Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
11,300 |
For \( n=1,2,3, \ldots \), let \( a_{n}=n^{2}+100 \), and let \( d_{n} \) denote the greatest common divisor of \( a_{n} \) and \( a_{n+1} \). Find the maximum value of \( d_{n} \) as \( n \) ranges over all positive integers.
|
401
| 88.28125 |
11,301 |
In triangle \( ABC \) with sides \( AB = 8 \), \( AC = 4 \), \( BC = 6 \), the angle bisector \( AK \) is drawn, and on the side \( AC \) a point \( M \) is marked such that \( AM : CM = 3 : 1 \). Point \( N \) is the intersection point of \( AK \) and \( BM \). Find \( AN \).
|
\frac{18\sqrt{6}}{11}
| 3.125 |
11,302 |
Find the sum of all real roots \( x \) of the equation \( \left(2^{x}-4\right)^{3}+\left(4^{x}-2\right)^{3}=\left(4^{x}+2^{x}-6\right)^{3} \).
|
3.5
| 0 |
11,303 |
In any finite grid of squares, some shaded and some not, for each unshaded square, record the number of shaded squares horizontally or vertically adjacent to it; this grid's *score* is the sum of all numbers recorded this way. Deyuan shades each square in a blank $n\times n$ grid with probability $k$ ; he notices that the expected value of the score of the resulting grid is equal to $k$ , too! Given that $k > 0.9999$ , find the minimum possible value of $n$ .
*Proposed by Andrew Wu*
|
51
| 25 |
11,304 |
In \\(\triangle ABC\\), let the sides opposite to angles \\(A\\), \\(B\\), and \\(C\\) be \\(a\\), \\(b\\), and \\(c\\) respectively. Let vector \\( \overrightarrow{m}=(\cos A+ \sqrt {2},\sin A)\\) and vector \\( \overrightarrow{n}=(-\sin A,\cos A)\\). If \\(| \overrightarrow{m}+ \overrightarrow{n}|=2\\),
\\((1)\\) find the magnitude of angle \\(A\\);
\\((2)\\) if \\(b=4 \sqrt {2}\\) and \\(c= \sqrt {2}a\\), find the area of \\(\triangle ABC\\).
|
16
| 37.5 |
11,305 |
Given the function $f(x)=2\cos \left(2x+ \frac{2\pi}{3}\right)+ \sqrt{3}\sin 2x$.
$(1)$ Find the smallest positive period and the maximum value of the function $f(x)$;
$(2)$ Let $\triangle ABC$ have internal angles $A$, $B$, and $C$ respectively. If $f\left(\frac{C}{2}\right)=- \frac{1}{2}$ and $AC=1$, $BC=3$, find the value of $\sin A$.
|
\frac{3 \sqrt{21}}{14}
| 66.40625 |
11,306 |
Consider the quadratic equation $2x^2 - 5x + m = 0$. Find the value of $m$ such that the sum of the roots of the equation is maximized while ensuring that the roots are real and rational.
|
\frac{25}{8}
| 24.21875 |
11,307 |
The graph of the function $y=g(x)$ is given. For all $x > 5$, it holds that $g(x) > 0.5$. The function $g(x)$ is defined as $g(x) = \frac{x^2}{Ax^2 + Bx + C}$ where $A$, $B$, and $C$ are integers. The vertical asymptotes of $g$ are at $x = -3$ and $x = 4$, and the horizontal asymptote is such that $y = 1/A < 1$. Find $A + B + C$.
|
-24
| 89.84375 |
11,308 |
With the rapid development of the "Internet + transportation" model, "shared bicycles" have appeared successively in many cities. In order to understand the satisfaction of users in a certain area with the services provided, a certain operating company randomly surveyed 10 users and obtained satisfaction ratings of 92, 84, 86, 78, 89, 74, 83, 77, 89.
$(1)$ Calculate the sample mean $\overline{x}$ and variance $s^{2}$;
$(2)$ Under condition (1), if the user's satisfaction rating is between $({\overline{x}-s,\overline{x}+s})$, then the satisfaction level is "$A$ grade". Estimate the percentage of users in the area whose satisfaction level is "$A$ grade".
