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|---|---|---|---|
28,200 | Simplify $\sqrt[3]{1+27} \cdot \sqrt[3]{1+\sqrt[3]{27}} \cdot \sqrt{4}$. | 2 \cdot \sqrt[3]{112} | 14.0625 |
28,201 | Farmer Tim is lost in the densely-forested Cartesian plane. Starting from the origin he walks a sinusoidal path in search of home; that is, after $t$ minutes he is at position $(t,\sin t)$ .
Five minutes after he sets out, Alex enters the forest at the origin and sets out in search of Tim. He walks in such a way that after he has been in the forest for $m$ minutes, his position is $(m,\cos t)$ .
What is the greatest distance between Alex and Farmer Tim while they are walking in these paths? | 3\sqrt{3} | 5.46875 |
28,202 | In the trapezoid \(ABCD\), the bases are given as \(AD = 4\) and \(BC = 1\), and the angles at \(A\) and \(D\) are \(\arctan 2\) and \(\arctan 3\) respectively.
Find the radius of the circle inscribed in triangle \(CBE\), where \(E\) is the intersection point of the diagonals of the trapezoid. | \frac{18}{25 + 2 \sqrt{130} + \sqrt{445}} | 0.78125 |
28,203 | Given the inequality about $x$, $2\log_2^2x - 5\log_2x + 2 \leq 0$, the solution set is $B$.
1. Find set $B$.
2. If $x \in B$, find the maximum and minimum values of $f(x) = \log_2 \frac{x}{8} \cdot \log_2 (2x)$. | -4 | 34.375 |
28,204 | Given that both $α$ and $β$ are acute angles, and $\cos(α+β)= \frac{\sin α}{\sin β}$, find the maximum value of $\tan α$. | \frac{ \sqrt {2}}{4} | 0 |
28,205 | The highest power of 2 that is a factor of \(15.13^{4} - 11^{4}\) needs to be determined. | 32 | 10.9375 |
28,206 | In the diagram, $R$ is on $QS$ and $QR=8$.
Also, $PR=12$, $\angle PRQ=120^{\circ}$, and $\angle RPS=90^{\circ}$.
What is the area of $\triangle QPS$? | $96 \sqrt{3}$ | 0 |
28,207 | Let $p = 101$ and let $S$ be the set of $p$ -tuples $(a_1, a_2, \dots, a_p) \in \mathbb{Z}^p$ of integers. Let $N$ denote the number of functions $f: S \to \{0, 1, \dots, p-1\}$ such that
- $f(a + b) + f(a - b) \equiv 2\big(f(a) + f(b)\big) \pmod{p}$ for all $a, b \in S$ , and
- $f(a) = f(b)$ whenever all components of $a-b$ are divisible by $p$ .
Compute the number of positive integer divisors of $N$ . (Here addition and subtraction in $\mathbb{Z}^p$ are done component-wise.)
*Proposed by Ankan Bhattacharya* | 5152 | 0 |
28,208 | (Geometry Proof Exercise) From a point A outside a circle ⊙O with radius 2, draw a line intersecting ⊙O at points C and D. A line segment AB is tangent to ⊙O at B. Given that AC=4 and AB=$4 \sqrt {2}$, find $tan∠DAB$. | \frac { \sqrt {2}}{4} | 0 |
28,209 | Two teams, Team A and Team B, are playing in a basketball finals series that uses a "best of seven" format (the first team to win four games wins the series and the finals end). Based on previous game results, Team A's home and away schedule is arranged as "home, home, away, away, home, away, home". The probability of Team A winning at home is 0.6, and the probability of winning away is 0.5. The results of each game are independent of each other. Calculate the probability that Team A wins the series with a 4:1 score. | 0.18 | 14.0625 |
28,210 | Given a geometric sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, and $a_1=2$, if $\frac {S_{6}}{S_{2}}=21$, then the sum of the first five terms of the sequence $\{\frac {1}{a_n}\}$ is
A) $\frac {1}{2}$ or $\frac {11}{32}$
B) $\frac {1}{2}$ or $\frac {31}{32}$
C) $\frac {11}{32}$ or $\frac {31}{32}$
D) $\frac {11}{32}$ or $\frac {5}{2}$ | \frac {31}{32} | 12.5 |
28,211 | Given the function $f\left( x \right)={x}^{2}+{\left( \ln 3x \right)}^{2}-2a(x+3\ln 3x)+10{{a}^{2}}(a\in \mathbf{R})$, determine the value of the real number $a$ for which there exists ${{x}_{0}}$ such that $f\left( {{x}_{0}} \right)\leqslant \dfrac{1}{10}$. | \frac{1}{30} | 2.34375 |
28,212 | There are 3 different pairs of shoes in a shoe cabinet. If one shoe is picked at random from the left shoe set of 6 shoes, and then another shoe is picked at random from the right shoe set of 6 shoes, calculate the probability that the two shoes form a pair. | \frac{1}{3} | 0.78125 |
28,213 | Given a parameterized curve $ C: x\equal{}e^t\minus{}e^{\minus{}t},\ y\equal{}e^{3t}\plus{}e^{\minus{}3t}$ .
