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24,400 | Let \( X \), \( Y \), and \( Z \) be nonnegative integers such that \( X+Y+Z = 15 \). What is the maximum value of
\[ X \cdot Y \cdot Z + X \cdot Y + Y \cdot Z + Z \cdot X? \] | 200 | 96.875 |
24,401 | Let $b_1, b_2, \ldots$ be a sequence determined by the rule $b_n= \frac{b_{n-1}}{3}$ if $b_{n-1}$ is divisible by 3, and $b_n = 2b_{n-1} + 2$ if $b_{n-1}$ is not divisible by 3. Determine how many positive integers $b_1 \le 3000$ are such that $b_1$ is less than each of $b_2$, $b_3$, and $b_4$. | 2000 | 66.40625 |
24,402 | A circle is circumscribed about an equilateral triangle with side lengths of $12$ units each. Calculate the area of the circle, and then find the perimeter of the triangle. | 36 | 67.1875 |
24,403 | Given that $x$ is a root of the equation $x^{2}+x-6=0$, simplify $\frac{x-1}{\frac{2}{{x-1}}-1}$ and find its value. | \frac{8}{3} | 45.3125 |
24,404 | The diagonal of a square is 10 inches and the diameter of a circle is also 10 inches. Calculate:
1. By how many square inches is the area of the circle greater than the area of the square?
2. By how many inches is the circumference of the circle greater than the perimeter of the square? Express your answers as decimals to the nearest tenth. | 3.1 | 5.46875 |
24,405 | Let the set of three-digit numbers composed of \\(0\\), \\(1\\), \\(2\\), and \\(3\\) without repeating digits be \\(A\\). If a number is randomly selected from \\(A\\), the probability that the number is exactly even is. | \dfrac{5}{9} | 98.4375 |
24,406 | Form a three-digit number without repeating digits using 1, 2, 3, 4, where the number of odd numbers is a certain number. | 12 | 80.46875 |
24,407 | What is the base $2$ representation of $125_{10}$? | 1111101_2 | 93.75 |
24,408 | Given that $\cos \alpha =\dfrac{\sqrt{5}}{5}$ and $\sin (\alpha -\beta )=\dfrac{\sqrt{10}}{10}$, calculate the value of $\cos \beta$. | \dfrac{\sqrt{2}}{2} | 53.90625 |
24,409 | It is very boring to look at a black-and-white clock face, so Clive painted the number 12 red exactly at noon and decided to paint the current hour red every 57 hours.
a) How many numbers on the clock face will be painted?
b) How many numbers will be red if Clive paints them every 1913th hour? | 12 | 76.5625 |
24,410 | The equation of the directrix of the parabola $y^{2}=6x$ is $x=\frac{3}{2}$. | -\dfrac{3}{2} | 78.125 |
24,411 | Given that the sequence $\{a_n\}$ is an arithmetic sequence and satisfies $a_1=1$, $a_3=7$, let $S_n$ be the sum of the first $n$ terms of the sequence $\{(-1)^n a_n\}$. Find the value of $S_{2017}$. | -3025 | 72.65625 |
24,412 | Let $A$ be a set of positive integers satisfying the following : $a.)$ If $n \in A$ , then $n \le 2018$ . $b.)$ If $S \subset A$ such that $|S|=3$ , then there exists $m,n \in S$ such that $|n-m| \ge \sqrt{n}+\sqrt{m}$ What is the maximum cardinality of $A$ ? | 44 | 82.03125 |
24,413 | A parallelogram has side lengths of 10, 12, $10y-2$, and $4x+6$. Determine the value of $x+y$. | 2.7 | 43.75 |
24,414 | Given that $E, U, L, S, R, T$ represent the numbers $1, 2, 3, 4, 5, 6$ (with each letter representing a different number), and satisfying the following conditions:
(1) $E + U + L = 6$
(2) $S + R + U + T = 18$
(3) $U \times T = 15$
(4) $S \times L = 8$
What is the six-digit number $\overline{EULSRT}$? | 132465 | 86.71875 |
24,415 | Given that $F\_1$ and $F\_2$ are the left and right foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$, and point $P$ is a point on the right branch of the hyperbola. $M$ is the incenter of $\triangle PF\_1F\_2$, satisfying $S\_{\triangle MPF\_1} = S\_{\triangle MPF\_2} + \lambda S\_{\triangle MF\_1F\_2}$. If the eccentricity of this hyperbola is $3$, then $\lambda = \_\_\_\_\_\_$.
