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32,200 | In the convex pentagon $ABCDE$, $\angle A = \angle B = 120^{\circ}$, $EA = AB = BC = 2$, and $CD = DE = 4$. The area of $ABCDE$ is | $7 \sqrt{3}$ | 0 |
32,201 | Given \( \frac{1}{3} \leqslant a \leqslant 1 \), if \( f(x)=a x^{2}-2 x+1 \) attains its maximum value \( M(a) \) and minimum value \( N(a) \) on the interval \([1,3]\), and let \( g(a)=M(a)-N(a) \), then the minimum value of \( g(a) \) is \(\quad\) . | \frac{1}{2} | 14.0625 |
32,202 | Given that $4^{-1} \equiv 57 \pmod{119}$, find $64^{-1} \pmod{119}$, as a residue modulo 119. (Give an answer between 0 and 118, inclusive.) | 29 | 3.90625 |
32,203 | Given a function defined on the set of positive integers as follows:
\[ f(n) = \begin{cases}
n - 3, & \text{if } n \geq 1000 \\
f[f(n + 7)], & \text{if } n < 1000
\end{cases} \]
What is the value of \( f(90) \)? | 999 | 57.03125 |
32,204 | Suppose six points are taken inside or on a rectangle with dimensions $1 \times 2$. Let $b$ be the smallest possible number with the property that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than $b$. Calculate the value of $b$. | \frac{\sqrt{5}}{2} | 71.875 |
32,205 | The volume of a sphere is $72\pi$ cubic inches. A cylinder has the same height as the diameter of the sphere. The radius of the cylinder is equal to the radius of the sphere. Calculate the total surface area of the sphere plus the total surface area (including the top and bottom) of the cylinder. | 90\pi \sqrt[3]{4} | 20.3125 |
32,206 | Let $\mathbf{a}$ and $\mathbf{b}$ be vectors, and let $\mathbf{m}$ be the midpoint of $2\mathbf{a}$ and $\mathbf{b}$. Given $\mathbf{m} = \begin{pmatrix} -1 \\ 5 \end{pmatrix}$ and $\mathbf{a} \cdot \mathbf{b} = 10$, find $\|\mathbf{a}\|^2 + \|\mathbf{b}\|^2.$ | 16 | 0 |
32,207 | The greatest common divisor of 30 and some number between 70 and 90 is 6. What is the number? | 78 | 84.375 |
32,208 | The germination rate of cotton seeds is $0.9$, and the probability of developing into strong seedlings is $0.6$,
$(1)$ If two seeds are sown per hole, the probability of missing seedlings in this hole is _______; the probability of having no strong seedlings in this hole is _______.
$(2)$ If three seeds are sown per hole, the probability of having seedlings in this hole is _______; the probability of having strong seedlings in this hole is _______. | 0.936 | 14.84375 |
32,209 | Given that $α∈(0, \dfrac{π}{2})$ and $β∈(\dfrac{π}{2},π)$, with $\cos β=-\dfrac{1}{3}$ and $\sin (α+β)=\dfrac{7}{9}$.
(1) Find the value of $\sin α$;
(2) Find the value of $\sin (2α+β)$. | \dfrac{10\sqrt{2}}{27} | 13.28125 |
32,210 | Given the function $f(x)=2\sqrt{3}\sin x\cos x+2\cos^{2}x-1$.
(I) Find the axis of symmetry and the center of symmetry of $f(x)$;
(II) Find the maximum and minimum values of $f(x)$ on the interval $\left[-\frac{\pi }{6}, \frac{\pi }{4}\right]$. | -1 | 67.96875 |
32,211 | We define a function $g(x)$ such that $g(12)=37$, and if there exists an integer $a$ such that $g(a)=b$, then $g(b)$ is defined and follows these rules:
1. $g(b)=3b+1$ if $b$ is odd
2. $g(b)=\frac{b}{2}$ if $b$ is even.
What is the smallest possible number of integers in the domain of $g$? | 23 | 7.03125 |
32,212 | Reading material: In class, the teacher explained the following two problems on the blackboard:
Problem 1: Calculate: $77.7\times 11-77.7\times 5+77.7\times 4$.
Solution: $77.7\times 11-77.7\times 5+77.7\times 4=77.7\times \left(11-5+4\right)=777$.
This problem uses the distributive property of multiplication.
Problem 2: Calculate $(\frac{1}{2}×\frac{3}{2})×(\frac{2}{3}×\frac{4}{3})×(\frac{3}{4}×\frac{5}{4})\times …×(\frac{18}{19}×\frac{20}{19})$.
