $2$: Thousands digit is odd $5$, one possibility, then there are $5$ choices for the units digit, with $8$ digits for the hundreds and $7$ for the tens place. This gives $1 \cdot 8 \cdot 7 \cdot 5= 280$ possibilities. Together, the solution is $448 + 280 = \boxed{728}$.	728	deepscale	6,566
At a hypothetical school, there are three departments in the faculty of sciences: biology, physics and chemistry. Each department has three male and one female professor. A committee of six professors is to be formed containing three men and three women, and each department must be represented by two of its members. Every committee must include at least one woman from the biology department. Find the number of possible committees that can be formed subject to these requirements.		27	deepscale	16,268
There are $20$ students participating in an after-school program offering classes in yoga, bridge, and painting. Each student must take at least one of these three classes, but may take two or all three. There are $10$ students taking yoga, $13$ taking bridge, and $9$ taking painting. There are $9$ students taking at least two classes. How many students are taking all three classes?	Let's define the variables for the Venn Diagram: - $a$: Number of students taking exactly Bridge and Yoga. - $b$: Number of students taking exactly Bridge and Painting. - $c$: Number of students taking all three classes (Yoga, Bridge, and Painting). - $d$: Number of students taking exactly Yoga and Painting. We are given: - Total students taking at least two classes: $9$. - Total students taking Yoga: $10$. - Total students taking Bridge: $13$. - Total students taking Painting: $9$. From the information, we know: \[ a + b + c + d = 9 \] This equation represents the total number of students taking at least two classes. #### Calculating students taking only one class: 1. Students taking only Bridge: \[ \text{Total Bridge} - (a + b + c) = 13 - (a + b + c) \] Using $a + b + c + d = 9$, we find $a + b + c = 9 - d$. Thus: \[ 13 - (9 - d) = 4 + d \] 2. Students taking only Yoga: \[ \text{Total Yoga} - (a + c + d) = 10 - (a + c + d) \] Using $a + b + c + d = 9$, we find $a + c + d = 9 - b$. Thus: \[ 10 - (9 - b) = 1 + b \] 3. Students taking only Painting: \[ \text{Total Painting} - (b + c + d) = 9 - (b + c + d) \] Using $a + b + c + d = 9$, we find $b + c + d = 9 - a$. Thus: \[ 9 - (9 - a) = a \] #### Total students taking exactly one class: Adding the equations for students taking only one class: \[ (4 + d) + (1 + b) + a = 11 \] \[ 5 + a + b + d = 11 \] Using $a + b + c + d = 9$, substitute $9 - c$ for $a + b + d$: \[ 5 + (9 - c) = 11 \] \[ 14 - c = 11 \] \[ c = 3 \] Thus, the number of students taking all three classes is $\boxed{\textbf{(C)}\ 3}$.	3	deepscale	2,772
Complex numbers \( p, q, r \) form an equilateral triangle with a side length of 24 in the complex plane. If \( \|p + q + r\| = 48 \), find \( \|pq + pr + qr\| \).		768	deepscale	20,753
A point \( M \) is chosen on the diameter \( AB \). Points \( C \) and \( D \), lying on the circumference on one side of \( AB \), are chosen such that \(\angle AMC=\angle BMD=30^{\circ}\). Find the diameter of the circle given that \( CD=12 \).		8\sqrt{3}	deepscale	28,506
For a finite sequence \(P = \left(p_1, p_2, \cdots, p_n\right)\), the Cesaro sum (named after the mathematician Cesaro) is defined as \(\frac{1}{n}(S_1 + S_2 + \cdots + S_n)\), where \(S_k = p_1 + p_2 + \cdots + p_k\) for \(1 \leq k \leq n\). If a sequence \(\left(p_1, p_2, \cdots, p_{99}\right)\) of 99 terms has a Cesaro sum of 1000, then the Cesaro sum of the 100-term sequence \(\left(1, p_1, p_2, \cdots, p_{99}\right)\) is ( ).		991	deepscale	16,270
What is the sum of all positive integer solutions less than or equal to $30$ to the congruence $15(3x-4) \equiv 30 \pmod{10}$?		240	deepscale	17,479
Given three forces in space, $\overrightarrow {F_{1}}$, $\overrightarrow {F_{2}}$, and $\overrightarrow {F_{3}}$, each with a magnitude of 2, and the angle between any two of them is 60°, the magnitude of their resultant force $\overrightarrow {F}$ is ______.		2 \sqrt {6}	deepscale	19,375
Let $a$ and $b$ be angles such that $\sin (a + b) = \frac{3}{4}$ and $\sin (a - b) = \frac{1}{2}.$ Find $\frac{\tan a}{\tan b}.$		5	deepscale	39,749
In the interval \([-6, 6]\), an element \(x_0\) is arbitrarily chosen. If the slope of the tangent line to the parabola \(y = x^2\) at \(x = x_0\) has an angle of inclination \(\alpha\), find the probability that \(\alpha \in \left[ \frac{\pi}{4}, \frac{3\pi}{4} \right]\).		\frac{11}{12}	deepscale	17,325
Given that \( x \) and \( y \) are positive integers such that \( 56 \leq x + y \leq 59 \) and \( 0.9 < \frac{x}{y} < 0.91 \), find the value of \( y^2 - x^2 \).		177	deepscale	11,389
Square $IJKL$ is contained within square $WXYZ$ such that each side of $IJKL$ can be extended to pass through a vertex of $WXYZ$. The side length of square $WXYZ$ is $\sqrt{98}$, and $WI = 2$. What is the area of the inner square $IJKL$? A) $62$ B) $98 - 4\sqrt{94}$ C) $94 - 4\sqrt{94}$ D) $98$ E) $100$		98 - 4\sqrt{94}	deepscale	13,118
The triangle $ABC$ is isosceles with $AB=BC$ . The point F on the side $[BC]$ and the point $D$ on the side $AC$ are the feets of the the internals bisectors drawn from $A$ and altitude drawn from $B$ respectively so that $AF=2BD$ . Fine the measure of the angle $ABC$ .		36	deepscale	24,718
Given a "double-single" number is a three-digit number made up of two identical digits followed by a different digit, calculate the number of double-single numbers between 100 and 1000.		81	deepscale	20,331
Find the number of positive integer solutions to $n^{x}+n^{y}=n^{z}$ with $n^{z}<2001$.	If $n=1$, the relation can not hold, so assume otherwise. If $x>y$, the left hand side factors as $n^{y}\left(n^{x-y}+1\right)$ so $n^{x-y}+1$ is a power of $n$. But it leaves a remainder of 1 when divided by $n$ and is greater than 1, a contradiction. We reach a similar contradiction if $y>x$. So $y=x$ and $2 n^{x}=n^{z}$, so 2 is a power of $n$ and $n=2$. So all solutions are of the form $2^{x}+2^{x}=2^{x+1}$, which holds for all $x$. $2^{x+1}<2001$ implies $x<11$, so there are 10 solutions.	10	deepscale	3,646
Let $f(r)=\sum_{j=2}^{2008} \frac{1}{j^{r}}=\frac{1}{2^{r}}+\frac{1}{3^{r}}+\cdots+\frac{1}{2008^{r}}$. Find $\sum_{k=2}^{\infty} f(k)$.	We change the order of summation: $$\sum_{k=2}^{\infty} \sum_{j=2}^{2008} \frac{1}{j^{k}}=\sum_{j=2}^{2008} \sum_{k=2}^{\infty} \frac{1}{j^{k}}=\sum_{j=2}^{2008} \frac{1}{j^{2}\left(1-\frac{1}{j}\right)}=\sum_{j=2}^{2008} \frac{1}{j(j-1)}=\sum_{j=2}^{2008}\left(\frac{1}{j-1}-\frac{1}{j}\right)=1-\frac{1}{2008}=\frac{2007}{2008}$$	\frac{2007}{2008}	deepscale	3,291
What is the largest integer that must divide the product of any $4$ consecutive integers?		24	deepscale	38,395
Let $a$ and $b$ each be chosen at random from the set $\{1, 2, 3, \ldots, 40\}$. Additionally, let $c$ and $d$ also be chosen at random from the same set. Calculate the probability that the integer $2^c + 5^d + 3^a + 7^b$ has a units digit of $8$.		\frac{3}{16}	deepscale	27,133
Squares $ABCD$ and $EFGH$ are congruent, $AB=10$, and $G$ is the center of square $ABCD$. The area of the region in the plane covered by these squares is	1. Identify the given information: We are given two congruent squares $ABCD$ and $EFGH$ with side length $AB = 10$. The point $G$ is the center of square $ABCD$. 2. Calculate the area of each square: - The area of square $ABCD$ is $AB^2 = 10^2 = 100$. - Since $EFGH$ is congruent to $ABCD$, its area is also $100$. 3. Calculate the total area if there were no overlap: - If there were no overlap, the total area would be $100 + 100 = 200$. 4. Determine the overlap: - Since $G$ is the center of $ABCD$, it lies at the midpoint of the diagonal of $ABCD$. - The length of the diagonal of $ABCD$ is $AB \sqrt{2} = 10\sqrt{2}$. - Therefore, $BG$, being half the diagonal, is $5\sqrt{2}$. 5. Analyze triangle $ABG$: - $\triangle ABG$ is formed by the line segment $AB$, $BG$, and $AG$. - Since $G$ is the midpoint of the diagonal, and the diagonals of a square bisect each other at right angles, $\triangle ABG$ is a right isosceles triangle with $\angle BGA = 90^\circ$. 6. Calculate the area of $\triangle ABG$: - The legs of $\triangle ABG$ are each $5\sqrt{2}$. - The area of $\triangle ABG$ is $\frac{1}{2} \times \text{leg} \times \text{leg} = \frac{1}{2} \times (5\sqrt{2})^2 = \frac{1}{2} \times 50 = 25$. 7. Calculate the area of the region covered by the squares: - Subtract the area of $\triangle ABG$ from the total area of the two squares to account for the overlap. - The area of the region is $200 - 25 = 175$. Thus, the area of the region in the plane covered by these squares is $\boxed{\textbf{(E)}\ 175}$.	175	deepscale	633
Given that the last initial of Mr. and Mrs. Alpha's baby's monogram is 'A', determine the number of possible monograms in alphabetical order with no letter repeated.		300	deepscale	30,426
Given $\sin\alpha= \frac {2 \sqrt {2}}{3}$, $\cos(\alpha+\beta)=- \frac {1}{3}$, and $\alpha, \beta\in(0, \frac {\pi}{2})$, determine the value of $\sin(\alpha-\beta)$.		\frac {10 \sqrt {2}}{27}	deepscale	17,026
In acute triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $b=2$, $B= \frac{π}{3}$, and $c\sin A= \sqrt{3}a\cos C$, find the area of $\triangle ABC$.		\sqrt{3}	deepscale	17,111
If \( A \) is a positive integer such that \( \frac{1}{1 \times 3} + \frac{1}{3 \times 5} + \cdots + \frac{1}{(A+1)(A+3)} = \frac{12}{25} \), find the value of \( A \).		22	deepscale	11,997
The sequence \(\{a_n\}\) has consecutive terms \(a_n\) and \(a_{n+1}\) as the roots of the equation \(x^2 - c_n x + \left(\frac{1}{3}\right)^n = 0\), with initial term \(a_1 = 2\). Find the sum of the infinite series \(c_1, c_2, \cdots, c_n, \cdots\).		\frac{9}{2}	deepscale	15,451
Given two circles $O\_1$: $(x-2)^2 + y^2 = 16$ and $O\_2$: $x^2 + y^2 = r^2 (0 < r < 2)$, a moving circle $M$ is tangent to both circles $O\_1$ and $O\_2$. The locus of the center of the moving circle $M$ forms two ellipses, with eccentricities $e\_1$ and $e\_2 (e\_1 > e\_2)$. Find the minimum value of $e\_1 + 2e\_2$.		\frac{3+2\sqrt{2}}{4}	deepscale	26,482
Two cars, Car A and Car B, start from point A and point B, respectively, and move towards each other simultaneously. They meet after 3 hours, after which Car A turns back towards point A and Car B continues moving forward. Once Car A reaches point A, it turns towards point B again and meets Car B half an hour later. How many minutes does Car B take to travel from point A to point B?		432	deepscale	12,928
Suppose $\cos S = 0.5$ in the diagram below. What is $ST$? [asy] pair P,S,T; P = (0,0); S = (6,0); T = (0,6*tan(acos(0.5))); draw(P--S--T--P); draw(rightanglemark(S,P,T,18)); label("$P$",P,SW); label("$S$",S,SE); label("$T$",T,N); label("$10$",S/2,S); [/asy]		20	deepscale	23,494
Let $f(x)$ have a domain of $R$, $f(x+1)$ be an odd function, and $f(x+2)$ be an even function. When $x\in [1,2]$, $f(x)=ax^{2}+b$. If $f(0)+f(3)=6$, then calculate the value of $f\left(\frac{9}{2}\right)$.		\frac{5}{2}	deepscale	31,392