Find the area bounded by the curve $ C$ , the $ x$ axis and two lines $ x\equal{}\pm 1$ . | \frac{5\sqrt{5}}{2} | 0.78125 |
28,214 | A point $Q$ lies inside the triangle $\triangle DEF$ such that lines drawn through $Q$ parallel to the sides of $\triangle DEF$ divide it into three smaller triangles $u_1$, $u_2$, and $u_3$ with areas $16$, $25$, and $36$ respectively. Determine the area of $\triangle DEF$. | 77 | 6.25 |
28,215 | The minimum positive period and the minimum value of the function $y=2\sin(2x+\frac{\pi}{6})+1$ are \_\_\_\_\_\_ and \_\_\_\_\_\_, respectively. | -1 | 72.65625 |
28,216 | How many 5-letter words with at least one consonant can be constructed from the letters $A$, $B$, $C$, $D$, $E$, $F$, $G$, and $I$? Each letter can be used more than once, and $B$, $C$, $D$, $F$, $G$ are consonants. | 32525 | 0 |
28,217 | What is the least positive integer $n$ such that $7350$ is a factor of $n!$? | 10 | 1.5625 |
28,218 | Given an ellipse $M: \frac{x^2}{a^2} + \frac{y^2}{3} = 1 (a > 0)$ with one of its foci at $F(-1, 0)$. Points $A$ and $B$ are the left and right vertices of the ellipse's major axis, respectively. A line $l$ passes through $F$ and intersects the ellipse at distinct points $C$ and $D$.
1. Find the equation of the ellipse $M$;
2. When the line $l$ has an angle of $45^{\circ}$, find the length of the line segment $CD$;
3. Let $S_1$ and $S_2$ represent the areas of triangles $\Delta ABC$ and $\Delta ABD$, respectively. Find the maximum value of $|S_1 - S_2|$. | \sqrt{3} | 0 |
28,219 | Find the greatest common divisor of $8!$ and $(6!)^2.$ | 5760 | 80.46875 |
28,220 | The Greenhill Soccer Club has 25 players, including 4 goalies. During an upcoming practice, the team plans to have a competition in which each goalie will try to stop penalty kicks from every other player, including the other goalies. How many penalty kicks are required for every player to have a chance to kick against each goalie? | 96 | 39.0625 |
28,221 | A dark drawer contains $90$ red socks, $70$ green socks, $50$ blue socks, and $30$ purple socks. Someone randomly selects socks without seeing their color. What is the smallest number of socks that they must pick to guarantee at least $12$ pairs? A pair of socks consists of two socks of the same color. | 27 | 1.5625 |
28,222 | $A,B,C$ are points in the plane such that $\angle ABC=90^\circ$ . Circles with diameters $BA$ and $BC$ meet at $D$ . If $BA=20$ and $BC=21$ , then the length of segment $BD$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ?
*Ray Li* | 449 | 71.09375 |
28,223 | Each two-digit is number is coloured in one of $k$ colours. What is the minimum value of $k$ such that, regardless of the colouring, there are three numbers $a$ , $b$ and $c$ with different colours with $a$ and $b$ having the same units digit (second digit) and $b$ and $c$ having the same tens digit (first digit)? | 11 | 19.53125 |
28,224 | Let \( M = \{1, 2, \cdots, 10\} \) and let \( T \) be a collection of certain two-element subsets of \( M \), such that for any two different elements \(\{a, b\} \) and \(\{x, y\} \) in \( T \), the condition \( 11 \nmid (ax + by)(ay + bx) \) is satisfied.