Solution: The original expression $=\frac{1}{2}×(\frac{3}{2}×\frac{2}{3})×(\frac{4}{3}×\frac{3}{4})×…×(\frac{19}{18}×\frac{18}{19})×\frac{20}{19}=\frac{1}{2}×\frac{20}{19}=\frac{10}{19}$.
This problem uses the commutative and associative properties of multiplication.
Attempt to solve: (1) Calculate: $99\times 118\frac{4}{5}+99×(-\frac{1}{5})-99×18\frac{3}{5}$.
Application of solution: (2) Subtract $\frac{1}{2}$ from $24$, then subtract $\frac{1}{3}$ from the remaining, then subtract $\frac{1}{4}$ from the remaining, and so on, until subtracting the remaining $\frac{1}{24}$. What is the final result?
(3) Given a rational number $a\neq 1$, calling $\frac{1}{1-a}$ the reciprocal difference of $a$, such as the reciprocal difference of $a=2$ is $\frac{1}{1-2}=-1$, the reciprocal difference of $-1$ is $\frac{1}{1-(-1)}=\frac{1}{2}$. If $a_{1}=-2$, $a_{1}$'s reciprocal difference is $a_{2}$, $a_{2}$'s reciprocal difference is $a_{3}$, $a_{3}$'s reciprocal difference is $a_{4}$, and so on. Find the value of $a_{1}+a_{2}+a_{3}-2a_{4}-2a_{5}-2a_{6}+3a_{7}+3a_{8}+3a_{9}$. | -\frac{1}{3} | 32.03125 |
32,213 | The clock shows 00:00, and the hour and minute hands coincide. Considering this coincidence to be number 0, determine after what time interval (in minutes) they will coincide the 19th time. If necessary, round the answer to two decimal places following the rounding rules. | 1243.64 | 10.9375 |
32,214 | Evaluate $$\lceil\sqrt{5}\rceil + \lceil\sqrt{6}\rceil + \lceil\sqrt{7}\rceil + \cdots + \lceil\sqrt{49}\rceil$$Note: For a real number $x,$ $\lceil x \rceil$ denotes the smallest integer that is greater than or equal to $x.$ | 245 | 0 |
32,215 | Given the function $$f(x)=4\sin(x- \frac {π}{6})\cos x+1$$.
(Ⅰ) Find the smallest positive period of f(x);
(Ⅱ) Find the maximum and minimum values of f(x) in the interval $$\[-\frac {π}{4}, \frac {π}{4}\]$$ . | -2 | 70.3125 |
32,216 | How many positive 3-digit numbers are multiples of 30, but not of 75? | 24 | 25 |
32,217 | A right rectangular prism has 6 faces, 12 edges, and 8 vertices. A new pyramid is to be constructed using one of the rectangular faces as the base. Calculate the maximum possible sum of the number of exterior faces, vertices, and edges of the combined solid (prism and pyramid). | 34 | 28.90625 |
32,218 | If the maximum and minimum values of the exponential function $f(x) = a^x$ on the interval $[1, 2]$ differ by $\frac{a}{2}$, then find the value of $a$. | \frac{3}{2} | 56.25 |
32,219 | The value of x that satisfies the equation \( x^{x^x} = 2 \) is calculated. | \sqrt{2} | 1.5625 |
32,220 | If a function $f(x)$ satisfies both (1) for any $x$ in the domain, $f(x) + f(-x) = 0$ always holds; and (2) for any $x_1, x_2$ in the domain where $x_1 \neq x_2$, the inequality $\frac{f(x_1) - f(x_2)}{x_1 - x_2} < 0$ always holds, then the function $f(x)$ is called an "ideal function." Among the following three functions: (1) $f(x) = \frac{1}{x}$; (2) $f(x) = x + 1$; (3) $f(x) = \begin{cases} -x^2 & \text{if}\ x \geq 0 \\ x^2 & \text{if}\ x < 0 \end{cases}$; identify which can be called an "ideal function" by their respective sequence numbers. | (3) | 0 |
32,221 | A steak initially at a temperature of 5°C is put into an oven. After 15 minutes, its temperature reaches 45°C. After another 15 minutes, its temperature is 77°C. The oven maintains a constant temperature. The steak changes temperature at a rate proportional to the difference between its temperature and that of the oven. Find the oven temperature. | 205 | 11.71875 |
32,222 | Given the functions $f(x)=x^{2}-2x+2$ and $g(x)=-x^{2}+ax+b- \frac {1}{2}$, one of their intersection points is $P$. The tangent lines $l_{1}$ and $l_{2}$ to the functions $f(x)$ and $g(x)$ at point $P$ are perpendicular. Find the maximum value of $ab$. | \frac{9}{4} | 3.125 |
32,223 |
Compute the definite integral:
$$
\int_{0}^{\pi} 2^{4} \cdot \sin ^{6}\left(\frac{x}{2}\right) \cos ^{2}\left(\frac{x}{2}\right) d x
$$ | \frac{5\pi}{8} | 75 |