Reference data: $\sqrt{30}≈5.48, \sqrt{33}≈5.74, \sqrt{35}≈5.92$.
|
50\%
| 7.03125 |
11,309 |
What is the smallest positive integer \( n \) such that \( 5n \equiv 105 \pmod{24} \)?
|
21
| 55.46875 |
11,310 |
Given that the sum of the first $n$ terms of a geometric sequence ${a_{n}}$ is $S_{n}$, if $a_{3}=4$, $S_{3}=12$, find the common ratio.
|
-\frac{1}{2}
| 60.9375 |
11,311 |
Given that \( A \), \( B \), and \( C \) are any three non-collinear points on a plane, and point \( O \) is inside \( \triangle ABC \) such that:
\[
\angle AOB = \angle BOC = \angle COA = 120^\circ.
\]
Find the maximum value of \( \frac{OA + OB + OC}{AB + BC + CA} \).
|
\frac{\sqrt{3}}{3}
| 75 |
11,312 |
In a class of 50 students, the math scores $\xi$ ($\xi \in \mathbb{N}$) follow a normal distribution $N(100, 10^2)$. It is known that $P(90 \leq \xi \leq 100) = 0.3$. Estimate the number of students whose math scores are above 110.
|
10
| 40.625 |
11,313 |
Given the sets \( A = \{(x, y) \mid ax + y = 1, x, y \in \mathbb{Z}\} \), \( B = \{(x, y) \mid x + ay = 1, x, y \in \mathbb{Z}\} \), and \( C = \{(x, y) \mid x^2 + y^2 = 1\} \), find the value of \( a \) when \( (A \cup B) \cap C \) is a set with four elements.
|
-1
| 45.3125 |
11,314 |
In triangle \( \triangle ABC \), \( M \) is the midpoint of side \( AC \), \( D \) is a point on side \( BC \) such that \( AD \) is the angle bisector of \( \angle BAC \), and \( P \) is the point of intersection of \( AD \) and \( BM \). Given that \( AB = 10 \, \text{cm} \), \( AC = 30 \, \text{cm} \), and the area of triangle \( \triangle ABC \) is \( 100 \, \text{cm}^2 \), calculate the area of triangle \( \triangle ABP \).
|
20
| 13.28125 |
11,315 |
Given the set \( A = \{1, 2, 3, \ldots, 1002\} \), Petya and Vasya play a game. Petya names a number \( n \), and Vasya then selects a subset of \( A \) consisting of \( n \) elements. Vasya wins if there are no two coprime numbers in the subset he selects; otherwise, Petya wins. What is the minimum \( n \) that Petya must name to guarantee a win?
|
502
| 96.09375 |
11,316 |
Determine the value of $\frac{3b^{-1} - \frac{b^{-1}}{3}}{b^2}$ when $b = \tfrac{1}{3}$.
|
72
| 73.4375 |
11,317 |
Given $a\ln a=be^{b}$, where $b > 0$, find the maximum value of $\frac{b}{{{a^2}}}$
|
\frac{1}{2e}
| 46.875 |
11,318 |
Given \( x_{1}, x_{2}, x_{3} \in [0, 12] \),
\[ x_{1} x_{2} x_{3} = \left(\left(12 - x_{1}\right)\left(12 - x_{2}\right)\left(12 - x_{3}\right)\right)^{2}. \]
Find the maximum value of \( f = x_{1} x_{2} x_{3} \).
|
729
| 16.40625 |
11,319 |
Calculate the limit of the numerical sequence:
$$\lim _{n \rightarrow \infty} \frac{\sqrt[4]{2+n^{5}}-\sqrt{2 n^{3}+3}}{(n+\sin n) \sqrt{7 n}}$$
|
-\sqrt{\frac{2}{7}}
| 1.5625 |
11,320 |
Given that $\{a_n\}$ is a geometric sequence with a common ratio of $q$, and $a_m$, $a_{m+2}$, $a_{m+1}$ form an arithmetic sequence.
