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21,200 | The vertex of the parabola $y^2 = 4x$ is $O$, and the coordinates of point $A$ are $(5, 0)$. A line $l$ with an inclination angle of $\frac{\pi}{4}$ intersects the line segment $OA$ (but does not pass through points $O$ and $A$) and intersects the parabola at points $M$ and $N$. The maximum area of $\triangle AMN$ is __________. | 8\sqrt{2} | 42.1875 |
21,201 | Cara is sitting at a circular table with six friends. Assume there are three males and three females among her friends. How many different possible pairs of people could Cara sit between if each pair must include at least one female friend? | 12 | 70.3125 |
21,202 | Given an ellipse $C:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with foci $F_{1}$ and $F_{2}$, point $A$ lies on $C$, point $B$ lies on the $y$-axis, and satisfies $\overrightarrow{A{F}_{1}}⊥\overrightarrow{B{F}_{1}}$, $\overrightarrow{A{F}_{2}}=\frac{2}{3}\overrightarrow{{F}_{2}B}$. What is the eccentricity of $C$? | \frac{\sqrt{5}}{5} | 12.5 |
21,203 | Given $\sin 2\alpha = 3\sin 2\beta$, calculate the value of $\frac {\tan(\alpha-\beta)}{\tan(\alpha +\beta )}$. | \frac{1}{2} | 30.46875 |
21,204 | Given real numbers $x$ and $y$ satisfying $x^{2}+y^{2}-4x-2y-4=0$, calculate the maximum value of $x-y$. | 1+3\sqrt{2} | 90.625 |
21,205 | Given that $a,b$ are constants, and $a \neq 0$, $f\left( x \right)=ax^{2}+bx$, $f\left( 2 \right)=0$.
(1) If the equation $f\left( x \right)-x=0$ has a unique real root, find the analytic expression of the function $f\left( x \right)$;
(2) When $a=1$, find the maximum and minimum values of the function $f\left( x \right)$ on the interval $\left[-1,2 \right]$. | -1 | 84.375 |
21,206 | In triangle $ABC$, with $BC=15$, $AC=10$, and $\angle A=60^\circ$, find $\cos B$. | \frac{\sqrt{6}}{3} | 28.90625 |
21,207 | Let \[Q(x) = (3x^3 - 27x^2 + gx + h)(4x^3 - 36x^2 + ix + j),\] where \(g, h, i, j\) are real numbers. Suppose that the set of all complex roots of \(Q(x)\) is \(\{1, 2, 6\}\). Find \(Q(7).\) | 10800 | 39.84375 |
21,208 | If a positive four-digit number's thousand digit \\(a\\), hundred digit \\(b\\), ten digit \\(c\\), and unit digit \\(d\\) satisfy the relation \\((a-b)(c-d) < 0\\), then it is called a "Rainbow Four-Digit Number", for example, \\(2012\\) is a "Rainbow Four-Digit Number". How many "Rainbow Four-Digit Numbers" are there among the positive four-digit numbers? (Answer with a number directly) | 3645 | 96.09375 |
21,209 | Find the smallest positive integer $b$ for which $x^2 + bx + 2023$ factors into a product of two polynomials, each with integer coefficients. | 136 | 82.8125 |
21,210 | Determine the value of the expression
\[\log_3 (64 + \log_3 (64 + \log_3 (64 + \cdots))),\]
assuming it is positive. | 3.8 | 0 |
21,211 | The graph of $y = \frac{p(x)}{q(x)}$ is shown, where $p(x)$ and $q(x)$ are quadratic polynomials. The horizontal asymptote is $y = 2$, and the vertical asymptote is $x = -3$. There is a hole in the graph at $x=4$.
Find $\frac{p(5)}{q(5)}$ if the graph passes through $(2,0)$. | \frac{3}{4} | 79.6875 |
21,212 | Given the function $f(x)=\sin (2x+ \frac {\pi}{6})+\cos 2x$.
