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32,800 | Given that square PQRS has dimensions 3 × 3, points T and U are located on side QR such that QT = TU = UR = 1, and points V and W are positioned on side RS such that RV = VW = WS = 1, find the ratio of the shaded area to the unshaded area. | 2:1 | 0 |
32,801 | A stack of logs has 15 logs on the bottom row, and each successive row has two fewer logs, ending with five special logs at the top. How many total logs are in the stack, and how many are the special logs? | 60 | 32.8125 |
32,802 | In triangle $\triangle ABC$, $a+b=11$. Choose one of the following two conditions as known, and find:<br/>$(Ⅰ)$ the value of $a$;<br/>$(Ⅱ)$ $\sin C$ and the area of $\triangle ABC$.<br/>Condition 1: $c=7$, $\cos A=-\frac{1}{7}$;<br/>Condition 2: $\cos A=\frac{1}{8}$, $\cos B=\frac{9}{16}$.<br/>Note: If both conditions 1 and 2 are answered separately, the first answer will be scored. | \frac{15\sqrt{7}}{4} | 4.6875 |
32,803 | In a certain academic knowledge competition, where the total score is 100 points, if the scores (ξ) of the competitors follow a normal distribution (N(80,σ^2) where σ > 0), and the probability that ξ falls within the interval (70,90) is 0.8, then calculate the probability that it falls within the interval [90,100]. | 0.1 | 16.40625 |
32,804 | Evaluate the product $\frac{1}{2} \times \frac{3}{2} \times \frac{4}{3} \times \frac{5}{4} \times \cdots \times \frac{2010}{2009}$. | 502.5 | 0 |
32,805 | Given the numbers 1, 2, 3, 4, 5, and 6, calculate the probability that three randomly selected numbers can be arranged in an arithmetic sequence in a certain order. | \frac{3}{10} | 81.25 |
32,806 | How many natural numbers between 200 and 400 are divisible by 8? | 26 | 43.75 |
32,807 | Given that the function $f(x)$ is an odd function defined on $\mathbb{R}$, and when $x \in [0,1]$, $f(x)=x^{3}$. Also, for all $x \in \mathbb{R}$, $f(x)=f(2-x)$. Determine the value of $f(2017.5)$. | -\frac{1}{8} | 2.34375 |
32,808 | For positive real numbers $a,$ $b,$ $c,$ and $d,$ compute the maximum value of
\[\frac{abcd(a + b + c + d)}{(a + b)^2 (c + d)^2}.\] | \frac{1}{4} | 76.5625 |
32,809 | Determine the smallest possible positive integer \( n \) with the following property: For all positive integers \( x \), \( y \), and \( z \) such that \( x \mid y^{3} \) and \( y \mid z^{3} \) and \( z \mid x^{3} \), it always holds that \( x y z \mid (x+y+z)^{n} \). | 13 | 22.65625 |
32,810 | Each positive integer number $n \ ge 1$ is assigned the number $p_n$ which is the product of all its non-zero digits. For example, $p_6 = 6$ , $p_ {32} = 6$ , $p_ {203} = 6$ . Let $S = p_1 + p_2 + p_3 + \dots + p_ {999}$ . Find the largest prime that divides $S $ . | 103 | 96.875 |
32,811 | Eddy draws $6$ cards from a standard $52$ -card deck. What is the probability that four of the cards that he draws have the same value? | 3/4165 | 2.34375 |
32,812 | Given that the total amount of money originally owned by Moe, Loki, and Nick was $72, and each of Loki, Moe, and Nick gave Ott$\, 4, determine the fractional part of the group's money that Ott now has. | \frac{1}{6} | 89.84375 |
32,813 | Find all reals $ k$ such that
\[ a^3 \plus{} b^3 \plus{} c^3 \plus{} d^3 \plus{} 1\geq k(a \plus{} b \plus{} c \plus{} d)
\]
holds for all $ a,b,c,d\geq \minus{} 1$ .