(Ⅰ) Find the value of $q$;
(Ⅱ) Let the sum of the first $n$ terms of the sequence $\{a_n\}$ be $S_n$. Determine whether $S_m$, $S_{m+2}$, $S_{m+1}$ form an arithmetic sequence and explain the reason.
|
-\frac{1}{2}
| 18.75 |
11,321 |
The problem involves finding the value of the expressions $\lg 2 + \lg 5$ and $4(-100)^4$.
|
400000000
| 97.65625 |
11,322 |
Let the function \( f(x) = x^2 - x + 1 \). Define \( f^{(n)}(x) \) as follows:
$$
f^{(1)}(x) = f(x), \quad f^{(n)}(x) = f\left(f^{(n-1)}(x)\right).
$$
Let \( r_{n} \) be the arithmetic mean of all the roots of \( f^{(n)}(x) = 0 \). Find \( r_{2015} \).
|
\frac{1}{2}
| 69.53125 |
11,323 |
A modified sign pyramid with five levels, where a cell gets a "+" if the two cells below it have the same sign, and it gets a "-" if the two cells below it have different signs. If a "-" is to be at the top of the pyramid, calculate the number of possible ways to fill the five cells in the bottom row.
|
16
| 52.34375 |
11,324 |
Using an electric stove with a power of $P=500 \mathrm{W}$, a certain amount of water is heated. When the electric stove is turned on for $t_{1}=1$ minute, the water temperature increases by $\Delta T=2^{\circ} \mathrm{C}$, and after turning off the heater, the temperature decreases to the initial value in $t_{2}=2$ minutes. Determine the mass of the heated water, assuming the thermal power losses are constant. The specific heat capacity of water is $c_{B}=4200$ J/kg$\cdot{ }^{\circ} \mathrm{C}$.
|
2.38
| 0.78125 |
11,325 |
Convert from kilometers to miles. In the problem 3.125, the Fibonacci numeral system was introduced as being useful when converting distances from kilometers to miles or vice versa.
Suppose we want to find out how many miles are in 30 kilometers. For this, we represent the number 30 in the Fibonacci numeral system:
$$
30=21+8+1=F_{8}+F_{6}+F_{2}=(1010001)_{\mathrm{F}}
$$
Now we need to shift each number one position to the right, obtaining
$$
F_{7}+F_{5}+F_{1}=13+5+1=19=(101001)_{\mathrm{F}}
$$
So, the estimated result is 19 miles. (The correct result is approximately 18.46 miles.) Similarly, conversions from miles to kilometers are done.
Explain why this algorithm works. Verify that it gives a rounded number of miles in $n$ kilometers for all $n \leqslant 100$, differing from the correct answer by less than $2 / 3$ miles.
|
19
| 12.5 |
11,326 |
In the expansion of $(x+y)(x-y)^{5}$, the coefficient of $x^{2}y^{4}$ is ____ (provide your answer as a number).
|
-5
| 80.46875 |
11,327 |
A shipbuilding company has an annual shipbuilding capacity of 20 ships. The output function of building $x$ ships is $R(x) = 3700x + 45x^2 - 10x^3$ (unit: ten thousand yuan), and the cost function is $C(x) = 460x + 5000$ (unit: ten thousand yuan). In economics, the marginal function $Mf(x)$ of a function $f(x)$ is defined as $Mf(x) = f(x+1) - f(x)$.
(1) Find the profit function $P(x)$ and the marginal profit function $MP(x)$; (Hint: Profit = Output - Cost)
(2) How many ships should be built annually to maximize the company's annual profit?
|
12
| 82.8125 |
11,328 |
Let $S$ and $S_{1}$ respectively be the midpoints of edges $AD$ and $B_{1}C_{1}$. A rotated cube is denoted by $A^{\prime}B^{\prime}C^{\prime}D^{\prime}A_{1}^{\prime}B_{1}^{\prime}C_{1}^{\prime}D_{1}^{\prime}$. The common part of the original cube and the rotated one is a polyhedron consisting of a regular quadrilateral prism $EFGHE_{1}F_{1}G_{1}H_{1}$ and two regular quadrilateral pyramids $SEFGH$ and $S_{1}E_{1}F_{1}G_{1}H_{1}$.
