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26,100 | Given that the sum of the binomial coefficients in the expansion of $(5x- \frac{1}{\sqrt{x}})^n$ is 64, determine the constant term in its expansion. | 375 | 7.03125 |
26,101 | Consider the two points \(A(4,1)\) and \(B(2,5)\). For each point \(C\) with positive integer coordinates, we define \(d_C\) to be the shortest distance needed to travel from \(A\) to \(C\) to \(B\) moving only horizontally and/or vertically. The positive integer \(N\) has the property that there are exactly 2023 points \(C(x, y)\) with \(x > 0\) and \(y > 0\) and \(d_C = N\). What is the value of \(N\)? | 12 | 0 |
26,102 | A light flashes in one of three different colors: red, green, and blue. Every $3$ seconds, it flashes green. Every $5$ seconds, it flashes red. Every $7$ seconds, it flashes blue. If it is supposed to flash in two colors at once, it flashes the more infrequent color. How many times has the light flashed green after $671$ seconds? | 154 | 0 |
26,103 | Given \( m = n^{4} + x \), where \( n \) is a natural number and \( x \) is a two-digit positive integer, what value of \( x \) will make \( m \) a composite number? | 64 | 0 |
26,104 | Let $a,$ $b,$ and $c$ be nonnegative numbers such that $a^2 + b^2 + c^2 = 1.$ Find the maximum value of
\[3ab \sqrt{2} + 6bc.\] | 4.5 | 0 |
26,105 | In an isosceles triangle \(ABC\) (\(AB = BC\)), the angle bisectors \(AM\) and \(BK\) intersect at point \(O\). The areas of triangles \(BOM\) and \(COM\) are 25 and 30, respectively. Find the area of triangle \(ABC\). | 110 | 39.84375 |
26,106 | Determine the number of ways to arrange the letters of the word PERSEVERANCE. | 9,979,200 | 0 |
26,107 | In the convex quadrilateral \(ABCD\): \(AB = AC = AD = BD\) and \(\angle BAC = \angle CBD\). Find \(\angle ACD\). | 60 | 22.65625 |
26,108 | From the 2015 natural numbers between 1 and 2015, what is the maximum number of numbers that can be found such that their product multiplied by 240 is a perfect square? | 134 | 0 |
26,109 | A fair six-sided die with faces numbered 1, 2, 3, 4, 5, and 6 is rolled twice. Let $a$ and $b$ denote the outcomes of the first and second rolls, respectively.
(1) Find the probability that the line $ax + by + 5 = 0$ is tangent to the circle $x^2 + y^2 = 1$.
(2) Find the probability that the segments with lengths $a$, $b$, and $5$ form an isosceles triangle. | \frac{7}{18} | 5.46875 |
26,110 | A king traversed a $9 \times 9$ chessboard, visiting each square exactly once. The king's route is not a closed loop and may intersect itself. What is the maximum possible length of such a route if the length of a move diagonally is $\sqrt{2}$ and the length of a move vertically or horizontally is 1? | 16 + 64 \sqrt{2} | 0 |
26,111 | In convex quadrilateral $ABCD$ , $\angle ADC = 90^\circ + \angle BAC$ . Given that $AB = BC = 17$ , and $CD = 16$ , what is the maximum possible area of the quadrilateral?
*Proposed by Thomas Lam* | 529/2 | 0 |
26,112 | Arrange 1, 2, 3, a, b, c in a row such that letter 'a' is not at either end and among the three numbers, exactly two are adjacent. The probability is $\_\_\_\_\_\_$. | \frac{2}{5} | 7.03125 |
26,113 | Given the function $f(x) = x^3 - 3x^2 - 9x + 2$,
(I) Determine the intervals of monotonicity for the function $f(x)$;
(II) Find the minimum value of function $f(x)$ on the interval [-2, 2]. | -20 | 70.3125 |
26,114 | Define the sequence $ (a_p)_{p\ge0}$ as follows: $ a_p\equal{}\displaystyle\frac{\binom p0}{2\cdot 4}\minus{}\frac{\binom p1}{3\cdot5}\plus{}\frac{\binom p2}{4\cdot6}\minus{}\ldots\plus{}(\minus{}1)^p\cdot\frac{\binom pp}{(p\plus{}2)(p\plus{}4)}$ .