(Note: $S\_{\triangle MPF\_1}$, $S\_{\triangle MPF\_2}$, $S\_{\triangle MF\_1F\_2}$ represent the area of $\triangle MPF\_1$, $\triangle MPF\_2$, $\triangle MF\_1F\_2$ respectively.) | \frac{1}{3} | 60.9375 |
24,416 | What is the largest possible value of \(| |a_1 - a_2| - a_3| - \ldots - a_{1990}|\), where \(a_1, a_2, \ldots, a_{1990}\) is a permutation of \(1, 2, 3, \ldots, 1990\)? | 1989 | 45.3125 |
24,417 | In the rectangular coordinate system xOy, a polar coordinate system is established with the origin O of the rectangular coordinate system as the pole and the positive semi-axis of the x-axis as the polar axis. The parametric equations of the line l are given by $$\begin{cases} x= \frac {1}{2}+ \frac {1}{2}t \\ y= \frac { \sqrt {3}}{2}t\end{cases}$$ (where t is the parameter), and the polar equation of the curve C is given by $$ρ= \frac {2cosθ}{sin^{2}\theta }$$.
(1) Find the ordinary equation of line l and the rectangular coordinate equation of curve C;
(2) Given the fixed point P ($$\frac {1}{2}$$, 0), let A and B be the intersection points of line l and curve C. Find the value of |PA| + |PB|. | \frac{8}{3} | 78.90625 |
24,418 | Given triangle $ABC$ with angles $A$, $B$, $C$ and their respective opposite sides $a$, $b$, $c$, if $\cos B= \frac{1}{4}, b=3$, and $\sin C=2\sin A$, find the area of triangle $ABC$. | \frac{9\sqrt{15}}{16} | 62.5 |
24,419 | The function $f(x) = |\log_3 x|$ has a range of $[0,1]$ on the interval $[a, b]$. Find the minimum value of $b - a$. | \frac{2}{3} | 10.15625 |
24,420 | A rectangular room measures 15-feet by 8-feet and has a triangular extension with a base of 8-feet and a height of 5-feet. How many square yards of carpet are needed to cover the entire floor of the room, including the triangular extension? | 16 | 21.09375 |
24,421 | If the solution set of the system of linear inequalities in one variable $x$, $\left\{{\begin{array}{l}{x-1≥2x+1}\\{2x-1＜a}\end{array}}\right.$, is $x\leqslant -2$, and the solution of the fractional equation in variable $y$, $\frac{{y-1}}{{y+1}}=\frac{a}{{y+1}}-2$, is negative, then the sum of all integers $a$ that satisfy the conditions is ______. | -8 | 0.78125 |
24,422 | Given $\sin \left(\alpha+ \frac {\pi}{3}\right)=- \frac {4}{5}$, and $- \frac {\pi}{2} < \alpha < 0$, find $\cos \alpha=$ ______. | \frac {3-4 \sqrt {3}}{10} | 0 |
24,423 | If $-1 < a < 0$, find the maximum value of the inequality $\frac{2}{a} - \frac{1}{1+a}$. | -3 - 2\sqrt{2} | 26.5625 |
24,424 | Points P and Q are on a circle of radius 7 and the chord PQ=8. Point R is the midpoint of the minor arc PQ. Calculate the length of the line segment PR. | \sqrt{98 - 14\sqrt{33}} | 35.15625 |
24,425 | Solve the equations
​
(1) $x^{2}-4x+3=0$
​
(2) $(x+1)(x-2)=4$
​
(3) $3x(x-1)=2-2x$
​
(4) $2x^{2}-4x-1=0$ | \frac{2- \sqrt{6}}{2} | 0 |
