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26,300 | A sequence begins with 3, and each subsequent term is triple the sum of all preceding terms. Determine the first term in the sequence that exceeds 15000. | 36864 | 12.5 |
26,301 | In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is $\begin{cases} x=- \dfrac { \sqrt {3}}{2}t \\ y=1+ \dfrac {1}{2}t\end{cases}$ (where $t$ is the parameter), with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar coordinate system is established. The equation of curve $C_{1}$ is $\rho= \dfrac {36}{4 \sqrt {3}\sin \theta-12\cos \theta-\rho}$. Let $M(6,0)$ be a fixed point, and $N$ be a moving point on curve $C_{1}$, with $Q$ being the midpoint of $MN$.
$(1)$ Find the Cartesian coordinate equation of the trajectory $C_{2}$ of point $Q$;
$(2)$ Given that the intersection of line $l$ with the $x$-axis is point $P$, and its intersections with curve $C_{2}$ are points $A$ and $B$. If the midpoint of $AB$ is $D$, find the length of $|PD|$. | \dfrac {3+ \sqrt {3}}{2} | 0 |
26,302 | Given the function $f(x)=\frac{{1-x}}{{2x+2}}$.
$(1)$ Find the solution set of the inequality $f(2^{x})-2^{x+1}+2 \gt 0$;
$(2)$ If the function $g(x)$ satisfies $2f(2^{x})\cdot g(x)=2^{-x}-2^{x}$, and for any $x$ $(x\neq 0)$, the inequality $g(2x)+3\geqslant k\cdot [g(x)-2]$ always holds, find the maximum value of the real number $k$. | \frac{7}{2} | 1.5625 |
26,303 | For how many real numbers $a$ does the quadratic equation $x^2 + ax + 12a = 0$ have only integer roots for $x$? | 15 | 5.46875 |
26,304 | Let $a, b, c$ be positive real numbers such that $3a + 4b + 2c = 3$. Find the minimum value of
\[
\frac{1}{2a + b} + \frac{1}{a + 3c} + \frac{1}{4b + c}.
\] | 1.5 | 0 |
26,305 | Cory has $4$ apples, $2$ oranges, and $1$ banana. If Cory eats one piece of fruit per day for a week, and must consume at least one apple before any orange, how many different orders can Cory eat these fruits? The fruits within each category are indistinguishable. | 105 | 8.59375 |
26,306 | The base radius of a circular television tower is 15 meters. Now, it is planned to plant a 5-meter-wide ring-shaped lawn around it.
(1) How many square meters of lawn are needed?
(2) If each square meter of lawn costs 500 yuan, how much money is needed to plant this lawn at least? | 274750 | 1.5625 |
26,307 | Evaluate the first three digits to the right of the decimal point in the decimal representation of $\left(10^{987} + 1\right)^{8/3}$ using the Binomial Expansion. | 666 | 53.90625 |
26,308 | A line with a slope of $-3$ intersects the positive $x$-axis at $A$ and the positive $y$-axis at $B$. A second line intersects the $x$-axis at $C(10,0)$ and the $y$-axis at $D$. The lines intersect at $E(5,5)$. What is the area of the shaded quadrilateral $OBEC$? | 25 | 1.5625 |
26,309 | Find distinct digits to replace the letters \(A, B, C, D\) such that the following division in the decimal system holds:
$$
\frac{ABC}{BBBB} = 0,\overline{BCDB \, BCDB \, \ldots}
$$
(in other words, the quotient should be a repeating decimal). | 219 | 0 |
26,310 | How many positive integers less than $1000$ are either a perfect cube or a perfect square? | 37 | 80.46875 |
26,311 | When f(t)=0.1, solve for t in the function model: $f(t)=\frac{1}{{1+{e^{-0.22(3t-40)}}}$. | 10 | 0.78125 |
26,312 | Given the function $f(x)=x^{3}+ax^{2}+bx+c$, it reaches a maximum value of $7$ when $x=-1$, and it reaches a minimum value when $x=3$. Find the values of $a$, $b$, $c$, and this minimum value. | -25 | 6.25 |