*Edited by orl.* | \frac{3}{4} | 69.53125 |
32,814 | Points $A$, $B$, and $C$ form an isosceles triangle with $AB = AC$. Points $M$, $N$, and $O$ are the midpoints of sides $AB$, $BC$, and $CA$ respectively. Find the number of noncongruent triangles that can be drawn using any three of these six points as vertices. | 10 | 1.5625 |
32,815 | A circle touches the sides of an angle at points \( A \) and \( B \). The distance from a point \( C \) lying on the circle to the line \( AB \) is 6. Find the sum of the distances from point \( C \) to the sides of the angle, given that one of these distances is nine times less than the other. | 12 | 24.21875 |
32,816 | If $\cos (π+α)=- \frac { \sqrt {10}}{5}$ and $α∈(- \frac {π}{2},0)$, find the value of $\tan ( \frac {3π}{2}+α)$. | - \frac { \sqrt {6}}{3} | 0.78125 |
32,817 | Given a sequence $\left\{ a_n \right\}$ satisfying $a_1=4$ and $a_1+a_2+\cdots +a_n=a_{n+1}$, and $b_n=\log_{2}a_n$, calculate the value of $\frac{1}{b_1b_2}+\frac{1}{b_2b_3}+\cdots +\frac{1}{b_{2017}b_{2018}}$. | \frac{3025}{4036} | 0 |
32,818 | 1. Given $$\cos\left(\alpha+ \frac {\pi}{6}\right)-\sin\alpha= \frac {3 \sqrt {3}}{5}$$, find the value of $$\sin\left(\alpha+ \frac {5\pi}{6}\right)$$;
2. Given $$\sin\alpha+\sin\beta= \frac {1}{2}$$ and $$\cos\alpha+\cos\beta= \frac {\sqrt {2}}{2}$$, find the value of $$\cos(\alpha-\beta)$$. | -\frac {5}{8} | 81.25 |
32,819 | In quadrilateral $ABCD$ , diagonals $AC$ and $BD$ intersect at $O$ . If the area of triangle $DOC$ is $4$ and the area of triangle $AOB$ is $36$ , compute the minimum possible value of the area of $ABCD$ . | 80 | 22.65625 |
32,820 | Given that point $A(1,1)$ is a point on the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$, and $F\_1$, $F\_2$ are the two foci of the ellipse such that $|AF\_1|+|AF\_2|=4$.
(I) Find the standard equation of the ellipse;
(II) Find the equation of the tangent line to the ellipse that passes through $A(1,1)$;
(III) Let points $C$ and $D$ be two points on the ellipse such that the slopes of lines $AC$ and $AD$ are complementary. Determine whether the slope of line $CD$ is a constant value. If it is, find the value; if not, explain the reason. | \frac{1}{3} | 4.6875 |
32,821 | A city uses a lottery system for assigning car permits, with 300,000 people participating in the lottery and 30,000 permits available each month.
1. If those who win the lottery each month exit the lottery, and those who do not win continue in the following month's lottery, with an additional 30,000 new participants added each month, how long on average does it take for each person to win a permit?
2. Under the conditions of part (1), if the lottery authority can control the proportion of winners such that in the first month of each quarter the probability of winning is $\frac{1}{11}$, in the second month $\frac{1}{10}$, and in the third month $\frac{1}{9}$, how long on average does it take for each person to win a permit? | 10 | 25.78125 |
32,822 | In chess, there are two types of minor pieces, the bishop and the knight. A bishop may move along a diagonal, as long as there are no pieces obstructing its path. A knight may jump to any lattice square $\sqrt{5}$ away as long as it isn't occupied.
One day, a bishop and a knight were on squares in the same row of an infinite chessboard, when a huge meteor storm occurred, placing a meteor in each square on the chessboard independently and randomly with probability $p$ . Neither the bishop nor the knight were hit, but their movement may have been obstructed by the meteors.
The value of $p$ that would make the expected number of valid squares that the bishop can move to and the number of squares that the knight can move to equal can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a, b$ . Compute $100a + b$ .
*Proposed by Lewis Chen* | 102 | 20.3125 |
32,823 | Given the polar coordinate system with the origin $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis. Point $A(4, \frac{5\pi}{4})$ is known. The polar coordinate equation of curve $E$ is $ρ=ρcos^2θ+\sqrt{2}acosθ (a > 0)$. A perpendicular line $l$ is drawn through point $A$ intersecting curve $E$ at points $B$ and $C$ when $θ=\frac{3\pi}{4} (\rho \in R)$.
$(1)$ Write down the rectangular coordinate equations of curve $E$ and line $l$.