(I) Find the interval of monotonic increase for the function $f(x)$;
(II) In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$. Given that $f(A)= \frac { \sqrt {3}}{2}$, $a=2$, and $B= \frac {\pi}{3}$, find the area of $\triangle ABC$. | \frac {3+ \sqrt {3}}{2} | 0 |
21,213 | What is the smallest multiple of 7 that is greater than -50? | -49 | 90.625 |
21,214 | If the two foci of a hyperbola are respectively $F_1(-3,0)$ and $F_2(3,0)$, and one asymptotic line is $y=\sqrt{2}x$, calculate the length of the chord that passes through the foci and is perpendicular to the $x$-axis. | 4\sqrt{3} | 92.96875 |
21,215 | In a specific year, a "prime date" occurs when both the month and the day are prime numbers. Determine the total number of prime dates in a non-leap year where February has 28 days, and March, May, and July have 31 days, while November has 30 days. | 52 | 44.53125 |
21,216 | Given the function $f(x)=2\sin \omega x\cos \omega x-2\sqrt{3}\cos^{2}\omega x+\sqrt{3}$ ($\omega > 0$), and the distance between two adjacent axes of symmetry of the graph of $y=f(x)$ is $\frac{\pi}{2}$.
(Ⅰ) Find the interval of monotonic increase for the function $f(x)$;
(Ⅱ) Given that in $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, angle $C$ is acute, and $f(C)=\sqrt{3}$, $c=3\sqrt{2}$, $\sin B=2\sin A$, find the area of $\triangle ABC$. | 3\sqrt{3} | 80.46875 |
21,217 | Given the function $f(x)=ax^{2}+bx+c(a\\neq 0)$, its graph intersects with line $l$ at two points $A(t,t^{3}-t)$, $B(2t^{2}+3t,t^{3}+t^{2})$, where $t\\neq 0$ and $t\\neq -1$. Find the value of $f{{'}}(t^{2}+2t)$. | \dfrac {1}{2} | 8.59375 |
21,218 | Given that Xiao Ming's elder brother was born in a year that is a multiple of 19, calculate his age in 2013. | 18 | 23.4375 |
21,219 | A coordinate paper is folded once such that the point \((0,2)\) overlaps with the point \((4,0)\). If the point \((7,3)\) overlaps with the point \((m, n)\), what is the value of \(m+n\)? | 6.8 | 0 |
21,220 | Triangle $DEF$ has side lengths $DE = 15$, $EF = 39$, and $FD = 36$. Rectangle $WXYZ$ has vertex $W$ on $\overline{DE}$, vertex $X$ on $\overline{DF}$, and vertices $Y$ and $Z$ on $\overline{EF}$. In terms of the side length $WX = \theta$, the area of $WXYZ$ can be expressed as the quadratic polynomial \[Area(WXYZ) = \gamma \theta - \delta \theta^2.\]
Then the coefficient $\delta = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. | 229 | 11.71875 |
21,221 | Mrs. Everett recorded the performance of her students in a chemistry test. However, due to a data entry error, 5 students who scored 60% were mistakenly recorded as scoring 70%. Below is the corrected table after readjusting these students. Using the data, calculate the average percent score for these $150$ students.
\begin{tabular}{|c|c|}
\multicolumn{2}{c}{}\\\hline
\textbf{$\%$ Score}&\textbf{Number of Students}\\\hline
100&10\\\hline
95&20\\\hline
85&40\\\hline
70&40\\\hline
60&20\\\hline
55&10\\\hline
45&10\\\hline
\end{tabular} | 75.33 | 14.0625 |
21,222 | Given Lucy starts with an initial term of 8 in her sequence, where each subsequent term is generated by either doubling the previous term and subtracting 2 if a coin lands on heads, or halving the previous term and subtracting 2 if a coin lands on tails, determine the probability that the fourth term in Lucy's sequence is an integer. | \frac{3}{4} | 12.5 |
21,223 | Gwen, Eli, and Kat take turns flipping a coin in their respective order. The first one to flip heads wins. What is the probability that Kat will win? | \frac{1}{7} | 85.9375 |
21,224 | Simplify the expression $\frac{15b^4 - 45b^3}{75b^2}$ when $b=2$. | -\frac{2}{5} | 96.875 |