24,426 | A biologist wishes to estimate the fish population in a protected area. Initially, on March 1st, she captures and tags 80 fish, then releases them back into the water. Four months later, on July 1st, she captures another 90 fish for a follow-up study, finding that 4 of these are tagged. For her estimation, she assumes that 30% of these fish have left the area by July 1st due to various environmental factors, and that an additional 50% of the fish in the July sample weren't in the area on March 1st due to new arrivals. How many fish does she estimate were in the area on March 1st? | 900 | 5.46875 |
24,427 | A solid has a triangular base with sides of lengths $s$, $s$, $s \sqrt{2}$. Two opposite vertices of the triangle extend vertically upward by a height $h$ where $h = 3s$. Given $s = 2\sqrt{2}$, what is the volume of this solid? | 24\sqrt{2} | 27.34375 |
24,428 | How many solutions does the equation \[ \frac{(x-1)(x-2)(x-3) \dotsm (x-150)}{(x-1^3)(x-2^3)(x-3^3) \dotsm (x-150^3)} = 0 \] have for \(x\)? | 145 | 84.375 |
24,429 | Calculate the value of $1.000 + 0.101 + 0.011 + 0.001$. | 1.113 | 57.8125 |
24,430 | Suppose the function $f(x)-f(2x)$ has derivative $5$ at $x=1$ and derivative $7$ at $x=2$ . Find the derivative of $f(x)-f(4x)$ at $x=1$ . | 19 | 96.09375 |
24,431 | What is the area enclosed by the graph of $|x| + |3y| = 9$? | 54 | 85.9375 |
24,432 | Given a triangle $ \triangle ABC $ with sides $ a $, $ b $, and $ c $ opposite angles $ A $, $ B $, and $ C $, respectively, consider vectors $ \overrightarrow{m} = (a, \sqrt{3}b) $ and $ \overrightarrow{n} = (\cos A, \sin B) $ being parallel.
(I) Find the angle $ A $.
(II) If $ a = \sqrt{7} $ and the area of $ \triangle ABC $ is $ \frac{3\sqrt{3}}{2} $, determine the perimeter of the triangle. | 5 + \sqrt{7} | 42.1875 |
24,433 | Determine the number of solutions to the equation
\[\tan (10 \pi \cos \theta) = \cot (10 \pi \sin \theta)\]
where $\theta \in (0, 2 \pi).$ | 56 | 40.625 |
24,434 | A rectangular cake with dimensions $4$ inches, $3$ inches, and $2$ inches (length, width, height respectively) is iced on the sides, the top, and the bottom. The cake is cut from the top center vertex across to the midpoint of the bottom edge on the opposite side face, creating one triangular piece. If the top center vertex is $T$, and the midpoint on the opposite bottom edge is denoted as $M$, find the volume of the cake on one side of the cut ($c$ cubic inches) and the area of icing on this piece ($s$ square inches). Calculate the sum $c+s$. | 38 | 14.84375 |
24,435 | Given sets \( A = \{ x \mid 5x - a \leq 0 \} \) and \( B = \{ x \mid 6x - b > 0 \} \), where \( a, b \in \mathbf{N} \), and \( A \cap B \cap \mathbf{N} = \{ 2, 3, 4 \} \), the number of integer pairs \((a, b)\) is: | 30 | 55.46875 |
24,436 | In the binomial expansion of $(x-1)^n$ ($n\in\mathbb{N}_{+}$), if only the 5th binomial coefficient is the largest, find the constant term in the binomial expansion of $(2\sqrt{x}-\frac{1}{\sqrt{x}})^n$. | 1120 | 89.0625 |
24,437 | From the natural numbers 1 to 100, each time we take out two different numbers so that their sum is greater than 100, how many different ways are there to do this? | 2500 | 89.0625 |
24,438 | Let $\alpha$, $\beta$, $\gamma$ be the roots of the cubic polynomial $x^3 - 3x - 2 = 0.$ Find
\[
\alpha(\beta - \gamma)^2 + \beta(\gamma - \alpha)^2 + \gamma(\alpha - \beta)^2.