The side length of the base of each pyramid is 1, and its height is $\frac{1}{2}$, making its volume $\frac{1}{6}$. The volume of the prism is $\sqrt{2}-1$.
|
\sqrt{2} - \frac{2}{3}
| 60.9375 |
11,329 |
In the expansion of $({\frac{1}{x}-\sqrt{x}})^{10}$, determine the coefficient of $x^{2}$.
|
45
| 96.09375 |
11,330 |
Richard and Shreyas are arm wrestling against each other. They will play $10$ rounds, and in each round, there is exactly one winner. If the same person wins in consecutive rounds, these rounds are considered part of the same “streak”. How many possible outcomes are there in which there are strictly more than $3$ streaks? For example, if we denote Richard winning by $R$ and Shreyas winning by $S,$ $SSRSSRRRRR$ is one such outcome, with $4$ streaks.
|
932
| 12.5 |
11,331 |
A circle with center $O$ and equation $x^2 + y^2 = 1$ passes through point $P(-1, \sqrt{3})$. Two tangents are drawn from $P$ to the circle, touching the circle at points $A$ and $B$ respectively. Find the length of the chord $|AB|$.
|
\sqrt{3}
| 53.90625 |
11,332 |
Let \( S = \{1, 2, \cdots, 2005\} \), and \( A \subseteq S \) with \( |A| = 31 \). Additionally, the sum of all elements in \( A \) is a multiple of 5. Determine the number of such subsets \( A \).
|
\frac{1}{5} \binom{2005}{31}
| 1.5625 |
11,333 |
The ratio of the magnitudes of two angles of a triangle is 2, and the difference in lengths of the sides opposite these angles is 2 cm; the length of the third side of the triangle is 5 cm. Calculate the area of the triangle.
|
\frac{15 \sqrt{7}}{4}
| 7.03125 |
11,334 |
Let \(x\), \(y\), and \(z\) be complex numbers such that:
\[
xy + 3y = -9, \\
yz + 3z = -9, \\
zx + 3x = -9.
\]
Find all possible values of \(xyz\).
|
27
| 42.1875 |
11,335 |
If the graph of the function $f(x)=\sin 2x+\cos 2x$ is translated to the left by $\varphi (\varphi > 0)$ units, and the resulting graph is symmetric about the $y$-axis, then find the minimum value of $\varphi$.
|
\frac{\pi}{8}
| 70.3125 |
11,336 |
The product of two of the four roots of the quartic equation \( x^4 - 18x^3 + kx^2 + 200x - 1984 = 0 \) is -32. Determine the value of \( k \).
|
86
| 45.3125 |
11,337 |
Find the minimum value of the expression \(\frac{13 x^{2}+24 x y+13 y^{2}+16 x+14 y+68}{\left(9-x^{2}-8 x y-16 y^{2}\right)^{5 / 2}}\). Round the answer to the nearest hundredth if needed.
|
0.26
| 0 |
11,338 |
Given an arithmetic sequence $\{a_{n}\}$ with a common difference of $\frac{{2π}}{3}$, let $S=\{\cos a_{n}|n\in N^{*}\}$. If $S=\{a,b\}$, find the value of $ab$.
|
-\frac{1}{2}
| 50 |
11,339 |
Vitya and Masha were born in the same year in June. Find the probability that Vitya is at least one day older than Masha.
|
29/60
| 72.65625 |
11,340 |
A container with a capacity of 100 liters is filled with pure alcohol. After pouring out a portion of the alcohol, the container is filled with water. The mixture is then stirred thoroughly, and an amount of liquid equal to the first portion poured out is poured out again. The container is filled with water once more. At this point, the volume of water in the container is three times the volume of pure alcohol. How many liters of pure alcohol were poured out the first time?
|
50
| 65.625 |
11,341 |
If \( a(x+1)=x^{3}+3x^{2}+3x+1 \), find \( a \) in terms of \( x \).
If \( a-1=0 \), then the value of \( x \) is \( 0 \) or \( b \). What is \( b \) ?
If \( p c^{4}=32 \), \( p c=b^{2} \), and \( c \) is positive, what is the value of \( c \) ?