Find $ \lim_{n\to\infty}(a_0\plus{}a_1\plus{}\ldots\plus{}a_n)$ . | 1/3 | 3.125 |
26,115 | Given that $17^{-1} \equiv 31 \pmod{53}$, find $36^{-1} \pmod{53}$ as a residue modulo 53 (i.e., a value between 0 and 52 inclusive). | 22 | 58.59375 |
26,116 |
The ferry "Yi Rong" travels at a speed of 40 kilometers per hour. On odd days, it travels downstream from point $A$ to point $B$, while on even days, it travels upstream from point $B$ to point $A$ (with the water current speed being 24 kilometers per hour). On one odd day, when the ferry reached the midpoint $C$, it lost power and drifted downstream to point $B$. The captain found that the total time taken that day was $\frac{43}{18}$ times the usual time for an odd day.
On another even day, the ferry again lost power as it reached the midpoint $C$. While drifting, the repair crew spent 1 hour repairing the ferry, after which it resumed its journey to point $A$ at twice its original speed. The captain observed that the total time taken that day was exactly the same as the usual time for an even day. What is the distance between points $A$ and $B$ in kilometers? | 192 | 11.71875 |
26,117 | Let tetrahedron $ABCD$ have $AD=BC=30$, $AC=BD=40$, and $AB=CD=50$. For any point $X$ in space, suppose $g(X)=AX+BX+CX+DX$. Determine the least possible value of $g(X)$, expressed as $p\sqrt{q}$ where $p$ and $q$ are positive integers with $q$ not divisible by the square of any prime. Report the sum $p+q$. | 101 | 19.53125 |
26,118 | How many integers $-15 \leq n \leq 10$ satisfy $(n-1)(n+3)(n + 7) < 0$? | 11 | 65.625 |
26,119 | A square with an area of 40 is inscribed in a semicircle. If another square is inscribed in a full circle with the same radius, what is the area of this square? | 80 | 28.125 |
26,120 | Given Aniyah has a $5 \times 7$ index card and if she shortens the length of one side of this card by $2$ inches, the card would have an area of $21$ square inches, determine the area of the card in square inches if she shortens the length of the other side by $2$ inches. | 25 | 69.53125 |
26,121 |
The standard enthalpy of formation (ΔH_f°) of a substance is equal to the heat effect of the formation reaction of 1 mole of the substance from simple substances in their standard states (at 1 atm pressure and a given temperature). Therefore, it is necessary to find the heat effect of the reaction:
$$
\underset{\text {graphite}}{6 \mathrm{C}(\kappa)} + 3 \mathrm{H}_{2}(g) = \mathrm{C}_{6} \mathrm{H}_{6} (l) + \mathrm{Q}_{\text{formation}}\left(\mathrm{C}_{6} \mathrm{H}_{6}(l)\right) (4)
$$
According to Hess's law, the heat effect of the reaction depends only on the types and states of the reactants and products and does not depend on the path of the transition.
Hess's law allows dealing with thermochemical equations like algebraic expressions, i.e., based on it, by combining equations of reactions with known heat effects, one can calculate the unknown heat effect of the overall reaction.
Thus, we obtain:
$$
\mathrm{C}_{2}\mathrm{H}_{2}(g) = \underset{\text {graphite}}{2 \mathrm{C}(\kappa)} + \mathrm{H}_{2}(g) + 226.7 \text { kJ; } \quad -3
$$
$$
\begin{array}{lll}
3 \mathrm{C}_{2} \mathrm{H}_{2}(g) & = \mathrm{C}_{6} \mathrm{H}_{6}(l) + 631.1 \text { kJ;} & 1 \\
\mathrm{C}_{6} \mathrm{H}_{6}(l) & = \mathrm{C}_{6} \mathrm{H}_{6}(l) - 33.9 \text { kJ;} & 1
\end{array}
$$
$$
\underset{\text {graphite}}{6 \mathrm{C}(\kappa)} + 3 \mathrm{H}_{2}(g) = \mathrm{C}_{6} \mathrm{H}_{6}(l) + \mathrm{Q}_{\text{formation}}\left(\mathrm{C}_{6} \mathrm{H}_{6}(l)\right);
$$
$$
\mathrm{Q}_{\text{formation}}\left(\mathrm{C}_{6} \mathrm{H}_{6}(l)\right) = 226.7 \cdot (-3) + 631.1 - 33.9 = -82.9 \text{ kJ/mol}.