21,225 | If $\det \mathbf{B} = -3,$ then find $\det (\mathbf{B}^5).$ | -243 | 96.09375 |
21,226 | Given the sequence $\{a\_n\}$ that satisfies $a\_n-(-1)^{n}a\_{n-1}=n$ $(n\geqslant 2)$, and $S\_n$ is the sum of the first $n$ terms of the sequence, find the value of $S\_{40}$. | 440 | 35.9375 |
21,227 | A mineralogist is hosting a competition to guess the age of an ancient mineral sample. The age is provided by the digits 2, 2, 3, 3, 5, and 9, with the condition that the age must start with an odd number. | 120 | 42.1875 |
21,228 | Let \(p\) and \(q\) be relatively prime positive integers such that \(\dfrac pq = \dfrac1{2^1} + \dfrac2{4^2} + \dfrac3{2^3} + \dfrac4{4^4} + \dfrac5{2^5} + \dfrac6{4^6} + \cdots\), where the numerators always increase by 1, and the denominators alternate between powers of 2 and 4, with exponents also increasing by 1 for each subsequent term. Compute \(p+q\). | 169 | 21.875 |
21,229 | There exist constants $b_1,$ $b_2,$ $b_3,$ $b_4$ such that
\[\sin^4 \theta = b_1 \sin \theta + b_2 \sin 2 \theta + b_3 \sin 3 \theta + b_4 \sin 4 \theta\]for all angles $\theta.$ Find $b_1^2 + b_2^2 + b_3^2 + b_4^2.$ | \frac{17}{64} | 30.46875 |
21,230 | Let a line passing through the origin \\(O\\) intersect a circle \\((x-4)^{2}+y^{2}=16\\) at point \\(P\\), and let \\(M\\) be the midpoint of segment \\(OP\\). Establish a polar coordinate system with the origin \\(O\\) as the pole and the positive half-axis of \\(x\\) as the polar axis.
\\((\\)Ⅰ\\()\\) Find the polar equation of the trajectory \\(C\\) of point \\(M\\);
\\((\\)Ⅱ\\()\\) Let the polar coordinates of point \\(A\\) be \\((3, \dfrac {π}{3})\\), and point \\(B\\) lies on curve \\(C\\). Find the maximum area of \\(\\triangle OAB\\). | 3+ \dfrac {3}{2} \sqrt {3} | 0 |
21,231 | Given that there exists a real number \( a \) such that the equation \( x \sqrt{a(x-a)} + y \sqrt{a(y-a)} = \sqrt{|\log(x-a) - \log(a-y)|} \) holds in the real number domain, find the value of \( \frac{3x^2 + xy - y^2}{x^2 - xy + y^2} \). | \frac{1}{3} | 10.9375 |
21,232 | A corporation plans to expand its sustainability team to include specialists in three areas: energy efficiency, waste management, and water conservation. The company needs 95 employees to specialize in energy efficiency, 80 in waste management, and 110 in water conservation. It is known that 30 employees will specialize in both energy efficiency and waste management, 35 in both waste management and water conservation, and 25 in both energy efficiency and water conservation. Additionally, 15 employees will specialize in all three areas. How many specialists does the company need to hire at minimum? | 210 | 83.59375 |
21,233 | What is the sum of all positive integers $n$ such that $\text{lcm}(2n, n^2) = 14n - 24$ ?
| 17 | 50.78125 |
21,234 | Given an ellipse $C:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with foci $F_{1}$ and $F_{2}$, point $A$ lies on $C$, point $B$ lies on the $y$-axis, and satisfies $\overrightarrow{AF_{1}}⊥\overrightarrow{BF_{1}}$, $\overrightarrow{AF_{2}}=\frac{2}{3}\overrightarrow{F_{2}B}$. Determine the eccentricity of $C$. | \frac{\sqrt{5}}{5} | 15.625 |
21,235 | In the Cartesian coordinate system $(xOy)$, the parametric equation of line $l$ is given by $\begin{cases}x=1+t\cos a \\ y= \sqrt{3}+t\sin a\end{cases} (t\text{ is a parameter})$, where $0\leqslant α < π$. In the polar coordinate system with $O$ as the pole and the positive half of the $x$-axis as the polar axis, the curve $C_{1}$ is defined by $ρ=4\cos θ$. The line $l$ is tangent to the curve $C_{1}$.