32,224 | At exactly noon, Anna Kuzminichna looked out the window and saw that Klava, the shop assistant of the countryside shop, was leaving for a break. At two minutes past one, Anna Kuzminichna looked out the window again and saw that no one was in front of the closed shop. Klava was absent for exactly 10 minutes, and when she came back, she found Ivan and Foma in front of the door, with Foma having apparently arrived after Ivan. Find the probability that Foma had to wait no more than 4 minutes for the shop to reopen. | 0.75 | 0 |
32,225 | $(1)$ $f(n)$ is a function defined on the set of positive integers, satisfying:<br/>① When $n$ is a positive integer, $f(f(n))=4n+9$;<br/>② When $k$ is a non-negative integer, $f(2^{k})=2^{k+1}+3$. Find the value of $f(1789)$.<br/>$(2)$ The function $f$ is defined on the set of ordered pairs of positive integers, and satisfies the following properties:<br/>① $f(x,x)=x$;<br/>② $f(x,y)=f(y,x)$;<br/>③ $(x+y)f(x,y)=yf(x,x+y)$. Find $f(14,52)$. | 364 | 7.8125 |
32,226 | Let \( n = 2^{31} \times 3^{19} \times 5^7 \). How many positive integer divisors of \( n^2 \) are less than \( n \) but do not divide \( n \)? | 13307 | 0 |
32,227 | A triangle has sides of lengths 30, 70, and 80. When an altitude is drawn to the side of length 80, the longer segment of this side that is intercepted by the altitude is: | 65 | 52.34375 |
32,228 | Let $p = 2027$ be the smallest prime greater than $2018$ , and let $P(X) = X^{2031}+X^{2030}+X^{2029}-X^5-10X^4-10X^3+2018X^2$ . Let $\mathrm{GF}(p)$ be the integers modulo $p$ , and let $\mathrm{GF}(p)(X)$ be the set of rational functions with coefficients in $\mathrm{GF}(p)$ (so that all coefficients are taken modulo $p$ ). That is, $\mathrm{GF}(p)(X)$ is the set of fractions $\frac{P(X)}{Q(X)}$ of polynomials with coefficients in $\mathrm{GF}(p)$ , where $Q(X)$ is not the zero polynomial. Let $D\colon \mathrm{GF}(p)(X)\to \mathrm{GF}(p)(X)$ be a function satisfying \[
D\left(\frac fg\right) = \frac{D(f)\cdot g - f\cdot D(g)}{g^2}
\]for any $f,g\in \mathrm{GF}(p)(X)$ with $g\neq 0$ , and such that for any nonconstant polynomial $f$ , $D(f)$ is a polynomial with degree less than that of $f$ . If the number of possible values of $D(P(X))$ can be written as $a^b$ , where $a$ , $b$ are positive integers with $a$ minimized, compute $ab$ .
*Proposed by Brandon Wang* | 4114810 | 19.53125 |
32,229 | Find the probability that the chord $\overline{AB}$ does not intersect with chord $\overline{CD}$ when four distinct points, $A$, $B$, $C$, and $D$, are selected from 2000 points evenly spaced around a circle. | \frac{2}{3} | 39.0625 |
32,230 | The base of a triangle is $80$ , and one side of the base angle is $60^\circ$ . The sum of the lengths of the other two sides is $90$ . The shortest side is: | 17 | 10.9375 |
32,231 | There are a total of $400$ machine parts. If person A works alone for $1$ day, and then person A and person B work together for $2$ days, there will still be $60$ parts unfinished. If both work together for $3$ days, they can produce $20$ parts more than needed. How many parts can each person make per day? | 80 | 0 |
32,232 | Given that point $M$ represents the number $9$ on the number line.<br/>$(1)$ If point $N$ is first moved $4$ units to the left and then $6$ units to the right to reach point $M$, then the number represented by point $N$ is ______.<br/>$(2)$ If point $M$ is moved $4$ units on the number line, then the number represented by point $M$ is ______. | 13 | 38.28125 |
32,233 | Point $M$ lies on the diagonal $BD$ of parallelogram $ABCD$ such that $MD = 3BM$ . Lines $AM$ and $BC$ intersect in point $N$ . What is the ratio of the area of triangle $MND$ to the area of parallelogram $ABCD$ ? | 3/8 | 27.34375 |
32,234 | Two balls, one blue and one orange, are randomly and independently tossed into bins numbered with positive integers. For each ball, the probability that it is tossed into bin $k$ is $3^{-k}$ for $k = 1, 2, 3,...$. What is the probability that the blue ball is tossed into a higher-numbered bin than the orange ball?