26,313 | Let $m=n^{4}+x$, where $n \in \mathbf{N}$ and $x$ is a two-digit positive integer. Which value of $x$ ensures that $m$ is always a composite number? | 64 | 5.46875 |
26,314 | Let $\min \{a, b\}$ denote the smaller value between $a$ and $b$. When the positive numbers $x$ and $y$ vary, $t = \min \left\{ x, \frac{y}{x^{2}+y^{2}} \right\}$ also varies. What is the maximum value of $t$? | 1/2 | 12.5 |
26,315 | Given that the area of $\triangle ABC$ is $\frac{1}{2}$, $AB=1$, $BC=\sqrt{2}$, determine the value of $AC$. | \sqrt{5} | 14.0625 |
26,316 | The Chinese mathematician Qin Jiushao (circa 1202-1261) from the Southern Song Dynasty proposed Qin Jiushao's algorithm for polynomial evaluation in his work "Mathematical Book in Nine Chapters." The provided diagram illustrates an example of using Qin Jiushao's algorithm to evaluate a polynomial. If the input values are $n=5, v=1, x=2$, what does the flowchart calculate? | $2^{5}+2^{4}+2^{3}+2^{2}+2+1$ | 0 |
26,317 | Find the smallest positive integer \( n \) such that:
1. \( n \) has exactly 144 distinct positive divisors.
2. There are ten consecutive integers among the positive divisors of \( n \). | 110880 | 33.59375 |
26,318 | What is the sum of all the integers between -20.5 and -1.5? | -210 | 89.84375 |
26,319 | Determine the value of \(a\) if \(a\) and \(b\) are integers such that \(x^3 - x - 1\) is a factor of \(ax^{19} + bx^{18} + 1\). | 2584 | 0 |
26,320 | Let $\overline{AB}$ be a diameter in a circle of radius $10$. Let $\overline{CD}$ be a chord in the circle that intersects $\overline{AB}$ at a point $E$ such that $BE=4$ and $\angle AEC = 30^{\circ}.$ Compute the sum $CE^2 + DE^2$. | 200 | 8.59375 |
26,321 | The "Tuning Day Method" is a procedural algorithm for seeking precise fractional representations of numbers. Suppose the insufficient approximation and the excessive approximation of a real number $x$ are $\dfrac{b}{a}$ and $\dfrac{d}{c}$ ($a,b,c,d \in \mathbb{N}^*$) respectively, then $\dfrac{b+d}{a+c}$ is a more accurate insufficient approximation or excessive approximation of $x$. Given that $\pi = 3.14159…$, and the initial values are $\dfrac{31}{10} < \pi < \dfrac{16}{5}$, determine the more accurate approximate fractional value of $\pi$ obtained after using the "Tuning Day Method" three times. | \dfrac{22}{7} | 19.53125 |
26,322 | Given that the polar coordinate equation of curve $C$ is $ρ=2\cos θ$, and the polar coordinate equation of line $l$ is $ρ\sin (θ+ \frac {π}{6})=m$. If line $l$ and curve $C$ have exactly one common point, find the value of the real number $m$. | \frac{3}{2} | 41.40625 |
26,323 | Evaluate the sum $$\lceil\sqrt{5}\rceil + \lceil\sqrt{6}\rceil + \lceil\sqrt{7}\rceil + \cdots + \lceil\sqrt{39}\rceil$$
Note: For a real number $x,$ $\lceil x \rceil$ denotes the smallest integer that is greater than or equal to $x.$ | 175 | 0.78125 |
26,324 | A group of n friends wrote a math contest consisting of eight short-answer problem $S_1, S_2, S_3, S_4, S_5, S_6, S_7, S_8$ , and four full-solution problems $F_1, F_2, F_3, F_4$ . Each person in the group correctly solved exactly 11 of the 12 problems. We create an 8 x 4 table. Inside the square located in the $i$ th row and $j$ th column, we write down the number of people who correctly solved both problem $S_i$ and $F_j$ . If the 32 entries in the table sum to 256, what is the value of n? | 32 | 31.25 |
26,325 | In a local election based on proportional representation, each voter is given a ballot with 10 candidate names. The voter must mark No. 1 against the candidate for whom they cast their first vote, and they may optionally fill out No. 2 through No. 10 for their subsequent preferences.