$$ | -82.9 | 56.25 |
26,122 | Select 5 volunteers from 8 candidates, including A and B, to participate in community service activities from Monday to Friday, with one person arranged for each day, and each person participating only once. If at least one of A and B must participate, and when both A and B participate, their service dates cannot be adjacent, then the number of different arrangements is ______ (Answer in digits). | 5040 | 48.4375 |
26,123 | Given that the function $f(x)$ is an even function on $(-\infty, +\infty)$, and $f(5+x) = f(5-x)$, if $f(x)$ only equals $0$ at $f(1)=0$ within the interval $[0,5]$, determine the number of zeros of $f(x)$ in the interval $[-2012, 2012]$. | 806 | 53.125 |
26,124 | In the 3rd grade, the boys wear blue swim caps, and the girls wear red swim caps. The male sports commissioner says, "I see 1 more blue swim cap than 4 times the number of red swim caps." The female sports commissioner says, "I see 24 more blue swim caps than red swim caps." Based on the sports commissioners' statements, calculate the total number of students in the 3rd grade. | 37 | 2.34375 |
26,125 | What is the maximum number of finite roots that the equation
$$
\left|x - a_{1}\right| + \ldots + |x - a_{50}| = \left|x - b_{1}\right| + \ldots + |x - b_{50}|
$$
can have, where $a_{1}, a_{2}, \ldots, a_{50}, b_{1}, b_{2}, \ldots, b_{50}$ are distinct numbers? | 49 | 77.34375 |
26,126 | Given \( x \) satisfies \(\log _{5 x} 2 x = \log _{625 x} 8 x\), find the value of \(\log _{2} x\). | \frac{\ln 5}{2 \ln 2 - 3 \ln 5} | 0.78125 |
26,127 | Eight students from a university are preparing to carpool for a trip. There are two students from each grade level—freshmen, sophomores, juniors, and seniors—divided into two cars, Car A and Car B, with each car seating four students. The seating arrangement of the four students in the same car is not considered. However, the twin sisters, who are freshmen, must ride in the same car. The number of ways for Car A to have exactly two students from the same grade is _______. | 24 | 23.4375 |
26,128 | $N$ students are seated at desks in an $m \times n$ array, where $m, n \ge 3$ . Each student shakes hands with the students who are adjacent horizontally, vertically or diagonally. If there are $1020 $ handshakes, what is $N$ ? | 280 | 28.125 |
26,129 | Define $\phi_n(x)$ to be the number of integers $y$ less than or equal to $n$ such that $\gcd(x,y)=1$ . Also, define $m=\text{lcm}(2016,6102)$ . Compute $$ \frac{\phi_{m^m}(2016)}{\phi_{m^m}(6102)}. $$ | 339/392 | 0.78125 |
26,130 | If five pairwise coprime distinct integers \( a_{1}, a_{2}, \cdots, a_{5} \) are randomly selected from \( 1, 2, \cdots, n \) and there is always at least one prime number among them, find the maximum value of \( n \). | 48 | 2.34375 |
26,131 | Every day at noon, a scheduled ship departs from Moscow to Astrakhan and from Astrakhan to Moscow. The ship traveling from Moscow takes exactly four days to reach Astrakhan, then stays there for two days, and at noon two days after its arrival in Astrakhan, it departs back to Moscow. The ship traveling from Astrakhan takes exactly five days to reach Moscow and, after resting for two days in Moscow, departs back to Astrakhan. How many ships must be operational on the Moscow - Astrakhan - Moscow route under the described travel conditions? | 13 | 34.375 |
26,132 | A truck delivered 4 bags of cement. They are stacked in the truck. A worker can carry one bag at a time either from the truck to the gate or from the gate to the shed. The worker can carry the bags in any order, each time taking the top bag, carrying it to the respective destination, and placing it on top of the existing stack (if there are already bags there). If given a choice to carry a bag from the truck or from the gate, the worker randomly chooses each option with a probability of 0.5. Eventually, all the bags end up in the shed.
a) (7th grade level, 1 point). What is the probability that the bags end up in the shed in the reverse order compared to how they were placed in the truck?