\( P \) is an operation such that \( P(A \cdot B) = P(A) + P(B) \).
\( P(A) = y \) means \( A = 10^{y} \). If \( d = A \cdot B \), \( P(A) = 1 \) and \( P(B) = c \), find \( d \).
|
1000
| 55.46875 |
11,342 |
In triangle $ABC$, let $a$, $b$, $c$ be the lengths of the sides opposite to angles $A$, $B$, $C$ respectively, and it is given that $b = a \cos C + \frac{\sqrt{3}}{3} c \sin A$.
(i) Find the measure of angle $A$.
(ii) If the area of $\triangle ABC$ is $\sqrt{3}$ and the median to side $AB$ is $\sqrt{2}$, find the lengths of sides $b$ and $c$.
|
2\sqrt{2}
| 6.25 |
11,343 |
Let
$$
\frac{1}{1+\frac{1}{1+\frac{1}{1+\ddots-\frac{1}{1}}}}=\frac{m}{n}
$$
where \(m\) and \(n\) are coprime natural numbers, and there are 1988 fraction lines on the left-hand side of the equation. Calculate the value of \(m^2 + mn - n^2\).
|
-1
| 44.53125 |
11,344 |
In a division equation, the dividend is 2016 greater than the divisor, the quotient is 15, and the remainder is 0. What is the dividend?
|
2160
| 57.03125 |
11,345 |
Given an equilateral triangle \( \triangle ABC \) with a side length of 1,
\[
\overrightarrow{AP} = \frac{1}{3}(\overrightarrow{AB} + \overrightarrow{AC}), \quad \overrightarrow{AQ} = \overrightarrow{AP} + \frac{1}{2}\overrightarrow{BC}.
\]
Find the area of \( \triangle APQ \).
|
\frac{\sqrt{3}}{12}
| 36.71875 |
11,346 |
Given \( f(x) = x^{5} + a_{1} x^{4} + a_{2} x^{3} + a_{3} x^{2} + a_{4} x + a_{5} \), and \( f(m) = 2017m \) for \( m = 1, 2, 3, 4 \), calculate \( f(10) - f(-5) \).
|
75615
| 17.96875 |
11,347 |
Find the area of trapezoid \(ABCD\) with a side \(BC = 5\), where the distances from vertices \(A\) and \(D\) to the line \(BC\) are 3 and 7 respectively.
|
25
| 68.75 |
11,348 |
Six consecutive prime numbers have sum \( p \). Given that \( p \) is also a prime, determine all possible values of \( p \).
|
41
| 92.96875 |
11,349 |
Calculate: $-\sqrt{4}+|-\sqrt{2}-1|+(\pi -2013)^{0}-(\frac{1}{5})^{0}$.
|
\sqrt{2} - 1
| 5.46875 |
11,350 |
The constant term in the expansion of $( \sqrt {x}+ \frac {2}{x^{2}})^{n}$ is \_\_\_\_\_\_ if only the sixth term of the binomial coefficient is the largest.
|
180
| 77.34375 |
11,351 |
Given that the focus of the parabola $y=x^{2}$ is $F$, a line passing through point $F$ intersects the parabola at points $A$ and $B$. If $|AB|=4$, find the distance from the midpoint of chord $AB$ to the $x$-axis.
|
\frac{7}{4}
| 78.125 |
11,352 |
Given that $|\overrightarrow{a}|=5$, $|\overrightarrow{b}|=3$, and $\overrightarrow{a}\cdot\overrightarrow{b}=-12$, find the projection of vector $\overrightarrow{a}$ onto vector $\overrightarrow{b}$.
|
-4
| 3.125 |
11,353 |
In a round-robin chess tournament, 30 players are participating. To achieve the 4th category norm, a player needs to score 60% of the possible points. What is the maximum number of players who can achieve the category norm by the end of the tournament?
|
24
| 48.4375 |
11,354 |
Find the sum of all three-digit natural numbers that do not contain the digit 0 or the digit 5.