$(2)$ If $|AB|$, $|BC|$, and $|AC|$ form a geometric progression, find the value of the real number $a$. | a = 1 + \sqrt{5} | 34.375 |
32,824 | Let ellipse $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ have left and right foci $F_{1}$, $F_{2}$, and line $l$ passing through point $F_{1}$. If the point $P$, which is the symmetric point of $F_{2}$ with respect to line $l$, lies exactly on ellipse $C$, and $\overrightarrow{F_1P} \cdot \overrightarrow{F_1F_2} = \frac{1}{2}a^2$, find the eccentricity of $C$. | \frac{1}{2} | 13.28125 |
32,825 | How many $5-$ digit positive numbers which contain only odd numbers are there such that there is at least one pair of consecutive digits whose sum is $10$ ? | 1845 | 0 |
32,826 | Triangle $PQR$ has positive integer side lengths with $PQ=PR$. Let $J$ be the intersection of the bisectors of $\angle Q$ and $\angle R$. Suppose $QJ=10$. Find the smallest possible perimeter of $\triangle PQR$. | 120 | 0 |
32,827 | Given the hyperbola $C$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ $(a>0, b>0)$ with the left vertex $A$, $P$ is a point on one of the asymptotes of $C$, and $Q$ is the intersection point of $AP$ and the other asymptote of $C$. If the slope of the line $AP$ is $1$ and $A$ is the trisection point of $PQ$, then the eccentricity of $C$ is ____. | \frac{\sqrt{10}}{3} | 39.84375 |
32,828 | If triangle $PQR$ has sides of length $PQ = 8,$ $PR = 7,$ and $QR = 5,$ then calculate
\[\frac{\cos \frac{P - Q}{2}}{\sin \frac{R}{2}} - \frac{\sin \frac{P - Q}{2}}{\cos \frac{R}{2}}.\] | \frac{5}{7} | 0 |
32,829 | How many integers can be expressed in the form: $\pm 1 \pm 2 \pm 3 \pm 4 \pm \cdots \pm 2018$ ? | 2037172 | 32.03125 |
32,830 | Four families visit a tourist spot that has four different routes available for exploration. Calculate the number of scenarios in which exactly one route is not visited by any of the four families. | 144 | 94.53125 |
32,831 | An isosceles trapezoid has sides labeled as follows: \(AB = 25\) units, \(BC = 12\) units, \(CD = 11\) units, and \(DA = 12\) units. Compute the length of the diagonal \( AC \). | \sqrt{419} | 28.125 |
32,832 | Let $[x]$ denote the largest integer not greater than the real number $x$. Define $A=\left[\frac{7}{8}\right]+\left[\frac{7^{2}}{8}\right]+\cdots+\left[\frac{7^{2016}}{8}\right]$. Find the remainder when $A$ is divided by 50. | 42 | 57.8125 |
32,833 | Determine the number of numbers between $1$ and $3000$ that are integer multiples of $5$ or $7$, but not $35$. | 943 | 0.78125 |
32,834 | Beginning with a $3 \mathrm{~cm}$ by $3 \mathrm{~cm}$ by $3 \mathrm{~cm}$ cube, a $1 \mathrm{~cm}$ by $1 \mathrm{~cm}$ by $1 \mathrm{~cm}$ cube is cut from one corner and a $2 \mathrm{~cm}$ by $2 \mathrm{~cm}$ by $2 \mathrm{~cm}$ cube is cut from the opposite corner. In $\mathrm{cm}^{2}$, what is the surface area of the resulting solid? | 54 | 38.28125 |
32,835 | In triangle $\triangle ABC$, the opposite sides of angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\tan A = 2\tan B$, $b = \sqrt{2}$, find the value of $a$ when the area of $\triangle ABC$ is maximized. | \sqrt{5} | 5.46875 |
32,836 | Given vectors $a=(2\cos \alpha,\sin ^{2}\alpha)$ and $b=(2\sin \alpha,t)$, where $\alpha\in\left( 0,\frac{\pi}{2} \right)$ and $t$ is a real number.
$(1)$ If $a-b=\left( \frac{2}{5},0 \right)$, find the value of $t$;
$(2)$ If $t=1$ and $a\cdot b=1$, find the value of $\tan \left( 2\alpha+\frac{\pi}{4} \right)$. | \frac{23}{7} | 13.28125 |
32,837 | Given that the batch of rice contains 1512 bushels and a sample of 216 grains contains 27 grains of wheat, calculate the approximate amount of wheat mixed in this batch of rice. | 189 | 97.65625 |
32,838 | Modify the constants in the problem: Let's alter the probability such that when a coin is flipped four times, the probability of having exactly two heads is $\frac{1}{12}$. Given that the coin's probability of landing on heads is less than $\frac{1}{2}$, find the probability that the coin lands on heads.