b) (7th grade level, 1 point). What is the probability that the bag that was second from the bottom in the truck ends up as the bottom bag in the shed? | \frac{1}{8} | 6.25 |
26,133 | What are the last $2$ digits of the number $2018^{2018}$ when written in base $7$ ? | 44 | 78.125 |
26,134 | For how many integer values of $a$ does the equation $$x^2 + ax + 12a = 0$$ have integer solutions for $x$? | 16 | 7.03125 |
26,135 | Given a sequence $1$, $1$, $3$, $1$, $3$, $5$, $1$, $3$, $5$, $7$, $1$, $3$, $5$, $7$, $9$, $\ldots$, where the first term is $1$, the next two terms are $1$, $3$, and the next three terms are $1$, $3$, $5$, and so on. Let $S_{n}$ denote the sum of the first $n$ terms of this sequence. Find the smallest positive integer value of $n$ such that $S_{n} > 400$. | 59 | 0.78125 |
26,136 | Evaluate the expression:
\[4(1+4(1+4(1+4(1+4(1+4(1+4(1+4(1))))))))\] | 87380 | 0.78125 |
26,137 | Given that $\alpha$ is an obtuse angle and $\beta$ is also an obtuse angle with $\cos\alpha = -\frac{2\sqrt{5}}{5}$ and $\sin\beta = \frac{\sqrt{10}}{10}$, find the value of $\alpha + \beta$. | \frac{7\pi}{4} | 13.28125 |
26,138 | Using toothpicks of equal length, a rectangular grid is constructed. The grid measures 25 toothpicks in height and 15 toothpicks in width. Additionally, there is an internal horizontal partition at every fifth horizontal line starting from the bottom. Calculate the total number of toothpicks used. | 850 | 6.25 |
26,139 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $\sqrt{3}b\cos A=a\cos B$.
$(1)$ Find the angle $A$;
$(2)$ If $a= \sqrt{2}$ and $\frac{c}{a}= \frac{\sin A}{\sin B}$, find the perimeter of $\triangle ABC$. | 3 \sqrt{2} | 41.40625 |
26,140 | Among the three-digit numbers formed by the digits 0, 1, 2, 3, 4, 5 without repetition, there are a total of numbers whose digits sum up to 9 (answer in digits). | 16 | 84.375 |
26,141 | Determine the value of
\[1002 + \frac{1}{3} \left( 1001 + \frac{1}{3} \left( 1000 + \dots + \frac{1}{3} \left( 3 + \frac{1}{3} \cdot 2 \right) \right) \dotsb \right).\] | 1502.25 | 4.6875 |
26,142 | How many pairs of positive integers have greatest common divisor 5! and least common multiple 50! ? | 16384 | 12.5 |
26,143 | Let the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ be $S_n$, and it satisfies $S_{2016} > 0$, $S_{2017} < 0$. For any positive integer $n$, we have $|a_n| \geqslant |a_k|$. Determine the value of $k$. | 1009 | 39.0625 |
26,144 | Given the function $f(x) = x^2 - (2t + 1)x + t \ln x$ where $t \in \mathbb{R}$,
(1) If $t = 1$, find the extreme values of $f(x)$.
(2) Let $g(x) = (1 - t)x$, and suppose there exists an $x_0 \in [1, e]$ such that $f(x_0) \geq g(x_0)$ holds. Find the maximum value of the real number $t$. | \frac{e(e - 2)}{e - 1} | 0 |
26,145 | Let $x, y, z$ be real numbers such that:
\begin{align*}
y+z & = 16, \\
z+x & = 18, \\
x+y & = 20.
\end{align*}
Find $\sqrt{xyz(x+y+z)}$. | 9\sqrt{77} | 0.78125 |
26,146 |
A circle \( K \) goes through the vertices \( A \) and \( C \) of triangle \( ABC \). The center of circle \( K \) lies on the circumcircle of triangle \( ABC \). Circle \( K \) intersects side \( AB \) at point \( M \). Find the angle \( BAC \) if \( AM : AB = 2 : 7 \) and \( \angle B = \arcsin \frac{4}{5} \). | 45 | 0 |
26,147 | A building has seven rooms numbered 1 through 7 on one floor, with various doors connecting these rooms. The doors can be either one-way or two-way. Additionally, there is a two-way door between room 1 and the outside, and there is a treasure in room 7. Design the arrangement of rooms and doors such that:
(a) It is possible to enter room 1, reach the treasure in room 7, and return outside.
(b) The minimum number of steps required to achieve this (each step involving walking through a door) is as large as possible. | 14 | 16.40625 |
26,148 | In a Cartesian coordinate system, the parametric equation for curve $C_1$ is
$$
\left\{ \begin{aligned}
x &= 2\cos\alpha, \\
y &= \sqrt{2}\sin\alpha
\end{aligned} \right.
$$
with $\alpha$ as the parameter. Using the origin as the pole, the positive half of the $x$-axis as the polar axis, and the same unit length as the Cartesian coordinate system, establish a polar coordinate system. The polar equation for curve $C_2$ is $\rho = \cos\theta$.