How many different ways can a voter mark their ballot? | 3628800 | 12.5 |
26,326 | In the diagram, three circles of radius 2 with centers $X$, $Y$, and $Z$ are tangent to one another and to the sides of $\triangle DEF$. Each circle touches two sides of the triangle.
Find the perimeter of triangle $DEF$. | 12\sqrt{3} | 0 |
26,327 | Let $\mathbf{a}, \mathbf{b},$ and $\mathbf{c}$ be vectors such that $\|\mathbf{a}\| = 2,$ $\|\mathbf{b}\| = 3,$ and $\|\mathbf{c}\| = 6,$ and
\[\mathbf{a} + 2\mathbf{b} + \mathbf{c} = \mathbf{0}.\]
Compute $\mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c} + \mathbf{b} \cdot \mathbf{c}.$ | -19 | 3.90625 |
26,328 | Given that $|\vec{a}| = 1$ and $\vec{b} = (1, \sqrt{3})$, and that $(\vec{b} - \vec{a}) \perp \vec{a}$, find the angle between vector $\vec{a}$ and vector $\vec{b}$. | \frac{\pi}{3} | 87.5 |
26,329 | Let $\mathbf{a} = \begin{pmatrix} 4 \\ 2 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -7 \\ 3 \\ 1 \end{pmatrix}.$ Find the area of the triangle with vertices $\mathbf{0},$ $\mathbf{a},$ and $\mathbf{b}.$ | 13 | 10.15625 |
26,330 | How many of the 729 smallest positive integers written in base 9 use 7 or 8 (or both) as a digit? | 386 | 7.8125 |
26,331 | The numbers from 1 to 9 are arranged in the cells of a $3 \times 3$ table such that the sum of the numbers on one diagonal is 7, and on the other diagonal, it is 21. What is the sum of the numbers in the five shaded cells?
 | 25 | 54.6875 |
26,332 | Two semicircles of diameters 2 and 4 are placed such that the smaller is inside the larger one and their diameters are along the same line. A square of side 1 is inscribed in the larger semicircle such that one of its sides is along the diameter of the semicircles. Calculate the area of the lune formed by the space in the smaller semicircle that is outside the larger semicircle and subtract the area covered by the square. | \frac{3}{2}\pi - 1 | 0 |
26,333 | Given the function $f(x)=\sin \left( \omega x- \frac{\pi }{6} \right)+\sin \left( \omega x- \frac{\pi }{2} \right)$, where $0 < \omega < 3$. It is known that $f\left( \frac{\pi }{6} \right)=0$.
(1) Find $\omega$;
(2) Stretch the horizontal coordinates of each point on the graph of the function $y=f(x)$ to twice its original length (the vertical coordinates remain unchanged), then shift the resulting graph to the left by $\frac{\pi }{4}$ units to obtain the graph of the function $y=g(x)$. Find the minimum value of $g(x)$ on $\left[ -\frac{\pi }{4},\frac{3\pi }{4} \right]$. | -\frac{\sqrt{3}}{2} | 0.78125 |
26,334 | Given the function $f(x)=2m\sin x-2\cos ^{2}x+ \frac{m^{2}}{2}-4m+3$, and the minimum value of the function $f(x)$ is $(-7)$, find the value of the real number $m$. | 10 | 12.5 |
26,335 | If $$\sin\theta= \frac {3}{5}$$ and $$\frac {5\pi}{2}<\theta<3\pi$$, then $$\sin \frac {\theta}{2}$$ equals \_\_\_\_\_\_. | -\frac {3 \sqrt {10}}{10} | 0 |
26,336 |
Through points \( A(0, 14) \) and \( B(0, 4) \), two parallel lines are drawn. The first line, passing through point \( A \), intersects the hyperbola \( y = \frac{1}{x} \) at points \( K \) and \( L \). The second line, passing through point \( B \), intersects the hyperbola \( y = \frac{1}{x} \) at points \( M \) and \( N \).