(1) Convert the polar coordinate equation of curve $C_{1}$ to Cartesian coordinates and determine the value of $α$;
(2) Given point $Q(2,0)$, line $l$ intersects with curve $C_{2}$: $x^{2}+\frac{{{y}^{2}}}{3}=1$ at points $A$ and $B$. Calculate the area of triangle $ABQ$. | \frac{6 \sqrt{2}}{5} | 0 |
21,236 | Evaluate the expression \((25 + 15)^2 - (25^2 + 15^2 + 150)\). | 600 | 39.0625 |
21,237 | Two circles have radius $2$ and $3$ , and the distance between their centers is $10$ . Let $E$ be the intersection of their two common external tangents, and $I$ be the intersection of their two common internal tangents. Compute $EI$ .
(A *common external tangent* is a tangent line to two circles such that the circles are on the same side of the line, while a *common internal tangent* is a tangent line to two circles such that the circles are on opposite sides of the line).
*Proposed by Connor Gordon)* | 24 | 6.25 |
21,238 | Given vectors $\overrightarrow{m}=(\sqrt{3}\cos x,-\cos x)$ and $\overrightarrow{n}=(\cos (x-\frac{π}{2}),\cos x)$, satisfying the function $f\left(x\right)=\overrightarrow{m}\cdot \overrightarrow{n}+\frac{1}{2}$.
$(1)$ Find the interval on which $f\left(x\right)$ is monotonically increasing on $[0,\frac{π}{2}]$.
$(2)$ If $f\left(\alpha \right)=\frac{5}{13}$, where $\alpha \in [0,\frac{π}{4}]$, find the value of $\cos 2\alpha$. | \frac{12\sqrt{3}-5}{26} | 58.59375 |
21,239 | Given $f(x)$ be a differentiable function, and $\lim_{\Delta x \to 0} \frac{{f(1)-f(1-2\Delta x)}}{{\Delta x}}=-1$, determine the slope of the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$. | -\frac{1}{2} | 55.46875 |
21,240 | How many distinct arrangements of the letters in the word "balloon" are there? | 1260 | 46.09375 |
21,241 | The rodent control task force went into the woods one day and caught $200$ rabbits and $18$ squirrels. The next day they went into the woods and caught $3$ fewer rabbits and two more squirrels than the day before. Each day they went into the woods and caught $3$ fewer rabbits and two more squirrels than the day before. This continued through the day when they caught more squirrels than rabbits. Up through that day how many rabbits did they catch in all? | 5491 | 39.84375 |
21,242 | A, B, and C are three people passing a ball to each other. The first pass is made by A, who has an equal chance of passing the ball to either of the other two people. After three passes, the probability that the ball is still with A is _______. | \frac{1}{4} | 56.25 |
21,243 | Given that the function $f(x)$ satisfies: $4f(x)f(y)=f(x+y)+f(x-y)$ $(x,y∈R)$ and $f(1)= \frac{1}{4}$, find $f(2014)$. | - \frac{1}{4} | 73.4375 |
21,244 | Let \( M = 35 \cdot 36 \cdot 65 \cdot 280 \). Calculate the ratio of the sum of the odd divisors of \( M \) to the sum of the even divisors of \( M \). | 1:62 | 0 |
21,245 | Let the function $f(x)=(x+a)\ln x$, $g(x)= \frac {x^{2}}{e^{x}}$, it is known that the tangent line of the curve $y=f(x)$ at the point $(1,f(1))$ is parallel to the line $2x-y=0$.
(Ⅰ) If the equation $f(x)=g(x)$ has a unique root within $(k,k+1)$ $(k\in\mathbb{N})$, find the value of $k$.
(Ⅱ) Let the function $m(x)=\min\{f(x),g(x)\}$ (where $\min\{p,q\}$ represents the smaller value between $p$ and $q$), find the maximum value of $m(x)$. | \frac {4}{e^{2}} | 0 |
21,246 | Given that $a>0$, the minimum value of the function $f(x) = e^{x-a} - \ln(x+a) - 1$ $(x>0)$ is 0. Determine the range of values for the real number $a$. | \{\frac{1}{2}\} | 0 |
21,247 | Calculate
(1) Use a simplified method to calculate $2017^{2}-2016 \times 2018$;
(2) Given $a+b=7$ and $ab=-1$, find the values of $(a+b)^{2}$ and $a^{2}-3ab+b^{2}$. | 54 | 92.96875 |
21,248 | In triangle $\triangle ABC$, it is known that the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, $c$, with $a=\sqrt{6}$, $\sin ^{2}B+\sin ^{2}C=\sin ^{2}A+\frac{2\sqrt{3}}{3}\sin A\sin B\sin C$. Choose one of the following conditions to determine whether triangle $\triangle ABC$ exists. If it exists, find the area of $\triangle ABC$; if it does not exist, please explain the reason.