(1) Find the general equation for curve $C_1$ and the Cartesian coordinate equation for curve $C_2$;
(2) If $P$ and $Q$ are any points on the curves $C_1$ and $C_2$, respectively, find the minimum value of $|PQ|$. | \frac{\sqrt{7} - 1}{2} | 13.28125 |
26,149 | What code will be produced for this message in the new encoding where the letter А is replaced by 21, the letter Б by 122, and the letter В by 1? | 211221121 | 0.78125 |
26,150 | From the numbers 1, 2, 3, 4, 5, two numbers are randomly selected to be the base and the true number (antilogarithm) of a logarithm, respectively. The total number of different logarithmic values that can be obtained is ___. | 13 | 0.78125 |
26,151 | In a regular tetrahedron ABCD with an edge length of 2, G is the centroid of triangle BCD, and M is the midpoint of line segment AG. The surface area of the circumscribed sphere of the tetrahedron M-BCD is __________. | 6\pi | 60.9375 |
26,152 | A person flips $2010$ coins at a time. He gains one penny every time he flips a prime number of heads, but must stop once he flips a non-prime number. If his expected amount of money gained in dollars is $\frac{a}{b}$ , where $a$ and $b$ are relatively prime, compute $\lceil\log_{2}(100a+b)\rceil$ .
*Proposed by Lewis Chen* | 2017 | 5.46875 |
26,153 | What is the circumference of the region defined by the equation $x^2+y^2 - 10 = 3y - 6x + 3$? | \pi \sqrt{73} | 0.78125 |
26,154 | There are two circles: one centered at point \(A\) with a radius of 5, and another centered at point \(B\) with a radius of 15. Their common internal tangent touches the circles at points \(C\) and \(D\) respectively. The lines \(AB\) and \(CD\) intersect at point \(E\). Find \(CD\) if \(BE = 39\). | 48 | 12.5 |
26,155 | The letter T is formed by placing a $2\:\text{inch} \times 6\:\text{inch}$ rectangle vertically and a $2\:\text{inch} \times 4\:\text{inch}$ rectangle horizontally across the top center of the vertical rectangle. What is the perimeter of the T, in inches? | 24 | 24.21875 |
26,156 | Alli rolls a standard 8-sided die twice. What is the probability of rolling integers that differ by 3 on her first two rolls? Express your answer as a common fraction. | \dfrac{1}{8} | 3.90625 |
26,157 | A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape. The parallel sides of the trapezoid have lengths $18$ and $30$ meters. What fraction of the yard is occupied by the flower beds?
A) $\frac{1}{6}$
B) $\frac{1}{5}$
C) $\frac{1}{4}$
D) $\frac{1}{3}$
E) $\frac{1}{2}$ | \frac{1}{5} | 42.96875 |
26,158 |
Let \( c_{n}=11 \ldots 1 \) be a number in which the decimal representation contains \( n \) ones. Then \( c_{n+1}=10 \cdot c_{n}+1 \). Therefore:
\[ c_{n+1}^{2}=100 \cdot c_{n}^{2} + 22 \ldots 2 \cdot 10 + 1 \]
For example,
\( c_{2}^{2}=11^{2}=(10 \cdot 1+1)^{2}=100+2 \cdot 10+1=121 \),
\( c_{3}^{2} = 111^{2} = 100 \cdot 11^{2} + 220 + 1 = 12100 + 220 + 1 = 12321 \),
\( c_{4}^{2} = 1111^{2} = 100 \cdot 111^{2} + 2220 + 1 = 1232100 + 2220 + 1 = 1234321 \), etc.
We observe that in all listed numbers \( c_{2}^{2}, c_{3}^{2}, c_{4}^{2} \), the digit with respect to which these numbers are symmetric (2 in the case of \( c_{2}^{2}, 3 \) in the case of \( c_{3}^{2}, 4 \) in the case of \( c_{4}^{2} \)) coincides with the number of ones in the number that was squared.
The given number \( c=123456787654321 \) is also symmetric with respect to the digit 8, which suggests that it might be the square of the number \( c_{8} = 11111111 \). This can be verified by performing multiplication by columns or using the recursive relation. | 11111111 | 80.46875 |
26,159 | Given an ellipse $C: \frac{y^{2}}{a^{2}}+ \frac{x^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity of $\frac{\sqrt{2}}{2}$ and the sum of the distances from a point on the ellipse to the two foci is $2\sqrt{2}$. A line $l$ with slope $k(k\neq 0)$ passes through the upper focus of the ellipse and intersects the ellipse at points $P$ and $Q$. The perpendicular bisector of segment $PQ$ intersects the $y$-axis at point $M(0,m)$.
