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|---|---|---|---|
30,900 | The hypotenuse of a right triangle whose legs are consecutive even numbers is 50 units. What is the sum of the lengths of the two legs? | 70 | 70.3125 |
30,901 | Given triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $\sqrt{3}\sin B + 2\cos^2\frac{B}{2} = 3$ and $\frac{\cos B}{b} + \frac{\cos C}{c} = \frac{\sin A \sin B}{6\sin C}$, find the area of the circumcircle of $\triangle ABC$. | 16\pi | 28.125 |
30,902 | Given that $689\Box\Box\Box20312 \approx 69$ billion (rounded), find the number of ways to fill in the three-digit number. | 500 | 46.875 |
30,903 | The school club sells 200 tickets for a total of $2500. Some tickets are sold at full price, and the rest are sold for one-third the price of the full-price tickets. Determine the amount of money raised by the full-price tickets. | 1250 | 13.28125 |
30,904 | In the independent college admissions process, a high school has obtained 5 recommendation spots, with 2 for Tsinghua University, 2 for Peking University, and 1 for Fudan University. Both Peking University and Tsinghua University require the participation of male students. The school selects 3 male and 2 female students as candidates for recommendation. The total number of different recommendation methods is ( ). | 24 | 2.34375 |
30,905 | The number of trees in a park must be more than 80 and fewer than 150. The number of trees is 2 more than a multiple of 4, 3 more than a multiple of 5, and 4 more than a multiple of 6. How many trees are in the park? | 98 | 0 |
30,906 | The value of $\tan {75}^{{o}}$ is $\dfrac{\sqrt{6}+\sqrt{2}}{4}$. | 2+\sqrt{3} | 71.09375 |
30,907 | Let $\mathcal{P}_1$ and $\mathcal{P}_2$ be two parabolas with distinct directrices $\ell_1$ and $\ell_2$ and distinct foci $F_1$ and $F_2$ respectively. It is known that $F_1F_2||\ell_1||\ell_2$ , $F_1$ lies on $\mathcal{P}_2$ , and $F_2$ lies on $\mathcal{P}_1$ . The two parabolas intersect at distinct points $A$ and $B$ . Given that $F_1F_2=1$ , the value of $AB^2$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ . Find $100m+n$ .
[i]Proposed by Yannick Yao | 1504 | 0 |
30,908 | Professors Alpha, Beta, Gamma, and Delta choose their chairs so that each professor will be between two students. Given that there are 13 chairs in total, determine the number of ways these four professors can occupy their chairs. | 1680 | 2.34375 |
30,909 | The polynomial $P$ is a quadratic with integer coefficients. For every positive integer $n$ , the integers $P(n)$ and $P(P(n))$ are relatively prime to $n$ . If $P(3) = 89$ , what is the value of $P(10)$ ? | 859 | 0.78125 |
30,910 | A pyramid with a square base has a base edge of 20 cm and a height of 40 cm. Two smaller similar pyramids are cut away from the original pyramid: one has an altitude that is one-third the original, and another that is one-fifth the original altitude, stacked atop the first smaller pyramid. What is the volume of the remaining solid as a fraction of the volume of the original pyramid? | \frac{3223}{3375} | 21.09375 |
30,911 | Define a $\it{great\ word}$ as a sequence of letters that consists only of the letters $D$, $E$, $F$, and $G$ --- some of these letters may not appear in the sequence --- and in which $D$ is never immediately followed by $E$, $E$ is never immediately followed by $F$, $F$ is never immediately followed by $G$, and $G$ is never immediately followed by $D$. How many six-letter great words are there? | 972 | 0.78125 |
30,912 | In 1900, a reader asked the following question in 1930: He knew a person who died at an age that was $\frac{1}{29}$ of the year of his birth. How old was this person in 1900? | 44 | 11.71875 |