Find the maximum number of elements in \( T \). | 25 | 95.3125 |
28,225 | In the Cartesian coordinate system $xOy$, a line segment of length $\sqrt{2}+1$ has its endpoints $C$ and $D$ sliding on the $x$-axis and $y$-axis, respectively. It is given that $\overrightarrow{CP} = \sqrt{2} \overrightarrow{PD}$. Let the trajectory of point $P$ be curve $E$.
(I) Find the equation of curve $E$;
(II) A line $l$ passing through point $(0,1)$ intersects curve $E$ at points $A$ and $B$, and $\overrightarrow{OM} = \overrightarrow{OA} + \overrightarrow{OB}$. When point $M$ is on curve $E$, find the area of quadrilateral $OAMB$. | \frac{\sqrt{6}}{2} | 9.375 |
28,226 | Given points $A(2, 0)$, $B(0, 2)$, $C(\cos\alpha, \sin\alpha)$ and $0 < \alpha < \pi$:
1. If $|\vec{OA} + \vec{OC}| = \sqrt{7}$, find the angle between $\vec{OB}$ and $\vec{OC}$.
2. If $\vec{AC} \perp \vec{BC}$, find the value of $\cos\alpha$. | \frac{1 + \sqrt{7}}{4} | 7.03125 |
28,227 | Find an axis of symmetry for the function $f(x) = \cos(2x + \frac{\pi}{6})$. | \frac{5\pi}{12} | 0.78125 |
28,228 | Suppose 9 people are arranged in a line randomly. What is the probability that person A is in the middle, and persons B and C are adjacent? | \frac{1}{42} | 0 |
28,229 | Let the random variable $\xi$ follow the normal distribution $N(1, \sigma^2)$ ($\sigma > 0$). If $P(0 < \xi < 1) = 0.4$, then find the value of $P(\xi > 2)$. | 0.2 | 0 |
28,230 | Given that $min\{ a,b\}$ represents the smaller value between the real numbers $a$ and $b$, and the vectors $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ satisfy $(\vert\overrightarrow{a}\vert=1,\vert\overrightarrow{b}\vert=2,\overrightarrow{a}\cdot\overrightarrow{b}=0,\overrightarrow{c}=\lambda\overrightarrow{a}+\mu\overrightarrow{b}(\lambda+\mu=1))$, find the maximum value of $min\{\overrightarrow{c}\cdot\overrightarrow{a}, \overrightarrow{c}\cdot\overrightarrow{b}\}$ and the value of $\vert\overrightarrow{c}\vert$. | \frac{2\sqrt{5}}{5} | 92.96875 |
28,231 | Two congruent 30-60-90 triangles are placed such that one triangle is translated 2 units vertically upwards, while their hypotenuses originally coincide when not translated. The hypotenuse of each triangle is 10. Calculate the area common to both triangles when one is translated. | 25\sqrt{3} - 10 | 6.25 |
28,232 | All of the roots of $x^3+ax^2+bx+c$ are positive integers greater than $2$ , and the coefficients satisfy $a+b+c+1=-2009$ . Find $a$ | -58 | 62.5 |
28,233 | Determine the constants $\alpha$ and $\beta$ such that $\frac{x-\alpha}{x+\beta} = \frac{x^2 - 64x + 975}{x^2 + 99x - 2200}$. What is $\alpha+\beta$? | 138 | 0 |
28,234 | Given the function $f(x)=\begin{cases} 2^{x}, & x < 0 \\ f(x-1)+1, & x\geqslant 0 \end{cases}$, calculate the value of $f(2)$. | \dfrac{5}{2} | 0 |
28,235 | A ferry boat shuttles tourists to an island every half-hour from 10 AM to 3 PM, with 100 tourists on the first trip and 2 fewer tourists on each successive trip. Calculate the total number of tourists taken to the island that day. | 990 | 3.90625 |
28,236 | Under normal circumstances, for people aged between 18 and 38 years old, the regression equation for weight $y$ (kg) based on height $x$ (cm) is $y=0.72x-58.5$. Zhang Honghong, who is neither fat nor thin, has a height of 1.78 meters. His weight should be around \_\_\_\_\_ kg. | 70 | 10.9375 |
28,237 | Consider a revised dataset given in the following stem-and-leaf plot, where $7|1$ represents 71:
