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26,200 | How many ordered pairs of integers $(m,n)$ are there such that $m$ and $n$ are the legs of a right triangle with an area equal to a prime number not exceeding $80$ ? | 87 | 51.5625 |
26,201 | A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is preparing 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\% | 66.40625 |
26,202 | The hyperbola $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \ (a > 0, b > 0)$ has an asymptote that is perpendicular to the line $x + 2y + 1 = 0$. Let $F_1$ and $F_2$ be the foci of $C$, and let $A$ be a point on the hyperbola such that $|F_1A| = 2|F_2A|$. Find $\cos \angle AF_2F_1$. | \frac{\sqrt{5}}{5} | 37.5 |
26,203 | A circle touches the extensions of two sides \( AB \) and \( AD \) of a square \( ABCD \) with a side length of 4 cm. From point \( C \), two tangents are drawn to this circle. Find the radius of the circle if the angle between the tangents is \( 60^{\circ} \). | 4 (\sqrt{2} + 1) | 3.125 |
26,204 | A square is divided into $25$ unit squares by drawing lines parallel to the sides of the square. Some diagonals of unit squares are drawn from such that two diagonals do not share points. What is the maximum number diagonals that can be drawn with this property? | 12 | 7.8125 |
26,205 | Find the smallest natural number \( n \) such that both \( n^2 \) and \( (n+1)^2 \) contain the digit 7. | 27 | 5.46875 |
26,206 | The wholesale department operates a product with a wholesale price of 500 yuan per unit and a gross profit margin of 4%. The inventory capital is 80% borrowed from the bank at a monthly interest rate of 4.2‰, and the storage and operating cost is 0.30 yuan per unit per day. Determine the maximum average storage period for the product without incurring a loss. | 56 | 6.25 |
26,207 | Let $a_1, a_2, \ldots$ be a sequence determined by the rule $a_n = \frac{a_{n-1}}{2}$ if $a_{n-1}$ is even and $a_n = 3a_{n-1} + 1$ if $a_{n-1}$ is odd. For how many positive integers $a_1 \le 3000$ is it true that $a_1$ is less than each of $a_2$, $a_3$, $a_4$, and $a_5$? | 750 | 0.78125 |
26,208 | There is an equilateral triangle $ABC$ on the plane. Three straight lines pass through $A$ , $B$ and $C$ , respectively, such that the intersections of these lines form an equilateral triangle inside $ABC$ . On each turn, Ming chooses a two-line intersection inside $ABC$ , and draws the straight line determined by the intersection and one of $A$ , $B$ and $C$ of his choice. Find the maximum possible number of three-line intersections within $ABC$ after 300 turns.
*Proposed by usjl* | 45853 | 0 |
26,209 | Two adjacent faces of a tetrahedron, which are equilateral triangles with side length 1, form a dihedral angle of 60 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto a plane containing the given edge. | \frac{\sqrt{3}}{4} | 10.9375 |
26,210 | When \( \frac{1}{2222} \) is expressed as a decimal, what is the sum of the first 50 digits after the decimal point? | 90 | 32.03125 |
26,211 | Given numbers $5, 6, 7, 8, 9, 10, 11, 12, 13$ are written in a $3\times3$ array, with the condition that two consecutive numbers must share an edge. If the sum of the numbers in the four corners is $32$, calculate the number in the center of the array. | 13 | 1.5625 |
26,212 | If I have a $5\times 5$ chess board, in how many ways can I place five distinct pawns on the board such that each column and row of the board contains no more than one pawn? | 14400 | 1.5625 |
26,213 | On the ray $(0,+\infty)$ of the number line, there are several (more than two) segments of length 1. For any two different segments, you can select one number from each so that these numbers differ exactly by a factor of 2. The left end of the leftmost segment is the number $a$, and the right end of the rightmost segment is the number $b$. What is the maximum value that the quantity $b-a$ can take? | 5.5 | 0 |