\] | -18 | 74.21875 |
24,439 | Given an ellipse $C$: $\frac{{x}^{2}}{3}+{y}^{2}=1$ with left focus and right focus as $F_{1}$ and $F_{2}$ respectively. The line $y=x+m$ intersects $C$ at points $A$ and $B$. If the area of $\triangle F_{1}AB$ is twice the area of $\triangle F_{2}AB$, find the value of $m$. | -\frac{\sqrt{2}}{3} | 41.40625 |
24,440 | The product of two whole numbers is 24. Calculate the smallest possible sum of these two numbers. | 10 | 100 |
24,441 | Given a random variable $\xi$ follows the normal distribution $N(0, \sigma^2)$. If $P(\xi > 2) = 0.023$, calculate $P(-2 \leq \xi \leq 2)$. | 0.954 | 92.1875 |
24,442 | Find the coefficient of $x^5$ in the expansion of $(1+2x-3x^2)^6$. | -168 | 60.15625 |
24,443 | A point $Q$ is chosen inside $\triangle DEF$ such that lines drawn through $Q$ parallel to $\triangle DEF$'s sides decompose it into three smaller triangles $u_1$, $u_2$, and $u_3$, which have areas $3$, $12$, and $15$ respectively. Determine the area of $\triangle DEF$. | 30 | 26.5625 |
24,444 | How many tetrahedrons can be formed using the vertices of a regular triangular prism? | 12 | 29.6875 |
24,445 | Given a number $x$ is randomly selected from the interval $[-1,1]$, calculate the probability that the value of $\sin \frac{\pi x}{4}$ falls between $-\frac{1}{2}$ and $\frac{\sqrt{2}}{2}$. | \frac{5}{6} | 78.90625 |
24,446 | Solve the equations:
(1) $x^2-4x+3=0$
(2) $4(2y-5)^2=(3y-1)^2$． | \frac{11}{7} | 1.5625 |
24,447 | A student, Leo, needs to earn 30 study points for a special credit. For the first 6 points, he needs to complete 1 project each. For the next 6 points, he needs 2 projects each; for the next 6 points, 3 projects each, and so on. Determine the minimum number of projects Leo needs to complete to earn 30 study points. | 90 | 10.15625 |
24,448 | In $\triangle ABC$, if $AB=2$, $AC=\sqrt{2}BC$, find the maximum value of $S_{\triangle ABC}$. | 2\sqrt{2} | 30.46875 |
24,449 | A line with a slope of $2$ passing through the right focus of the ellipse $\frac{x^2}{5} + \frac{y^2}{4} = 1$ intersects the ellipse at points $A$ and $B$. If $O$ is the origin, then the area of $\triangle OAB$ is \_\_\_\_\_\_. | \frac{5}{3} | 60.9375 |
24,450 | Given a geometric sequence $\{a_n\}$ with a sum of the first $n$ terms as $S_n$, and $S_{10}:S_5 = 1:2$, find the value of $\frac{S_5 + S_{10} + S_{15}}{S_{10} - S_5}$. | -\frac{9}{2} | 89.84375 |
24,451 | Given positive numbers $x$, $y$ satisfying $xy= \frac{x-y}{x+3y}$, find the maximum value of $y$. | \frac{1}{3} | 34.375 |
24,452 | If the integers $m,n,k$ hold the equation $221m+247n+323k=2001$ , what is the smallest possible value of $k$ greater than $100$ ? | 111 | 0.78125 |
24,453 | Calculate $(42 \div (12 - 10 + 3))^{2} \cdot 7$. | 493.92 | 19.53125 |
24,454 | Thirty 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 $\frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $\log_2 q.$ | 409 | 45.3125 |
24,455 | Given $\sin \theta + \cos \theta = \frac{1}{5}$, with $\theta \in (0,\pi)$.
$(1)$ Find the value of $\tan \theta$;
$(2)$ Find the value of $\frac{1-2\sin \theta \cos \theta}{\cos^2 \theta - \sin^2 \theta}$. | -7 | 78.90625 |
24,456 | Given the function $f\left(x\right)=4\cos x\sin \left(x-\frac{π}{3}\right)+a$ with a maximum value of $2$.
(1) Find the value of $a$ and the minimum positive period of the function $f\left(x\right)$;
(2) In $\triangle ABC$, if $A < B$, and $f\left(A\right)=f\left(B\right)=1$, find the value of $\frac{BC}{AB}$. | \sqrt{2} | 40.625 |
24,457 | Given $f(\alpha) = \frac{\sin(\pi - \alpha)\cos(\pi + \alpha)\sin(-\alpha + \frac{3\pi}{2})}{\cos(-\alpha)\cos(\alpha + \frac{\pi}{2})}$.