An ant starts at the origin of a coordinate plane. Each minute, it either walks one unit to the right or one unit up, but it will never move in the same direction more than twice in the row. In how many different ways can it get to the point $(5,5)$ ?	We can change the ant's sequence of moves to a sequence $a_{1}, a_{2}, \ldots, a_{10}$, with $a_{i}=0$ if the $i$-th step is up, and $a_{i}=1$ if the $i$-th step is right. We define a subsequence of moves $a_{i}, a_{i+1}, \ldots, a_{j}$, ( $i \leq j$ ) as an up run if all terms of the subsequence are equal to 0 , and $a_{i-1}$ and $a_{j+1}$ either do not exist or are not equal to 0 , and define a right run similarly. In a sequence of moves, up runs and right runs alternate, so the number of up rights can differ from the number of right runs by at most one. Now let $f(n)$ denote the number of sequences $a_{1}, a_{2}, \ldots, a_{n}$ where $a_{i} \in\{1,2\}$ for $1 \leq i \leq n$, and $a_{1}+a_{2}+\cdots+a_{n}=5$. (In essence, we are splitting the possible 5 up moves into up runs, and we are doing the same with the right moves). We can easily compute that $f(3)=3, f(4)=4, f(5)=1$, and $f(n)=0$ otherwise. For each possible pair of numbers of up runs and right runs, we have two choices of which type of run is first. Our answer is then $2\left(f(3)^{2}+f(3) f(4)+f(4)^{2}+f(4) f(5)+f(5)^{2}\right)=2(9+12+16+4+1)=84$.	84	deepscale	5,178
Find the shortest distance from the line $3 x+4 y=25$ to the circle $x^{2}+y^{2}=6 x-8 y$.	The circle is $(x-3)^{2}+(y+4)^{2}=5^{2}$. The center $(3,-4)$ is a distance of $$ \frac{\|3 \cdot 3+4 \cdot-4-25\|}{\sqrt{3^{2}+4^{2}}}=\frac{32}{5} $$ from the line, so we subtract 5 for the radius of the circle and get $7 / 5$.	7 / 5	deepscale	3,332
Maximum difference in weights of two bags is achieved by taking the largest and smallest possible values of the two different brands. Given the weights of three brands of flour are $\left(25\pm 0.1\right)kg$, $\left(25\pm 0.2\right)kg$, and $\left(25\pm 0.3\right)kg$, calculate the maximum possible difference in weights.		0.6	deepscale	24,483
Let $a, b, c$ be positive real numbers such that $a \leq b \leq c \leq 2 a$. Find the maximum possible value of $$\frac{b}{a}+\frac{c}{b}+\frac{a}{c}$$	Fix the values of $b, c$. By inspecting the graph of $$f(x)=\frac{b}{x}+\frac{x}{c}$$ we see that on any interval the graph attains its maximum at an endpoint. This argument applies when we fix any two variables, so it suffices to check boundary cases in which $b=a$ or $b=c$, and $c=b$ or $c=2 a$. All pairs of these conditions determine the ratio between $a, b, c$, except $b=c$ and $c=b$, in which case the boundary condition on $a$ tells us that $a=b$ or $2 a=b=c$. In summary, these cases are $$(a, b, c) \in\{(a, a, a),(a, a, 2 a),(a, 2 a, 2 a)\}$$ The largest value achieved from any of these three is $\frac{7}{2}$.	\frac{7}{2}	deepscale	4,754
Determine the set of all real numbers $p$ for which the polynomial $Q(x)=x^{3}+p x^{2}-p x-1$ has three distinct real roots.	First, we note that $x^{3}+p x^{2}-p x-1=(x-1)(x^{2}+(p+1)x+1)$. Hence, $x^{2}+(p+1)x+1$ has two distinct roots. Consequently, the discriminant of this equation must be positive, so $(p+1)^{2}-4>0$, so either $p>1$ or $p<-3$. However, the problem specifies that the quadratic must have distinct roots (since the original cubic has distinct roots), so to finish, we need to check that 1 is not a double root-we will do this by checking that 1 is not a root of $x^{2}+(p+1)x+1$ for any value $p$ in our range. But this is clear, since $1+(p+1)+1=0 \Rightarrow p=-3$, which is not in the aforementioned range. Thus, our answer is all $p$ satisfying $p>1$ or $p<-3$.	p>1 \text{ and } p<-3	deepscale	4,726
Let $n$ be the integer such that $0 \le n < 31$ and $3n \equiv 1 \pmod{31}$. What is $\left(2^n\right)^3 - 2 \pmod{31}$? Express your answer as an integer from $0$ to $30$, inclusive.		6	deepscale	37,845
What is the number of units in the distance between $(2,5)$ and $(-6,-1)$?		10	deepscale	33,281
Let $\mathrm {Q}$ be the product of the roots of $z^8+z^6+z^4+z^3+z+1=0$ that have a positive imaginary part, and suppose that $\mathrm {Q}=s(\cos{\phi^{\circ}}+i\sin{\phi^{\circ}})$, where $0<s$ and $0\leq \phi <360$. Find $\phi$.		180	deepscale	26,853
What is the smallest integral value of $k$ such that $2x(kx-4)-x^2+6=0$ has no real roots?	1. Expand and simplify the given quadratic equation: \[ 2x(kx-4) - x^2 + 6 = 0 \] Expanding the terms: \[ 2kx^2 - 8x - x^2 + 6 = 0 \] Combine like terms: \[ (2k-1)x^2 - 8x + 6 = 0 \] 2. Condition for no real roots: A quadratic equation $ax^2 + bx + c = 0$ has no real roots if its discriminant $b^2 - 4ac$ is negative. For the equation $(2k-1)x^2 - 8x + 6 = 0$, we identify $a = 2k-1$, $b = -8$, and $c = 6$. 3. Calculate the discriminant and set it to be less than zero: \[ \text{Discriminant} = (-8)^2 - 4(2k-1)(6) \] Simplify the expression: \[ 64 - 4(2k-1)(6) = 64 - (48k - 24) = 88 - 48k \] Set the discriminant to be less than zero: \[ 88 - 48k < 0 \] 4. Solve the inequality for $k$: \[ 88 - 48k < 0 \implies 88 < 48k \implies \frac{88}{48} < k \implies \frac{11}{6} < k \] Since $k$ must be an integer, the smallest integer greater than $\frac{11}{6}$ is $2$. 5. Conclusion: The smallest integral value of $k$ such that the quadratic equation has no real roots is $2$. \[ \boxed{\text{B}} \]	2	deepscale	512
Let $p$ be the largest prime with 2010 digits. What is the smallest positive integer $k$ such that $p^2 - k$ is divisible by 12?		k = 1	deepscale	38,360
Starting at 1:00 p.m., Jorge watched three movies. The first movie was 2 hours and 20 minutes long. He took a 20 minute break and then watched the second movie, which was 1 hour and 45 minutes long. He again took a 20 minute break and then watched the last movie, which was 2 hours and 10 minutes long. At what time did the final movie end?	Starting at 1:00 p.m., Jorge watches a movie that is 2 hours and 20 minutes long. This first movie ends at 3:20 p.m. Then, Jorge takes a 20 minute break. This break ends at 3:40 p.m. Then, Jorge watches a movie that is 1 hour and 45 minutes long. After 20 minutes of this movie, it is 4:00 p.m. and there is still 1 hour and 25 minutes left in the movie. This second movie thus ends at 5:25 p.m. Then, Jorge takes a 20 minute break which ends at 5:45 p.m. Finally, Jorge watches a movie that is 2 hours and 10 minutes long. This final movie ends at 7:55 p.m.	7:55 \text{ p.m.}	deepscale	5,903
Given 365 cards, in which distinct numbers are written. We may ask for any three cards, the order of numbers written in them. Is it always possible to find out the order of all 365 cards by 2000 such questions?		1691	deepscale	12,681
In a certain country, there are 100 senators, each of whom has 4 aides. These senators and aides serve on various committees. A committee may consist either of 5 senators, of 4 senators and 4 aides, or of 2 senators and 12 aides. Every senator serves on 5 committees, and every aide serves on 3 committees. How many committees are there altogether?	If each senator gets a point for every committee on which she serves, and every aide gets $1 / 4$ point for every committee on which he serves, then the 100 senators get 500 points altogether, and the 400 aides get 300 points altogether, for a total of 800 points. On the other hand, each committee contributes 5 points, so there must be $800 / 5=160$ committees.	160	deepscale	4,054
\(\triangle ABC\) is isosceles with base \(AC\). Points \(P\) and \(Q\) are respectively in \(CB\) and \(AB\) and such that \(AC=AP=PQ=QB\). The number of degrees in \(\angle B\) is:	1. Identify the properties of the triangles: Given that $\triangle ABC$ is isosceles with base $AC$, and $AC = AP = PQ = QB$, we can infer that $\triangle APQ$ and $\triangle BQP$ are also isosceles. Additionally, $\triangle APC$ and $\triangle BQC$ are isosceles because $AP = AC$ and $BQ = BC$. 2. Analyze $\triangle BQP$: Since $\triangle BQP$ is isosceles with $PQ = QB$, the base angles are equal. Let $\angle BPQ = \angle BQP = b$. By the Triangle Angle Sum Property, the sum of angles in a triangle is $180^\circ$. Therefore, \[ \angle BPQ + \angle BQP + \angle PQB = 180^\circ \implies b + b + \angle PQB = 180^\circ \implies \angle PQB = 180^\circ - 2b. \] 3. Analyze $\triangle APQ$: Since $\triangle APQ$ is isosceles with $AP = PQ$, the base angles are equal. Let $\angle PAQ = \angle PQA = 2b$ (since $\angle PQA$ is an exterior angle to $\triangle BQP$ and equal to $2b$). Again, using the Triangle Angle Sum Property, \[ \angle PAQ + \angle PQA + \angle QPA = 180^\circ \implies 2b + 2b + \angle QPA = 180^\circ \implies \angle QPA = 180^\circ - 4b. \] 4. Analyze $\triangle APC$: Since $\triangle APC$ is isosceles with $AP = AC$, the base angles are equal. Let $\angle PAC = \angle PCA = b$. Since $\angle APC$ is an exterior angle to $\triangle APQ$, \[ \angle APC = 180^\circ - \angle QPA = 180^\circ - (180^\circ - 4b) = 4b. \] 5. Analyze $\triangle ABC$: Since $\triangle ABC$ is isosceles with $AB = BC$, the base angles are equal. Let $\angle BAC = \angle BCA = 3b$. Using the Triangle Angle Sum Property, \[ \angle BAC + \angle BCA + \angle B = 180^\circ \implies 3b + 3b + b = 180^\circ \implies 7b = 180^\circ. \] 6. Solve for $b$: Solving for $b$, \[ b = \frac{180^\circ}{7} = 25\frac{5}{7}^\circ. \] 7. Conclusion: Therefore, the measure of $\angle B$ in $\triangle ABC$ is $\boxed{25\frac{5}{7}^\circ}$, corresponding to choice $\textbf{(A)}\ 25\frac{5}{7}$.	25\frac{5}{7}	deepscale	2,455
Given that the sum of two numbers is 15 and their product is 36, find the sum of their reciprocals and the sum of their squares.		153	deepscale	23,272
Given $f(x)= \sqrt {3}\sin \dfrac {x}{4}\cos \dfrac {x}{4}+ \cos ^{2} \dfrac {x}{4}+ \dfrac {1}{2}$. (1) Find the period of $f(x)$; (2) In $\triangle ABC$, sides $a$, $b$, and $c$ correspond to angles $A$, $B$, and $C$ respectively, and satisfy $(2a-c)\cos B=b\cos C$, find the value of $f(B)$.		\dfrac{\sqrt{3}}{2} + 1	deepscale	21,109
Find the area of the $MNRK$ trapezoid with the lateral side $RK = 3$ if the distances from the vertices $M$ and $N$ to the line $RK$ are $5$ and $7$ , respectively.		18	deepscale	28,320
An assembly line produces, on average, 85% first grade products. How many products need to be sampled so that, with a probability of 0.997, the deviation of the proportion of first grade products from 0.85 in absolute value does not exceed 0.01?		11475	deepscale	15,545
Find a six-digit number where the first digit is 6 times less than the sum of all the digits to its right, and the second digit is 6 times less than the sum of all the digits to its right.		769999	deepscale	7,409
There are 12 sprinters in an international track event final, including 5 Americans. Gold, silver, and bronze medals are awarded to the first, second, and third place finishers respectively. Determine the number of ways the medals can be awarded if no more than two Americans receive medals.		1260	deepscale	18,860
In an opaque bag, there are four small balls labeled with the Chinese characters "阳", "过", "阳", and "康" respectively. Apart from the characters, the balls are indistinguishable. Before each draw, the balls are thoroughly mixed.<br/>$(1)$ If one ball is randomly drawn from the bag, the probability that the character on the ball is exactly "康" is ________; <br/>$(2)$ If two balls are drawn from the bag by person A, using a list or a tree diagram, find the probability $P$ that one ball has the character "阳" and the other ball has the character "康".		\frac{1}{3}	deepscale	10,589
If the numbers $x$ and $y$ are inversely proportional and when the sum of $x$ and $y$ is 54, $x$ is three times $y$, find the value of $y$ when $x = 5$.		109.35	deepscale	9,584
Given that the roots of the polynomial $81x^3 - 162x^2 + 81x - 8 = 0$ are in arithmetic progression, find the difference between the largest and smallest roots.		\frac{4\sqrt{6}}{9}	deepscale	14,237
If $k$ and $\ell$ are positive 4-digit integers such that $\gcd(k,\ell)=3$, what is the smallest possible value for $\mathop{\text{lcm}}[k,\ell]$?		335{,}670	deepscale	37,916