\begin{tabular}{|c|c|}\hline
\textbf{Tens} & \textbf{Units} \\ \hline
2 & $0 \hspace{2mm} 0 \hspace{2mm} 1 \hspace{2mm} 1 \hspace{2mm} 2$ \\ \hline
3 & $3 \hspace{2mm} 6 \hspace{2mm} 6 \hspace{2mm} 7$ \\ \hline
4 & $3 \hspace{2mm} 5 \hspace{2mm} 7 \hspace{2mm} 9$ \\ \hline
6 & $2 \hspace{2mm} 4 \hspace{2mm} 5 \hspace{2mm} 6 \hspace{2mm} 8$ \\ \hline
7 & $1 \hspace{2mm} 3 \hspace{2mm} 5 \hspace{2mm} 9$ \\ \hline
\end{tabular}
What is the positive difference between the median and the mode of the new dataset? | 23 | 0 |
28,238 | Given $|\vec{a}|=1$, $|\vec{b}|= \sqrt{2}$, and $\vec{a} \perp (\vec{a} - \vec{b})$, find the angle between the vectors $\vec{a}$ and $\vec{b}$. | \frac{\pi}{4} | 100 |
28,239 | As shown in the picture, the knight can move to any of the indicated squares of the $8 \times 8$ chessboard in 1 move. If the knight starts from the position shown, find the number of possible landing positions after 20 consecutive moves. | 32 | 3.125 |
28,240 | Given that a tetrahedron $ABCD$ is inscribed in a sphere $O$, and $AD$ is the diameter of the sphere $O$. If triangles $\triangle ABC$ and $\triangle BCD$ are equilateral triangles with side length 1, calculate the volume of tetrahedron $ABCD$. | \frac{\sqrt{3}}{12} | 7.03125 |
28,241 | Given the tower function $T(n)$ defined by $T(1) = 3$ and $T(n + 1) = 3^{T(n)}$ for $n \geq 1$, calculate the largest integer $k$ for which $\underbrace{\log_3\log_3\log_3\ldots\log_3B}_{k\text{ times}}$ is defined, where $B = (T(2005))^A$ and $A = (T(2005))^{T(2005)}$. | 2005 | 31.25 |
28,242 | A marathon of 42 km started at 11:30 AM and the winner finished at 1:45 PM on the same day. What was the average speed of the winner, in km/h? | 18.6 | 0 |
28,243 | What is the least positive integer with exactly $12$ positive factors? | 96 | 0 |
28,244 | Let $ABCD$ be a square with side length $16$ and center $O$ . Let $\mathcal S$ be the semicircle with diameter $AB$ that lies outside of $ABCD$ , and let $P$ be a point on $\mathcal S$ so that $OP = 12$ . Compute the area of triangle $CDP$ .
*Proposed by Brandon Wang* | 120 | 3.125 |
28,245 | A parking lot in Flower Town is a square with $7 \times 7$ cells, each of which can accommodate a car. The parking lot is enclosed by a fence, and one of the corner cells has an open side (this is the gate). Cars move along paths that are one cell wide. Neznaika was asked to park as many cars as possible in such a way that any car can exit while the others remain parked. Neznaika parked 24 cars as shown in the diagram. Try to arrange the cars differently so that more can fit. | 28 | 1.5625 |
28,246 | A 10 by 10 checkerboard has alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (horizontal and vertical) and containing at least 6 black squares, can be drawn on the checkerboard? | 140 | 12.5 |
28,247 | Given that the random variable $X$ follows a normal distribution $N(2,\sigma^{2})$, and its normal distribution density curve is the graph of the function $f(x)$, and $\int_{0}^{2} f(x)dx=\dfrac{1}{3}$, calculate $P(x > 4)$. | \dfrac{1}{3} | 32.8125 |
28,248 | Monsieur and Madame Dubois are traveling from Paris to Deauville, where their children live. Each is driving their own car. They depart together and arrive in Deauville at the same time. However, Monsieur Dubois spent on stops one-third of the time during which his wife continued driving, while Madame Dubois spent on stops one-quarter of the time during which her husband was driving.