A) $\frac{1}{8}$
B) $\frac{1}{9}$
C) $\frac{1}{16}$
D) $\frac{7}{16}$
E) $\frac{3}{8}$ | \frac{7}{16} | 10.15625 |
32,235 | Given two arithmetic sequences $\{a_n\}$ and $\{b_n\}$, the sum of the first $n$ terms of each sequence is denoted as $S_n$ and $T_n$ respectively. If $$\frac {S_{n}}{T_{n}}= \frac {7n+45}{n+3}$$, and $$\frac {a_{n}}{b_{2n}}$$ is an integer, then the value of $n$ is \_\_\_\_\_\_. | 15 | 7.8125 |
32,236 | A three-digit $\overline{abc}$ number is called *Ecuadorian* if it meets the following conditions: $\bullet$ $\overline{abc}$ does not end in $0$ . $\bullet$ $\overline{abc}$ is a multiple of $36$ . $\bullet$ $\overline{abc} - \overline{cba}$ is positive and a multiple of $36$ .
Determine all the Ecuadorian numbers. | 864 | 31.25 |
32,237 | Simplify first, then evaluate: $(a-2b)(a^2+2ab+4b^2)-a(a-5b)(a+3b)$, where $a$ and $b$ satisfy $a^2+b^2-2a+4b=-5$. | 120 | 61.71875 |
32,238 | For $x > 0$, find the maximum value of $f(x) = 1 - 2x - \frac{3}{x}$ and the value of $x$ at which it occurs. | \frac{\sqrt{6}}{2} | 24.21875 |
32,239 | Let $f : Q \to Q$ be a function satisfying the equation $f(x + y) = f(x) + f(y) + 2547$ for all rational numbers $x, y$ . If $f(2004) = 2547$ , find $f(2547)$ . | 2547 | 3.90625 |
32,240 | Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth. | 0.298 | 0 |
32,241 | Add $256_{7} + 463_{7} + 132_7$. Express your answer in base $7$. | 1214_{7} | 0 |
32,242 | What is the sum of the fractions of the form $\frac{2}{n(n+2)}$, where $n$ takes on odd positive integers from 1 to 2011? Express your answer as a decimal to the nearest thousandth. | 0.999 | 94.53125 |
32,243 | Suppose that $x_1+1 = x_2+2 = x_3+3 = \cdots = x_{1000}+1000 = x_1 + x_2 + x_3 + \cdots + x_{1000} + 1001$. Find the value of $\left\lfloor |S| \right\rfloor$, where $S = \sum_{n=1}^{1000} x_n$. | 501 | 24.21875 |
32,244 | A clock takes $7$ seconds to strike $9$ o'clock starting precisely from $9:00$ o'clock. If the interval between each strike increases by $0.2$ seconds as time progresses, calculate the time it takes to strike $12$ o'clock. | 12.925 | 46.875 |
32,245 | Given $\overrightarrow{a}=( \sqrt {2},m)(m > 0)$, $\overrightarrow{b}=(\sin x,\cos x)$, and the maximum value of the function $f(x)= \overrightarrow{a} \cdot \overrightarrow{b}$ is $2$.
1. Find $m$ and the smallest positive period of the function $f(x)$;
2. In $\triangle ABC$, $f(A- \frac {\pi}{4})+f(B- \frac {\pi}{4})=12 \sqrt {2}\sin A\sin B$, where $A$, $B$, $C$ are the angles opposite to sides $a$, $b$, $c$ respectively, and $C= \frac {\pi}{3}$, $c= \sqrt {6}$, find the area of $\triangle ABC$. | \frac { \sqrt {3}}{4} | 0 |
32,246 | Determine the value of the sum \[ \sum_{n=0}^{332} (-1)^{n} {1008 \choose 3n} \] and find the remainder when the sum is divided by $500$. | 54 | 1.5625 |
32,247 | Given the functions $f(x)=\log_{a}x$ and $g(x)=\log_{a}(2x+t-2)$, where $a > 0$ and $a\neq 1$, $t\in R$.
(1) If $0 < a < 1$, and $x\in[\frac{1}{4},2]$ such that $2f(x)\geqslant g(x)$ always holds, find the range of values for the real number $t$;
(2) If $t=4$, and $x\in[\frac{1}{4},2]$ such that the minimum value of $F(x)=2g(x)-f(x)$ is $-2$, find the value of the real number $a$. | a=\frac{1}{5} | 0.78125 |
32,248 | $(1)$ Given the function $f(x) = |x+1| + |2x-4|$, find the solution to $f(x) \geq 6$;<br/>$(2)$ Given positive real numbers $a$, $b$, $c$ satisfying $a+2b+4c=8$, find the minimum value of $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$. | \frac{11+6\sqrt{2}}{8} | 14.84375 |
32,249 | Let $ABCD$ be a unit square. For any interior points $M,N$ such that the line $MN$ does not contain a vertex of the square, we denote by $s(M,N)$ the least area of the triangles having their vertices in the set of points $\{ A,B,C,D,M,N\}$ . Find the least number $k$ such that $s(M,N)\le k$ , for all points $M,N$ .