(1) Find the standard equation of the ellipse;
(2) Find the range of values for $m$;
(3) Express the area $S$ of triangle $\Delta MPQ$ in terms of $m$, and find the maximum value of the area $S$. | \frac{3\sqrt{6}}{16} | 2.34375 |
26,160 | The polar coordinate equation of curve C is given by C: ρ² = $\frac{12}{5 - \cos(2\theta)}$, and the parametric equations of line l are given by $\begin{cases} x = 1 + \frac{\sqrt{2}}{2}t \\ y = \frac{\sqrt{2}}{2}t \end{cases}$ (where t is the parameter).
1. Write the rectangular coordinate equation of C and the standard equation of l.
2. Line l intersects curve C at two points A and B. Let point M be (0, -1). Calculate the value of $\frac{|MA| + |MB|}{|MA| \cdot |MB|}$. | \frac{4\sqrt{3}}{3} | 4.6875 |
26,161 | How many of the natural numbers from 1 to 800, inclusive, contain the digit 7 at least once? | 62 | 0 |
26,162 | Given an ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ with its left and right foci being $F_1$ and $F_2$ respectively, and its eccentricity $e = \dfrac{\sqrt{2}}{2}$, the length of the minor axis is $2$.
$(1)$ Find the equation of the ellipse;
$(2)$ Point $A$ is a moving point on the ellipse (not the endpoints of the major axis), the extension line of $AF_2$ intersects the ellipse at point $B$, and the extension line of $AO$ intersects the ellipse at point $C$. Find the maximum value of the area of $\triangle ABC$. | \sqrt{2} | 19.53125 |
26,163 | How many positive odd integers greater than 1 and less than $200$ are square-free? | 79 | 0 |
26,164 | If four coins are tossed at the same time, what is the probability of getting exactly three heads and one tail? Follow this by rolling a six-sided die. What is the probability that the die shows a number greater than 4? | \frac{1}{12} | 95.3125 |
26,165 | What is the radius of the smallest circle inside which every planar closed polygonal line with a perimeter of 1 can be enclosed? | \frac{1}{4} | 3.90625 |
26,166 | Half of the yellow flowers are tulips, one third of the blue flowers are daisies, and seven tenths of the flowers are yellow. Find the percentage of flowers that are daisies. | 45\% | 3.125 |
26,167 | Petya invented four distinct natural numbers and wrote down all their pairwise sums on the board. Below those, he wrote all their sums taken three at a time. It turned out that the sum of the two largest pairwise sums and the two smallest sums from those taken three at a time (a total of four sums) is 2017. Find the largest possible value of the sum of the four numbers that Petya invented. | 1006 | 3.125 |
26,168 | What is the greatest possible sum of the digits in the base-nine representation of a positive integer less than $3000$? | 24 | 2.34375 |
26,169 | There is a settlement $C$ (a point) located at the intersection of roads $A$ and $B$ (straight lines). Sasha walks along road $A$ towards $C$, taking 45 steps per minute with a step length of 60 cm. At the start, Sasha is 290 m away from $C$. Dania walks along road $B$ towards $C$ at a rate of 55 steps per minute with a step length of 65 cm, and at the start of their movement, Dania is 310 m away from $C$. Each person continues walking along their road without stopping after passing point $C$. We record the moments in time when both Dania and Sasha have taken whole numbers of steps. Find the minimum possible distance between them (along the roads) at such moments in time. Determine the number of steps each of them has taken by the time this minimum distance is achieved. | 57 | 0 |
26,170 | Given that $a$ is a positive real number and $b$ is an integer between $1$ and $201$, inclusive, find the number of ordered pairs $(a,b)$ such that $(\log_b a)^{2023}=\log_b(a^{2023})$. | 603 | 53.125 |
26,171 | Given a triangle with integral sides and an isosceles perimeter of 11, calculate the area of the triangle. | \frac{5\sqrt{2.75}}{2} | 0 |
26,172 |
There are 5 students taking a test, and each student's score ($a, b, c, d, e$) is an integer between 0 and 100 inclusive. It is known that $a \leq b \leq c \leq d \leq e$. If the average score of the 5 students is $p$, then the median score $c$ is at least $\qquad$ . | 40 | 18.75 |
26,173 | At a dog show, each dog was assigned a sequential number from 1 to 24. Due to health reasons, one of the dogs was unable to participate in the competition. It turns out that among the remaining 23 dogs, one has a number equal to the arithmetic mean of the remaining dogs' numbers. What was the number assigned to the dog that could not participate in the show? If there are multiple solutions, list these numbers in ascending order without spaces. | 124 | 0 |
26,174 | Given the function f(x) = a^x (a > 0, a ≠ 1).