What is the value of \( \frac{AL - AK}{BN - BM} \)? | 3.5 | 2.34375 |
26,337 | Given circle $M$: $(x+1)^{2}+y^{2}=1$, and circle $N$: $(x-1)^{2}+y^{2}=9$, a moving circle $P$ is externally tangent to circle $M$ and internally tangent to circle $N$. The trajectory of the center of circle $P$ is curve $C$.
$(1)$ Find the equation of $C$.
$(2)$ Let $l$ be a line tangent to both circle $P$ and circle $M$, and $l$ intersects curve $C$ at points $A$ and $B$. When the radius of circle $P$ is the longest, find $|AB|$. | \dfrac {18}{7} | 0 |
26,338 | In triangle $\triangle ABC$, $A+B=5C$, $\sin \left(A-C\right)=2\sin B$.
$(1)$ Find $A$;
$(2)$ If $CM=2\sqrt{7}$ and $M$ is the midpoint of $AB$, find the area of $\triangle ABC$. | 4\sqrt{3} | 13.28125 |
26,339 | A workshop has 11 workers, of which 5 are fitters, 4 are turners, and the remaining 2 master workers can act as both fitters and turners. If we need to select 4 fitters and 4 turners to repair a lathe from these 11 workers, there are __ different methods for selection. | 185 | 3.90625 |
26,340 | Given the sets \( M = \{x, y, \lg(xy)\} \) and \( N = \{0, |x|, y\} \) with \( M = N \), find the value of \(\left(x + \frac{1}{y}\right) + \left(x^{2} + \frac{1}{y^{2}}\right) + \left(x^{3} + \frac{1}{y^{3}}\right) + \cdots + \left(x^{2001} + \frac{1}{y^{2001}}\right) \). | -2 | 4.6875 |
26,341 | Determine the largest value of $S$ such that any finite collection of small squares with a total area $S$ can always be placed inside a unit square $T$ in such a way that no two of the small squares share an interior point. | \frac{1}{2} | 35.9375 |
26,342 | There exist $r$ unique nonnegative integers $n_1 > n_2 > \cdots > n_r$ and $r$ unique integers $a_k$ ($1\le k\le r$) with each $a_k$ either $1$ or $- 1$ such that \[a_13^{n_1} + a_23^{n_2} + \cdots + a_r3^{n_r} = 1025.\]
Find $n_1 + n_2 + \cdots + n_r$. | 17 | 14.0625 |
26,343 | The region shown is bounded by the arcs of circles having a radius 8 units, with each arc having a central angle measure of 45 degrees and intersecting at points of tangency. Determine the area of this region, which can be expressed in the form $a\sqrt{b} + c\pi$ square units, where $\sqrt{b}$ is a radical in simplest form. What is the value of $a + b + c$? | -22 | 0 |
26,344 | Given the ranges $-6 \leq x \leq -2$ and $0 \leq y \leq 4$, determine the largest possible value of the expression $\frac{x+y}{x}$. | \frac{1}{3} | 65.625 |
26,345 | In the convex quadrilateral $ABCD$, the sum of $AB+BD+DC$ is at most 2, and the area of the quadrilateral is $1/2$. What can be the length of diagonal $AC$? | \sqrt{2} | 12.5 |
26,346 | The numbers \(1, 2, 3, \ldots, 400\) are written on 400 cards. Two players, \(A\) and \(B\), play the following game:
1. In the first step, \(A\) takes 200 cards for themselves.
2. \(B\) then takes 100 cards from both the remaining 200 cards and the 200 cards that \(A\) has, totaling 200 cards for themselves, and leaves the remaining 200 cards for \(A\).
3. In the next step, \(A\) takes 100 cards from both players' cards, totaling 200 cards for themselves, leaving the remaining 200 cards for \(B\).