*Dinu Șerbănescu* | 1/8 | 56.25 |
32,250 | Given that line l: x - y + 1 = 0 is tangent to the parabola C with focus F and equation y² = 2px (p > 0).
(I) Find the equation of the parabola C;
(II) The line m passing through point F intersects parabola C at points A and B. Find the minimum value of the sum of the distances from points A and B to line l. | \frac{3\sqrt{2}}{2} | 2.34375 |
32,251 | There are \(100\) countries participating in an olympiad. Suppose \(n\) is a positive integers such that each of the \(100\) countries is willing to communicate in exactly \(n\) languages. If each set of \(20\) countries can communicate in exactly one common language, and no language is common to all \(100\) countries, what is the minimum possible value of \(n\)? | 20 | 10.15625 |
32,252 | Given the line $l$: $x-2y+2=0$ passes through the left focus $F\_1$ and one vertex $B$ of an ellipse. Determine the eccentricity of the ellipse. | \frac{2\sqrt{5}}{5} | 91.40625 |
32,253 | Suppose Lucy picks a letter at random from the extended set of characters 'ABCDEFGHIJKLMNOPQRSTUVWXYZ0123456789'. What is the probability that the letter she picks is in the word 'MATHEMATICS123'? | \frac{11}{36} | 61.71875 |
32,254 | Given a family of sets \(\{A_{1}, A_{2}, \ldots, A_{n}\}\) that satisfies the following conditions:
(1) Each set \(A_{i}\) contains exactly 30 elements;
(2) For any \(1 \leq i < j \leq n\), the intersection \(A_{i} \cap A_{j}\) contains exactly 1 element;
(3) The intersection \(A_{1} \cap A_{2} \cap \ldots \cap A_{n} = \varnothing\).
Find the maximum number \(n\) of such sets. | 871 | 20.3125 |
32,255 | China Unicom charges for mobile phone calls with two types of packages: Package $A$ (monthly fee of $15$ yuan, call fee of $0.1 yuan per minute) and Package $B$ (monthly fee of $0$ yuan, call fee of $0.15 yuan per minute). Let $y_{1}$ represent the monthly bill for Package $A$ (in yuan), $y_{2}$ represent the monthly bill for Package $B$ (in yuan), and $x$ represent the monthly call duration in minutes. <br/>$(1)$ Express the functions of $y_{1}$ with respect to $x$ and $y_{2}$ with respect to $x$. <br/>$(2)$ For how long should the monthly call duration be such that the charges for Packages $A$ and $B$ are the same? <br/>$(3)$ In what scenario is Package $A$ more cost-effective? | 300 | 32.03125 |
32,256 | The area of an isosceles obtuse triangle is 8, and the median drawn to one of its equal sides is $\sqrt{37}$. Find the cosine of the angle at the vertex. | -\frac{3}{5} | 16.40625 |
32,257 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. If $a^2 + b^2 = 2017c^2$, calculate the value of $\frac{\tan C}{\tan A} + \frac{\tan C}{\tan B}$. | \frac{1}{1008} | 23.4375 |
32,258 | Given that Mary is 30% older than Sally, and Sally is 50% younger than Danielle, and the sum of their ages is 45 years, determine Mary's age on her next birthday. | 14 | 57.8125 |
32,259 | If each digit of a four-digit natural number $M$ is not $0$, and the five times of the digit in the thousandth place is equal to the sum of the digits in the hundredth, tenth, and unit places, then this four-digit number is called a "modest number". For example, for the four-digit number $2163$, since $5\times 2=1+6+3$, $2163$ is a "modest number". Another example is the four-digit number $3147$, since $5\times 3\neq 1+4+7$, $3147$ is not a "modest number". If the four-digit number $\overline{467x}$ is a "modest number", then $x=$______. If a "modest number" $M=\overline{abcd}$ (where $d$ is even), rearrange the tens and units digits of the "modest number" $M$ to the front of the thousand and hundred digits to form a new four-digit number $M'=\overline{cdab}$. Define $G(M)=\frac{M-M'}{99}$. If $G(M)$ is divisible by $11$, and $\overline{abc}$ is divisible by $3$, then the maximum value of $M$ is ______. | 3816 | 83.59375 |
32,260 | Given the parametric equations of curve C as $$\begin{cases} x=2\cos\theta \\ y= \sqrt {3}\sin\theta \end{cases}(\theta\text{ is the parameter})$$, in the same Cartesian coordinate system, the points on curve C are transformed by the coordinate transformation $$\begin{cases} x'= \frac {1}{2}x \\ y'= \frac {1}{ \sqrt {3}}y \end{cases}$$ to obtain curve C'. With the origin as the pole and the positive half-axis of x as the polar axis, a polar coordinate system is established.