$(1)$ Simplify $f(\alpha)$;
$(2)$ If $\alpha$ is an angle in the third quadrant, and $\cos(\alpha - \frac{3\pi}{2}) = \frac{1}{5}$, find the value of $f(\alpha)$; | \frac{2\sqrt{6}}{5} | 90.625 |
24,458 | Given that $\cos\alpha = \frac{1}{7}$, $\cos(\alpha-\beta) = \frac{13}{14}$, and $0<\beta<\alpha<\frac{\pi}{2}$,
(1) find the value of $\tan 2\alpha$;
(2) determine $\beta$. | \frac{\pi}{3} | 53.90625 |
24,459 | In the geometric sequence $\{a_n\}$, it is given that $a_1 + a_4 + a_7 = 2$, and $a_3 + a_6 + a_9 = 18$. Find the sum of the first 9 terms, $S_9$, of the sequence $\{a_n\}$. | 26 | 78.125 |
24,460 | Given $x^{2}+y^{2}=4$, find the minimum value of $\sqrt{2-y}+\sqrt{5-2x}$. | \sqrt{5} | 3.90625 |
24,461 | Determine the area of the circle described by the graph of the equation
\[r = 4 \cos \theta - 3 \sin \theta.\] | \frac{25\pi}{4} | 94.53125 |
24,462 | Form a three-digit number using the digits 4, 5, and 6. The probability that the number formed is a common multiple of 3 and 5 is ____. | \frac{1}{3} | 59.375 |
24,463 | The Engan alphabet of a fictional region contains 15 letters: A, B, C, D, E, F, G, H, I, J, K, L, M, N, O. Suppose license plates are to be formed with six letters using only the letters in the Engan alphabet. How many license plates of six letters are possible that begin with either A or B, end with O, cannot contain the letter I, and have no letters that repeat? | 34320 | 44.53125 |
24,464 | A die has faces numbered $1$, $2$, $3$, $3$, $4$, and $4$. Another die has faces numbered $2$, $3$, $5$, $6$, $7$, and $8$. Determine the probability that the sum of the top two numbers when both dice are rolled will be either $6$, $8$, or $10$.
A) $\frac{1}{36}$
B) $\frac{1}{18}$
C) $\frac{11}{36}$
D) $\frac{5}{36}$
E) $\frac{1}{6}$ | \frac{11}{36} | 87.5 |
24,465 | Given the graph of $y=\sqrt{2}\sin 3x$, calculate the horizontal shift required to obtain the graph of $y=\sin 3x+\cos 3x$. | \dfrac{\pi}{12} | 22.65625 |
24,466 | Consider a game involving two standard decks of 52 cards each mixed together, making a total of 104 cards. Each deck has 13 ranks and 4 suits, with the suits retaining their colors as in a standard deck. What is the probability that the top card of this combined deck is the King of $\diamondsuit$? | \frac{1}{52} | 32.8125 |
24,467 | Given that every high school in the town of Pythagoras sent a team of 3 students to a math contest, and Andrea's score was the median among all students, and hers was the highest score on her team, and Andrea's teammates Beth and Carla placed 40th and 75th, respectively, calculate the number of schools in the town. | 25 | 21.09375 |
24,468 | If $x$ and $y$ are real numbers, and $x^{2}+2xy-y^{2}=7$, find the minimum value of $x^{2}+y^{2}$. | \frac{7\sqrt{2}}{2} | 7.03125 |
24,469 | In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a=3$, and $\left(a+b\right)\sin B=\left(\sin A+\sin C\right)\left(a+b-c\right)$.<br/>$(1)$ Find angle $A$;<br/>$(2)$ If $acosB+bcosA=\sqrt{3}$, find the area of $\triangle ABC$. | \frac{3\sqrt{3}}{2} | 14.84375 |
24,470 | Find the remainder when $5^{5^{5^5}}$ is divided by 500. | 125 | 99.21875 |