If $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = a \cdot d - b \cdot c$, what is the value of $\begin{vmatrix} 3 & 4 \\ 1 & 2 \end{vmatrix}$?	Given the determinant formula for a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the determinant is calculated as: \[ \text{det} = ad - bc \] For the specific matrix $\begin{pmatrix} 3 & 4 \\ 1 & 2 \end{pmatrix}$, we identify the elements as: - $a = 3$ - $b = 4$ - $c = 1$ - $d = 2$ Using the determinant formula: 1. Calculate $ad$: \[ ad = 3 \times 2 = 6 \] 2. Calculate $bc$: \[ bc = 4 \times 1 = 4 \] 3. Subtract $bc$ from $ad$ to find the determinant: \[ \text{det} = ad - bc = 6 - 4 = 2 \] Thus, the value of $\begin{pmatrix} 3 & 4 \\ 1 & 2 \end{pmatrix}$ is $2$. $\boxed{\text{E}}$	$2$	deepscale	2,510
Given in $\triangle ABC$, $\tan A$ and $\tan B$ are the two real roots of the equation $x^2 + ax + 4 = 0$: (1) If $a = -8$, find the value of $\tan C$; (2) Find the minimum value of $\tan C$, and specify the corresponding values of $\tan A$ and $\tan B$.		\frac{4}{3}	deepscale	28,715
Given that the sum of the first three terms of a geometric sequence $\{a_n\}$ is $3$ and the sum of the first nine terms is $39$, calculate the value of the sum of the first six terms.		12	deepscale	21,411
Suppose you have $6$ red shirts, $7$ green shirts, $9$ pairs of pants, $10$ blue hats, and $10$ red hats, all distinct. How many outfits can you make consisting of one shirt, one pair of pants, and one hat, if neither the hat nor the pants can match the shirt in color?		1170	deepscale	11,588
50 students from fifth to ninth grade collectively posted 60 photos on Instagram, with each student posting at least one photo. All students in the same grade (parallel) posted an equal number of photos, while students from different grades posted different numbers of photos. How many students posted exactly one photo?		46	deepscale	9,597
What is equivalent to $\sqrt{\frac{x}{1-\frac{x-1}{x}}}$ when $x < 0$?	To solve the problem, we first simplify the expression inside the square root: \[ \sqrt{\frac{x}{1-\frac{x-1}{x}}} \] 1. Simplify the denominator: \[ 1 - \frac{x-1}{x} = 1 - \left(\frac{x}{x} - \frac{1}{x}\right) = 1 - 1 + \frac{1}{x} = \frac{1}{x} \] 2. Substitute back into the original expression: \[ \sqrt{\frac{x}{\frac{1}{x}}} = \sqrt{x^2} = \|x\| \] Since $x < 0$, $\|x\| = -x$. 3. Now, we compare this result with the given options: - $\mathrm{(A) \ } -x$ - $\mathrm{(B) \ } x$ - $\mathrm{(C) \ } 1$ - $\mathrm{(D) \ } \sqrt{\frac{x}{2}}$ - $\mathrm{(E) \ } x\sqrt{-1}$ Given that $x < 0$, $\|x\| = -x$ matches with option $\mathrm{(A)}$. Thus, the equivalent expression for $\sqrt{\frac{x}{1-\frac{x-1}{x}}}$ when $x < 0$ is $\boxed{\textbf{(A)} -x}$.	-x	deepscale	284
Compute the number of ways there are to assemble 2 red unit cubes and 25 white unit cubes into a $3 \times 3 \times 3$ cube such that red is visible on exactly 4 faces of the larger cube. (Rotations and reflections are considered distinct.)	We do casework on the two red unit cubes; they can either be in a corner, an edge, or the center of the face. - If they are both in a corner, they must be adjacent - for each configuration, this corresponds to an edge, of which there are 12. - If one is in the corner and the other is at an edge, we have 8 choices to place the corner. For the edge, the red edge square has to go on the boundary of the faces touching the red corner square, and there are six places here. Thus, we get $8 \cdot 6=48$ configurations. - If one is a corner and the other is in the center of a face, we again have 8 choices for the corner and 3 choices for the center face (the faces not touching the red corner). This gives $8 \cdot 3=24$ options. - We have now completed the cases with a red corner square! Now suppose we have two edges: If we chose in order, we have 12 choices for the first cube. For the second cube, we must place the edge so it covers two new faces, and thus we have five choices. Since we could have picked these edges in either order, we divide by two to avoid overcounting, and we have $12 \cdot 5 / 2=30$ in this case. Now, since edges and faces only cover at most 2 and 1 face respectively, no other configuration works. Thus we have all the cases, and we add: $12+48+24+30=114$.	114	deepscale	3,783
The equations $x^3 + Ax + 10 = 0$ and $x^3 + Bx^2 + 50 = 0$ have two roots in common. Then the product of these common roots can be expressed in the form $a \sqrt[b]{c},$ where $a,$ $b,$ and $c$ are positive integers, when simplified. Find $a + b + c.$		12	deepscale	36,944
Let $\pi$ be a uniformly random permutation of the set $\{1,2, \ldots, 100\}$. The probability that $\pi^{20}(20)=$ 20 and $\pi^{21}(21)=21$ can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$. (Here, $\pi^{k}$ means $\pi$ iterated $k$ times.)	We look at the cycles formed by $\pi$ Let $\operatorname{ord}_{\pi}(n)$ denote the smallest $m$ such that $\pi^{m}(n)=n$. In particular, the condition implies that $\operatorname{ord}_{\pi}(20) \mid 20$ and $\operatorname{ord}_{\pi}(21) \mid 21$. Claim 1. 20 and 21 cannot be in the same cycle. Proof. If 20 and 21 were in the same cycle, then $x=\operatorname{ord}_{\pi}(20)=\operatorname{ord}_{\pi}(21)$ for some $x$. Then $x>1$ since the cycle contains both 20 and 21, but $x\|20, x\| 21$ implies $x=1$, a contradiction. Claim 2. The probability that $a=\operatorname{ord}_{\pi}(20), b=\operatorname{ord}_{\pi}(21)$ for some fixed $a, b$ such that $a+b \leq 100$ is $\frac{1}{99 \cdot 100}$. Proof. We can just count these permutations. We first choose $a-1$ elements of $[100] \backslash\{20,21\}$ to be in the cycle of 20, then we similarly choose $b-1$ to be in the cycle of 21. We then have $(a-1)$! ways to reorder within the cycle of $20,(b-1)$! ways to reorder within the cycle of 21, and $(100-a-b)$! ways to permute the remaining elements. The total number of ways is just $$\frac{98!}{(a-1)!(b-1)!(100-a-b)!} \cdot(a-1)!(b-1)!(100-a-b)!=98!$$ so the probability this happens is just $\frac{98!}{100!}=\frac{1}{9900}$. Now, since $\operatorname{ord}_{\pi}(20) \mid 20$ and $\operatorname{ord}_{\pi}(21) \mid 21$, we have 6 possible values for $\operatorname{ord}_{\pi}(20)$ and 4 for $\operatorname{ord}_{\pi}(21)$, so in total we have a $\frac{6 \cdot 4}{9900}=\frac{2}{825}$ probability that the condition is satisfied.	1025	deepscale	3,986
Harry Potter is creating an enhanced magical potion called "Elixir of Life" (this is a very potent sleeping potion composed of powdered daffodil root and wormwood infusion. The concentration of the "Elixir of Life" is the percentage of daffodil root powder in the entire potion). He first adds a certain amount of wormwood infusion to the regular "Elixir of Life," making its concentration $9 \%$. If he then adds the same amount of daffodil root powder, the concentration of the "Elixir of Life" becomes $23 \%$. What is the concentration of the regular "Elixir of Life"? $\qquad$ $\%$.		11	deepscale	7,959
On an infinite tape, numbers are written in a row. The first number is one, and each subsequent number is obtained from the previous one by adding the smallest nonzero digit of its decimal representation. How many digits are in the decimal representation of the number that is in the $9 \cdot 1000^{1000}$-th position in this row?		3001	deepscale	15,962
Find the ordered pair $(x,y)$ if \begin{align} x+y&=(5-x)+(5-y),\\ x-y&=(x-1)+(y-1). \end{align}		(4,1)	deepscale	33,447
In triangle \( ABC \) with sides \( AB = 8 \), \( AC = 4 \), \( BC = 6 \), the angle bisector \( AK \) is drawn, and on the side \( AC \) a point \( M \) is marked such that \( AM : CM = 3 : 1 \). Point \( N \) is the intersection point of \( AK \) and \( BM \). Find \( AN \).		\frac{18\sqrt{6}}{11}	deepscale	11,306
Given that the function $f(x)$ defined on $\mathbb{R}$ satisfies $f(4)=2-\sqrt{3}$, and for any $x$, $f(x+2)=\frac{1}{-f(x)}$, find $f(2018)$.		-2-\sqrt{3}	deepscale	22,534
Given vectors $\overrightarrow{m}=(\sin x,-1)$ and $\overrightarrow{n}=(\sqrt{3}\cos x,-\frac{1}{2})$, and the function $f(x)=(\overrightarrow{m}+\overrightarrow{n})\cdot\overrightarrow{m}$. - (I) Find the interval where $f(x)$ is monotonically decreasing; - (II) Given $a$, $b$, and $c$ are respectively the sides opposite to angles $A$, $B$, and $C$ in $\triangle ABC$, with $A$ being an acute angle, $a=2\sqrt{3}$, $c=4$, and $f(A)$ is exactly the maximum value of $f(x)$ on the interval $\left[0, \frac{\pi}{2}\right]$, find $A$, $b$, and the area $S$ of $\triangle ABC$.		2\sqrt{3}	deepscale	21,696
Find the largest positive integer $N$ so that the number of integers in the set $\{1,2,\dots,N\}$ which are divisible by 3 is equal to the number of integers which are divisible by 5 or 7 (or both).		65	deepscale	18,881
If \( p \) and \( q \) are positive integers, \(\max (p, q)\) is the maximum of \( p \) and \( q \) and \(\min (p, q)\) is the minimum of \( p \) and \( q \). For example, \(\max (30,40)=40\) and \(\min (30,40)=30\). Also, \(\max (30,30)=30\) and \(\min (30,30)=30\). Determine the number of ordered pairs \((x, y)\) that satisfy the equation $$ \max (60, \min (x, y))=\min (\max (60, x), y) $$ where \(x\) and \(y\) are positive integers with \(x \leq 100\) and \(y \leq 100\).		4100	deepscale	14,037
How many 5-digit positive numbers contain only odd numbers and have at least one pair of consecutive digits whose sum is 10?		1845	deepscale	17,656
How many three-digit numbers are there in which each digit is greater than the digit to its right?		84	deepscale	11,520
In the geometric sequence $\{a_n\}$ with a common ratio greater than $1$, $a_3a_7=72$, $a_2+a_8=27$, calculate $a_{12}$.		96	deepscale	8,980
There are two hourglasses - one for 7 minutes and one for 11 minutes. An egg needs to be boiled for 15 minutes. How can you measure this amount of time using the hourglasses?		15	deepscale	11,486
Point $(x,y)$ is chosen randomly from the rectangular region with vertices at $(0,0)$, $(3036,0)$, $(3036,3037)$, and $(0,3037)$. What is the probability that $x > 3y$? Express your answer as a common fraction.		\frac{506}{3037}	deepscale	16,986
The average age of 8 people in Room C is 35. The average age of 6 people in Room D is 30. Calculate the average age of all people when the two groups are combined.		\frac{460}{14}	deepscale	29,004
How many $4$-digit positive integers (that is, integers between $1000$ and $9999$, inclusive) having only even digits are divisible by $5?$	1. Identify the range and conditions: We are looking for 4-digit integers between 1000 and 9999, inclusive, that have only even digits and are divisible by 5. 2. Digits must be even: The possible even digits are 0, 2, 4, 6, and 8. 3. Divisibility by 5: For a number to be divisible by 5, its units digit must be either 0 or 5. Since we are restricted to even digits, the units digit must be 0. 4. Thousands digit: The thousands digit must be one of 2, 4, 6, or 8, as it cannot be 0 (otherwise, it would not be a 4-digit number). This gives us 4 choices for the thousands digit. 5. Middle two digits: Each of the middle two digits (hundreds and tens) can independently be any of the 5 even digits (0, 2, 4, 6, 8). This gives us 5 choices for each of these digits. 6. Calculate the total number of such numbers: - 1 choice for the units digit (0). - 5 choices for the hundreds digit. - 5 choices for the tens digit. - 4 choices for the thousands digit. Multiplying these choices together gives the total number of valid numbers: \[ 1 \times 5 \times 5 \times 4 = 100 \] 7. Conclusion: There are 100 such numbers that meet all the conditions. Thus, the number of 4-digit positive integers having only even digits and divisible by 5 is $\boxed{\textbf{(B) } 100}$.	100	deepscale	1,062