26,214 | Point \( M \) belongs to the edge \( CD \) of the parallelepiped \( ABCDA_1B_1C_1D_1 \), where \( CM: MD = 1:2 \). Construct the section of the parallelepiped with a plane passing through point \( M \) parallel to the lines \( DB \) and \( AC_1 \). In what ratio does this plane divide the diagonal \( A_1C \) of the parallelepiped? | 1 : 11 | 0 |
26,215 | How many natural numbers greater than 10 but less than 100 are relatively prime to 21? | 51 | 4.6875 |
26,216 | Given the function $f(x)=-\frac{1}{2}x^{2}+x$ with a domain that contains an interval $[m,n]$, and its range on this interval is $[3m,3n]$. Find the value of $m+n$. | -4 | 35.15625 |
26,217 | A right triangle has perimeter $2008$ , and the area of a circle inscribed in the triangle is $100\pi^3$ . Let $A$ be the area of the triangle. Compute $\lfloor A\rfloor$ . | 31541 | 55.46875 |
26,218 | In a meeting room, the first row has a total of 8 seats. Now 3 people are seated, and the requirement is that there should be empty seats to the left and right of each person. Calculate the number of different seating arrangements. | 24 | 9.375 |
26,219 | Find the sum $$\frac{3^1}{9^1 - 1} + \frac{3^2}{9^2 - 1} + \frac{3^3}{9^3 - 1} + \frac{3^4}{9^4 - 1} + \cdots.$$ | \frac{1}{2} | 32.03125 |
26,220 | Let $S(M)$ denote the sum of the digits of a positive integer $M$ written in base $10$ . Let $N$ be the smallest positive integer such that $S(N) = 2013$ . What is the value of $S(5N + 2013)$ ? | 18 | 5.46875 |
26,221 | Consider a rectangle $ABCD$ with $BC = 2 \cdot AB$ . Let $\omega$ be the circle that touches the sides $AB$ , $BC$ , and $AD$ . A tangent drawn from point $C$ to the circle $\omega$ intersects the segment $AD$ at point $K$ . Determine the ratio $\frac{AK}{KD}$ .
*Proposed by Giorgi Arabidze, Georgia* | 1/2 | 10.9375 |
26,222 | A square is divided into 2016 triangles, with no vertex of any triangle lying on the sides or inside any other triangle. The sides of the square are sides of some of the triangles in the division. How many total points, which are the vertices of the triangles, are located inside the square? | 1007 | 14.84375 |
26,223 | Find the minimum value of the function \( f(x) = 3^x - 9^x \) for real numbers \( x \). | \frac{1}{4} | 2.34375 |
26,224 | Given the function $y=\cos \left(x+ \frac {\pi}{3}\right)$, derive the horizontal shift of the graph of the function $y=\sin x$. | \frac {5\pi}{6} | 29.6875 |
26,225 | Anna flips an unfair coin 10 times. The coin has a $\frac{1}{3}$ probability of coming up heads and a $\frac{2}{3}$ probability of coming up tails. What is the probability that she flips exactly 7 tails? | \frac{5120}{19683} | 10.15625 |
26,226 | The axial section of a cone is an equilateral triangle with a side length of 1. Find the radius of the sphere that is tangent to the axis of the cone, its base, and its lateral surface. | \frac{\sqrt{3} - 1}{4} | 30.46875 |
26,227 | Find the sum of all the positive integers which have at most three not necessarily distinct prime factors where the primes come from the set $\{ 2, 3, 5, 7 \}$ . | 1932 | 91.40625 |
26,228 | A cube with edge length 1 can freely flip inside a regular tetrahedron with edge length $a$. Find the minimum value of $a$. | 3\sqrt{2} | 3.90625 |
26,229 | When \( N \) takes all values from 1, 2, 3, ..., to 2015, how many numbers of the form \( 3^n + n^3 \) are divisible by 7? | 288 | 85.9375 |
26,230 | The height of a right-angled triangle, dropped to the hypotenuse, divides this triangle into two triangles. The distance between the centers of the inscribed circles of these triangles is 1. Find the radius of the inscribed circle of the original triangle. | \frac{\sqrt{2}}{2} | 5.46875 |
26,231 | Given the function $f(x) = \sin x + \cos x$.