(I) If $f(1) + f(-1) = \frac{5}{2}$, find the value of f(2) + f(-2).
(II) If the difference between the maximum and minimum values of the function f(x) on [-1, 1] is $\frac{8}{3}$, find the value of the real number a. | \frac{1}{3} | 25 |
26,175 | Consider the region \(B\) in the complex plane consisting of all points \(z\) such that both \(\frac{z}{50}\) and \(\frac{50}{\overline{z}}\) have real and imaginary parts between 0 and 1, inclusive. Find the area of \(B\). | 2500 - 312.5 \pi | 4.6875 |
26,176 | How many unique numbers can you get by multiplying two or more distinct members of the set $\{1,2,3,5,7\}$ together? | 11 | 0.78125 |
26,177 | A sequence of integers has a mode of 32, a mean of 22, a smallest number of 10, and a median of \( m \). If \( m \) is replaced by \( m+10 \), the new sequence has a mean of 24 and a median of \( m+10 \). If \( m \) is replaced by \( m-8 \), the new sequence has a median of \( m-4 \). What is the value of \( m \)? | 20 | 2.34375 |
26,178 | Three lines are drawn parallel to the sides of a triangle through a point inside it, dividing the triangle into six parts: three triangles and three quadrilaterals. The areas of all three inner triangles are equal. Determine the range within which the ratio of the area of each inner triangle to the area of the original triangle can lie. | 1/9 | 2.34375 |
26,179 | Given the function $f\left(x\right)=x^{3}+ax^{2}+bx+2$ has an extremum of $7$ at $x=-1$.<br/>$(1)$ Find the intervals where $f\left(x\right)$ is monotonic;<br/>$(2)$ Find the extremum of $f\left(x\right)$ on $\left[-2,4\right]$. | -25 | 35.15625 |
26,180 | Compute
\[
\sin^6 0^\circ + \sin^6 1^\circ + \sin^6 2^\circ + \dots + \sin^6 90^\circ.
\] | \frac{229}{8} | 20.3125 |
26,181 | Given an geometric sequence \\(\{a_n\}\) with a common ratio less than \\(1\\), the sum of the first \\(n\\) terms is \\(S_n\\), and \\(a_1 = \frac{1}{2}\\), \\(7a_2 = 2S_3\\).
\\((1)\\) Find the general formula for the sequence \\(\{a_n\}\).
\\((2)\\) Let \\(b_n = \log_2(1-S_{n+1})\\). If \\(\frac{1}{{b_1}{b_3}} + \frac{1}{{b_3}{b_5}} + \ldots + \frac{1}{{b_{2n-1}}{b_{2n+1}}} = \frac{5}{21}\\), find \\(n\\). | 10 | 4.6875 |
26,182 | Given the function $f(x) = |2x+1| + |3x-2|$, and the solution set of the inequality $f(x) \leq 5$ is $\left\{x \mid -\frac{4a}{5} \leq x \leq \frac{3a}{5}\right\}$, where $a, b \in \mathbb{R}$.
1. Find the values of $a$ and $b$;
2. For any real number $x$, the inequality $|x-a| + |x+b| \geq m^2 - 3m$ holds, find the maximum value of the real number $m$. | \frac{3 + \sqrt{21}}{2} | 8.59375 |
26,183 | The maximum value of the real number $k$ for which the inequality $\sqrt{x-3}+\sqrt{6-x} \geqslant k$ has a solution with respect to $x$ is: | $\sqrt{6}$ | 0 |
26,184 | A sequence of natural numbers $\left\{x_{n}\right\}$ is constructed according to the following rules:
$$
x_{1}=a, x_{2}=b, x_{n+2}=x_{n}+x_{n+1}, \text{ for } n \geq 1.
$$
It is known that some term in the sequence is 1000. What is the smallest possible value of $a+b$? | 10 | 81.25 |
26,185 | In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. Given $a^{2}-c^{2}=b^{2}- \frac {8bc}{5}$, $a=6$, $\sin B= \frac {4}{5}$.
(I) Find the value of $\sin A$;
(II) Find the area of $\triangle ABC$. | \frac {168}{25} | 2.34375 |
26,186 | Positive integers \(a\), \(b\), \(c\), and \(d\) satisfy \(a > b > c > d\), \(a + b + c + d = 2200\), and \(a^2 - b^2 + c^2 - d^2 = 2200\). Find the number of possible values of \(a\). | 548 | 2.34375 |
26,187 | (1) The definite integral $\int_{-1}^{1}(x^{2}+\sin x)dx=$ ______.