This process continues until \(B\) completes the 200th step. At the end, the sums of the numbers on the cards held by \(A\) and \(B\) are calculated as \(C_A\) and \(C_B\) respectively. Then, player \(A\) pays player \(B\) the difference \(C_B - C_A\).
What is the maximum difference that \(B\) can achieve if both players play optimally? | 20000 | 40.625 |
26,347 | Given a tetrahedron $ABCD$, each edge is colored red or blue with equal probability. What is the probability that point $A$ can reach point $B$ through red edges within the tetrahedron $ABCD$? | \frac{3}{4} | 5.46875 |
26,348 | The function \(f(x) = 5x^2 - 15x - 2\) has a minimum value when x is negative. | -13.25 | 0.78125 |
26,349 | Fill each cell in the given grid with a number from 1 to 4 so that no number repeats within any row or column. Each "L" shaped block spans two rows and two columns. The numbers inside the circles on the line indicate the sum of the numbers in the two adjacent cells (as shown in the provided example, where the third row, from left to right, is 2, 3, 1). What is the four-digit number formed by the two numbers in the bottom row of the given grid, in left to right order? | 2143 | 0 |
26,350 | In circle $O$, $\overline{EB}$ is a diameter and the line $\overline{DC}$ is parallel to $\overline{EB}$. The line $\overline{AB}$ intersects the circle again at point $F$ such that $\overline{AB}$ is parallel to $\overline{ED}$. If angles $AFB$ and $ABF$ are in the ratio 3:2, find the degree measure of angle $BCD$. | 72 | 0 |
26,351 | How many four-digit positive integers are multiples of 7? | 1285 | 0 |
26,352 | A typical triangle is defined as one in which each angle is at least \(30^{\circ}\).
A triangle is identified using the following procedure. The perimeter of a circle is divided into 180 equal arcs, and 3 distinct points are randomly chosen from these division points. (Any three points are chosen with equal probability.) What is the probability that the selected points form a typical triangle?
What is the probability of selecting a typical triangle if the vertices can be chosen arbitrarily on the perimeter of the circle? | 0.25 | 0 |
26,353 | In a blackboard, it's written the following expression $ 1-2-2^2-2^3-2^4-2^5-2^6-2^7-2^8-2^9-2^{10}$ We put parenthesis by different ways and then we calculate the result. For example: $ 1-2-\left(2^2-2^3\right)-2^4-\left(2^5-2^6-2^7\right)-2^8-\left( 2^9-2^{10}\right)= 403$ and $ 1-\left(2-2^2 \left(-2^3-2^4 \right)-\left(2^5-2^6-2^7\right)\right)- \left(2^8- 2^9 \right)-2^{10}= -933$ How many different results can we obtain? | 1024 | 9.375 |
26,354 | A circle can be inscribed in a trapezoid with bases of lengths 3 and 5, and a circle can also be circumscribed about it. Calculate the area of the pentagon formed by the radii of the inscribed circle that are perpendicular to the legs of the trapezoid, its smaller base, and the corresponding segments of the legs. | \frac{3 \sqrt{15}}{2} | 5.46875 |
26,355 | Calculate the value for the expression $\sqrt{25\sqrt{15\sqrt{9}}}$. | 5\sqrt{15} | 0 |
26,356 | Let \(a\), \(b\), \(c\) be real numbers such that \(9a^2 + 4b^2 + 25c^2 = 1\). Find the maximum value of
\[
10a + 3b + 5c.
\] | \sqrt{134} | 2.34375 |
26,357 |
The hour and minute hands of a clock move continuously and at constant speeds. A moment of time $X$ is called interesting if there exists such a moment $Y$ (the moments $X$ and $Y$ do not necessarily have to be different), so that the hour hand at moment $Y$ will be where the minute hand is at moment $X$, and the minute hand at moment $Y$ will be where the hour hand is at moment $X$. How many interesting moments will there be from 00:01 to 12:01? | 143 | 17.1875 |
26,358 | Given the arithmetic sequence {a<sub>n</sub>} satisfies a<sub>3</sub> − a<sub>2</sub> = 3, a<sub>2</sub> + a<sub>4</sub> = 14.