Daniel worked for 50 hours per week for 10 weeks during the summer, earning \$6000. If he wishes to earn an additional \$8000 during the school year which lasts for 40 weeks, how many fewer hours per week must he work compared to the summer if he receives the same hourly wage?

33.33

deepscale

20,926

In the Cartesian coordinate system $xOy$, with the origin as the pole and the positive half-axis of the $x$-axis as the polar axis, the polar coordinate equation of the curve $C_{1}$ is $\rho \cos \theta = 4$. $(1)$ Let $M$ be a moving point on the curve $C_{1}$, point $P$ lies on the line segment $OM$, and satisfies $|OP| \cdot |OM| = 16$. Find the rectangular coordinate equation of the locus $C_{2}$ of point $P$. $(2)$ Suppose the polar coordinates of point $A$ are $({2, \frac{π}{3}})$, point $B$ lies on the curve $C_{2}$. Find the maximum value of the area of $\triangle OAB$.

2 + \sqrt{3}

deepscale

22,528

Compute $104 \times 96$.

9984

deepscale

21,496

Let $a, b \in \mathbb{R}$. If the line $l: ax+y-7=0$ is transformed by the matrix $A= \begin{bmatrix} 3 & 0 \\ -1 & b\end{bmatrix}$, and the resulting line is $l′: 9x+y-91=0$. Find the values of the real numbers $a$ and $b$.

13

deepscale

19,414

Find the number of ordered quadruples $(a, b, c, d)$ where each of $a, b, c,$ and $d$ are (not necessarily distinct) elements of $\{1, 2, 3, 4, 5, 6, 7\}$ and $3abc + 4abd + 5bcd$ is even. For example, $(2, 2, 5, 1)$ and $(3, 1, 4, 6)$ satisfy the conditions.

2017

deepscale

30,281

John recently bought a used car for $\$5000$ for his pizza delivery job. He gets $\$10$ for each pizza he delivers, but he has to spend $\$3$ on gas for each pizza he delivers. What is the minimum whole number of pizzas John must deliver in order to earn back the money he spent on the car he bought?

715

deepscale

38,720

Let $x$, $y$, and $z$ be real numbers greater than $1$, and let $z$ be the geometric mean of $x$ and $y$. The minimum value of $\frac{\log z}{4\log x} + \frac{\log z}{\log y}$ is \_\_\_\_\_\_.

\frac{9}{8}

deepscale

21,786

In a class of 120 students, the teacher recorded the following scores for an exam. Calculate the average score for the class. \begin{tabular}{|c|c|} \multicolumn{2}{c}{}\\\hline \textbf{Score (\%)}&\textbf{Number of Students}\\\hline 95&12\\\hline 85&24\\\hline 75&30\\\hline 65&20\\\hline 55&18\\\hline 45&10\\\hline 35&6\\\hline \end{tabular}

69.83

deepscale

22,579

Given $a$ and $b$ are the roots of the equation $x^2-2cx-5d = 0$, and $c$ and $d$ are the roots of the equation $x^2-2ax-5b=0$, where $a,b,c,d$ are distinct real numbers, find $a+b+c+d$.

30

deepscale

25,928

If $5(\cos a + \cos b) + 4(\cos a \cos b + 1) = 0,$ then find all possible values of \[\tan \frac{a}{2} \tan \frac{b}{2}.\]Enter all the possible values, separated by commas.

3,-3

deepscale

39,697

Determine the following number: \[ \frac{12346 \cdot 24689 \cdot 37033 + 12347 \cdot 37034}{12345^{2}} \]

74072

deepscale

15,979

Triangles $\triangle ABC$ and $\triangle DEC$ share side $BC$. Given that $AB = 7\ \text{cm}$, $AC = 15\ \text{cm}$, $EC = 9\ \text{cm}$, and $BD = 26\ \text{cm}$, what is the least possible integral number of centimeters in $BC$?

17

deepscale

27,301

Determine the number of ways to arrange the letters of the word PERSEVERANCE.

19,958,400

deepscale

26,816

How many four-digit numbers greater than 2999 can be formed such that the product of the middle two digits exceeds 5?

4970

deepscale

34,887

A four-meter gas pipe has rusted in two places. Determine the probability that all three resulting pieces can be used as connections to gas stoves, given that regulations require a stove to be at least 1 meter away from the main gas pipeline.