|
284160
| 80.46875 |
11,355 |
If \( a + x^2 = 2015 \), \( b + x^2 = 2016 \), \( c + x^2 = 2017 \), and \( abc = 24 \), find the value of \( \frac{a}{bc} + \frac{b}{ac} + \frac{c}{ab} - \frac{1}{a} - \frac{1}{b} - \frac{1}{c} \).
|
1/8
| 11.71875 |
11,356 |
How many rectangles can be formed where each vertex is a point on a 4x4 grid of equally spaced points?
|
36
| 9.375 |
11,357 |
The foci of the ellipse \(\frac{x^2}{25} + \frac{y^2}{b^2} = 1\) and the foci of the hyperbola
\[\frac{x^2}{196} - \frac{y^2}{121} = \frac{1}{49}\] coincide. Find \(b^2\).
|
\frac{908}{49}
| 89.84375 |
11,358 |
How many distinct four-digit numbers composed of the digits $1$, $2$, $3$, and $4$ are even?
|
12
| 97.65625 |
11,359 |
Evaluate the integral $$\int_{ -2 }^{ 2 }$$($$\sqrt {16-x^{2}}$$+sinx)dx=\_\_\_\_\_\_
|
4\sqrt{3} + \frac{8\pi}{3}
| 2.34375 |
11,360 |
In $\triangle ABC$, the sides opposite to $\angle A$, $\angle B$, and $\angle C$ are $a$, $b$, and $c$ respectively. Given that $a=1$, $b=1$, and $c= \sqrt{2}$, then $\sin A= \_\_\_\_\_\_$.
|
\frac{\sqrt{2}}{2}
| 58.59375 |
11,361 |
Pete liked the puzzle; he decided to glue it together and hang it on the wall. In one minute, he glued together two pieces (initial or previously glued). As a result, the entire puzzle was assembled into one complete picture in 2 hours. How much time would it take to assemble the picture if Pete glued three pieces together each minute instead of two?
|
60
| 28.125 |
11,362 |
Given \( x \) and \( y \) are in \( (0, +\infty) \), and \(\frac{19}{x} + \frac{98}{y} = 1\). What is the minimum value of \( x + y \)?
|
117 + 14\sqrt{38}
| 0 |
11,363 |
In right triangle $PQR$, where $PQ=8$, $QR=15$, and $\angle Q = 90^\circ$. Points $M$ and $N$ are midpoints of $\overline{PQ}$ and $\overline{PR}$ respectively; $\overline{QN}$ and $\overline{MR}$ intersect at point $Z$. Compute the ratio of the area of triangle $PZN$ to the area of quadrilateral $QZMR$.
|
\frac{1}{2}
| 25 |
11,364 |
Find the remainder when 53! is divided by 59.
|
30
| 18.75 |
11,365 |
Given $\theta \in (0, \frac{\pi}{2})$, and $\sin\theta = \frac{4}{5}$, find the value of $\cos\theta$ and $\sin(\theta + \frac{\pi}{3})$.
|
\frac{4 + 3\sqrt{3}}{10}
| 47.65625 |
11,366 |
A group of children, numbering between 50 and 70, attended a spring math camp. To celebrate Pi Day (March 14), they decided to give each other squares if they were just acquaintances and circles if they were friends. Andrey noted that each boy received 3 circles and 8 squares, and each girl received 2 squares and 9 circles. Katya discovered that the total number of circles and squares given out was the same. How many children attended the camp?
|
60
| 94.53125 |
11,367 |
The diagram shows a square \(PQRS\). The arc \(QS\) is a quarter circle. The point \(U\) is the midpoint of \(QR\) and the point \(T\) lies on \(SR\). The line \(TU\) is a tangent to the arc \(QS\). What is the ratio of the length of \(TR\) to the length of \(UR\)?
|
4:3
| 0 |
11,368 |
The mean of one set of four numbers is 15, and the mean of a separate set of eight numbers is 20. What is the mean of the set of all twelve numbers?
|
\frac{55}{3}
| 27.34375 |
11,369 |
Determine the number of possible values for \( m \) such that the lengths of the sides of a triangle are \( \ln 20 \), \( \ln 60 \), and \( \ln m \), and the triangle has a positive area.