① The length of the median $AD$ of side $BC$ is $\frac{\sqrt{10}}{2}$
② $b+c=2\sqrt{3}$
③ $\cos B=-\frac{3}{5}$. | \frac{\sqrt{3}}{2} | 19.53125 |
21,249 | A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is planning to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered. | 12\% | 64.84375 |
21,250 | The minimum value of \\(f(x)=\sin x+\cos x-\sin x\cos x\\) is | - \frac{1}{2}- \sqrt{2} | 53.90625 |
21,251 | A plane intersects a right circular cylinder of radius $3$ forming an ellipse. If the major axis of the ellipse is $75\%$ longer than the minor axis, the length of the major axis is: | 10.5 | 77.34375 |
21,252 | Among 7 students, 6 are to be arranged to participate in social practice activities in two communities on Saturday, with 3 people in each community. How many different arrangements are there? (Answer with a number) | 140 | 3.125 |
21,253 | Given the actual lighthouse's cylindrical base is 60 meters high, and the spherical top's volume is approximately 150,000 liters, and the miniature model's top holds around 0.15 liters, determine the height of Lara’s model lighthouse, in centimeters. | 60 | 63.28125 |
21,254 | How many numbers are in the list $-58, -51, -44, \ldots, 71, 78$? | 20 | 88.28125 |
21,255 | Given the sequence $\{a_n\}$, its first term is $7$, and $a_n= \frac{1}{2}a_{n-1}+3(n\geqslant 2)$, find the value of $a_6$. | \frac{193}{32} | 38.28125 |
21,256 | Regarding the value of \\(\pi\\), the history of mathematics has seen many creative methods for its estimation, such as the famous Buffon's Needle experiment and the Charles' experiment. Inspired by these, we can also estimate the value of \\(\pi\\) through designing the following experiment: ask \\(200\\) students, each to randomly write down a pair of positive real numbers \\((x,y)\\) both less than \\(1\\); then count the number of pairs \\((x,y)\\) that can form an obtuse triangle with \\(1\\) as the third side, denoted as \\(m\\); finally, estimate the value of \\(\pi\\) based on the count \\(m\\). If the result is \\(m=56\\), then \\(\pi\\) can be estimated as \_\_\_\_\_\_ (expressed as a fraction). | \dfrac {78}{25} | 14.0625 |
21,257 | Determine the smallest integer $k$ such that $k>1$ and $k$ has a remainder of $3$ when divided by any of $11,$ $4,$ and $3.$ | 135 | 74.21875 |
21,258 | A copper cube with an edge length of $l = 5 \text{ cm}$ is heated to a temperature of $t_{1} = 100^{\circ} \text{C}$. Then, it is placed on ice, which has a temperature of $t_{2} = 0^{\circ} \text{C}$. Determine the maximum depth the cube can sink into the ice. The specific heat capacity of copper is $c_{\text{s}} = 400 \text{ J/(kg}\cdot { }^{\circ} \text{C})$, the latent heat of fusion of ice is $\lambda = 3.3 \times 10^{5} \text{ J/kg}$, the density of copper is $\rho_{m} = 8900 \text{ kg/m}^3$, and the density of ice is $\rho_{n} = 900 \text{ kg/m}^3$. (10 points) | 0.06 | 3.125 |
21,259 | Pentagon $ANDD'Y$ has $AN \parallel DY$ and $AY \parallel D'N$ with $AN = D'Y$ and $AY = DN$ . If the area of $ANDY$ is 20, the area of $AND'Y$ is 24, and the area of $ADD'$ is 26, the area of $ANDD'Y$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ . Find $m+n$ .
*Proposed by Andy Xu* | 71 | 22.65625 |
21,260 | Two cylindrical poles, with diameters of $10$ inches and $30$ inches respectively, are placed side by side and bound together with a wire. Calculate the length of the shortest wire that will go around both poles.