**A)** $\frac{12 + \sqrt{96 + 48\sqrt{2}}}{24}$
**B)** $\frac{12 - \sqrt{96 + 48\sqrt{2}}}{24}$
**C)** $\frac{12 + \sqrt{48 - 12\sqrt{2}}}{24}$
**D)** $\frac{12 - \sqrt{48 - 12\sqrt{2}}}{24}$
**E)** $\frac{12 - \sqrt{96 - 48\sqrt{2}}}{24}$ | \frac{12 - \sqrt{96 + 48\sqrt{2}}}{24} | 10.9375 |
32,839 | Given that the numbers $1, 4, 7, 10, 13$ are placed in five squares such that the sum of the three numbers in the horizontal row equals the sum of the three numbers in the vertical column, determine the largest possible value for the horizontal or vertical sum. | 24 | 75.78125 |
32,840 | A ball is dropped from a height of $128$ meters, and each time it hits the ground, it bounces back to half of its original height. When it hits the ground for the $9$th time, the total distance it has traveled is ______ meters. | 383 | 12.5 |
32,841 | Carl drove continuously from 7:30 a.m. until 2:15 p.m. of the same day and covered a distance of 234 miles. What was his average speed in miles per hour? | \frac{936}{27} | 0 |
32,842 | Sarah and John run on a circular track. Sarah runs counterclockwise and completes a lap every 120 seconds, while John runs clockwise and completes a lap every 75 seconds. Both start from the same line at the same time. A photographer standing inside the track takes a picture at a random time between 15 minutes and 16 minutes after they begin to run. The picture shows one-third of the track, centered on the starting line. What is the probability that both Sarah and John are in the picture?
A) $\frac{1}{24}$
B) $\frac{1}{16}$
C) $\frac{1}{12}$
D) $\frac{1}{8}$
E) $\frac{1}{6}$ | \frac{1}{12} | 10.9375 |
32,843 | The side length of square \(ABCD\) is 4. Point \(E\) is the midpoint of \(AB\), and point \(F\) is a moving point on side \(BC\). Triangles \(\triangle ADE\) and \(\triangle DCF\) are folded up along \(DE\) and \(DF\) respectively, making points \(A\) and \(C\) coincide at point \(A'\). Find the maximum distance from point \(A'\) to plane \(DEF\). | \frac{4\sqrt{5}}{5} | 3.90625 |
32,844 | If $m$ is a root of the equation $4^{x+ \frac {1}{2}}-9\cdot2^{x}+4=0$, then the eccentricity of the conic section $x^{2}+ \frac {y^{2}}{m}=1$ is \_\_\_\_\_\_. | \sqrt {2} | 0 |
32,845 | For points P and Q on the curve $y = 1 - x^2$, which are situated on opposite sides of the y-axis, find the minimum area of the triangle formed by the tangents at P and Q and the x-axis. | \frac{8 \sqrt{3}}{9} | 5.46875 |
32,846 | Suppose that the roots of $x^3 + 2x^2 + 5x - 8 = 0$ are $p$, $q$, and $r$, and that the roots of $x^3 + ux^2 + vx + w = 0$ are $p+q$, $q+r$, and $r+p$. Find $w$. | 18 | 64.0625 |
32,847 | Two cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a spade and the second card is a king? | \frac{1}{52} | 28.125 |
32,848 | Let $p,$ $q,$ $r,$ $s$ be distinct real numbers such that the roots of $x^2 - 12px - 13q = 0$ are $r$ and $s,$ and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q.$ Find the value of $p + q + r + s.$ | 2028 | 39.84375 |
32,849 | Let \(a\) and \(b\) be two natural numbers. If the remainder of the product \(a \cdot b\) divided by 15 is 1, then \(b\) is called the multiplicative inverse of \(a\) modulo 15. Based on this definition, find the sum of all multiplicative inverses of 7 modulo 15 that lie between 100 and 200. | 1036 | 3.90625 |