|
1196
| 58.59375 |
11,370 |
In triangle $ABC$, if $a=2$, $c=2\sqrt{3}$, and $\angle A=30^\circ$, then the area of $\triangle ABC$ is equal to __________.
|
\sqrt{3}
| 73.4375 |
11,371 |
If $x+\frac1x = -5$, what is $x^5+\frac1{x^5}$?
|
-2525
| 96.09375 |
11,372 |
In $ xy$ plane, find the minimum volume of the solid by rotating the region boubded by the parabola $ y \equal{} x^2 \plus{} ax \plus{} b$ passing through the point $ (1,\ \minus{} 1)$ and the $ x$ axis about the $ x$ axis
|
\frac{16\pi}{15}
| 84.375 |
11,373 |
Given rectangle $R_1$ with one side $4$ inches and area $24$ square inches. Rectangle $R_2$ with diagonal $17$ inches is similar to $R_1$. Find the area of $R_2$ in square inches.
|
\frac{433.5}{3.25}
| 0 |
11,374 |
Twenty-five people who always tell the truth or always lie are standing in a queue. The man at the front of the queue says that everyone behind him always lies. Everyone else says that the person immediately in front of them always lies. How many people in the queue always lie?
|
13
| 52.34375 |
11,375 |
Given $f(x) = \begin{cases} x^{2}+1 & (x>0) \\ 2f(x+1) & (x\leq 0) \end{cases}$, find $f(2)$ and $f(-2)$.
|
16
| 92.96875 |
11,376 |
Given a function $f(x)$ defined on $R$ such that $f(2x+2)=-f(2x)$. If $f(x)=4x+3$ when $x\in(\frac{1}{2},\frac{5}{4})$, then $f(2023)=\_\_\_\_\_\_$.
|
-7
| 24.21875 |
11,377 |
Given complex numbers \( z, z_{1}, z_{2} \left( z_{1} \neq z_{2} \right) \) such that \( z_{1}^{2}=z_{2}^{2}=-2-2 \sqrt{3} \mathrm{i} \), and \(\left|z-z_{1}\right|=\left|z-z_{2}\right|=4\), find \(|z|=\ \ \ \ \ .\)
|
2\sqrt{3}
| 17.1875 |
11,378 |
In a chess tournament, each pair of players plays exactly one game. The winner of each game receives 2 points, the loser receives 0 points, and in case of a draw, both players receive 1 point each. Four scorers have recorded the total score of the tournament, but due to negligence, the scores recorded by each are different: 1979, 1980, 1984, and 1985. After verification, it is found that one of the scorers has recorded the correct total score. How many players participated in the tournament?
|
45
| 70.3125 |
11,379 |
Given the sequence $\{a_n\}$ satisfies $\{a_1=2, a_2=1,\}$ and $\frac{a_n \cdot a_{n-1}}{a_{n-1}-a_n}=\frac{a_n \cdot a_{n+1}}{a_n-a_{n+1}}(n\geqslant 2)$, determine the $100^{\text{th}}$ term of the sequence $\{a_n\}$.
|
\frac{1}{50}
| 43.75 |
11,380 |
Given that $1 \leq x \leq 2$, calculate the number of different integer values of the expression $10-10|2x-3|$.
|
11
| 82.03125 |
11,381 |
Using the differential, calculate to an accuracy of 0.01 the increment of the function \( y = x \sqrt{x^{2} + 5} \) at \( x = 2 \) and \( \Delta x = 0.2 \).
|
0.87
| 73.4375 |
11,382 |
What is the area enclosed by the graph of $|2x| + |5y| = 10$?
|
20
| 99.21875 |
11,383 |
The Aeroflot cashier must deliver tickets to five groups of tourists. Three of these groups live in the hotels "Druzhba," "Russia," and "Minsk." The cashier will be given the address of the fourth group by the tourists from "Russia," and the address of the fifth group by the tourists from "Minsk." In how many ways can the cashier choose the order of visiting the hotels to deliver the tickets?