What is the ratio of the average speeds of each of their cars? | 8/9 | 0 |
28,249 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$. It is known that $c\sin A= \sqrt {3}a\cos C$.
(I) Find $C$;
(II) If $c= \sqrt {7}$ and $\sin C+\sin (B-A)=3\sin 2A$, find the area of $\triangle ABC$. | \frac {3 \sqrt {3}}{4} | 0 |
28,250 | Given vectors $\overrightarrow{a}=(1, -2)$ and $\overrightarrow{b}=(3, 4)$, the projection of vector $\overrightarrow{a}$ in the direction of vector $\overrightarrow{b}$ is ______. | -1 | 0 |
28,251 | If the seven digits 1, 1, 3, 5, 5, 5, and 9 are arranged to form a seven-digit positive integer, what is the probability that the integer is divisible by 25? | \frac{1}{14} | 7.03125 |
28,252 | How many of the numbers from the set $\{1,\ 2,\ 3,\ldots,\ 100\}$ have a perfect square factor other than one? | 41 | 0 |
28,253 | A dice is thrown twice. Let $a$ be the number of dots that appear on the first throw and $b$ be the number of dots that appear on the second throw. Given the system of equations $\begin{cases} ax+by=2 \\\\ 2x+y=3\\end{cases}$.
(I) Find the probability that the system of equations has only one solution.
(II) If each solution of the system of equations corresponds to a point $P(x,y)$ in the Cartesian plane, find the probability that point $P$ lies in the fourth quadrant. | \frac{7}{12} | 0.78125 |
28,254 | Given a certain DNA fragment consists of 500 base pairs, with A+T making up 34% of the total number of bases, calculate the total number of free cytosine deoxyribonucleotide molecules required when this DNA fragment is replicated twice. | 1320 | 4.6875 |
28,255 | In an isosceles trapezoid, the larger base is equal to the sum of the smaller base and the length of the altitude, and every diagonal is equal to the length of the smaller base plus half the altitude. Find the ratio of the smaller base to the larger base.
A) $\frac{1}{2}$
B) $\frac{\sqrt{5}}{2}$
C) $\frac{2+\sqrt{5}}{4}$
D) $\frac{2-\sqrt{5}}{2}$
E) $\frac{3-\sqrt{5}}{2}$ | \frac{2-\sqrt{5}}{2} | 0.78125 |
28,256 | How many positive odd integers greater than 1 and less than $200$ are square-free? | 79 | 0 |
28,257 | If point \( P \) is the circumcenter of \(\triangle ABC\) and \(\overrightarrow{PA} + \overrightarrow{PB} + \lambda \overrightarrow{PC} = \mathbf{0}\), where \(\angle C = 120^\circ\), then find the value of the real number \(\lambda\). | -1 | 21.09375 |
28,258 | Let $\triangle ABC$ be a triangle in the plane, and let $D$ be a point outside the plane of $\triangle ABC$, so that $DABC$ is a pyramid whose faces are all triangles.
Suppose that every edge of $DABC$ has length $20$ or $45$, but no face of $DABC$ is equilateral. Then what is the surface area of $DABC$? | 40 \sqrt{1925} | 0 |
28,259 | A row consists of 10 chairs, but chair #5 is broken and cannot be used. Mary and James each sit in one of the available chairs, choosing their seats at random from the remaining chairs. What is the probability that they don't sit next to each other? | \frac{5}{6} | 3.125 |
28,260 | In triangle \(ABC\), angle bisectors \(AA_{1}\), \(BB_{1}\), and \(CC_{1}\) are drawn. \(L\) is the intersection point of segments \(B_{1}C_{1}\) and \(AA_{1}\), \(K\) is the intersection point of segments \(B_{1}A_{1}\) and \(CC_{1}\). Find the ratio \(LM: MK\) if \(M\) is the intersection point of angle bisector \(BB_{1}\) with segment \(LK\), and \(AB: BC: AC = 2: 3: 4\). (16 points) | 11/12 | 2.34375 |
28,261 | Let $N \ge 5$ be given. Consider all sequences $(e_1,e_2,...,e_N)$ with each $e_i$ equal to $1$ or $-1$ . Per move one can choose any five consecutive terms and change their signs. Two sequences are said to be similar if one of them can be transformed into the other in finitely many moves. Find the maximum number of pairwise non-similar sequences of length $N$ .