(I) Find the general term formula for {a<sub>n</sub>};
(II) Let S<sub>n</sub> be the sum of the first n terms of the geometric sequence {b<sub>n</sub>}. If b<sub>2</sub> = a<sub>2</sub>, b<sub>4</sub> = a<sub>6</sub>, find S<sub>7</sub>. | -86 | 3.90625 |
26,359 | In triangle $XYZ$, points $X'$, $Y'$, and $Z'$ are on the sides $YZ$, $ZX$, and $XY$, respectively. Given that lines $XX'$, $YY'$, and $ZZ'$ are concurrent at point $P$, and that $\frac{XP}{PX'}+\frac{YP}{PY'}+\frac{ZP}{PZ'}=100$, find the product $\frac{XP}{PX'}\cdot \frac{YP}{PY'}\cdot \frac{ZP}{PZ'}$. | 102 | 0 |
26,360 | Given the cubic function $f(x) = x^3 - 9x^2 + 20x - 4$, and the arithmetic sequence $\{a_n\}$ with $a_5 = 3$, find the value of $f(a_1) + f(a_2) + ... + f(a_9)$. | 18 | 19.53125 |
26,361 | In a regular quadrilateral pyramid $P-ABCD$, if $\angle APC = 60^{\circ}$, find the cosine of the dihedral angle between planes $A-PB-C$. | $-\frac{1}{7}$ | 0 |
26,362 | From the digits 0, 1, 2, 3, 4, 5, 6, select 2 even numbers and 1 odd number to form a three-digit number without repeating digits. The number of such three-digit numbers that are divisible by 5 is ____. (Answer with a number) | 27 | 2.34375 |
26,363 | How many four-digit positive integers are multiples of 7? | 1286 | 99.21875 |
26,364 | Select two distinct integers, $m$ and $n$, randomly from the set $\{3,4,5,6,7,8,9,10,11,12\}$. What is the probability that $3mn - m - n$ is a multiple of $5$? | \frac{2}{9} | 14.0625 |
26,365 | A line initially 1 inch long grows according to a new law, where the first term is the initial length and an additional linear term is introduced.
\[1+\frac{1}{3}\sqrt{3}+2\cdot\frac{1}{9}+3\cdot\frac{1}{27}\sqrt{3}+4\cdot\frac{1}{81}+\cdots\]
Determine the limit of the length of the line if the growth process continues indefinitely.
A) $\frac{1}{72}(4+\sqrt{3})$
B) $\frac{1}{3}(4+\sqrt{3})$
C) $\sqrt{3}$
D) $\infty$ | \frac{1}{3}(4+\sqrt{3}) | 58.59375 |
26,366 | Let \( n = 2^{25} 3^{17} \). How many positive integer divisors of \( n^2 \) are less than \( n \) but do not divide \( n \)? | 424 | 84.375 |
26,367 | Let $ABCDEF$ be a regular hexagon. Points $P$ and $Q$ on tangents to its circumcircle at $A$ and $D$ respectively are such that $PQ$ touches the minor arc $EF$ of this circle. Find the angle between $PB$ and $QC$ . | 30 | 1.5625 |
26,368 | Define a natural number to be "super-nice" if it is either the product of exactly three distinct prime numbers or the fourth power of a prime number. What is the sum of the first eight super-nice numbers? | 520 | 0.78125 |
26,369 | Given that $S_{n}=\sin \frac{\pi}{7}+\sin \frac{2\pi}{7}+\cdots+\sin \frac{n\pi}{7}$, determine the number of positive values in the sequence $S_{1}$, $S_{2}$, …, $S_{100}$. | 86 | 2.34375 |
26,370 | What is the tenth number in the row of Pascal's triangle that has 100 numbers? | \binom{99}{9} | 0 |
26,371 | How many ways can 8 teaching positions be allocated to three schools, given that each school receives at least one position, and School A receives at least two positions? | 10 | 12.5 |
26,372 | One day, School A bought 56 kilograms of fruit candy at 8.06 yuan per kilogram. A few days later, School B also needed to buy the same 56 kilograms of fruit candy, but it happened that there was a promotional event, and the price of fruit candy was reduced by 0.56 yuan per kilogram. Additionally, they received 5% extra fruit candy for free. How much less did School B spend compared to School A? | 51.36 | 0 |
26,373 | Given circle $O$, points $E$ and $F$ are located such that $E$ and $F$ are on opposite sides of the diameter $\overline{AB}$. If $\angle AOE = 60^\circ$ and $\angle BOF = 30^\circ$, find the ratio of the area of sector $EOF$ to the area of the circle. | \frac{3}{4} | 13.28125 |
26,374 | Compute $\tan\left(\frac{\pi}{9}\right)\tan\left(\frac{2\pi}{9}\right)\tan\left(\frac{4\pi}{9}\right)$. | \sqrt{3} | 75.78125 |
26,375 | Let $M$ be a subset of $\{1,2,3... 2011\}$ satisfying the following condition:
For any three elements in $M$ , there exist two of them $a$ and $b$ such that $a|b$ or $b|a$ .