32,850 | Choose any four distinct digits $w, x, y, z$ and form the four-digit numbers $wxyz$ and $zyxw$. What is the greatest common divisor of the numbers of the form $wxyz + zyxw + wxyz \cdot zyxw$? | 11 | 4.6875 |
32,851 | What is the area and perimeter of the smallest square that can contain a circle with a radius of 6? | 48 | 56.25 |
32,852 | We wrote an even number in binary. By removing the trailing $0$ from this binary representation, we obtain the ternary representation of the same number. Determine the number! | 10 | 12.5 |
32,853 | The integers \( r \) and \( k \) are randomly selected, where \(-5 < r < 10\) and \(0 < k < 10\). What is the probability that the division \( r \div k \) results in \( r \) being a square number? Express your answer as a common fraction. | \frac{8}{63} | 12.5 |
32,854 | Use the bisection method to find an approximate zero of the function $f(x) = \log x + x - 3$, given that approximate solutions (accurate to 0.1) are $\log 2.5 \approx 0.398$, $\log 2.75 \approx 0.439$, and $\log 2.5625 \approx 0.409$. | 2.6 | 69.53125 |
32,855 | Express $0.3\overline{45}$ as a common fraction. | \frac{83}{110} | 0 |
32,856 | Compute $\tan 40^\circ + 4 \sin 40^\circ.$ | \sqrt{3} | 0 |
32,857 | Find a constant \( A > 0 \) such that for all real \( x \geqslant A \), we have
\[ x^{3} + x + 1 \leq \frac{x^{4}}{1000000} \] | 1000000 | 75 |
32,858 | A function $g$ is defined by $g(z) = (3 + 2i) z^2 + \beta z + \delta$ for all complex numbers $z$, where $\beta$ and $\delta$ are complex numbers and $i^2 = -1$. Suppose that $g(1)$ and $g(-i)$ are both real. What is the smallest possible value of $|\beta| + |\delta|$? | 2\sqrt{2} | 29.6875 |
32,859 | Given a sequence $\{a_n\}$ where each term is a positive number and satisfies the relationship $a_{n+1}^2 = ta_n^2 +(t-1)a_na_{n+1}$, where $n\in \mathbb{N}^*$.
(1) If $a_2 - a_1 = 8$, $a_3 = a$, and the sequence $\{a_n\}$ is unique:
① Find the value of $a$.
② Let another sequence $\{b_n\}$ satisfy $b_n = \frac{na_n}{4(2n+1)2^n}$. Is there a positive integer $m, n$ ($1 < m < n$) such that $b_1, b_m, b_n$ form a geometric sequence? If it exists, find all possible values of $m$ and $n$; if it does not exist, explain why.
(2) If $a_{2k} + a_{2k-1} + \ldots + a_{k+1} - (a_k + a_{k-1} + \ldots + a_1) = 8$, with $k \in \mathbb{N}^*$, determine the minimum value of $a_{2k+1} + a_{2k+2} + \ldots + a_{3k}$. | 32 | 34.375 |
32,860 | From vertex $A$ of an equilateral triangle $ABC$ , a ray $Ax$ intersects $BC$ at point $D$ . Let $E$ be a point on $Ax$ such that $BA =BE$ . Calculate $\angle AEC$ . | 30 | 11.71875 |
32,861 | In the diagram, the numbers 1 to 10 are placed around a circle. Sandy crosses out 1, then 4, and then 7. Continuing in a clockwise direction, she crosses out every third number of those remaining, until only two numbers are left. The sum of these two numbers is: | 10 | 11.71875 |
32,862 | Given the function $f(x)=1-2\sin ^{2}(x+ \frac {π}{8})+2\sin (x+ \frac {π}{8})\cos (x+ \frac {π}{8})$.