(Ⅰ) Find the polar equation of curve C';
(Ⅱ) If a line l passing through point $$A\left( \frac {3}{2},\pi\right)$$ (in polar coordinates) with a slope angle of $$\frac {\pi}{6}$$ intersects curve C' at points M and N, and the midpoint of chord MN is P, find the value of $$\frac {|AP|}{|AM|\cdot |AN|}$$. | \frac {3 \sqrt {3}}{5} | 0 |
32,261 | Add $123.4567$ to $98.764$ and round your answer to the nearest hundredth. Then, subtract $0.02$ from the rounded result. | 222.20 | 29.6875 |
32,262 | The function $y=(m^2-m-1)x^{m^2-3m-3}$ is a power function, and it is an increasing function on the interval $(0, +\infty)$. Find the value of $m$. | -1 | 8.59375 |
32,263 | Let the number $x$ . Using multiply and division operations of any 2 given or already given numbers we can obtain powers with natural exponent of the number $x$ (for example, $x\cdot x=x^{2}$ , $x^{2}\cdot x^{2}=x^{4}$ , $x^{4}: x=x^{3}$ , etc). Determine the minimal number of operations needed for calculating $x^{2006}$ . | 17 | 25 |
32,264 | Given a trapezoid with labeled points as shown in the diagram, the point of intersection of the extended non-parallel sides is labeled as \(E\), and the point of intersection of the diagonals is labeled as \(F\). Similar right triangles \(BFC\) and \(DFA\) have corresponding sides \(x, y\) in the first triangle and \(4x, 4y\) in the second triangle.
The height of the trapezoid \(h\) is given by the sum of the heights of triangles \(BFC\) and \(AFD\):
\[ h = x y + \frac{4x \cdot 4y}{4} = 5xy \]
The area of the trapezoid is \(\frac{15}{16}\) of the area of triangle \(AED\):
\[ \frac{1}{2} AC \cdot BD = \frac{15}{16} \cdot \frac{1}{2} \cdot AE \cdot ED \cdot \sin 60^\circ = \frac{15 \sqrt{3}}{64} \cdot \frac{4}{3} AB \cdot \frac{4}{3} CD = \frac{15 \sqrt{3}}{36} AB \cdot CD \]
From this, we have:
\[ \frac{25}{2} xy = \frac{15 \sqrt{3}}{36} \sqrt{x^2 + 16y^2} \cdot \sqrt{y^2 + 16x^2} \]
Given that \(x^2 + y^2 = 1\), find:
\[ \frac{30}{\sqrt{3}} xy = \sqrt{1 + 15y^2} \cdot \sqrt{1 + 15x^2} \]
By simplifying:
\[ 300 x^2 y^2 = 1 + 15(x^2 + y^2) + 225 x^2 y^2 \]
\[ 75 x^2 y^2 = 16 \]
\[ 5xy = \frac{4}{\sqrt{3}} \] | \frac{4}{\sqrt{3}} | 30.46875 |
32,265 | Isosceles trapezoid $ABCD$ has side lengths $AB = 6$ and $CD = 12$ , while $AD = BC$ . It is given that $O$ , the circumcenter of $ABCD$ , lies in the interior of the trapezoid. The extensions of lines $AD$ and $BC$ intersect at $T$ . Given that $OT = 18$ , the area of $ABCD$ can be expressed as $a + b\sqrt{c}$ where $a$ , $b$ , and $c$ are positive integers where $c$ is not divisible by the square of any prime. Compute $a+b+c$ .