(1) If $f(x) = 2f(-x)$, find the value of $\frac{\cos^2x - \sin x\cos x}{1 + \sin^2x}$;
(2) Find the maximum value and the intervals of monotonic increase for the function $F(x) = f(x) \cdot f(-x) + f^2(x)$. | \frac{6}{11} | 0.78125 |
26,232 | In Yang's number theory class, Michael K, Michael M, and Michael R take a series of tests. Afterwards, Yang makes the following observations about the test scores:
(a) Michael K had an average test score of $90$ , Michael M had an average test score of $91$ , and Michael R had an average test score of $92$ .
(b) Michael K took more tests than Michael M, who in turn took more tests than Michael R.
(c) Michael M got a higher total test score than Michael R, who in turn got a higher total test score than Michael K. (The total test score is the sum of the test scores over all tests)
What is the least number of tests that Michael K, Michael M, and Michael R could have taken combined?
*Proposed by James Lin* | 413 | 55.46875 |
26,233 | The coefficient of $x^{3}$ in the expanded form of $(1+x-x^{2})^{10}$ is given by the binomial coefficient $\binom{10}{3}(-1)^{7} + \binom{10}{4}(-1)^{6}$. Calculate the value of this binomial coefficient expression. | 30 | 50.78125 |
26,234 | Given the ellipse C: $mx^2+3my^2=1$ ($m>0$) with a major axis length of $2\sqrt{6}$, and O as the origin.
(1) Find the equation of ellipse C and its eccentricity.
(2) Let point A be (3,0), point B be on the y-axis, and point P be on ellipse C, with point P on the right side of the y-axis. If $BA=BP$, find the minimum value of the area of quadrilateral OPAB. | 3\sqrt{3} | 0.78125 |
26,235 | A ball is dropped from a height of 150 feet and rebounds to three-fourths of the distance it fell on each bounce. How many feet will the ball have traveled when it hits the ground the fifth time? | 765.234375 | 7.8125 |
26,236 | In the diagram, $\triangle ABC$ is right-angled at $A,$ with $AB=45$ and $AC=108.$ The point $D$ is on $BC$ so that $AD$ is perpendicular to $BC.$ Determine the length of $AD$ and the ratio of the areas of triangles $ABD$ and $ADC$. | 5:12 | 0 |
26,237 | Let $(a_1,a_2,a_3,\ldots,a_{15})$ be a permutation of $(1,2,3,\ldots,15)$ for which
$a_1>a_2>a_3>a_4>a_5>a_6>a_7 \mathrm{\ and \ } a_7<a_8<a_9<a_{10}<a_{11}<a_{12}<a_{13}<a_{14}<a_{15}.$
An example of such a permutation is $(7,6,5,4,3,2,1,8,9,10,11,12,13,14,15).$ Find the number of such permutations. | 3003 | 7.03125 |
26,238 | A nine-digit number is formed by repeating a three-digit number three times; for example, $256256256$. Determine the common factor that divides any number of this form exactly. | 1001001 | 41.40625 |
26,239 | A function $f: \N\rightarrow\N$ is circular if for every $p\in\N$ there exists $n\in\N,\ n\leq{p}$ such that $f^n(p)=p$ ( $f$ composed with itself $n$ times) The function $f$ has repulsion degree $k>0$ if for every $p\in\N$ $f^i(p)\neq{p}$ for every $i=1,2,\dots,\lfloor{kp}\rfloor$ . Determine the maximum repulsion degree can have a circular function.**Note:** Here $\lfloor{x}\rfloor$ is the integer part of $x$ . | 1/2 | 41.40625 |
26,240 | Find the remainder when $5^{2021}$ is divided by $17$. | 11 | 4.6875 |
26,241 | Given that Erin the ant starts at a given corner of a hypercube (4-dimensional cube) and crawls along exactly 15 edges in such a way that she visits every corner exactly once and then finds that she is unable to return along an edge to her starting point, determine the number of paths that Erin can follow to meet these conditions. | 24 | 0 |
26,242 | The probability of an event happening is $\frac{1}{2}$, find the relation between this probability and the outcome of two repeated experiments. | 50\% | 0 |
26,243 | As the Chinese New Year approaches, workers from a factory begin to go home to reunite with their families starting from Monday, January 17, 2011. If the number of workers leaving the factory each day is the same, and by January 31, 121 workers remain in the factory, while the total number of worker-days during this 15-day period is 2011 (one worker working for one day counts as one worker-day, and the day a worker leaves and any days after are not counted), with weekends (Saturdays and Sundays) being rest days and no one being absent, then by January 31, the total number of workers who have gone home for the New Year is ____. | 120 | 0.78125 |