| 16 | 1.5625 |
28,262 | For some real number $c,$ the graphs of the equation $y=|x-20|+|x+18|$ and the line $y=x+c$ intersect at exactly one point. What is $c$ ? | 18 | 46.875 |
28,263 | Determine the time in hours it will take to fill a 32,000 gallon swimming pool using three hoses that deliver 3 gallons of water per minute. | 59 | 0 |
28,264 | Calculate $x$ such that the sum \[1 \cdot 1979 + 2 \cdot 1978 + 3 \cdot 1977 + \dots + 1978 \cdot 2 + 1979 \cdot 1 = 1979 \cdot 990 \cdot x.\] | 661 | 0 |
28,265 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $b\sin^{2}\frac{A}{2}+a\sin^{2}\frac{B}{2}=\frac{C}{2}$.
1. If $c=2$, find the perimeter of $\triangle ABC$.
2. If $C=\frac{\pi}{3}$ and the area of $\triangle ABC$ is $2\sqrt{3}$, find $c$. | 2\sqrt{2} | 24.21875 |
28,266 | An acute isosceles triangle, $ABC$, is inscribed in a circle. Through $B$ and $C$, tangents to the circle are drawn, meeting at point $D$. If $\angle ABC = \angle ACB = 3 (\angle D$) and $\angle BAC = t \pi$ in radians, then find $t$.
[asy]
import graph;
unitsize(2 cm);
pair O, A, B, C, D;
O = (0,0);
A = dir(90);
B = dir(-30);
C = dir(210);
D = extension(B, B + rotate(90)*(B), C, C + rotate(90)*(C));
draw(Circle(O,1));
draw(A--B--C--cycle);
draw(B--D--C);
label("$A$", A, N);
label("$B$", B, SE);
label("$C$", C, SW);
label("$D$", D, S);
[/asy] | \frac{5}{11} | 7.03125 |
28,267 | Given that $21^{-1} \equiv 15 \pmod{61}$, find $40^{-1} \pmod{61}$, as a residue modulo 61. (Provide a number between 0 and 60, inclusive.) | 46 | 78.90625 |
28,268 | In triangle $PQR$, angle $R$ is a right angle and the altitude from $R$ meets $\overline{PQ}$ at $S$. The lengths of the sides of $\triangle PQR$ are integers, $PS=17^3$, and $\cos Q = a/b$, where $a$ and $b$ are relatively prime positive integers. Find $a+b$. | 18 | 2.34375 |
28,269 | A spider is on the edge of a ceiling of a circular room with a radius of 65 feet. The spider moves straight across the ceiling to the opposite edge, passing through the circle's center. It then moves directly to another point on the edge of the circle, not passing through the center. The final segment of the journey is straight back to the starting point and is 90 feet long. How many total feet did the spider travel during the entire journey? | 220 + 20\sqrt{22} | 29.6875 |
28,270 | Let $S$ be the set of integers between $1$ and $2^{40}$ whose binary expansions have exactly two $1$'s. If a number is chosen at random from $S,$ the probability that it is divisible by $15$ is $p/q,$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$ | 49 | 3.90625 |
28,271 | Let $T$ be a triangle with side lengths $1, 1, \sqrt{2}$. Two points are chosen independently at random on the sides of $T$. The probability that the straight-line distance between the points is at least $\dfrac{\sqrt{2}}{2}$ is $\dfrac{d-e\pi}{f}$, where $d$, $e$, and $f$ are positive integers with $\gcd(d,e,f)=1$. What is $d+e+f$? | 17 | 3.90625 |
28,272 | Given the ellipse $c_{1}$: $\frac{x^{2}}{8} + \frac{y^{2}}{4} = 1$ with left and right focal points $F_{1}$ and $F_{2}$, a line $l_{1}$ is drawn through point $F_{1}$ perpendicular to the x-axis. A line $l_{2}$ intersects $l_{1}$ perpendicularly at point $P$. The perpendicular bisector of the line segment $PF_{2}$ intersects $l_{2}$ at point $M$.