**A)** $20\sqrt{3} + 24\pi$
**B)** $20\sqrt{3} + \frac{70\pi}{3}$
**C)** $30\sqrt{3} + 22\pi$
**D)** $16\sqrt{3} + 25\pi$
**E)** $18\sqrt{3} + \frac{60\pi}{3}$ | 20\sqrt{3} + \frac{70\pi}{3} | 35.15625 |
21,261 | Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a Queen and the second card is a $\diamondsuit$? | \frac{52}{221} | 0 |
21,262 | In triangle $\triangle ABC$, $\angle BAC = \frac{π}{3}$, $D$ is the midpoint of $AB$, $P$ is a point on segment $CD$, and satisfies $\overrightarrow{AP} = t\overrightarrow{AC} + \frac{1}{3}\overrightarrow{AB}$. If $|\overrightarrow{BC}| = \sqrt{6}$, then the maximum value of $|\overrightarrow{AP}|$ is ______. | \sqrt{2} | 7.03125 |
21,263 | Let $F_{1}$ and $F_{2}$ be the two foci of the hyperbola $\dfrac {x^{2}}{4}- \dfrac {y^{2}}{b^{2}}=1$. Point $P$ is on the hyperbola and satisfies $\angle F_{1}PF_{2}=90^{\circ}$. If the area of $\triangle F_{1}PF_{2}$ is $2$, find the value of $b$. | \sqrt {2} | 0 |
21,264 | Given real numbers $a$, $b$, $c$, and $d$ satisfy $(b + 2a^2 - 6\ln a)^2 + |2c - d + 6| = 0$, find the minimum value of $(a - c)^2 + (b - d)^2$. | 20 | 7.03125 |
21,265 | Without using a calculator, find the largest prime factor of \( 17^4 + 2 \times 17^2 + 1 - 16^4 \). | 17 | 78.90625 |
21,266 | In the list where each integer $n$ appears $n$ times for $1 \leq n \leq 300$, find the median of the numbers. | 212 | 23.4375 |
21,267 | $x_{n+1}= \left ( 1+\frac2n \right )x_n+\frac4n$ , for every positive integer $n$ . If $x_1=-1$ , what is $x_{2000}$ ? | 2000998 | 2.34375 |
21,268 | In $\Delta ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given that $b=2$, $c=2\sqrt{2}$, and $C=\frac{\pi}{4}$, find the area of $\Delta ABC$. | \sqrt{3} +1 | 14.84375 |
21,269 | Given 50 feet of fencing, where 5 feet is used for a gate that does not contribute to the enclosure area, what is the greatest possible number of square feet in the area of a rectangular pen enclosed by the remaining fencing? | 126.5625 | 78.90625 |
21,270 | A conference hall is setting up seating for a workshop. Each row must contain $13$ chairs, and initially, there are $169$ chairs in total. The organizers expect $95$ participants to attend the workshop. To ensure all rows are completely filled with minimal empty seats, how many chairs should be removed? | 65 | 23.4375 |
21,271 | Suppose that $N$ is a three digit number divisible by $7$ such that upon removing its middle digit, the remaining two digit number is also divisible by $7$ . What is the minimum possible value of $N$ ?
*2019 CCA Math Bonanza Lightning Round #3.1* | 154 | 100 |
21,272 | For the inequality system about $y$ $\left\{\begin{array}{l}{2y-6≤3(y-1)}\\{\frac{1}{2}a-3y＞0}\end{array}\right.$, if it has exactly $4$ integer solutions, then the product of all integer values of $a$ that satisfy the conditions is ______. | 720 | 62.5 |
21,273 | (1) Given $\sin \left( \frac{\pi }{3}-\alpha \right)=\frac{1}{2}$, find the value of $\cos \left( \frac{\pi }{6}+\alpha \right)$;
(2) Given $\cos \left( \frac{5\pi }{12}+\alpha \right)=\frac{1}{3}$ and $-\pi < \alpha < -\frac{\pi }{2}$, find the value of $\cos \left( \frac{7\pi }{12}-\alpha \right)+\sin \left( \alpha -\frac{7\pi }{12} \right)$: | - \frac{1}{3}+ \frac{2 \sqrt{2}}{3} | 0.78125 |
21,274 | Given that point $(a, b)$ moves on the line $x + 2y + 3 = 0$, find the maximum or minimum value of $2^a + 4^b$. | \frac{\sqrt{2}}{2} | 55.46875 |
21,275 | How many different positive three-digit integers can be formed using only the digits in the set $\{1, 3, 4, 4, 7, 7, 7\}$ if no digit may be used more times than it appears in the given set of available digits? | 43 | 57.03125 |
21,276 | Distribute 5 students into two dormitories, A and B, with each dormitory accommodating at least 2 students. Find the number of distinct arrangements. | 20 | 59.375 |
21,277 | Given that $m$ is an integer and $0<3m<27$, what is the sum of all possible integer values of $m$? | 36 | 92.1875 |
21,278 | In the Cartesian coordinate system $xOy$, the parametric equation of curve $C$ is $\begin{cases} & x=3\cos \alpha \\ & y=\sin \alpha \end{cases}$ ($\alpha$ is the parameter), and in the polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, the polar equation of line $l$ is $\rho \sin (\theta -\dfrac{\pi }{4})=\sqrt{2}$.