24,471 | Given $f(x) = x^2$ and $g(x) = |x - 1|$, let $f_1(x) = g(f(x))$, $f_{n+1}(x) = g(f_n(x))$, calculate the number of solutions to the equation $f_{2015}(x) = 1$. | 2017 | 7.8125 |
24,472 | Determine the product of the solutions of the equation $-21 = -x^2 + 4x$. | -21 | 85.9375 |
24,473 | Liam has $x$ candies, Mia has three times as many candies as Liam, Noah has four times as many candies as Mia, and Olivia has six times the number of candies Noah has. If in total Liam, Mia, Noah, and Olivia have 468 candies, what is the value of $x?$ | \frac{117}{22} | 7.8125 |
24,474 | Solve the equations:
(1) $3x^2 -32x -48=0$
(2) $4x^2 +x -3=0$
(3) $(3x+1)^2 -4=0$
(4) $9(x-2)^2 =4(x+1)^2.$ | \frac{4}{5} | 0.78125 |
24,475 | Let \( A = (-4, -1), B = (-3, 2), C = (3, 2), \) and \( D = (4, -1) \). Suppose that point \( P \) satisfies
\[ PA + PD = PB + PC = 10. \]
Find the \( y \)-coordinate of \( P \), when simplified, can be expressed in the form \( \frac{-a + b\sqrt{c}}{d}, \) where \( a, b, c, d \) are positive integers. Additionally, ensure that the \( x \)-coordinate of \( P \) is greater than 0. | \frac{2}{7} | 1.5625 |
24,476 | Five friends all brought some cakes with them when they met. Each of them gave a cake to each of the others. They then ate all the cakes they had just been given. As a result, the total number of cakes they had between them decreased by half. How many cakes did the five friends have at the start? | 40 | 84.375 |
24,477 | Given positive real numbers $a$ and $b$ satisfying $a+b=1$, find the maximum value of $\dfrac {2a}{a^{2}+b}+ \dfrac {b}{a+b^{2}}$. | \dfrac {2 \sqrt {3}+3}{3} | 0 |
24,478 | 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 | 76.5625 |
24,479 | The function $f$ satisfies the condition $$ f (x + 1) = \frac{1 + f (x)}{1 - f (x)} $$ for all real $x$ , for which the function is defined. Determine $f(2012)$ , if we known that $f(1000)=2012$ . | 2012 | 98.4375 |
24,480 | In a 6×6 grid, park 3 identical red cars and 3 identical black cars such that there is only one car in each row and each column, with each car occupying one cell. There are ______ possible parking arrangements. (Answer with a number) | 14400 | 20.3125 |
24,481 | Find all positive integers $n$ such that the product of all positive divisors of $n$ is $24^{240}$ . | 24^5 | 0 |
24,482 | In a game, two wheels are present. The first wheel has six segments with numbers 1 through 6. The second wheel has four segments, numbered 1, 1, 2, and 2. The game is to spin both wheels and add the resulting numbers. The player wins if the sum is a number less than 5. What is the probability of winning the game?
A) $\frac{1}{6}$
B) $\frac{1}{4}$
C) $\frac{1}{3}$
D) $\frac{1}{2}$
E) $\frac{2}{3}$ | \frac{1}{3} | 51.5625 |
24,483 | Given acute angles $α$ and $β$ that satisfy: $\cos α= \frac {1}{3}$ and $\cos (α+β)=- \frac {1}{3}$, find $\cos (α-β)$. | \frac{23}{27} | 48.4375 |
24,484 | Consider the polynomials $P\left(x\right)=16x^4+40x^3+41x^2+20x+16$ and $Q\left(x\right)=4x^2+5x+2$ . If $a$ is a real number, what is the smallest possible value of $\frac{P\left(a\right)}{Q\left(a\right)}$ ?