26,244 | Find the smallest real number $p$ such that the inequality $\sqrt{1^2+1}+\sqrt{2^2+1}+...+\sqrt{n^2+1} \le \frac{1}{2}n(n+p)$ holds for all natural numbers $n$ . | 2\sqrt{2} - 1 | 7.8125 |
26,245 | Given that $21^{-1} \equiv 17 \pmod{53}$, find $32^{-1} \pmod{53}$, as a residue modulo 53. (Give a number between 0 and 52, inclusive.) | 36 | 68.75 |
26,246 | The vertical coordinate of the intersection point of the new graph obtained by shifting the graph of the quadratic function $y=x^{2}+2x+1$ $2$ units to the left and then $3$ units up is ______. | 12 | 37.5 |
26,247 | Given that $\sin \alpha \cos \alpha = \frac{1}{8}$, and $\alpha$ is an angle in the third quadrant. Find $\frac{1 - \cos^2 \alpha}{\cos(\frac{3\pi}{2} - \alpha) + \cos \alpha} + \frac{\sin(\alpha - \frac{7\pi}{2}) + \sin(2017\pi - \alpha)}{\tan^2 \alpha - 1}$. | \frac{\sqrt{5}}{2} | 3.125 |
26,248 | Given the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $e = \frac{\sqrt{2}}{2}$, and one of its vertices is at $(0, -1)$.
(Ⅰ) Find the equation of the ellipse $C$.
(Ⅱ) If there exist two distinct points $A$ and $B$ on the ellipse $C$ that are symmetric about the line $y = -\frac{1}{m}x + \frac{1}{2}$, find the maximum value of the area of $\triangle OAB$ ($O$ is the origin). | \frac{\sqrt{2}}{2} | 42.1875 |
26,249 | A certain school randomly selected several students to investigate the daily physical exercise time of students in the school. They obtained data on the daily physical exercise time (unit: minutes) and organized and described the data. Some information is as follows:
- $a$. Distribution of daily physical exercise time:
| Daily Exercise Time $x$ (minutes) | Frequency (people) | Percentage |
|-----------------------------------|--------------------|------------|
| $60\leqslant x \lt 70$ | $14$ | $14\%$ |
| $70\leqslant x \lt 80$ | $40$ | $m$ |
| $80\leqslant x \lt 90$ | $35$ | $35\%$ |
| $x\geqslant 90$ | $n$ | $11\%$ |
- $b$. The daily physical exercise time in the group $80\leqslant x \lt 90$ is: $80$, $81$, $81$, $81$, $82$, $82$, $83$, $83$, $84$, $84$, $84$, $84$, $84$, $85$, $85$, $85$, $85$, $85$, $85$, $85$, $85$, $86$, $87$, $87$, $87$, $87$, $87$, $88$, $88$, $88$, $89$, $89$, $89$, $89$, $89$.
Based on the above information, answer the following questions:
$(1)$ In the table, $m=$______, $n=$______.
$(2)$ If the school has a total of $1000$ students, estimate the number of students in the school who exercise for at least $80$ minutes per day.
$(3)$ The school is planning to set a time standard $p$ (unit: minutes) to commend students who exercise for at least $p$ minutes per day. If $25\%$ of the students are to be commended, what value can $p$ be? | 86 | 0 |
26,250 | You know that the Jones family has five children, and the Smith family has three children. Of the eight children you know that there are five girls and three boys. Let $\dfrac{m}{n}$ be the probability that at least one of the families has only girls for children. Given that $m$ and $n$ are relatively prime positive integers, find $m+ n$ . | 67 | 3.90625 |
26,251 | Given a $4 \times 4$ square grid partitioned into $16$ unit squares, each of which is painted white or black with a probability of $\frac{1}{2}$, determine the probability that the grid is entirely black after a $90^{\circ}$ clockwise rotation and any white square landing in a position previously occupied by a black square is repainted black. | \frac{1}{65536} | 40.625 |
26,252 | The largest value of the real number $k$ for which the inequality $\frac{1+\sin x}{2+\cos x} \geqslant k$ has a solution. | \frac{4}{3} | 3.125 |
26,253 | How many distinct five-digit positive integers are there such that the product of their digits equals 16? | 15 | 1.5625 |
26,254 | How many natural numbers greater than 10 but less than 100 are relatively prime to 21? | 53 | 3.125 |
26,255 | Given that the sequence $\{a_n\}$ is a geometric sequence, and $a_4 = e$, if $a_2$ and $a_7$ are the two real roots of the equation $$ex^2 + kx + 1 = 0, (k > 2\sqrt{e})$$ (where $e$ is the base of the natural logarithm),