$(1)$ Find the general equation of curve $C$ and the inclination angle of line $l$;
$(2)$ Let point $P(0,2)$, line $l$ intersects curve $C$ at points $A$ and $B$, find $|PA|+|PB|$． | \dfrac {18 \sqrt {2}}{5} | 0 |
21,279 | Given plane vectors $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ satisfy $\langle \overrightarrow{a}, \overrightarrow{b} \rangle = 60^\circ$, and $\{|\overrightarrow{a}|, |\overrightarrow{b}|, |\overrightarrow{c}|\} = \{1, 2, 3\}$, calculate the maximum value of $|\overrightarrow{a}+ \overrightarrow{b}+ \overrightarrow{c}|$. | \sqrt{7}+3 | 0.78125 |
21,280 | The symbol \( R_{k} \) represents an integer whose decimal representation consists of \( k \) consecutive 1s. For example, \( R_{3} = 111 \), \( R_{5} = 11111 \), and so on. If \( R_{4} \) divides \( R_{24} \), the quotient \( Q = \frac{R_{24}}{R_{4}} \) is an integer, and its decimal representation contains only the digits 1 and 0. How many zeros are in \( Q \)? | 15 | 33.59375 |
21,281 | Find the value of the algebraic cofactor of element $a$ in the determinant \\( \begin{vmatrix} 3 & a & 5 \\\\ 0 & -4 & 1 \\\\ -2 & 1 & 3\\end{vmatrix} \\). | -2 | 66.40625 |
21,282 | Given the quadratic function $f(x)=ax^{2}+(2b+1)x-a-2 (a,b \in R, a \neq 0)$ has at least one root in the interval $[3,4]$, calculate the minimum value of $a^{2}+b^{2}$. | \frac{1}{100} | 39.84375 |
21,283 | The new PUMaC tournament hosts $2020$ students, numbered by the following set of labels $1, 2, . . . , 2020$ . The students are initially divided up into $20$ groups of $101$ , with each division into groups equally likely. In each of the groups, the contestant with the lowest label wins, and the winners advance to the second round. Out of these $20$ students, we chose the champion uniformly at random. If the expected value of champion’s number can be written as $\frac{a}{b}$ , where $a, b$ are relatively prime integers, determine $a + b$ . | 2123 | 13.28125 |
21,284 | In $\triangle ABC$, $2\sin 2A\cos A-\sin 3A+\sqrt{3}\cos A=\sqrt{3}$.
(1) Find the measure of angle $A$;
(2) Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, if $a=1$ and $\sin A+\sin (B-C)=2\sin 2C$, find the area of $\triangle ABC$. | \frac{\sqrt{3}}{6} | 39.84375 |
21,285 | Given that $α∈(0,π)$, and $\sin α + \cos α = \frac{\sqrt{2}}{2}$, find the value of $\sin α - \cos α$. | \frac{\sqrt{6}}{2} | 89.0625 |
21,286 | There is a solid iron cone with a base radius of $3cm$ and a slant height of $5cm$. After melting it at high temperature and casting it into a solid iron sphere (without considering any loss), the radius of this iron sphere is _______ $cm$. | \sqrt[3]{9} | 89.0625 |
21,287 | If a worker receives a 30% cut in wages, calculate the percentage raise he needs to regain his original pay. | 42.857\% | 5.46875 |
21,288 | Calculate and simplify $\sqrt[4]{3^5 \cdot 5^3}$ to the form $c\sqrt[4]{d}$, where $c$ and $d$ are positive integers. What is $c+d$? | 378 | 95.3125 |
21,289 | Given the function $f(x)=\frac{1}{3}x^{3}+ax^{2}+bx$ $(a,b\in \mathbb{R})$ attains a maximum value of $9$ at $x=-3$.