1. Find the equation of the locus $C_{2}$ of point $M$.
2. Draw two mutually perpendicular lines $AC$ and $BD$ through point $F_{2}$, intersecting the ellipse at points $A$, $B$, $C$, and $D$. Find the minimum value of the area of the quadrilateral $ABCD$. | \frac{64}{9} | 0 |
28,273 | If the random variable $X$ follows a Bernoulli distribution with a success probability of $0.7$, and the random variable $Y$ follows a binomial distribution with $Y \sim B(10, 0.8)$, then $EX$, $DX$, $EY$, $DY$ are respectively ........, ........, ........, ........ | 1.6 | 1.5625 |
28,274 | I live on the ground floor of a ten-story building. Each friend of mine lives on a different floor. One day, I put the numbers $1, 2, \ldots, 9$ into a hat and drew them randomly, one by one. I visited my friends in the order in which I drew their floor numbers. On average, how many meters did I travel by elevator, if the distance between each floor is 4 meters, and I took the elevator from each floor to the next one drawn? | 440/3 | 0 |
28,275 | How many multiples of 5 are between 100 and 400? | 60 | 0 |
28,276 | Square $ABCD$ has side length $1$ ; circle $\Gamma$ is centered at $A$ with radius $1$ . Let $M$ be the midpoint of $BC$ , and let $N$ be the point on segment $CD$ such that $MN$ is tangent to $\Gamma$ . Compute $MN$ .
*2018 CCA Math Bonanza Individual Round #11* | \frac{5}{6} | 10.15625 |
28,277 | A six-digit number begins with digit 1 and ends with digit 7. If the digit in the units place is decreased by 1 and moved to the first place, the resulting number is five times the original number. Find this number. | 142857 | 11.71875 |
28,278 | Given a triangle \( \triangle ABC \) with sides \( a, b, c \) and corresponding medians \( m_a, m_b, m_c \), and angle bisectors \( w_a, w_b, w_c \). Let \( w_a \cap m_b = P \), \( w_b \cap m_c = Q \), and \( w_c \cap m_a = R \). Denote the area of \( \triangle PQR \) by \( \delta \) and the area of \( \triangle ABC \) by \( F \). Determine the smallest positive constant \( \lambda \) such that the inequality \( \frac{\delta}{F} < \lambda \) holds. | \frac{1}{6} | 64.0625 |
28,279 | The average score of 60 students is 72. After disqualifying two students whose scores are 85 and 90, calculate the new average score for the remaining class. | 71.47 | 21.875 |
28,280 | Five brothers divided their father's inheritance equally. The inheritance included three houses. Since it was not possible to split the houses, the three older brothers took the houses, and the younger brothers were given money: each of the three older brothers paid $2,000. How much did one house cost in dollars? | 3000 | 4.6875 |
28,281 | The sides of a non-degenerate isosceles triangle are \(x\), \(x\), and \(24\) units. How many integer values of \(x\) are possible? | 11 | 50.78125 |
28,282 | If 3913 were to be expressed as a sum of distinct powers of 2, what would be the least possible sum of the exponents of these powers? | 47 | 10.9375 |
28,283 | Given non-zero vectors $a=(-x, x)$ and $b=(2x+3, 1)$, where $x \in \mathbb{R}$.