1. Find the general formula for $\{a_n\}$.
2. Let $b_n = \ln a_n$, and $S_n$ be the sum of the first $n$ terms of the sequence $\{b_n\}$. When $S_n = n$, find the value of $n$.
3. For the sequence $\{b_n\}$ in (2), let $c_n = b_nb_{n+1}b_{n+2}$, and $T_n$ be the sum of the first $n$ terms of the sequence $\{c_n\}$. Find the maximum value of $T_n$ and the corresponding value of $n$. | n = 4 | 12.5 |
26,256 | Given the function $f(x) = \frac{1}{3}x^3 - 4x + 4$,
(I) Find the extreme values of the function;
(II) Find the maximum and minimum values of the function on the interval [-3, 4]. | -\frac{4}{3} | 41.40625 |
26,257 | The Songjiang tram project is in full swing. After the tram starts operating, it will bring convenience to the public's travel. It is known that after the opening of a certain route, the tram interval $t$ (unit: minutes) satisfies $2 \leq t \leq 20$. According to market survey calculations, the tram’s passenger capacity is related to the departure interval $t$. When $10 \leq t \leq 20$, the tram is fully loaded with a capacity of 400 people. When $2 \leq t < 10$, the passenger capacity decreases, and the number of passengers reduced is directly proportional to the square of $(10-t)$, and when the interval is 2 minutes, the passenger capacity is 272 people. Let the tram's passenger capacity be $p(t)$.
(1) Find the expression for $p(t)$ and the passenger capacity when the departure interval is 6 minutes;
(2) If the net income per minute of the line is $Q = \dfrac{6p(t)-1500}{t} - 60$ (unit: yuan), what is the departure interval that maximizes the line’s net income per minute? | 60 | 0.78125 |
26,258 | A school selects 4 teachers from 8 to teach in 4 remote areas at the same time (one person per area), where A and B cannot go together, and A and C can only go together or not go at all. The total number of different dispatch plans is \_\_\_\_\_\_ (answer in numbers). | 600 | 53.125 |
26,259 | Consider the system of equations:
\begin{align*}
8x - 5y &= a, \\
10y - 15x &= b.
\end{align*}
If this system has a solution \((x, y)\) where both \(x\) and \(y\) are nonzero, calculate \(\frac{a}{b}\), assuming \(b\) is nonzero. | \frac{8}{15} | 0 |
26,260 | In rectangle $ABCD$, $P$ is a point on side $\overline{BC}$ such that $BP = 20$ and $CP = 5.$ If $\tan \angle APD = 2,$ then find $AB.$ | 20 | 3.90625 |
26,261 | Eight students from a university are planning to carpool for a trip, with two students from each of the grades one, two, three, and four. How many ways are there to arrange the four students in car A, such that the last two students are from the same grade? | 24 | 0.78125 |
26,262 | Given the function $f(x) = \frac {a^{x}}{a^{x}+1}$ ($a>0$ and $a \neq 1$).
- (I) Find the range of $f(x)$.