(2) There are 2 red balls, 1 white ball, and 1 blue ball in a box. The probability of drawing two balls with at least one red ball is ______.
(3) Given the function $f(x)=\begin{cases}1-\log_{a}(x+2), & x\geqslant 0 \\ g(x), & x < 0\end{cases}$ is an odd function, then the root of the equation $g(x)=2$ is ______.
(4) Given the ellipse $M: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(0 < b < a < \sqrt{2}b)$, its foci are $F_{1}$ and $F_{2}$ respectively. Circle $N$ has $F_{2}$ as its center, and its minor axis length as the diameter. A tangent line to circle $N$ passing through point $F_{1}$ touches it at points $A$ and $B$. If the area of quadrilateral $F_{1}AF_{2}B$ is $S= \frac{2}{3}a^{2}$, then the eccentricity of ellipse $M$ is ______. | \frac{\sqrt{3}}{3} | 7.8125 |
26,188 | Two people, A and B, visit the "2011 Xi'an World Horticultural Expo" together. They agree to independently choose 4 attractions from numbered attractions 1 to 6 to visit, spending 1 hour at each attraction. Calculate the probability that they will be at the same attraction during their last hour. | \dfrac{1}{6} | 23.4375 |
26,189 | A rectangular prism has dimensions of 1 by 1 by 2. Calculate the sum of the areas of all triangles whose vertices are also vertices of this rectangular prism, and express the sum in the form $m + \sqrt{n} + \sqrt{p}$, where $m, n,$ and $p$ are integers. Find $m + n + p$. | 40 | 0.78125 |
26,190 | If the maximum value of the function $f(x)=a^{x} (a > 0, a \neq 1)$ on $[-2,1]$ is $4$, and the minimum value is $m$, what is the value of $m$? | \frac{1}{2} | 39.0625 |
26,191 | Given the line $l$: $x=my+1$ passes through the right focus $F$ of the ellipse $C$: $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a > b > 0)$, the focus of the parabola $x^{2}=4\sqrt{3}y$ is the upper vertex of the ellipse $C$, and the line $l$ intersects the ellipse $C$ at points $A$ and $B$.
1. Find the equation of the ellipse $C$.
2. If the line $l$ intersects the $y$-axis at point $M$, and $\overrightarrow{MA}=\lambda_{1}\overrightarrow{AF}, \overrightarrow{MB}=\lambda_{2}\overrightarrow{BF}$, is the value of $\lambda_{1}+\lambda_{2}$ a constant as $m$ varies? If so, find this constant. If not, explain why. | -\frac{8}{3} | 63.28125 |
26,192 | What is the maximum number of checkers that can be placed on an $8 \times 8$ board so that each one is being attacked? | 32 | 85.15625 |
26,193 | For how many values of $k$ is $18^{18}$ the least common multiple of the positive integers $6^9$, $9^9$, and $k$? | 19 | 3.90625 |
26,194 | Let the three-digit number \( n = abc \). If \( a, b, \) and \( c \) as the lengths of the sides can form an isosceles (including equilateral) triangle, then how many such three-digit numbers \( n \) are there? | 165 | 95.3125 |
26,195 | How many ordered pairs $(s, d)$ of positive integers with $4 \leq s \leq d \leq 2019$ are there such that when $s$ silver balls and $d$ diamond balls are randomly arranged in a row, the probability that the balls on each end have the same color is $\frac{1}{2}$ ? | 60 | 11.71875 |
26,196 | In $\triangle ABC$, $\angle ACB=60^{\circ}$, $BC > 1$, and $AC=AB+\frac{1}{2}$. When the perimeter of $\triangle ABC$ is at its minimum, the length of $BC$ is $\_\_\_\_\_\_\_\_\_\_$. | 1 + \frac{\sqrt{2}}{2} | 0 |
26,197 | In the numbers from $1$ to $2002$, the number of positive integers that contain exactly one digit $0$ is: | 414 | 0 |
26,198 | Solve the equations:<br/>$(1)2x\left(x-1\right)=1$;<br/>$(2)x^{2}+8x+7=0$. | -1 | 0 |
26,199 | Let \( f: \mathbb{N} \rightarrow \mathbb{N} \) be a function that satisfies
\[ f(1) = 2, \]
\[ f(2) = 1, \]
\[ f(3n) = 3f(n), \]
\[ f(3n + 1) = 3f(n) + 2, \]
\[ f(3n + 2) = 3f(n) + 1. \]
Find how many integers \( n \leq 2014 \) satisfy \( f(n) = 2n \). | 127 | 93.75 |
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