Determine the maximum value of $|M|$ where $|M|$ denotes the number of elements in $M$ | 18 | 0 |
26,376 | Let $f(x)$ be a polynomial with real, nonnegative coefficients. If $f(5) = 25$ and $f(20) = 1024$, find the largest possible value of $f(10)$. | 100 | 1.5625 |
26,377 | Let $ABCD$ be a rhombus of sides $AB = BC = CD= DA = 13$ . On the side $AB$ construct the rhombus $BAFC$ outside $ABCD$ and such that the side $AF$ is parallel to the diagonal $BD$ of $ABCD$ . If the area of $BAFE$ is equal to $65$ , calculate the area of $ABCD$ . | 120 | 29.6875 |
26,378 | In Cologne, there were three brothers who had 9 containers of wine. The first container had a capacity of 1 quart, the second contained 2 quarts, with each subsequent container holding one more quart than the previous one, so the last container held 9 quarts. The task is to divide the wine equally among the three brothers without transferring wine between the containers. | 15 | 14.0625 |
26,379 | Let \(a\), \(b\), and \(c\) be nonnegative real numbers such that \(a^2 + b^2 + c^2 = 1\). Find the maximum value of
\[2ab \sqrt{3} + 2ac.\] | \sqrt{3} | 10.9375 |
26,380 | Two individuals, A and B, travel from point $A$ to point $B$. A departs 48 minutes before B. When A has traveled $\frac{2}{3}$ of the total distance, B catches up to A. If B immediately returns to point $A$ after arriving at point $B$ at the same speed, B meets A 6 minutes after leaving point $B$. How many more minutes will A need to reach point $B$ after B catches up to A again? | 12 | 2.34375 |
26,381 | What is the largest prime factor of the sum of $1579$ and $5464$? | 7043 | 1.5625 |
26,382 | When three standard dice are tossed, the numbers $x, y, z$ are obtained. Find the probability that $xyz = 72$. | \frac{1}{24} | 35.9375 |
26,383 | The symbol $[x]$ represents the greatest integer less than or equal to the real number $x$. Find the solution to the equation $\left[3 x - 4 \frac{5}{6}\right] - 2 x - 1 = 0$. | 6.5 | 1.5625 |
26,384 | Given the hyperbola $C:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(a>0,b>0)$ with the right focus $F$, the upper endpoint of the imaginary axis $B$, points $P$ and $Q$ on the hyperbola, and point $M(-2,1)$ as the midpoint of segment $PQ$, where $PQ$ is parallel to $BF$. Find $e^{2}$. | \frac{\sqrt{2}+1}{2} | 0 |
26,385 | A room is built in the shape of the region between two semicircles with the same center and parallel diameters. The farthest distance between two points with a clear line of sight is $12$ m. What is the area (in $m^2$ ) of the room?