(1) Find the smallest positive period and the monotonically increasing interval of $f(x)$;
(2) Find the maximum and minimum values of $f(x)$ on the interval $[- \frac {π}{4}, \frac {3π}{8}]$. | -1 | 82.8125 |
32,863 | A bag contains 4 red, 3 blue, and 6 yellow marbles. What is the probability, expressed as a decimal, of drawing one red marble followed by one blue marble without replacement? | 0.076923 | 18.75 |
32,864 | Xiaoqiang conducts an experiment while drinking a beverage. He inserts a chopstick vertically into the bottom of the cup and measures the wetted part, which is exactly 8 centimeters. He then turns the chopstick around and inserts the other end straight into the bottom of the cup. He finds that the dry part of the chopstick is exactly half of the wetted part. How long is the chopstick? | 24 | 10.9375 |
32,865 | Given the curve $C$: $y^{2}=4x$ with a focus at point $F$, a line $l$ passes through point $F$ and intersects curve $C$ at points $P$ and $Q$. If the relationship $\overrightarrow{FP}+2\overrightarrow{FQ}=\overrightarrow{0}$ holds, calculate the area of triangle $OPQ$. | \frac{3\sqrt{2}}{2} | 18.75 |
32,866 | Given that there are two alloys with different percentages of copper, with alloy A weighing 40 kg and alloy B weighing 60 kg, a piece of equal weight is cut from each of these two alloys, and each cut piece is then melted together with the remaining part of the other alloy, determine the weight of the alloy cut. | 24 | 36.71875 |
32,867 | $a,b,c,d,e$ are equal to $1,2,3,4,5$ in some order, such that no two of $a,b,c,d,e$ are equal to the same integer. Given that $b \leq d, c \geq a,a \leq e,b \geq e,$ and that $d\neq5,$ determine the value of $a^b+c^d+e.$ | 628 | 89.84375 |
32,868 | In a chorus performance, there are 6 female singers (including 1 lead singer) and 2 male singers arranged in two rows.
(1) If there are 4 people per row, how many different arrangements are possible?
(2) If the lead singer stands in the front row and the male singers stand in the back row, with again 4 people per row, how many different arrangements are possible? | 5760 | 22.65625 |
32,869 | Let $ABC$ be a triangle, and $K$ and $L$ be points on $AB$ such that $\angle ACK = \angle KCL = \angle LCB$ . Let $M$ be a point in $BC$ such that $\angle MKC = \angle BKM$ . If $ML$ is the angle bisector of $\angle KMB$ , find $\angle MLC$ . | 30 | 36.71875 |
32,870 | When $\frac{1}{1001}$ is expressed as a decimal, what is the sum of the first 50 digits after the decimal point? | 216 | 20.3125 |
32,871 | The bottoms of two vertical poles are 20 feet apart on a flat ground. One pole is 8 feet tall and the other is 18 feet tall. Simultaneously, the ground between the poles is sloped, with the base of the taller pole being 2 feet higher than the base of the shorter pole due to the slope. Calculate the length in feet of a wire stretched from the top of the shorter pole to the top of the taller pole. | \sqrt{544} | 0 |
32,872 | In the sum shown, each of the letters \( D, O, G, C, A \), and \( T \) represents a different digit.
$$
\begin{array}{r}
D O G \\
+C A T \\
\hline 1000
\end{array}
$$
What is the value of \( D + O + G + C + A + T \)? | 28 | 44.53125 |
32,873 | One cube has each of its faces covered by one face of an identical cube, making a solid as shown. The volume of the solid is \(875 \ \text{cm}^3\). What, in \(\text{cm}^2\), is the surface area of the solid?
A) 750
B) 800
C) 875
D) 900
E) 1050 | 750 | 39.84375 |
32,874 | Given that the area of $\triangle ABC$ is $S$, and $\overrightarrow{AB} \cdot \overrightarrow{AC} = S$.