|
30
| 0 |
11,384 |
Given that \( x \) and \( y \) are positive integers such that \( 56 \leq x + y \leq 59 \) and \( 0.9 < \frac{x}{y} < 0.91 \), find the value of \( y^2 - x^2 \).
|
177
| 73.4375 |
11,385 |
Find the smallest period of the function \( y = \cos^{10} x + \sin^{10} x \).
|
\frac{\pi}{2}
| 36.71875 |
11,386 |
If the function $f(x) = x^2$ has a domain $D$ and its range is $\{0, 1, 2, 3, 4, 5\}$, how many such functions $f(x)$ exist? (Please answer with a number).
|
243
| 1.5625 |
11,387 |
What is the minimum number of points in which 5 different non-parallel lines, not passing through a single point, can intersect?
|
10
| 94.53125 |
11,388 |
Circle $A$ is tangent to circle $B$ at one point, and the center of circle $A$ lies on the circumference of circle $B$. The area of circle $A$ is $16\pi$ square units. Find the area of circle $B$.
|
64\pi
| 93.75 |
11,389 |
Consider a new infinite geometric series: $$\frac{7}{4} + \frac{28}{9} + \frac{112}{27} + \dots$$
Determine the common ratio of this series.
|
\frac{16}{9}
| 98.4375 |
11,390 |
Convert $110_{(5)}$ to binary.
|
11110_{(2)}
| 17.1875 |
11,391 |
In a triangle, one of the sides is equal to 6, the radius of the inscribed circle is 2, and the radius of the circumscribed circle is 5. Find the perimeter.
|
24
| 10.9375 |
11,392 |
In triangle \( \triangle ABC \), \( \angle BAC = 60^{\circ} \). The angle bisector of \( \angle BAC \), \( AD \), intersects \( BC \) at point \( D \). Given that \( \overrightarrow{AD} = \frac{1}{4} \overrightarrow{AC} + t \overrightarrow{AB} \) and \( AB = 8 \), find the length of \( AD \).
|
6 \sqrt{3}
| 8.59375 |
11,393 |
A regular triangle is inscribed in a circle with a diameter of $\sqrt{12}$. Another regular triangle is constructed on its height as a side, and a new circle is inscribed in this triangle. Find the radius of this circle.
|
\frac{3}{4}
| 75.78125 |
11,394 |
120 schools each send 20 people to form 20 teams, with each team having exactly 1 person from each school. Find the smallest positive integer \( k \) such that when \( k \) people are selected from each team, there will be at least 20 people from the same school among all the selected individuals.
|
115
| 13.28125 |
11,395 |
$ x$ and $ y$ are two distinct positive integers. What is the minimum positive integer value of $ (x + y^2)(x^2 - y)/(xy)$ ?
|
14
| 2.34375 |
11,396 |
For a bijective function $g : R \to R$ , we say that a function $f : R \to R$ is its superinverse if it satisfies the following identity $(f \circ g)(x) = g^{-1}(x)$ , where $g^{-1}$ is the inverse of $g$ . Given $g(x) = x^3 + 9x^2 + 27x + 81$ and $f$ is its superinverse, find $|f(-289)|$ .
|
10
| 75.78125 |
11,397 |
The sequence $\left\{x_{n}\right\}$ satisfies $x_{1}=1$, and for any $n \in \mathbb{Z}^{+}$, it holds that $x_{n+1}=x_{n}+3 \sqrt{x_{n}}+\frac{n}{\sqrt{x_{n}}}$. Find the value of $\lim _{n \rightarrow+\infty} \frac{n^{2}}{x_{n}}$.
|
\frac{4}{9}
| 70.3125 |
11,398 |
If point $P$ is any point on the curve $y=x^{2}-\ln x$, then the minimum distance from point $P$ to the line $y=x-2$ is ____.
|
\sqrt{2}
| 83.59375 |
11,399 |
At a conference with 35 businessmen, 18 businessmen drank coffee, 15 businessmen drank tea, and 8 businessmen drank juice. Six businessmen drank both coffee and tea, four drank both tea and juice, and three drank both coffee and juice. Two businessmen drank all three beverages. How many businessmen drank only one type of beverage?
|
21
| 78.90625 |
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