$(I)$ Find the values of $a$ and $b$;
$(II)$ Find the maximum and minimum values of the function $f(x)$ in the interval $[-3,3]$. | - \frac{5}{3} | 39.0625 |
21,290 |
A polynomial \( P(x) \) of degree 10 with a leading coefficient of 1 is given. The graph of \( y = P(x) \) lies entirely above the x-axis. The polynomial \( -P(x) \) was factored into irreducible factors (i.e., polynomials that cannot be represented as the product of two non-constant polynomials). It is known that at \( x = 2020 \), all the resulting irreducible polynomials take the value -3. Find \( P(2020) \). | 243 | 28.125 |
21,291 | Calculate using your preferred method:
(1) $42.67-(12.95-7.33)$
(2) $\left[8.4-8.4\times(3.12-3.7)\right]\div0.42$
(3) $5.13\times0.23+8.7\times0.513-5.13$
(4) $6.66\times222+3.33\times556$ | 3330 | 45.3125 |
21,292 | How many ways are there to put 7 balls in 2 boxes if the balls are distinguishable but the boxes are not? | 64 | 73.4375 |
21,293 | A triangle is divided into 1000 triangles. What is the maximum number of distinct points that can be vertices of these triangles? | 1002 | 14.84375 |
21,294 | Define the determinant $D_1$ = $|1|$ , the determinant $D_2$ = $|1 1|$ $|1 3|$
, and the determinant $D_3=$
|1 1 1|
|1 3 3|
|1 3 5|
.
In general, for positive integer n, let the determinant $D_n$ have 1s in every position of its first row and first
column, 3s in the remaining positions of the second row and second column, 5s in the remaining positions of the third row and third column, and so forth. Find the least n so that $D_n$ $\geq$ 2015. | 12 | 56.25 |
21,295 | Calculate: $|1-\sqrt{2}|+(\frac{1}{2})^{-2}-\left(\pi -2023\right)^{0}$. | \sqrt{2} + 2 | 89.0625 |
21,296 | Given the function $f(x)=x^{2}-2ax+{b}^{2}$, where $a$ and $b$ are real numbers.
(1) If $a$ is taken from the set $\{0,1,2,3\}$ and $b$ is taken from the set $\{0,1,2\}$, find the probability that the equation $f(x)=0$ has two distinct real roots.
(2) If $a$ is taken from the interval $[0,2]$ and $b$ is taken from the interval $[0,3]$, find the probability that the equation $f(x)=0$ has no real roots. | \frac{2}{3} | 39.84375 |
21,297 | In the Cartesian coordinate system $xOy$, a moving line $l$: $y=x+m$ intersects the parabola $C$: $x^2=2py$ ($p>0$) at points $A$ and $B$, and $\overrightarrow {OA}\cdot \overrightarrow {OB}=m^{2}-2m$.
1. Find the equation of the parabola $C$.
2. Let $P$ be the point where the line $y=x$ intersects $C$ (and $P$ is different from the origin), and let $D$ be the intersection of the tangent line to $C$ at $P$ and the line $l$. Define $t= \frac {|PD|^{2}}{|DA|\cdot |DB|}$. Is $t$ a constant value? If so, compute its value; otherwise, explain why it's not constant. | \frac{5}{2} | 28.125 |
21,298 | Nine lines parallel to the base of a triangle divide the other sides each into $10$ equal segments and the area into $10$ distinct parts. If the area of the largest of these parts is $38$, calculate the area of the original triangle. | 200 | 49.21875 |
21,299 | Find the sum of the squares of the solutions to
\[\left| x^2 - x + \frac{1}{2010} \right| = \frac{1}{2010}.\] | \frac{2008}{1005} | 35.9375 |
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