- (II) If the maximum value of $f(x)$ on the interval $[-1, 2]$ is $\frac {3}{4}$, find the value of $a$. | \frac {1}{3} | 23.4375 |
26,263 | The ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a > b > 0$) has an eccentricity of $e = \frac{2}{3}$. Points A and B lie on the ellipse and are not symmetrical with respect to the x-axis or the y-axis. The perpendicular bisector of segment AB intersects the x-axis at point P(1, 0). Let the midpoint of AB be C($x_0$, $y_0$). Find the value of $x_0$. | \frac{9}{4} | 29.6875 |
26,264 | Let \( p, q, r, s, \) and \( t \) be the roots of the polynomial
\[ x^5 + 10x^4 + 20x^3 + 15x^2 + 6x + 3 = 0. \]
Find the value of
\[ \frac{1}{pq} + \frac{1}{pr} + \frac{1}{ps} + \frac{1}{pt} + \frac{1}{qr} + \frac{1}{qs} + \frac{1}{qt} + \frac{1}{rs} + \frac{1}{rt} + \frac{1}{st}. \] | \frac{20}{3} | 3.125 |
26,265 | What is the maximum number of checkers that can be placed on a $6 \times 6$ board such that no three checkers (specifically, the centers of the cells they occupy) lie on the same straight line (at any angle)? | 12 | 50 |
26,266 | A 10x10 arrangement of alternating black and white squares has a black square $R$ in the second-bottom row and a white square $S$ in the top-most row. Given that a marker is initially placed at $R$ and can move to an immediately adjoining white square on the row above either to the left or right, and the path must consist of exactly 8 steps, calculate the number of valid paths from $R$ to $S$. | 70 | 32.03125 |
26,267 | In an isosceles triangle \( \triangle AMC \), \( AM = AC \), the median \( MV = CU = 12 \), and \( MV \perp CU \) at point \( P \). What is the area of \( \triangle AMC \)? | 96 | 31.25 |
26,268 | In a certain competition, the rules are as follows: among the 5 questions preset by the organizer, if a contestant can answer two consecutive questions correctly, they will stop answering and advance to the next round. Assuming the probability of a contestant correctly answering each question is 0.8, and the outcomes of answering each question are independent of each other, then the probability that the contestant will exactly answer 4 questions before advancing to the next round is | 0.128 | 2.34375 |
26,269 | Find the area of the region bounded by a function $y=-x^4+16x^3-78x^2+50x-2$ and the tangent line which is tangent to the curve at exactly two distinct points.
Proposed by Kunihiko Chikaya | 1296/5 | 0 |
26,270 | Write a twelve-digit number that is not a perfect cube. | 100000000000 | 37.5 |
26,271 | Find the smallest positive real number $\lambda$ such that for every numbers $a_1,a_2,a_3 \in \left[0, \frac{1}{2} \right]$ and $b_1,b_2,b_3 \in (0, \infty)$ with $\sum\limits_{i=1}^3a_i=\sum\limits_{i=1}^3b_i=1,$ we have $$ b_1b_2b_3 \le \lambda (a_1b_1+a_2b_2+a_3b_3). $$ | 1/8 | 1.5625 |
26,272 | From a deck of cards marked with 1, 2, 3, and 4, two cards are drawn consecutively. The probability of drawing the card with the number 4 on the first draw, the probability of not drawing it on the first draw but drawing it on the second, and the probability of drawing the number 4 at any point during the drawing process are, respectively: | \frac{1}{2} | 53.125 |
26,273 | Given the sets $M={x|m\leqslant x\leqslant m+ \frac {3}{4}}$ and $N={x|n- \frac {1}{3}\leqslant x\leqslant n}$, both of which are subsets of ${x|0\leqslant x\leqslant 1}$, what is the minimum "length" of the set $M\cap N$? (Note: The "length" of a set ${x|a\leqslant x\leqslant b}$ is defined as $b-a$.) | \frac{1}{12} | 69.53125 |
26,274 | When three standard dice are tossed, the numbers $x, y, z$ are obtained. Find the probability that $xyz = 72$. | \frac{7}{216} | 1.5625 |
26,275 | Let \( D \) be a point inside \( \triangle ABC \) such that \( \angle BAD = \angle BCD \) and \( \angle BDC = 90^\circ \). If \( AB = 5 \), \( BC = 6 \), and \( M \) is the midpoint of \( AC \), find the length of \( DM \). | \frac{\sqrt{11}}{2} | 0.78125 |
26,276 | Three students, A, B, and C, are playing badminton with the following rules:<br/>The player who loses two games in a row will be eliminated. Before the game, two players are randomly selected to play against each other, while the third player has a bye. The winner of each game will play against the player with the bye in the next round, and the loser will have the bye in the next round. This process continues until one player is eliminated. After one player is eliminated, the remaining two players will continue playing until one of them is eliminated, and the other player will be the ultimate winner of the game. A and B are selected to play first, while C has the bye. The probability of winning for each player in a game is $\frac{1}{2}$.<br/>$(1)$ Find the probability of A winning four consecutive games;<br/>$(2)$ Find the probability of needing a fifth game to be played;<br/>$(3)$ Find the probability of C being the ultimate winner. | \frac{7}{16} | 0 |
26,277 | What is the least positive multiple of 45 for which the product of its digits is also a positive multiple of 45? | 945 | 0 |
26,278 | The diagonals of a convex quadrilateral \(ABCD\) intersect at point \(E\). Given that \(AB = AD\) and \(CA\) is the bisector of angle \(C\), with \(\angle BAD = 140^\circ\) and \(\angle BEA = 110^\circ\), find angle \(CDB\). | 50 | 4.6875 |
26,279 | Given a triangle $ABC$ with sides $a$, $b$, $c$, and area $S$ satisfying $S=a^{2}-(b-c)^{2}$, and $b+c=8$.