1/8

deepscale

7,486

Let $S$ be the set of all nonzero real numbers. Let $f : S \to S$ be a function such that \[f(x) + f(y) = f(xyf(x + y))\]for all $x,$ $y \in S$ such that $x + y \neq 0.$ Let $n$ be the number of possible values of $f(4),$ and let $s$ be the sum of all possible values of $f(4).$ Find $n \times s.$

\frac{1}{4}

deepscale

36,346

Given $a\ln a=be^{b}$, where $b > 0$, find the maximum value of $\frac{b}{{{a^2}}}$

\frac{1}{2e}

deepscale

11,322

Define $a \Delta b = a^2 -b $. What is the value of $ (2^{4 \Delta13})\Delta(3^{3\Delta5})$

-17

deepscale

33,944

Determine the maximal size of a set of positive integers with the following properties: 1. The integers consist of digits from the set {1,2,3,4,5,6}. 2. No digit occurs more than once in the same integer. 3. The digits in each integer are in increasing order. 4. Any two integers have at least one digit in common (possibly at different positions). 5. There is no digit which appears in all the integers.

32

deepscale

13,386

In the trapezoid $ABCD$, the lengths of the bases $AD = 24 \text{ cm}$, $BC = 8 \text{ cm}$, and the diagonals $AC = 13 \text{ cm}$, $BD = 5\sqrt{17} \text{ cm}$ are known. Calculate the area of the trapezoid.

80

deepscale

13,548

Construct a square $A B C D$ with side length $6 \text{ cm}$. Construct a line $p$ parallel to the diagonal $A C$ passing through point $D$. Construct a rectangle $A C E F$ such that vertices $E$ and $F$ lie on the line $p$. Using the given information, calculate the area of rectangle $A C E F$.

36

deepscale

8,076

Let $ a, b \in \mathbb{N}$ with $ 1 \leq a \leq b,$ and $ M \equal{} \left[\frac {a \plus{} b}{2} \right].$ Define a function $ f: \mathbb{Z} \mapsto \mathbb{Z}$ by \[ f(n) \equal{} \begin{cases} n \plus{} a, & \text{if } n \leq M, \\ n \minus{} b, & \text{if } n >M. \end{cases} \] Let $ f^1(n) \equal{} f(n),$ $ f_{i \plus{} 1}(n) \equal{} f(f^i(n)),$ $ i \equal{} 1, 2, \ldots$ Find the smallest natural number $ k$ such that $ f^k(0) \equal{} 0.$

Let $ a, b \in \mathbb{N} $ with $ 1 \leq a \leq b $, and let $ M = \left\lfloor \frac{a + b}{2} \right\rfloor $. The function $ f: \mathbb{Z} \to \mathbb{Z} $ is defined as: \[ f(n) = \begin{cases} n + a, & \text{if } n \leq M, \\ n - b, & \text{if } n > M. \end{cases} \] We are required to find the smallest natural number $ k $ such that $ f^k(0) = 0 $, where $ f^1(n) = f(n) $ and for $ i \geq 1 $, $ f_{i+1}(n) = f(f^i(n)) $. ### Step-by-step Analysis: 1. **Initial Application of $ f $:** Start with $ n = 0 $. Since $ 0 \leq M $ (as $ M \geq 0 $), we apply the first case of the function: \[ f(0) = 0 + a = a. \] 2. **Subsequent Applications of $ f $:** Next, apply $ f $ to $ a $: \[ f(a) = a + a = 2a \quad \text{(since } a \leq M \text{ if } 2a \leq M \text{, which is usually true for small } a). \] Continue applying $ f $ on subsequent results until $ n > M $. As soon as $ n > M $ and since each application adds $ a $ until the threshold is reached, beyond this point, $ n $ will start decreasing by $ b $. 3. **Loop Detection:** The behavior of $ f $ may create a cycle due to the periodic nature of additions of $ a $ and subtractions of $ b $. To return precisely to zero, a full cycle must have a net effect of zero. This means the total sum of increases and decreases (by adding $ a $ and subtracting $ b $) must cancel out over some iterations. 4. **Formulating the Cycle Condition:** We need $ f^k(0) = 0 $, meaning: \[ j \cdot a = l \cdot b \quad \text{for some integers } j, l, \] or equivalently, the number of times $ a $ is added must equal multiples of $ b $. Hence, the smallest $ k $ is connected to multiples that sum to zero. 5. **Using $ \gcd(a, b) $:** The smallest $ k $ is determined by the period after which multiples of $ a $ transform into some form of $ b $-cycle to return to zero. For this, the period length is directly linked to the least common multiple cycles determined by: \[ k = \frac{a + b}{\gcd(a, b)}, \] where $ \gcd(a, b) $ ensures the minimal integer multiple cycle to zero. Thus, the smallest $ k $ satisfying $ f^k(0) = 0 $ is: \[ \boxed{\frac{a + b}{\gcd(a, b)}}. \]

\frac {a + b}{\gcd(a,b)}

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6,356

The year 2009 has a unique property: by rearranging the digits of the number 2009, it is impossible to form a smaller four-digit number (numbers do not start with zero). In which future year will this property first repeat again?

2022

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10,026

Given the ellipse $ x^{2}+ \frac {y^{2}}{b^{2}+1}=1(b > 0) $ has an eccentricity of $ \frac {\sqrt {10}}{10} $, determine the value of $ b $.

\frac{1}{3}

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12,252

How many even integers between 4000 and 7000 have four different digits?

The thousands digit is $\in \{4,5,6\}$. Case $1$: Thousands digit is even $4, 6$, two possibilities, then there are only $\frac{10}{2} - 1 = 4$ possibilities for the units digit. This leaves $8$ possible digits for the hundreds and $7$ for the tens places, yielding a total of $2 \cdot 8 \cdot 7 \cdot 4 = 448$. Case $2$: Thousands digit is odd $5$, one possibility, then there are $5$ choices for the units digit, with $8$ digits for the hundreds and $7$ for the tens place. This gives $1 \cdot 8 \cdot 7 \cdot 5= 280$ possibilities. Together, the solution is $448 + 280 = \boxed{728}$.

728

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6,566

At a hypothetical school, there are three departments in the faculty of sciences: biology, physics and chemistry. Each department has three male and one female professor. A committee of six professors is to be formed containing three men and three women, and each department must be represented by two of its members. Every committee must include at least one woman from the biology department. Find the number of possible committees that can be formed subject to these requirements.

27

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16,268

There are $20$ students participating in an after-school program offering classes in yoga, bridge, and painting. Each student must take at least one of these three classes, but may take two or all three. There are $10$ students taking yoga, $13$ taking bridge, and $9$ taking painting. There are $9$ students taking at least two classes. How many students are taking all three classes?

Let's define the variables for the Venn Diagram: - $a$: Number of students taking exactly Bridge and Yoga. - $b$: Number of students taking exactly Bridge and Painting. - $c$: Number of students taking all three classes (Yoga, Bridge, and Painting). - $d$: Number of students taking exactly Yoga and Painting. We are given: - Total students taking at least two classes: $9$. - Total students taking Yoga: $10$. - Total students taking Bridge: $13$. - Total students taking Painting: $9$. From the information, we know: \[ a + b + c + d = 9 \] This equation represents the total number of students taking at least two classes. #### Calculating students taking only one class: 1. **Students taking only Bridge**: \[ \text{Total Bridge} - (a + b + c) = 13 - (a + b + c) \] Using $a + b + c + d = 9$, we find $a + b + c = 9 - d$. Thus: \[ 13 - (9 - d) = 4 + d \] 2. **Students taking only Yoga**: \[ \text{Total Yoga} - (a + c + d) = 10 - (a + c + d) \] Using $a + b + c + d = 9$, we find $a + c + d = 9 - b$. Thus: \[ 10 - (9 - b) = 1 + b \] 3. **Students taking only Painting**: \[ \text{Total Painting} - (b + c + d) = 9 - (b + c + d) \] Using $a + b + c + d = 9$, we find $b + c + d = 9 - a$. Thus: \[ 9 - (9 - a) = a \] #### Total students taking exactly one class: Adding the equations for students taking only one class: \[ (4 + d) + (1 + b) + a = 11 \] \[ 5 + a + b + d = 11 \] Using $a + b + c + d = 9$, substitute $9 - c$ for $a + b + d$: \[ 5 + (9 - c) = 11 \] \[ 14 - c = 11 \] \[ c = 3 \] Thus, the number of students taking all three classes is $\boxed{\textbf{(C)}\ 3}$.

3

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2,772

Complex numbers $ p, q, r $ form an equilateral triangle with a side length of 24 in the complex plane. If $ |p + q + r| = 48 $, find $ |pq + pr + qr| $.

768

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20,753

A point $ M $ is chosen on the diameter $ AB $. Points $ C $ and $ D $, lying on the circumference on one side of $ AB $, are chosen such that $\angle AMC=\angle BMD=30^{\circ}$. Find the diameter of the circle given that $ CD=12 $.

8\sqrt{3}

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28,506

For a finite sequence $P = \left(p_1, p_2, \cdots, p_n\right)$, the Cesaro sum (named after the mathematician Cesaro) is defined as $\frac{1}{n}(S_1 + S_2 + \cdots + S_n)$, where $S_k = p_1 + p_2 + \cdots + p_k$ for $1 \leq k \leq n$. If a sequence $\left(p_1, p_2, \cdots, p_{99}\right)$ of 99 terms has a Cesaro sum of 1000, then the Cesaro sum of the 100-term sequence $\left(1, p_1, p_2, \cdots, p_{99}\right)$ is ( ).

991

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16,270

What is the sum of all positive integer solutions less than or equal to $30$ to the congruence $15(3x-4) \equiv 30 \pmod{10}$?

240

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17,479

Given three forces in space, $\overrightarrow {F_{1}}$, $\overrightarrow {F_{2}}$, and $\overrightarrow {F_{3}}$, each with a magnitude of 2, and the angle between any two of them is 60°, the magnitude of their resultant force $\overrightarrow {F}$ is ______.

2 \sqrt {6}

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19,375

Let $a$ and $b$ be angles such that $\sin (a + b) = \frac{3}{4}$ and $\sin (a - b) = \frac{1}{2}.$ Find $\frac{\tan a}{\tan b}.$

5

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39,749

In the interval $[-6, 6]$, an element $x_0$ is arbitrarily chosen. If the slope of the tangent line to the parabola $y = x^2$ at $x = x_0$ has an angle of inclination $\alpha$, find the probability that $\alpha \in \left[ \frac{\pi}{4}, \frac{3\pi}{4} \right]$.

\frac{11}{12}

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17,325

Given that $ x $ and $ y $ are positive integers such that $ 56 \leq x + y \leq 59 $ and $ 0.9 < \frac{x}{y} < 0.91 $, find the value of $ y^2 - x^2 $.

177

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11,389

Square $IJKL$ is contained within square $WXYZ$ such that each side of $IJKL$ can be extended to pass through a vertex of $WXYZ$. The side length of square $WXYZ$ is $\sqrt{98}$, and $WI = 2$. What is the area of the inner square $IJKL$? A) $62$ B) $98 - 4\sqrt{94}$ C) $94 - 4\sqrt{94}$ D) $98$ E) $100$

98 - 4\sqrt{94}

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13,118

The triangle $ABC$ is isosceles with $AB=BC$ . The point F on the side $[BC]$ and the point $D$ on the side $AC$ are the feets of the the internals bisectors drawn from $A$ and altitude drawn from $B$ respectively so that $AF=2BD$ . Fine the measure of the angle $ABC$ .