(I) Find the value of $\tan 2A$;
(II) If $\cos C = \frac{3}{5}$, and $|\overrightarrow{AC} - \overrightarrow{AB}| = 2$, find the area of $\triangle ABC$. | \frac{8}{5} | 32.8125 |
32,875 | If $a$ and $b$ are elements of the set ${ 1,2,3,4,5,6 }$ and $|a-b| \leqslant 1$, calculate the probability that any two people playing this game form a "friendly pair". | \dfrac{4}{9} | 50 |
32,876 | The polynomial $-5x^2-xy^4+2^6xy+3$ has terms, and the degree of this polynomial is . The coefficient of the highest degree term is . | -1 | 49.21875 |
32,877 | What is the remainder when \(2,468,135,790\) is divided by \(99\)? | 54 | 8.59375 |
32,878 | Let \( Q \) be a point chosen uniformly at random in the interior of the unit square with vertices at \( (0,0), (1,0), (1,1) \), and \( (0,1) \). Find the probability that the slope of the line determined by \( Q \) and the point \( \left(\frac{3}{4}, \frac{1}{4} \right) \) is less than or equal to \( -1 \). | \frac{1}{2} | 84.375 |
32,879 | How many $3$-digit positive integers have digits whose product equals $36$? | 21 | 25 |
32,880 | Solve the equations:
(1) $(x-2)^2=25$;
(2) $x^2+4x+3=0$;
(3) $2x^2+4x-1=0$. | \frac{-2-\sqrt{6}}{2} | 0 |
32,881 | The sequence $\{a_{n}\}$ satisfies $a_{1}+\frac{1}{2}{a_2}+\frac{1}{3}{a_3}+…+\frac{1}{n}{a_n}={a_{n+1}}-1$, $n\in N^{*}$, and $a_{1}=1$.<br/>$(1)$ Find the general formula for the sequence $\{a_{n}\}$;<br/>$(2)$ Let $S_{n}=a_{1}\cdot a_{n}+a_{2}\cdot a_{n-1}+a_{3}\cdot a_{n-2}+\ldots +a_{n}\cdot a_{1}$, $b_{n}=\frac{1}{{3{S_n}}}$, the sum of the first $n$ terms of the sequence $\{b_{n}\}$ is $T_{n}$, find the smallest positive integer $m$ such that $T_{n} \lt \frac{m}{{2024}}$ holds for any $n\in N*$.<br/>$(Reference formula: $1^{2}+2^{2}+3^{2}+\ldots +n^{2}=\frac{{n({n+1})({2n+1})}}{6}$, $n\in N*)$ | 1012 | 39.0625 |
32,882 | Let $p,$ $q,$ $r,$ $s$ be distinct real numbers such that the roots of $x^2 - 12px - 13q = 0$ are $r$ and $s,$ and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q.$ Find the value of $p + q + r + s.$ | 1716 | 0 |
32,883 | Ten positive integers include the numbers 3, 5, 8, 9, and 11. What is the largest possible value of the median of this list of ten positive integers? | 11 | 31.25 |
32,884 | A point $P$ is given on the curve $x^4+y^4=1$ . Find the maximum distance from the point $P$ to the origin. | \sqrt{2} | 30.46875 |
32,885 | Six people stand in a row, with exactly two people between A and B. Calculate the number of different ways for them to stand. | 144 | 27.34375 |
32,886 | A sector with central angle $60^\circ$ is cut from a circle of radius 10. Determine both the radius of the circle circumscribed about the sector and the inradius of the triangle formed by the sector. | \frac{5\sqrt{3}}{3} | 64.0625 |
32,887 | Given the curve $C$: $\begin{cases}x=2\cos \alpha \\ y= \sqrt{3}\sin \alpha\end{cases}$ ($\alpha$ is a parameter) and the fixed point $A(0, \sqrt{3})$, $F_1$ and $F_2$ are the left and right foci of this curve, respectively. Establish a polar coordinate system with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis.
$(1)$ Find the polar equation of the line $AF_2$;
$(2)$ A line passing through point $F_1$ and perpendicular to the line $AF_2$ intersects this conic curve at points $M$ and $N$. Find the value of $||MF_1|-|NF_1||$. | \frac{12\sqrt{3}}{13} | 1.5625 |
32,888 | 【Information Extraction】
In some cases, it is not necessary to calculate the result to remove the absolute value symbol. For example: $|6+7|=6+7$, $|6-7|=7-6$, $|7-6|=7-6$, $|-6-7|=6+7$.