$(1)$ Find $\cos A$;
$(2)$ Find the maximum value of $S$. | \frac{64}{17} | 82.8125 |
26,280 | Given Ms. Grace Swift leaves her house at 7:30 AM to reach her office, she arrives 5 minutes late at an average speed of 50 miles per hour, and 5 minutes early at an average speed of 70 miles per hour, determine the average speed she should maintain to arrive exactly on time. | 58 | 0 |
26,281 | Determine how many 4-digit numbers are mountain numbers, where mountain numbers are defined as having their middle two digits larger than any other digits in the number. For example, 3942 and 5732 are mountain numbers. | 240 | 4.6875 |
26,282 | Given that the sum of the first $n$ terms of the sequence ${a_n}$ is $S_n$, and $S_n=n^2$ ($n\in\mathbb{N}^*$).
1. Find $a_n$;
2. The function $f(n)$ is defined as $$f(n)=\begin{cases} a_{n} & \text{, $n$ is odd} \\ f(\frac{n}{2}) & \text{, $n$ is even}\end{cases}$$, and $c_n=f(2^n+4)$ ($n\in\mathbb{N}^*$), find the sum of the first $n$ terms of the sequence ${c_n}$, denoted as $T_n$.
3. Let $\lambda$ be a real number, for any positive integers $m$, $n$, $k$ that satisfy $m+n=3k$ and $m\neq n$, the inequality $S_m+S_n>\lambda\cdot S_k$ always holds, find the maximum value of the real number $\lambda$. | \frac{9}{2} | 48.4375 |
26,283 | Consider an ordinary $6$-sided die, with numbers from $1$ to $6$. How many ways can I paint three faces of a die so that the sum of the numbers on the painted faces is not exactly $9$? | 18 | 2.34375 |
26,284 | Given the set $X=\left\{1,2,3,4\right\}$, consider a function $f:X\to X$ where $f^1=f$ and $f^{k+1}=\left(f\circ f^k\right)$ for $k\geq1$. Determine the number of functions $f$ that satisfy $f^{2014}\left(x\right)=x$ for all $x$ in $X$. | 13 | 1.5625 |
26,285 | Two distinct numbers are selected simultaneously and at random from the set $\{1, 2, 3, 6, 9\}$. What is the probability that the smaller one divides the larger one? Express your answer as a common fraction. | \frac{3}{5} | 10.9375 |
26,286 | Calculate: \( 8.0 \dot{\dot{1}} + 7.1 \dot{2} + 6.2 \dot{3} + 5.3 \dot{4} + 4.4 \dot{5} + 3.5 \dot{6} + 2.6 \dot{7} + 1.7 \dot{8} = \) | 39.2 | 7.8125 |
26,287 | Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $\overrightarrow{a} \cdot (\overrightarrow{a} - 2\overrightarrow{b}) = 3$, and $|\overrightarrow{a}| = 1$, $\overrightarrow{b} = (1,1)$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{3\pi}{4} | 92.96875 |
26,288 | While there do not exist pairwise distinct real numbers $a,b,c$ satisfying $a^2+b^2+c^2 = ab+bc+ca$ , there do exist complex numbers with that property. Let $a,b,c$ be complex numbers such that $a^2+b^2+c^2 = ab+bc+ca$ and $|a+b+c| = 21$ . Given that $|a-b| = 2\sqrt{3}$ , $|a| = 3\sqrt{3}$ , compute $|b|^2+|c|^2$ .