36

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24,718

Given a "double-single" number is a three-digit number made up of two identical digits followed by a different digit, calculate the number of double-single numbers between 100 and 1000.

81

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20,331

Find the number of positive integer solutions to $n^{x}+n^{y}=n^{z}$ with $n^{z}<2001$.

If $n=1$, the relation can not hold, so assume otherwise. If $x>y$, the left hand side factors as $n^{y}\left(n^{x-y}+1\right)$ so $n^{x-y}+1$ is a power of $n$. But it leaves a remainder of 1 when divided by $n$ and is greater than 1, a contradiction. We reach a similar contradiction if $y>x$. So $y=x$ and $2 n^{x}=n^{z}$, so 2 is a power of $n$ and $n=2$. So all solutions are of the form $2^{x}+2^{x}=2^{x+1}$, which holds for all $x$. $2^{x+1}<2001$ implies $x<11$, so there are 10 solutions.

10

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3,646

Let $f(r)=\sum_{j=2}^{2008} \frac{1}{j^{r}}=\frac{1}{2^{r}}+\frac{1}{3^{r}}+\cdots+\frac{1}{2008^{r}}$. Find $\sum_{k=2}^{\infty} f(k)$.

We change the order of summation: $$\sum_{k=2}^{\infty} \sum_{j=2}^{2008} \frac{1}{j^{k}}=\sum_{j=2}^{2008} \sum_{k=2}^{\infty} \frac{1}{j^{k}}=\sum_{j=2}^{2008} \frac{1}{j^{2}\left(1-\frac{1}{j}\right)}=\sum_{j=2}^{2008} \frac{1}{j(j-1)}=\sum_{j=2}^{2008}\left(\frac{1}{j-1}-\frac{1}{j}\right)=1-\frac{1}{2008}=\frac{2007}{2008}$$

\frac{2007}{2008}

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3,291

What is the largest integer that must divide the product of any $4$ consecutive integers?

24

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38,395

Let $a$ and $b$ each be chosen at random from the set $\{1, 2, 3, \ldots, 40\}$. Additionally, let $c$ and $d$ also be chosen at random from the same set. Calculate the probability that the integer $2^c + 5^d + 3^a + 7^b$ has a units digit of $8$.

\frac{3}{16}

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27,133

Squares $ABCD$ and $EFGH$ are congruent, $AB=10$, and $G$ is the center of square $ABCD$. The area of the region in the plane covered by these squares is

1. **Identify the given information**: We are given two congruent squares $ABCD$ and $EFGH$ with side length $AB = 10$. The point $G$ is the center of square $ABCD$. 2. **Calculate the area of each square**: - The area of square $ABCD$ is $AB^2 = 10^2 = 100$. - Since $EFGH$ is congruent to $ABCD$, its area is also $100$. 3. **Calculate the total area if there were no overlap**: - If there were no overlap, the total area would be $100 + 100 = 200$. 4. **Determine the overlap**: - Since $G$ is the center of $ABCD$, it lies at the midpoint of the diagonal of $ABCD$. - The length of the diagonal of $ABCD$ is $AB \sqrt{2} = 10\sqrt{2}$. - Therefore, $BG$, being half the diagonal, is $5\sqrt{2}$. 5. **Analyze triangle $ABG$**: - $\triangle ABG$ is formed by the line segment $AB$, $BG$, and $AG$. - Since $G$ is the midpoint of the diagonal, and the diagonals of a square bisect each other at right angles, $\triangle ABG$ is a right isosceles triangle with $\angle BGA = 90^\circ$. 6. **Calculate the area of $\triangle ABG$**: - The legs of $\triangle ABG$ are each $5\sqrt{2}$. - The area of $\triangle ABG$ is $\frac{1}{2} \times \text{leg} \times \text{leg} = \frac{1}{2} \times (5\sqrt{2})^2 = \frac{1}{2} \times 50 = 25$. 7. **Calculate the area of the region covered by the squares**: - Subtract the area of $\triangle ABG$ from the total area of the two squares to account for the overlap. - The area of the region is $200 - 25 = 175$. Thus, the area of the region in the plane covered by these squares is $\boxed{\textbf{(E)}\ 175}$.

175

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633

Given that the last initial of Mr. and Mrs. Alpha's baby's monogram is 'A', determine the number of possible monograms in alphabetical order with no letter repeated.

300

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30,426

Given $\sin\alpha= \frac {2 \sqrt {2}}{3}$, $\cos(\alpha+\beta)=- \frac {1}{3}$, and $\alpha, \beta\in(0, \frac {\pi}{2})$, determine the value of $\sin(\alpha-\beta)$.

\frac {10 \sqrt {2}}{27}

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17,026

In acute triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $b=2$, $B= \frac{π}{3}$, and $c\sin A= \sqrt{3}a\cos C$, find the area of $\triangle ABC$.

\sqrt{3}

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17,111

If $ A $ is a positive integer such that $ \frac{1}{1 \times 3} + \frac{1}{3 \times 5} + \cdots + \frac{1}{(A+1)(A+3)} = \frac{12}{25} $, find the value of $ A $.

22

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11,997

The sequence $\{a_n\}$ has consecutive terms $a_n$ and $a_{n+1}$ as the roots of the equation $x^2 - c_n x + \left(\frac{1}{3}\right)^n = 0$, with initial term $a_1 = 2$. Find the sum of the infinite series $c_1, c_2, \cdots, c_n, \cdots$.

\frac{9}{2}

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15,451

Given two circles $O\_1$: $(x-2)^2 + y^2 = 16$ and $O\_2$: $x^2 + y^2 = r^2 (0 < r < 2)$, a moving circle $M$ is tangent to both circles $O\_1$ and $O\_2$. The locus of the center of the moving circle $M$ forms two ellipses, with eccentricities $e\_1$ and $e\_2 (e\_1 > e\_2)$. Find the minimum value of $e\_1 + 2e\_2$.

\frac{3+2\sqrt{2}}{4}

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26,482

Two cars, Car A and Car B, start from point A and point B, respectively, and move towards each other simultaneously. They meet after 3 hours, after which Car A turns back towards point A and Car B continues moving forward. Once Car A reaches point A, it turns towards point B again and meets Car B half an hour later. How many minutes does Car B take to travel from point A to point B?

432

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12,928

Suppose $\cos S = 0.5$ in the diagram below. What is $ST$? [asy] pair P,S,T; P = (0,0); S = (6,0); T = (0,6*tan(acos(0.5))); draw(P--S--T--P); draw(rightanglemark(S,P,T,18)); label("$P$",P,SW); label("$S$",S,SE); label("$T$",T,N); label("$10$",S/2,S); [/asy]

20

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23,494

Let $f(x)$ have a domain of $R$, $f(x+1)$ be an odd function, and $f(x+2)$ be an even function. When $x\in [1,2]$, $f(x)=ax^{2}+b$. If $f(0)+f(3)=6$, then calculate the value of $f\left(\frac{9}{2}\right)$.

\frac{5}{2}

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31,392

An ant starts at the origin of a coordinate plane. Each minute, it either walks one unit to the right or one unit up, but it will never move in the same direction more than twice in the row. In how many different ways can it get to the point $(5,5)$ ?

We can change the ant's sequence of moves to a sequence $a_{1}, a_{2}, \ldots, a_{10}$, with $a_{i}=0$ if the $i$-th step is up, and $a_{i}=1$ if the $i$-th step is right. We define a subsequence of moves $a_{i}, a_{i+1}, \ldots, a_{j}$, ( $i \leq j$ ) as an up run if all terms of the subsequence are equal to 0 , and $a_{i-1}$ and $a_{j+1}$ either do not exist or are not equal to 0 , and define a right run similarly. In a sequence of moves, up runs and right runs alternate, so the number of up rights can differ from the number of right runs by at most one. Now let $f(n)$ denote the number of sequences $a_{1}, a_{2}, \ldots, a_{n}$ where $a_{i} \in\{1,2\}$ for $1 \leq i \leq n$, and $a_{1}+a_{2}+\cdots+a_{n}=5$. (In essence, we are splitting the possible 5 up moves into up runs, and we are doing the same with the right moves). We can easily compute that $f(3)=3, f(4)=4, f(5)=1$, and $f(n)=0$ otherwise. For each possible pair of numbers of up runs and right runs, we have two choices of which type of run is first. Our answer is then $2\left(f(3)^{2}+f(3) f(4)+f(4)^{2}+f(4) f(5)+f(5)^{2}\right)=2(9+12+16+4+1)=84$.

84

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5,178

Find the shortest distance from the line $3 x+4 y=25$ to the circle $x^{2}+y^{2}=6 x-8 y$.

The circle is $(x-3)^{2}+(y+4)^{2}=5^{2}$. The center $(3,-4)$ is a distance of $$ \frac{|3 \cdot 3+4 \cdot-4-25|}{\sqrt{3^{2}+4^{2}}}=\frac{32}{5} $$ from the line, so we subtract 5 for the radius of the circle and get $7 / 5$.

7 / 5

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3,332

Maximum difference in weights of two bags is achieved by taking the largest and smallest possible values of the two different brands. Given the weights of three brands of flour are $\left(25\pm 0.1\right)kg$, $\left(25\pm 0.2\right)kg$, and $\left(25\pm 0.3\right)kg$, calculate the maximum possible difference in weights.

0.6

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24,483

Let $a, b, c$ be positive real numbers such that $a \leq b \leq c \leq 2 a$. Find the maximum possible value of $$\frac{b}{a}+\frac{c}{b}+\frac{a}{c}$$

Fix the values of $b, c$. By inspecting the graph of $$f(x)=\frac{b}{x}+\frac{x}{c}$$ we see that on any interval the graph attains its maximum at an endpoint. This argument applies when we fix any two variables, so it suffices to check boundary cases in which $b=a$ or $b=c$, and $c=b$ or $c=2 a$. All pairs of these conditions determine the ratio between $a, b, c$, except $b=c$ and $c=b$, in which case the boundary condition on $a$ tells us that $a=b$ or $2 a=b=c$. In summary, these cases are $$(a, b, c) \in\{(a, a, a),(a, a, 2 a),(a, 2 a, 2 a)\}$$ The largest value achieved from any of these three is $\frac{7}{2}$.

\frac{7}{2}

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4,754

Determine the set of all real numbers $p$ for which the polynomial $Q(x)=x^{3}+p x^{2}-p x-1$ has three distinct real roots.

First, we note that $x^{3}+p x^{2}-p x-1=(x-1)(x^{2}+(p+1)x+1)$. Hence, $x^{2}+(p+1)x+1$ has two distinct roots. Consequently, the discriminant of this equation must be positive, so $(p+1)^{2}-4>0$, so either $p>1$ or $p<-3$. However, the problem specifies that the quadratic must have distinct roots (since the original cubic has distinct roots), so to finish, we need to check that 1 is not a double root-we will do this by checking that 1 is not a root of $x^{2}+(p+1)x+1$ for any value $p$ in our range. But this is clear, since $1+(p+1)+1=0 \Rightarrow p=-3$, which is not in the aforementioned range. Thus, our answer is all $p$ satisfying $p>1$ or $p<-3$.

p>1 \text{ and } p<-3

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4,726

Let $n$ be the integer such that $0 \le n < 31$ and $3n \equiv 1 \pmod{31}$. What is $\left(2^n\right)^3 - 2 \pmod{31}$? Express your answer as an integer from $0$ to $30$, inclusive.