*Proposed by Andrew Wen* | 84 | 2.34375 |
32,266 | Compute: $\sin 187^{\circ}\cos 52^{\circ}+\cos 7^{\circ}\sin 52^{\circ}=\_\_\_\_\_\_ \cdot$ | \frac{\sqrt{2}}{2} | 63.28125 |
32,267 | Let \(ABC\) be a non-degenerate triangle and \(I\) the center of its incircle. Suppose that \(\angle A I B = \angle C I A\) and \(\angle I C A = 2 \angle I A C\). What is the value of \(\angle A B C\)? | 60 | 14.0625 |
32,268 | The function $g(x),$ defined for $0 \le x \le 1,$ has the following properties:
(i) $g(0) = 0.$
(ii) If $0 \le x < y \le 1,$ then $g(x) \le g(y).$
(iii) $g(1 - x) = 1 - g(x)$ for all $0 \le x \le 1.$
(iv) $g\left( \frac{2x}{5} \right) = \frac{g(x)}{3}$ for $0 \le x \le 1.$
Find $g\left( \frac{3}{5} \right).$ | \frac{2}{3} | 46.09375 |
32,269 | Given a parallelepiped $A B C D A_1 B_1 C_1 D_1$, points $M, N, K$ are midpoints of edges $A B$, $B C$, and $D D_1$ respectively. Construct the cross-sectional plane of the parallelepiped with the plane $MNK$. In what ratio does this plane divide the edge $C C_1$ and the diagonal $D B_1$? | 3:7 | 5.46875 |
32,270 | Calculate the value of $\cos 96^\circ \cos 24^\circ - \sin 96^\circ \sin 66^\circ$. | -\frac{1}{2} | 75 |
32,271 | Trapezoid $EFGH$ has base $EF = 24$ units and base $GH = 36$ units. Diagonals $EG$ and $FH$ intersect at point $Y$. If the area of trapezoid $EFGH$ is $360$ square units, what is the area of triangle $FYH$? | 86.4 | 9.375 |
32,272 | A mole has chewed a hole in a carpet in the shape of a rectangle with sides of 10 cm and 4 cm. Find the smallest size of a square patch that can cover this hole (a patch covers the hole if all points of the rectangle lie inside the square or on its boundary). | \sqrt{58} | 0 |
32,273 | Let $M$ be the greatest five-digit number whose digits have a product of $180$. Find the sum of the digits of $M$. | 20 | 29.6875 |
32,274 | Given an integer \( n \) with \( n \geq 2 \), determine the smallest constant \( c \) such that the inequality \(\sum_{1 \leq i \leq j \leq n} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leq c\left(\sum_{i=1}^{n} x_{i}\right)^{4}\) holds for all non-negative real numbers \( x_{1}, x_{2}, \cdots, x_{n} \). | \frac{1}{8} | 31.25 |
32,275 | A geometric sequence of positive integers has a first term of 3, and the fourth term is 243. Find the sixth term of the sequence. | 729 | 40.625 |
32,276 | A point $(x,y)$ is randomly and uniformly chosen inside the square with vertices (0,0), (0,3), (3,3), and (3,0). What is the probability that $x+y < 5$? | \dfrac{17}{18} | 18.75 |
32,277 | Given the function $f(x) = \sin x + \cos x$, where $x \in \mathbb{R}$,
- (I) Find the value of $f\left(\frac{\pi}{2}\right)$.
- (II) Determine the smallest positive period of the function $f(x)$.
- (III) Calculate the minimum value of the function $g(x) = f\left(x + \frac{\pi}{4}\right) + f\left(x + \frac{3\pi}{4}\right)$. | -2 | 80.46875 |
32,278 | What is the value of \(1.90 \frac{1}{1-\sqrt[4]{3}}+\frac{1}{1+\sqrt[4]{3}}+\frac{2}{1+\sqrt{3}}\)? | -2 | 0 |
32,279 | Given a square with four vertices and its center, find the probability that the distance between any two of these five points is less than the side length of the square. | \frac{2}{5} | 41.40625 |
32,280 | In a row of 10 chairs, Mary and James each choose their seats at random but are not allowed to sit in the first or the last chair (chairs #1 and #10). What is the probability that they do not sit next to each other? | \frac{3}{4} | 62.5 |
32,281 | Let $d_1 = a^2 + 2^a + a \cdot 2^{(a+1)/2} + a^3$ and $d_2 = a^2 + 2^a - a \cdot 2^{(a+1)/2} + a^3$. If $1 \le a \le 300$, how many integral values of $a$ are there such that $d_1 \cdot d_2$ is a multiple of $3$? | 100 | 18.75 |
32,282 | A person was asked how much he paid for a hundred apples and he answered the following:
- If a hundred apples cost 4 cents more, then for 1 dollar and 20 cents, he would get five apples less.