【Initial Experience】
$(1)$ According to the above rules, write the following expressions in the form without absolute value symbols (do not calculate the result):
① $|7-21|=$______;
② $|{-\frac{1}{2}+\frac{4}{5}}|=$______;
③ $|{\frac{7}{{17}}-\frac{7}{{18}}}|=\_\_\_\_\_\_.$
【Extended Application】
$(2)$ Calculate:
④ $|{\frac{1}{5}-\frac{{150}}{{557}}}|+|{\frac{{150}}{{557}}-\frac{1}{2}}|-|{-\frac{1}{2}}|$;
⑤ $|{\frac{1}{3}-\frac{1}{2}}|+|{\frac{1}{4}-\frac{1}{3}}|+|{\frac{1}{5}-\frac{1}{4}}|+…+|{\frac{1}{{2021}}-\frac{1}{{2020}}}|+|{\frac{1}{{2022}}-\frac{1}{{2021}}}|$. | \frac{505}{1011} | 40.625 |
32,889 | Let $S_{n}$ be the sum of the first $n$ terms of a geometric sequence $\{a_{n}\}$, and $2S_{3}=7a_{2}$. Determine the value of $\frac{{S}_{5}}{{a}_{2}}$. | \frac{31}{8} | 41.40625 |
32,890 | A positive integer is said to be "nefelibata" if, upon taking its last digit and placing it as the first digit, keeping the order of all the remaining digits intact (for example, 312 -> 231), the resulting number is exactly double the original number. Find the smallest possible nefelibata number. | 105263157894736842 | 53.125 |
32,891 | Given that $a\in (\frac{\pi }{2},\pi )$ and $\sin \alpha =\frac{1}{3}$,
(1) Find the value of $\sin 2\alpha$;
(2) If $\sin (\alpha +\beta )=-\frac{3}{5}$, $\beta \in (0,\frac{\pi }{2})$, find the value of $\sin \beta$. | \frac{6\sqrt{2}+4}{15} | 0.78125 |
32,892 | The three vertices of an inscribed triangle in a circle divide the circumference into three arcs of lengths $3$, $4$, and $5$. The area of this triangle is: | $\frac{9}{\pi^{2}}(\sqrt{3}+3)$ | 0 |
32,893 | Last year, Michael took 8 math tests and received 8 different scores, each an integer between 85 and 95, inclusive. After each test, he noticed that the average of his test scores was an integer. His score on the eighth test was 90. What was his score on the seventh test? | 90 | 2.34375 |
32,894 | At a school cafeteria, Sam wants to buy a lunch consisting of one main course, one beverage, and one snack. The table below lists Sam's options available in the cafeteria. How many different lunch combinations can Sam choose from?
\begin{tabular}{ |c | c | c | }
\hline \textbf{Main Courses} & \textbf{Beverages} & \textbf{Snacks} \\ \hline
Burger & Water & Apple \\ \hline
Pasta & Soda & Banana \\ \hline
Salad & Juice & \\ \hline
Tacos & & \\ \hline
\end{tabular} | 24 | 26.5625 |
32,895 | Jia, Yi, Bing, Ding, and Wu sit around a circular table to play cards. Jia has a fixed seat. If Yi and Ding cannot sit next to each other, how many different seating arrangements are possible? | 12 | 41.40625 |
32,896 | Given that the function $y=f(x)$ is an odd function defined on $R$, when $x\leqslant 0$, $f(x)=2x+x^{2}$. If there exist positive numbers $a$ and $b$ such that when $x\in[a,b]$, the range of $f(x)$ is $[\frac{1}{b}, \frac{1}{a}]$, find the value of $a+b$. | \frac{3+ \sqrt{5}}{2} | 39.0625 |
32,897 | The sum to infinity of the terms of an infinite geometric progression is 10. The sum of the first two terms is 7. Compute the first term of the progression. | 10\left(1 + \sqrt{\frac{3}{10}}\right) | 0.78125 |
32,898 | The sum of the first $n$ terms of the arithmetic sequences ${a_n}$ and ${b_n}$ are $S_n$ and $T_n$ respectively. If $$\frac {S_{n}}{T_{n}}= \frac {2n+1}{3n+2}$$, find the value of $$\frac {a_{3}+a_{11}+a_{19}}{b_{7}+b_{15}}$$. | \frac{129}{130} | 28.125 |
32,899 | Following the concept of a healthy, low-carbon lifestyle, an increasing number of people are renting bicycles for cycling tours. A particular bicycle rental point charges no fee for rentals that do not exceed two hours, and for rentals that exceed two hours, the charging standard is 2 yuan per hour (with fractions of an hour calculated as a full hour). Suppose two individuals, A and B, each rent a bicycle once. The probability that A and B return their bicycles within two hours is $\frac{1}{4}$ and $\frac{1}{2}$, respectively. The probability that they return their bicycles between two and three hours is $\frac{1}{2}$ and $\frac{1}{4}$, respectively. Neither A nor B will rent a bicycle for more than four hours.
(I) Calculate the probability that the bicycle rental fees paid by A and B are the same.
(II) Let $\xi$ be the random variable representing the sum of the bicycle rental fees paid by A and B. Find the distribution of $\xi$ and its mathematical expectation $E_{\xi}$. | \frac{7}{2} | 12.5 |
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