6

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37,845

What is the number of units in the distance between $(2,5)$ and $(-6,-1)$?

10

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33,281

Let $\mathrm {Q}$ be the product of the roots of $z^8+z^6+z^4+z^3+z+1=0$ that have a positive imaginary part, and suppose that $\mathrm {Q}=s(\cos{\phi^{\circ}}+i\sin{\phi^{\circ}})$, where $0<s$ and $0\leq \phi <360$. Find $\phi$.

180

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26,853

What is the smallest integral value of $k$ such that $2x(kx-4)-x^2+6=0$ has no real roots?

1. **Expand and simplify the given quadratic equation**: \[ 2x(kx-4) - x^2 + 6 = 0 \] Expanding the terms: \[ 2kx^2 - 8x - x^2 + 6 = 0 \] Combine like terms: \[ (2k-1)x^2 - 8x + 6 = 0 \] 2. **Condition for no real roots**: A quadratic equation $ax^2 + bx + c = 0$ has no real roots if its discriminant $b^2 - 4ac$ is negative. For the equation $(2k-1)x^2 - 8x + 6 = 0$, we identify $a = 2k-1$, $b = -8$, and $c = 6$. 3. **Calculate the discriminant and set it to be less than zero**: \[ \text{Discriminant} = (-8)^2 - 4(2k-1)(6) \] Simplify the expression: \[ 64 - 4(2k-1)(6) = 64 - (48k - 24) = 88 - 48k \] Set the discriminant to be less than zero: \[ 88 - 48k < 0 \] 4. **Solve the inequality for $k$**: \[ 88 - 48k < 0 \implies 88 < 48k \implies \frac{88}{48} < k \implies \frac{11}{6} < k \] Since $k$ must be an integer, the smallest integer greater than $\frac{11}{6}$ is $2$. 5. **Conclusion**: The smallest integral value of $k$ such that the quadratic equation has no real roots is $2$. \[ \boxed{\text{B}} \]

2

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512

Let $p$ be the largest prime with 2010 digits. What is the smallest positive integer $k$ such that $p^2 - k$ is divisible by 12?

k = 1

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38,360

Starting at 1:00 p.m., Jorge watched three movies. The first movie was 2 hours and 20 minutes long. He took a 20 minute break and then watched the second movie, which was 1 hour and 45 minutes long. He again took a 20 minute break and then watched the last movie, which was 2 hours and 10 minutes long. At what time did the final movie end?

Starting at 1:00 p.m., Jorge watches a movie that is 2 hours and 20 minutes long. This first movie ends at 3:20 p.m. Then, Jorge takes a 20 minute break. This break ends at 3:40 p.m. Then, Jorge watches a movie that is 1 hour and 45 minutes long. After 20 minutes of this movie, it is 4:00 p.m. and there is still 1 hour and 25 minutes left in the movie. This second movie thus ends at 5:25 p.m. Then, Jorge takes a 20 minute break which ends at 5:45 p.m. Finally, Jorge watches a movie that is 2 hours and 10 minutes long. This final movie ends at 7:55 p.m.

7:55 \text{ p.m.}

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5,903

Given 365 cards, in which distinct numbers are written. We may ask for any three cards, the order of numbers written in them. Is it always possible to find out the order of all 365 cards by 2000 such questions?

1691

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12,681

In a certain country, there are 100 senators, each of whom has 4 aides. These senators and aides serve on various committees. A committee may consist either of 5 senators, of 4 senators and 4 aides, or of 2 senators and 12 aides. Every senator serves on 5 committees, and every aide serves on 3 committees. How many committees are there altogether?

If each senator gets a point for every committee on which she serves, and every aide gets $1 / 4$ point for every committee on which he serves, then the 100 senators get 500 points altogether, and the 400 aides get 300 points altogether, for a total of 800 points. On the other hand, each committee contributes 5 points, so there must be $800 / 5=160$ committees.

160

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4,054

$\triangle ABC$ is isosceles with base $AC$. Points $P$ and $Q$ are respectively in $CB$ and $AB$ and such that $AC=AP=PQ=QB$. The number of degrees in $\angle B$ is:

1. **Identify the properties of the triangles**: Given that $\triangle ABC$ is isosceles with base $AC$, and $AC = AP = PQ = QB$, we can infer that $\triangle APQ$ and $\triangle BQP$ are also isosceles. Additionally, $\triangle APC$ and $\triangle BQC$ are isosceles because $AP = AC$ and $BQ = BC$. 2. **Analyze $\triangle BQP$**: Since $\triangle BQP$ is isosceles with $PQ = QB$, the base angles are equal. Let $\angle BPQ = \angle BQP = b$. By the Triangle Angle Sum Property, the sum of angles in a triangle is $180^\circ$. Therefore, \[ \angle BPQ + \angle BQP + \angle PQB = 180^\circ \implies b + b + \angle PQB = 180^\circ \implies \angle PQB = 180^\circ - 2b. \] 3. **Analyze $\triangle APQ$**: Since $\triangle APQ$ is isosceles with $AP = PQ$, the base angles are equal. Let $\angle PAQ = \angle PQA = 2b$ (since $\angle PQA$ is an exterior angle to $\triangle BQP$ and equal to $2b$). Again, using the Triangle Angle Sum Property, \[ \angle PAQ + \angle PQA + \angle QPA = 180^\circ \implies 2b + 2b + \angle QPA = 180^\circ \implies \angle QPA = 180^\circ - 4b. \] 4. **Analyze $\triangle APC$**: Since $\triangle APC$ is isosceles with $AP = AC$, the base angles are equal. Let $\angle PAC = \angle PCA = b$. Since $\angle APC$ is an exterior angle to $\triangle APQ$, \[ \angle APC = 180^\circ - \angle QPA = 180^\circ - (180^\circ - 4b) = 4b. \] 5. **Analyze $\triangle ABC$**: Since $\triangle ABC$ is isosceles with $AB = BC$, the base angles are equal. Let $\angle BAC = \angle BCA = 3b$. Using the Triangle Angle Sum Property, \[ \angle BAC + \angle BCA + \angle B = 180^\circ \implies 3b + 3b + b = 180^\circ \implies 7b = 180^\circ. \] 6. **Solve for $b$**: Solving for $b$, \[ b = \frac{180^\circ}{7} = 25\frac{5}{7}^\circ. \] 7. **Conclusion**: Therefore, the measure of $\angle B$ in $\triangle ABC$ is $\boxed{25\frac{5}{7}^\circ}$, corresponding to choice $\textbf{(A)}\ 25\frac{5}{7}$.

25\frac{5}{7}

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2,455

Given that the sum of two numbers is 15 and their product is 36, find the sum of their reciprocals and the sum of their squares.

153

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23,272

Given $f(x)= \sqrt {3}\sin \dfrac {x}{4}\cos \dfrac {x}{4}+ \cos ^{2} \dfrac {x}{4}+ \dfrac {1}{2}$. (1) Find the period of $f(x)$; (2) In $\triangle ABC$, sides $a$, $b$, and $c$ correspond to angles $A$, $B$, and $C$ respectively, and satisfy $(2a-c)\cos B=b\cos C$, find the value of $f(B)$.

\dfrac{\sqrt{3}}{2} + 1

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21,109

Find the area of the $MNRK$ trapezoid with the lateral side $RK = 3$ if the distances from the vertices $M$ and $N$ to the line $RK$ are $5$ and $7$ , respectively.

18

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28,320

An assembly line produces, on average, 85% first grade products. How many products need to be sampled so that, with a probability of 0.997, the deviation of the proportion of first grade products from 0.85 in absolute value does not exceed 0.01?

11475

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15,545

Find a six-digit number where the first digit is 6 times less than the sum of all the digits to its right, and the second digit is 6 times less than the sum of all the digits to its right.

769999

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7,409

There are 12 sprinters in an international track event final, including 5 Americans. Gold, silver, and bronze medals are awarded to the first, second, and third place finishers respectively. Determine the number of ways the medals can be awarded if no more than two Americans receive medals.

1260

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18,860

In an opaque bag, there are four small balls labeled with the Chinese characters "阳", "过", "阳", and "康" respectively. Apart from the characters, the balls are indistinguishable. Before each draw, the balls are thoroughly mixed.<br/>$(1)$ If one ball is randomly drawn from the bag, the probability that the character on the ball is exactly "康" is ________; <br/>$(2)$ If two balls are drawn from the bag by person A, using a list or a tree diagram, find the probability $P$ that one ball has the character "阳" and the other ball has the character "康".

\frac{1}{3}

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10,589

If the numbers $x$ and $y$ are inversely proportional and when the sum of $x$ and $y$ is 54, $x$ is three times $y$, find the value of $y$ when $x = 5$.

109.35

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9,584

Given that the roots of the polynomial $81x^3 - 162x^2 + 81x - 8 = 0$ are in arithmetic progression, find the difference between the largest and smallest roots.

\frac{4\sqrt{6}}{9}

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14,237

If $k$ and $\ell$ are positive 4-digit integers such that $\gcd(k,\ell)=3$, what is the smallest possible value for $\mathop{\text{lcm}}[k,\ell]$?

335{,}670

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37,916

If $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = a \cdot d - b \cdot c$, what is the value of $\begin{vmatrix} 3 & 4 \\ 1 & 2 \end{vmatrix}$?

Given the determinant formula for a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the determinant is calculated as: \[ \text{det} = ad - bc \] For the specific matrix $\begin{pmatrix} 3 & 4 \\ 1 & 2 \end{pmatrix}$, we identify the elements as: - $a = 3$ - $b = 4$ - $c = 1$ - $d = 2$ Using the determinant formula: 1. Calculate $ad$: \[ ad = 3 \times 2 = 6 \] 2. Calculate $bc$: \[ bc = 4 \times 1 = 4 \] 3. Subtract $bc$ from $ad$ to find the determinant: \[ \text{det} = ad - bc = 6 - 4 = 2 \] Thus, the value of $\begin{pmatrix} 3 & 4 \\ 1 & 2 \end{pmatrix}$ is $2$. $\boxed{\text{E}}$

$2$

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2,510

Given in $\triangle ABC$, $\tan A$ and $\tan B$ are the two real roots of the equation $x^2 + ax + 4 = 0$: (1) If $a = -8$, find the value of $\tan C$; (2) Find the minimum value of $\tan C$, and specify the corresponding values of $\tan A$ and $\tan B$.

\frac{4}{3}

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28,715

Given that the sum of the first three terms of a geometric sequence $\{a_n\}$ is $3$ and the sum of the first nine terms is $39$, calculate the value of the sum of the first six terms.

12

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21,411

Suppose you have $6$ red shirts, $7$ green shirts, $9$ pairs of pants, $10$ blue hats, and $10$ red hats, all distinct. How many outfits can you make consisting of one shirt, one pair of pants, and one hat, if neither the hat nor the pants can match the shirt in color?