 | 18\pi | 57.8125 |
26,386 | Esquecinaldo has a poor memory for storing numbers but an excellent memory for remembering sequences of operations. To remember his five-digit bank code, he can recall that:
1. The code has no repeated digits.
2. None of the digits are zero.
3. The first two digits form a power of 5.
4. The last two digits form a power of 2.
5. The middle digit is a multiple of 3.
6. The sum of all the digits is an odd number.
Now he doesn't need to memorize the number because he knows his code is the largest number that satisfies these conditions. What is this code? | 25916 | 25.78125 |
26,387 | Given sets $A=\{2, a-1\}$, $B=\{a^2-7, -1\}$, and $A\cap B=\{2\}$, find the real number $a$. | -3 | 17.1875 |
26,388 | Let $g$ be a function defined on the positive integers, such that
\[g(xy) = g(x) + g(y)\] for all positive integers $x$ and $y.$ Given $g(12) = 18$ and $g(48) = 26,$ find $g(600).$ | 36 | 0 |
26,389 | What is the smallest number of 3-cell L-shaped tiles that can be placed in an 8x8 square such that no more of these tiles can be placed in the square? | 11 | 0.78125 |
26,390 | For how many positive integers \(n\) less than or equal to \(50\) is \(n!\) evenly divisible by \(1 + 2 + \cdots + n\)? | 34 | 47.65625 |
26,391 | Given two arithmetic sequences $\{a\_n\}$ and $\{b\_n\}$, let $S\_n$ and $T\_n$ denote the sum of the first $n$ terms of each sequence, respectively. If $\frac{S\_n}{T\_n} = \frac{7n+2}{n+3}$, determine the value of $\frac{a\_5}{b\_5}$. | \frac{65}{12} | 50 |
26,392 | A solid box is 20 cm by 15 cm by 12 cm. A new solid is formed by removing a cube 4 cm on a side from each of the top four corners of this box. After that, four cubes, 2 cm on a side, are placed on each lower corner of the box. What percent of the original volume has been altered (either lost or gained)? | 6.22\% | 25.78125 |
26,393 | Eight distinct integers are picked at random from $\{1,2,3,\ldots,15\}$. What is the probability that, among those selected, the third smallest is $5$? | \frac{72}{307} | 0 |
26,394 | On Monday, 5 students in the class received A's in math, on Tuesday 8 students received A's, on Wednesday 6 students, on Thursday 4 students, and on Friday 9 students. None of the students received A's on two consecutive days. What is the minimum number of students that could have been in the class? | 14 | 1.5625 |
26,395 | If the solution set of the inequality system about $x$ is $\left\{\begin{array}{l}{x+1≤\frac{2x-5}{3}}\\{a-x＞1}\end{array}\right.$ is $x\leqslant -8$, and the solution of the fractional equation about $y$ is $4+\frac{y}{y-3}=\frac{a-1}{3-y}$ is a non-negative integer, then the sum of all integers $a$ that satisfy the conditions is ____. | 24 | 43.75 |
26,396 | Two linear functions \( f(x) \) and \( g(x) \) satisfy the properties that for all \( x \),
- \( f(x) + g(x) = 2 \)
- \( f(f(x)) = g(g(x)) \)
and \( f(0) = 2022 \). Compute \( f(1) \). | 2021 | 0.78125 |
26,397 | A number is called *6-composite* if it has exactly 6 composite factors. What is the 6th smallest 6-composite number? (A number is *composite* if it has a factor not equal to 1 or itself. In particular, 1 is not composite.)
*Ray Li.* | 441 | 43.75 |
26,398 | Let $ABC$ be a right triangle where $\measuredangle A = 90^\circ$ and $M\in (AB)$ such that $\frac{AM}{MB}=3\sqrt{3}-4$ . It is known that the symmetric point of $M$ with respect to the line $GI$ lies on $AC$ . Find the measure of $\measuredangle B$ . | 30 | 71.875 |
26,399 | In rectangle $ABCD$, $AB = 4$ and $BC = 8$. The rectangle is folded so that points $A$ and $C$ coincide, forming the pentagon $ABEFD$. What is the length of segment $EF$? Express your answer in simplest radical form. | 2\sqrt{5} | 25 |
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