<details><summary>Clarifications</summary>
- The problem should read $|a+b+c| = 21$ . An earlier version of the test read $|a+b+c| = 7$ ; that value is incorrect.
- $|b|^2+|c|^2$ should be a positive integer, not a fraction; an earlier version of the test read ``... for relatively prime positive integers $m$ and $n$ . Find $m+n$ .''
</details>
*Ray Li* | 132 | 0.78125 |
26,289 | In triangle $ABC$, side $AC = 900$ and side $BC = 600$. Points $K$ and $L$ are located on segment $AC$ and segment $AB$ respectively, such that $AK = CK$, and $CL$ is the angle bisector of $\angle C$. Let point $P$ be the intersection of line segments $BK$ and $CL$. Point $M$ is located on line $BK$ such that $K$ is the midpoint of segment $PM$. If $AM = 360$, find $LP$. | 144 | 7.8125 |
26,290 | Let $ a $ , $ b $ , $ c $ , $ d $ , $ (a + b + c + 18 + d) $ , $ (a + b + c + 18 - d) $ , $ (b + c) $ , and $ (c + d) $ be distinct prime numbers such that $ a + b + c = 2010 $ , $ a $ , $ b $ , $ c $ , $ d \neq 3 $ , and $ d \le 50 $ . Find the maximum value of the difference between two of these prime numbers. | 2067 | 25.78125 |
26,291 | The mode and median of the data $9.30$, $9.05$, $9.10$, $9.40$, $9.20$, $9.10$ are ______ and ______, respectively. | 9.15 | 25.78125 |
26,292 | A trapezoid \(ABCD\) (\(AD \parallel BC\)) and a rectangle \(A_{1}B_{1}C_{1}D_{1}\) are inscribed in a circle \(\Omega\) with a radius of 13 such that \(AC \parallel B_{1}D_{1}\) and \(BD \parallel A_{1}C_{1}\). Find the ratio of the areas of \(ABCD\) and \(A_{1}B_{1}C_{1}D_{1}\), given that \(AD = 24\) and \(BC = 10\). | \frac{1}{2} | 0 |
26,293 | Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $2500$ are not factorial tails? | 499 | 3.125 |
26,294 | Two chords \( PQ \) and \( PR \) are drawn in a circle with diameter \( PS \). The point \( T \) lies on \( PR \) and \( QT \) is perpendicular to \( PR \). The angle \(\angle QPR = 60^\circ\), \( PQ = 24 \text{ cm} \), and \( RT = 3 \text{ cm} \). What is the length of the chord \( QS \) in cm?
A) \( \sqrt{3} \)
B) 2
C) 3
D) \( 2\sqrt{3} \)
E) \( 3\sqrt{2} \) | 2\sqrt{3} | 33.59375 |
26,295 | In triangle $PQR$, $PQ = 8$, $QR = 15$, $PR = 17$, and $QS$ is the angle bisector. Find the length of $QS$. | \sqrt{87.04} | 0 |
26,296 | Let $a_0 = 3,$ $b_0 = 4,$ and
\[a_{n + 1} = \frac{a_n^2}{b_n} \quad \text{and} \quad b_{n + 1} = \frac{b_n^2}{a_n}\] for all $n \ge 0.$ Calculate $b_7 = \frac{4^p}{3^q}$ for some integers $p$ and $q.$ | (1094,1093) | 0 |
26,297 | Given non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2|\overrightarrow{b}|$, and $(\overrightarrow{a}-\overrightarrow{b})\bot \overrightarrow{b}$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is ______. | \frac{\pi}{3} | 96.09375 |
26,298 | Given \(0 \leqslant x \leqslant 2\), the function \(y=4^{x-\frac{1}{2}}-3 \cdot 2^{x}+5\) reaches its minimum value at? | \frac{1}{2} | 20.3125 |
26,299 | Given Erin has 4 sisters and 6 brothers, determine the product of the number of sisters and the number of brothers of her brother Ethan. | 30 | 0.78125 |
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