*2016 CCA Math Bonanza Team #6* | 4\sqrt{3} | 17.1875 |
24,485 | The expansion of (ax- \frac {3}{4x}+ \frac {2}{3})(x- \frac {3}{x})^{6} is given, and the sum of its coefficients is 16. Determine the coefficient of the x^{3} term in this expansion. | \frac{117}{2} | 42.96875 |
24,486 | How many positive three-digit integers with a $7$ in the units place are divisible by $21$? | 39 | 0 |
24,487 | Given that $S_{n}$ is the sum of the first $n$ terms of the sequence $\{a_{n}\}$, if ${a_1}=\frac{5}{2}$, and ${a_{n+1}}({2-{a_n}})=2$ for $n\in\mathbb{N}^*$, then $S_{22}=$____. | -\frac{4}{3} | 87.5 |
24,488 | Points $P, Q, R, S, T, U, V,$ and $W$ lie, in that order, on line $\overline{PW}$, dividing it into seven equal segments, each of length 1. Point $X$ is not on line $PW$. Points $Y$ and $Z$ lie on line segments $\overline{XR}$ and $\overline{XW}$ respectively. The line segments $\overline{YQ}, \overline{ZT},$ and $\overline{PX}$ are parallel. Determine the ratio $\frac{YQ}{ZT}$. | \frac{7}{6} | 0.78125 |
24,489 | The integer $n$ has exactly six positive divisors, and they are: $1<a<b<c<d<n$ . Let $k=a-1$ . If the $k$ -th divisor (according to above ordering) of $n$ is equal to $(1+a+b)b$ , find the highest possible value of $n$ . | 2009 | 2.34375 |
24,490 | If the average of a sample $m$, $4$, $6$, $7$ is $5$, then the variance of this sample is ______. | \frac{5}{2} | 0 |
24,491 | Simplify the product \[\frac{9}{3}\cdot\frac{15}{9}\cdot\frac{21}{15}\dotsm\frac{3n+6}{3n}\dotsm\frac{3003}{2997}.\] | 1001 | 68.75 |
24,492 | Select 4 students from 5 female and 4 male students to participate in a speech competition.
(1) If 2 male and 2 female students are to be selected, how many different selections are there?
(2) If at least 1 male and 1 female student must be selected, and male student A and female student B cannot be selected at the same time, how many different selections are there? | 99 | 58.59375 |
24,493 | A triangle has three different integer side lengths and a perimeter of 30 units. What is the maximum length of any one side? | 14 | 5.46875 |
24,494 | The sum of the first $15$ positive even integers is also the sum of six consecutive even integers. What is the smallest of these six integers? | 35 | 26.5625 |
24,495 | A rectangle has integer length and width, and a perimeter of 120 units. Determine the number of square units in the greatest possible area, given that one of the dimensions must be a prime number. | 899 | 46.875 |
24,496 | In the rectangular coordinate system $(xOy)$, with the coordinate origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis, establish a polar coordinate system. Consider the curve $C\_1$: $ρ^{2}-4ρ\cos θ+3=0$, $θ∈[0,2π]$, and the curve $C\_2$: $ρ= \frac {3}{4\sin ( \frac {π}{6}-θ)}$, $θ∈[0,2π]$.
(I) Find a parametric equation of the curve $C\_1$;
(II) If the curves $C\_1$ and $C\_2$ intersect at points $A$ and $B$, find the value of $|AB|$. | \frac { \sqrt {15}}{2} | 0 |
24,497 | Given that the sequence $\{a\_n\}$ is an arithmetic sequence with all non-zero terms, $S\_n$ denotes the sum of its first $n$ terms, and satisfies $a\_n^2 = S\_{2n-1}$ for all positive integers $n$. If the inequality $\frac{λ}{a\_{n+1}} \leqslant \frac{n + 8 \cdot (-1)^n}{2n}$ holds true for any positive integer $n$, determine the maximum value of the real number $λ$. | -\frac{21}{2} | 4.6875 |
24,498 | In $\triangle ABC$, $AB = BC = 2$, $\angle ABC = 120^\circ$. A point $P$ is outside the plane of $\triangle ABC$, and a point $D$ is on the line segment $AC$, such that $PD = DA$ and $PB = BA$. Find the maximum volume of the tetrahedron $PBCD$. | 1/2 | 9.375 |
24,499 | Susie Q has $2000 to invest. She invests part of the money in Alpha Bank, which compounds annually at 4 percent, and the remainder in Beta Bank, which compounds annually at 6 percent. After three years, Susie's total amount is $\$2436.29$. Determine how much Susie originally invested in Alpha Bank. | 820 | 13.28125 |
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