$(1)$ If $a \perp b$, find the value of $x$;
$(2)$ If $a \nparallel b$, find $|a - b|$. | 3\sqrt{2} | 5.46875 |
28,284 | Let $a$, $b$, $c$, $d$, and $e$ be positive integers with $a+b+c+d+e=2020$ and let $M$ be the largest of the sum $a+b$, $b+c$, $c+d$, and $d+e$. What is the smallest possible value of $M$? | 675 | 0 |
28,285 | Find the number of diagonals in a polygon with 120 sides, and calculate the length of one diagonal of this polygon assuming it is regular and each side length is 5 cm. | 10 | 18.75 |
28,286 | Paint both sides of a small wooden board. It takes 1 minute to paint one side, but you must wait 5 minutes for the paint to dry before painting the other side. How many minutes will it take to paint 6 wooden boards in total? | 12 | 1.5625 |
28,287 | In $\triangle ABC, AB = 10, BC = 8, CA = 7$ and side $BC$ is extended to a point $P$ such that $\triangle PAB$ is similar to $\triangle PCA$. The length of $PC$ is | \frac{56}{3} | 3.90625 |
28,288 | From a certain airport, 100 planes (1 command plane and 99 supply planes) take off at the same time. When their fuel tanks are full, each plane can fly 1000 kilometers. During the flight, planes can refuel each other, and after completely transferring their fuel, planes can land as planned. How should the flight be arranged to enable the command plane to fly as far as possible? | 100000 | 8.59375 |
28,289 | Find the smallest number $a$ such that a square of side $a$ can contain five disks of radius $1$ , so that no two of the disks have a common interior point. | 2 + 2\sqrt{2} | 46.09375 |
28,290 | Given that \(\alpha\) is an acute angle and \(\beta\) is an obtuse angle, and \(\sec (\alpha - 2\beta)\), \(\sec \alpha\), and \(\sec (\alpha + 2\beta)\) form an arithmetic sequence, find the value of \(\frac{\cos \alpha}{\cos \beta}\). | \sqrt{2} | 55.46875 |
28,291 | 2022 knights and liars are lined up in a row, with the ones at the far left and right being liars. Everyone except the ones at the extremes made the statement: "There are 42 times more liars to my right than to my left." Provide an example of a sequence where there is exactly one knight. | 48 | 22.65625 |
28,292 | Every card in a deck displays one of three symbols - star, circle, or square, each filled with one of three colors - red, yellow, or blue, and each color is shaded in one of three intensities - light, normal, or dark. The deck contains 27 unique cards, each representing a different symbol-color-intensity combination. A set of three cards is considered harmonious if:
i. Each card has a different symbol or all three have the same symbol.
ii. Each card has a different color or all three have the same color.
iii. Each card has a different intensity or all three have the same intensity.
Determine how many different harmonious three-card sets are possible. | 702 | 0 |
28,293 | Given an ellipse $$\frac {x^{2}}{a^{2}}$$ + $$\frac {y^{2}}{b^{2}}$$ = 1 with its right focus F, a line passing through the origin O intersects the ellipse C at points A and B. If |AF| = 2, |BF| = 4, and the eccentricity of the ellipse C is $$\frac {\sqrt {7}}{3}$$, calculate the area of △AFB. | 2\sqrt{3} | 3.90625 |
28,294 | What percent of square $PQRS$ is shaded? All angles in the diagram are right angles. [asy]
import graph;
defaultpen(linewidth(0.8));
xaxis(0,7,Ticks(1.0,NoZero));
yaxis(0,7,Ticks(1.0,NoZero));
fill((0,0)--(2,0)--(2,2)--(0,2)--cycle);
fill((3,0)--(5,0)--(5,5)--(0,5)--(0,3)--(3,3)--cycle);
fill((6,0)--(7,0)--(7,7)--(0,7)--(0,6)--(6,6)--cycle);
label("$P$",(0,0),SW);
label("$Q$",(0,7),N);
label("$R$",(7,7),NE);
label("$S$",(7,0),E);[/asy] | 67.35\% | 21.875 |
28,295 | Alexa wrote the first $16$ numbers of a sequence:
\[1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, …\]
Then she continued following the same pattern, until she had $2015$ numbers in total.
What was the last number she wrote?
| 1344 | 0.78125 |
28,296 | There are 8 students arranged in two rows, with 4 people in each row. If students A and B must be arranged in the front row, and student C must be arranged in the back row, then the total number of different arrangements is ___ (answer in digits). | 5760 | 56.25 |
28,297 | In a grade, Class 1, Class 2, and Class 3 each select two students (one male and one female) to form a group of high school students. Two students are randomly selected from this group to serve as the chairperson and vice-chairperson. Calculate the probability of the following events:
- The two selected students are not from the same class;
- The two selected students are from the same class;
- The two selected students are of different genders and not from the same class. | \dfrac{2}{5} | 41.40625 |
28,298 | Find the number of rearrangements of the letters in the word MATHMEET that begin and end with the same letter such as TAMEMHET. | 540 | 15.625 |
28,299 | Let \( n = \overline{abc} \) be a three-digit number, where \( a, b, \) and \( c \) are the digits of the number. If \( a, b, \) and \( c \) can form an isosceles triangle (including equilateral triangles), how many such three-digit numbers \( n \) are there? | 165 | 89.0625 |
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