How much did 100 apples cost? | 96 | 46.875 |
32,283 | Given a $4 \times 4$ grid with 16 unit squares, each painted white or black independently and with equal probability, find the probability that the entire grid becomes black after a 90° clockwise rotation, where any white square landing on a place previously occupied by a black square is repainted black. | \frac{1}{65536} | 50 |
32,284 | Four cats, four dogs, and four mice are placed in 12 cages. If a cat and a mouse are in the same column, the cat will meow non-stop; if a mouse is surrounded by two cats on both sides, the mouse will squeak non-stop; if a dog is flanked by a cat and a mouse, the dog will bark non-stop. In other cases, the animals remain silent. One day, the cages numbered 3, 4, 6, 7, 8, and 9 are very noisy, while the other cages are quiet. What is the sum of the cage numbers that contain the four dogs? | 28 | 3.125 |
32,285 | Solve the equations:
1. $2x^{2}+4x+1=0$ (using the method of completing the square)
2. $x^{2}+6x=5$ (using the formula method) | -3-\sqrt{14} | 5.46875 |
32,286 | There are $5$ people participating in a lottery, each drawing a ticket from a box containing $5$ tickets ($3$ of which are winning tickets) without replacement until all $3$ winning tickets have been drawn, ending the activity. The probability that the activity ends exactly after the $4$th person draws is $\_\_\_\_\_\_$. | \frac{3}{10} | 40.625 |
32,287 | The number of real roots of the equation $\frac{x}{100} = \sin x$ is:
(32nd United States of America Mathematical Olympiad, 1981) | 63 | 82.8125 |
32,288 | Given the function $f(x) = x^3 + ax^2 + bx + a^2$ has an extremum of 10 at $x = 1$, find the value of $f(2)$. | 18 | 60.9375 |
32,289 | Given the function $y=4^{x}-6\times2^{x}+8$, find the minimum value of the function and the value of $x$ when the minimum value is obtained. | -1 | 36.71875 |
32,290 | Given a regular triangular prism \(ABC-A_1B_1C_1\) with side edges and base edges all equal to 1, find the volume of the common part of the tetrahedra \(A_1ABC\), \(B_1ABC\), and \(C_1ABC\). | \frac{\sqrt{3}}{36} | 20.3125 |
32,291 | Solve the equations:<br/>$(1)x^{2}-10x-10=0$;<br/>$(2)3\left(x-5\right)^{2}=2\left(5-x\right)$. | \frac{13}{3} | 1.5625 |
32,292 | Let set $P=\{0, 2, 4, 6, 8\}$, and set $Q=\{m | m=100a_1+10a_2+a_3, a_1, a_2, a_3 \in P\}$. Determine the 68th term of the increasing sequence of elements in set $Q$. | 464 | 10.15625 |
32,293 | In a game, Jimmy and Jacob each randomly choose to either roll a fair six-sided die or to automatically roll a $1$ on their die. If the product of the two numbers face up on their dice is even, Jimmy wins the game. Otherwise, Jacob wins. The probability Jimmy wins $3$ games before Jacob wins $3$ games can be written as $\tfrac{p}{2^q}$ , where $p$ and $q$ are positive integers, and $p$ is odd. Find the remainder when $p+q$ is divided by $1000$ .
*Proposed by firebolt360* | 360 | 0.78125 |
32,294 | The function
$$
\begin{aligned}
y= & |x-1|+|2x-1|+|3x-1|+ \\
& |4x-1|+|5x-1|
\end{aligned}
$$
achieves its minimum value when the variable $x$ equals ______. | $\frac{1}{3}$ | 0 |
32,295 | When any two numbers are taken from the set {0, 1, 2, 3, 4, 5} to perform division, calculate the number of different sine values that can be obtained. | 10 | 7.8125 |
32,296 | The following is Xiao Liang's problem-solving process. Please read carefully and answer the following questions.
Calculate $({-15})÷({\frac{1}{3}-3-\frac{3}{2}})×6$.
Solution: Original expression $=({-15})÷({-\frac{{25}}{6}})×6\ldots \ldots $ First step $=\left(-15\right)\div \left(-25\right)\ldots \ldots $ Second step $=-\frac{3}{5}\ldots \ldots $ Third step
$(1)$ There are two errors in the solution process. The first error is in the ______ step, the mistake is ______. The second error is in the ______ step, the mistake is ______.
$(2)$ Please write down the correct solution process. | \frac{108}{5} | 83.59375 |
32,297 | Given that the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $60^{\circ}$, $\overrightarrow{a}=(2,)$, $|\overrightarrow{b}|=1$, calculate $|\overrightarrow{a}+2\overrightarrow{b}|$. | 2\sqrt{3} | 75.78125 |
32,298 | Add $12_8 + 157_8.$ Express your answer in base 8. | 171_8 | 74.21875 |
32,299 | 24 people participate in a training competition consisting of 12 rounds. After each round, every participant receives a certain score \( a_k \) based on their ranking \( k \) in that round, where \( a_{k} \in \mathbb{N}_+, k = 1, 2, \ldots, n, a_1 > a_2 > \cdots > a_n \). After all the rounds are completed, the overall ranking is determined based on the total score each person has accumulated over the 12 rounds. Find the smallest positive integer \( n \) such that no matter the ranking in the penultimate round, at least 2 participants have the potential to win the championship. | 13 | 9.375 |
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