1170

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11,588

50 students from fifth to ninth grade collectively posted 60 photos on Instagram, with each student posting at least one photo. All students in the same grade (parallel) posted an equal number of photos, while students from different grades posted different numbers of photos. How many students posted exactly one photo?

46

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9,597

What is equivalent to $\sqrt{\frac{x}{1-\frac{x-1}{x}}}$ when $x < 0$?

To solve the problem, we first simplify the expression inside the square root: \[ \sqrt{\frac{x}{1-\frac{x-1}{x}}} \] 1. Simplify the denominator: \[ 1 - \frac{x-1}{x} = 1 - \left(\frac{x}{x} - \frac{1}{x}\right) = 1 - 1 + \frac{1}{x} = \frac{1}{x} \] 2. Substitute back into the original expression: \[ \sqrt{\frac{x}{\frac{1}{x}}} = \sqrt{x^2} = |x| \] Since $x < 0$, $|x| = -x$. 3. Now, we compare this result with the given options: - $\mathrm{(A) \ } -x$ - $\mathrm{(B) \ } x$ - $\mathrm{(C) \ } 1$ - $\mathrm{(D) \ } \sqrt{\frac{x}{2}}$ - $\mathrm{(E) \ } x\sqrt{-1}$ Given that $x < 0$, $|x| = -x$ matches with option $\mathrm{(A)}$. Thus, the equivalent expression for $\sqrt{\frac{x}{1-\frac{x-1}{x}}}$ when $x < 0$ is $\boxed{\textbf{(A)} -x}$.

-x

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284

Compute the number of ways there are to assemble 2 red unit cubes and 25 white unit cubes into a $3 \times 3 \times 3$ cube such that red is visible on exactly 4 faces of the larger cube. (Rotations and reflections are considered distinct.)

We do casework on the two red unit cubes; they can either be in a corner, an edge, or the center of the face. - If they are both in a corner, they must be adjacent - for each configuration, this corresponds to an edge, of which there are 12. - If one is in the corner and the other is at an edge, we have 8 choices to place the corner. For the edge, the red edge square has to go on the boundary of the faces touching the red corner square, and there are six places here. Thus, we get $8 \cdot 6=48$ configurations. - If one is a corner and the other is in the center of a face, we again have 8 choices for the corner and 3 choices for the center face (the faces not touching the red corner). This gives $8 \cdot 3=24$ options. - We have now completed the cases with a red corner square! Now suppose we have two edges: If we chose in order, we have 12 choices for the first cube. For the second cube, we must place the edge so it covers two new faces, and thus we have five choices. Since we could have picked these edges in either order, we divide by two to avoid overcounting, and we have $12 \cdot 5 / 2=30$ in this case. Now, since edges and faces only cover at most 2 and 1 face respectively, no other configuration works. Thus we have all the cases, and we add: $12+48+24+30=114$.

114

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3,783

The equations $x^3 + Ax + 10 = 0$ and $x^3 + Bx^2 + 50 = 0$ have two roots in common. Then the product of these common roots can be expressed in the form $a \sqrt[b]{c},$ where $a,$ $b,$ and $c$ are positive integers, when simplified. Find $a + b + c.$

12

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36,944

Let $\pi$ be a uniformly random permutation of the set $\{1,2, \ldots, 100\}$. The probability that $\pi^{20}(20)=$ 20 and $\pi^{21}(21)=21$ can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$. (Here, $\pi^{k}$ means $\pi$ iterated $k$ times.)

We look at the cycles formed by $\pi$ Let $\operatorname{ord}_{\pi}(n)$ denote the smallest $m$ such that $\pi^{m}(n)=n$. In particular, the condition implies that $\operatorname{ord}_{\pi}(20) \mid 20$ and $\operatorname{ord}_{\pi}(21) \mid 21$. Claim 1. 20 and 21 cannot be in the same cycle. Proof. If 20 and 21 were in the same cycle, then $x=\operatorname{ord}_{\pi}(20)=\operatorname{ord}_{\pi}(21)$ for some $x$. Then $x>1$ since the cycle contains both 20 and 21, but $x|20, x| 21$ implies $x=1$, a contradiction. Claim 2. The probability that $a=\operatorname{ord}_{\pi}(20), b=\operatorname{ord}_{\pi}(21)$ for some fixed $a, b$ such that $a+b \leq 100$ is $\frac{1}{99 \cdot 100}$. Proof. We can just count these permutations. We first choose $a-1$ elements of $[100] \backslash\{20,21\}$ to be in the cycle of 20, then we similarly choose $b-1$ to be in the cycle of 21. We then have $(a-1)$! ways to reorder within the cycle of $20,(b-1)$! ways to reorder within the cycle of 21, and $(100-a-b)$! ways to permute the remaining elements. The total number of ways is just $$\frac{98!}{(a-1)!(b-1)!(100-a-b)!} \cdot(a-1)!(b-1)!(100-a-b)!=98!$$ so the probability this happens is just $\frac{98!}{100!}=\frac{1}{9900}$. Now, since $\operatorname{ord}_{\pi}(20) \mid 20$ and $\operatorname{ord}_{\pi}(21) \mid 21$, we have 6 possible values for $\operatorname{ord}_{\pi}(20)$ and 4 for $\operatorname{ord}_{\pi}(21)$, so in total we have a $\frac{6 \cdot 4}{9900}=\frac{2}{825}$ probability that the condition is satisfied.

1025

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3,986

Harry Potter is creating an enhanced magical potion called "Elixir of Life" (this is a very potent sleeping potion composed of powdered daffodil root and wormwood infusion. The concentration of the "Elixir of Life" is the percentage of daffodil root powder in the entire potion). He first adds a certain amount of wormwood infusion to the regular "Elixir of Life," making its concentration $9 \%$. If he then adds the same amount of daffodil root powder, the concentration of the "Elixir of Life" becomes $23 \%$. What is the concentration of the regular "Elixir of Life"? $\qquad$ $\%$.

11

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7,959

On an infinite tape, numbers are written in a row. The first number is one, and each subsequent number is obtained from the previous one by adding the smallest nonzero digit of its decimal representation. How many digits are in the decimal representation of the number that is in the $9 \cdot 1000^{1000}$-th position in this row?

3001

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15,962

Find the ordered pair $(x,y)$ if \begin{align*} x+y&=(5-x)+(5-y),\\ x-y&=(x-1)+(y-1). \end{align*}

(4,1)

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33,447

In triangle $ ABC $ with sides $ AB = 8 $, $ AC = 4 $, $ BC = 6 $, the angle bisector $ AK $ is drawn, and on the side $ AC $ a point $ M $ is marked such that $ AM : CM = 3 : 1 $. Point $ N $ is the intersection point of $ AK $ and $ BM $. Find $ AN $.

\frac{18\sqrt{6}}{11}

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11,306

Given that the function $f(x)$ defined on $\mathbb{R}$ satisfies $f(4)=2-\sqrt{3}$, and for any $x$, $f(x+2)=\frac{1}{-f(x)}$, find $f(2018)$.

-2-\sqrt{3}

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22,534

Given vectors $\overrightarrow{m}=(\sin x,-1)$ and $\overrightarrow{n}=(\sqrt{3}\cos x,-\frac{1}{2})$, and the function $f(x)=(\overrightarrow{m}+\overrightarrow{n})\cdot\overrightarrow{m}$. - (I) Find the interval where $f(x)$ is monotonically decreasing; - (II) Given $a$, $b$, and $c$ are respectively the sides opposite to angles $A$, $B$, and $C$ in $\triangle ABC$, with $A$ being an acute angle, $a=2\sqrt{3}$, $c=4$, and $f(A)$ is exactly the maximum value of $f(x)$ on the interval $\left[0, \frac{\pi}{2}\right]$, find $A$, $b$, and the area $S$ of $\triangle ABC$.

2\sqrt{3}

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21,696

Find the largest positive integer $N$ so that the number of integers in the set $\{1,2,\dots,N\}$ which are divisible by 3 is equal to the number of integers which are divisible by 5 or 7 (or both).

65

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18,881

If $ p $ and $ q $ are positive integers, $\max (p, q)$ is the maximum of $ p $ and $ q $ and $\min (p, q)$ is the minimum of $ p $ and $ q $. For example, $\max (30,40)=40$ and $\min (30,40)=30$. Also, $\max (30,30)=30$ and $\min (30,30)=30$. Determine the number of ordered pairs $(x, y)$ that satisfy the equation $$ \max (60, \min (x, y))=\min (\max (60, x), y) $$ where $x$ and $y$ are positive integers with $x \leq 100$ and $y \leq 100$.

4100

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14,037

How many 5-digit positive numbers contain only odd numbers and have at least one pair of consecutive digits whose sum is 10?

1845

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17,656

How many three-digit numbers are there in which each digit is greater than the digit to its right?

84

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11,520

In the geometric sequence $\{a_n\}$ with a common ratio greater than $1$, $a_3a_7=72$, $a_2+a_8=27$, calculate $a_{12}$.

96

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8,980

There are two hourglasses - one for 7 minutes and one for 11 minutes. An egg needs to be boiled for 15 minutes. How can you measure this amount of time using the hourglasses?

15

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11,486

Point $(x,y)$ is chosen randomly from the rectangular region with vertices at $(0,0)$, $(3036,0)$, $(3036,3037)$, and $(0,3037)$. What is the probability that $x > 3y$? Express your answer as a common fraction.

\frac{506}{3037}

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16,986

The average age of 8 people in Room C is 35. The average age of 6 people in Room D is 30. Calculate the average age of all people when the two groups are combined.

\frac{460}{14}

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29,004

How many $4$-digit positive integers (that is, integers between $1000$ and $9999$, inclusive) having only even digits are divisible by $5?$

1. **Identify the range and conditions**: We are looking for 4-digit integers between 1000 and 9999, inclusive, that have only even digits and are divisible by 5. 2. **Digits must be even**: The possible even digits are 0, 2, 4, 6, and 8. 3. **Divisibility by 5**: For a number to be divisible by 5, its units digit must be either 0 or 5. Since we are restricted to even digits, the units digit must be 0. 4. **Thousands digit**: The thousands digit must be one of 2, 4, 6, or 8, as it cannot be 0 (otherwise, it would not be a 4-digit number). This gives us 4 choices for the thousands digit. 5. **Middle two digits**: Each of the middle two digits (hundreds and tens) can independently be any of the 5 even digits (0, 2, 4, 6, 8). This gives us 5 choices for each of these digits. 6. **Calculate the total number of such numbers**: - 1 choice for the units digit (0). - 5 choices for the hundreds digit. - 5 choices for the tens digit. - 4 choices for the thousands digit. Multiplying these choices together gives the total number of valid numbers: \[ 1 \times 5 \times 5 \times 4 = 100 \] 7. **Conclusion**: There are 100 such numbers that meet all the conditions. Thus, the number of 4-digit positive integers having only even digits and divisible by 5 is $\boxed{\textbf{(B) } 100}$.

100

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1,062