30,913 | In the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length 1, point $E$ is on $A_{1} D_{1}$, point $F$ is on $C D$, and $A_{1} E = 2 E D_{1}$, $D F = 2 F C$. Find the volume of the triangular prism $B-F E C_{1}$. | \frac{5}{27} | 5.46875 |
30,914 | Let $a$ and $b$ be relatively prime positive integers such that $a/b$ is the maximum possible value of \[\sin^2x_1+\sin^2x_2+\sin^2x_3+\cdots+\sin^2x_{2007},\] where, for $1\leq i\leq 2007$ , $x_i$ is a nonnegative real number, and \[x_1+x_2+x_3+\cdots+x_{2007}=\pi.\] Find the value of $a+b$ . | 2008 | 2.34375 |
30,915 | Given the function $f(x)=e^{x}(x^{3}-3x+3)-ae^{x}-x$, where $e$ is the base of the natural logarithm, find the minimum value of the real number $a$ such that the inequality $f(x)\leqslant 0$ has solutions in the interval $x\in\[-2,+\infty)$. | 1-\frac{1}{e} | 3.90625 |
30,916 | Given that acute angles $\alpha$ and $\beta$ satisfy $\sin\alpha=\frac{4}{5}$ and $\cos(\alpha+\beta)=-\frac{12}{13}$, determine the value of $\cos \beta$. | -\frac{16}{65} | 0 |
30,917 | Find the sum of the $x$-coordinates of the solutions to the system of equations $y=|x^2-8x+12|$ and $y=4-x$. | 16 | 38.28125 |
30,918 | Points on a square with side length $ c$ are either painted blue or red. Find the smallest possible value of $ c$ such that how the points are painted, there exist two points with same color having a distance not less than $ \sqrt {5}$ . | $ \frac {\sqrt {10} }{2} $ | 0 |
30,919 | For all $m, n$ satisfying $1 \leqslant n \leqslant m \leqslant 5$, the number of distinct hyperbolas represented by the polar equation $\rho = \frac{1}{1 - c_{m}^{n} \cos \theta}$ is: | 15 | 0.78125 |
30,920 | Compute the determinant of the matrix:
\[
\begin{vmatrix} 2 & 4 & -2 \\ 0 & 3 & -1 \\ 5 & -1 & 2 \end{vmatrix}.
\] | 20 | 31.25 |
30,921 | A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(6,0)$, $(6,2)$, and $(0,2)$. What is the probability that $x + y < 3$? | \frac{1}{4} | 18.75 |
30,922 | Two numbers in the $4 \times 4$ grid can be swapped to create a Magic Square (in which all rows, all columns and both main diagonals add to the same total).
What is the sum of these two numbers?
A 12
B 15
C 22
D 26
E 28
\begin{tabular}{|c|c|c|c|}
\hline
9 & 6 & 3 & 16 \\
\hline
4 & 13 & 10 & 5 \\
\hline
14 & 1 & 8 & 11 \\
\hline
7 & 12 & 15 & 2 \\
\hline
\end{tabular} | 28 | 42.1875 |
30,923 | Compute \(\sqrt[4]{506250000}\). | 150 | 24.21875 |
30,924 | In the rectangular coordinate system, the parametric equation of line $l$ is given by $\begin{cases}x=1+t\cos a \\ y=1+t\sin a\end{cases}$ ($t$ is the parameter). In the polar coordinate system, the equation of circle $C$ is $\rho =4\cos \theta$.
(1) Find the rectangular coordinate equation of circle $C$.
(2) If $P(1,1)$, and circle $C$ intersects with line $l$ at points $A$ and $B$, find the maximum or minimum value of $|PA|+|PB|$. | 2\sqrt{2} | 65.625 |
30,925 | Find the number of ways to partition a set of $10$ elements, $S = \{1, 2, 3, . . . , 10\}$ into two parts; that is, the number of unordered pairs $\{P, Q\}$ such that $P \cup Q = S$ and $P \cap Q = \emptyset$ . | 511 | 84.375 |
30,926 | Three people, A, B, and C, are playing a game: each person chooses a real number from the interval $[0,1]$. The person whose chosen number is between the numbers chosen by the other two wins. A randomly chooses a number from the interval $[0,1]$, B randomly chooses a number from the interval $\left[\frac{1}{2}, \frac{2}{3}\right]$. To maximize his probability of winning, what number should C choose? | $\frac{13}{24}$ | 0 |
30,927 | Given that $n$ represents a positive integer less than $100$, determine the number of integers with an odd number of positive divisors and the number of integers with an even number of divisors. | 90 | 49.21875 |
30,928 | The sum of the following seven numbers is exactly 19: $a_1 = 2.56$ , $a_2 = 2.61$ , $a_3 = 2.65$ , $a_4 = 2.71$ , $a_5 = 2.79$ , $a_6 = 2.82$ , $a_7 = 2.86$ . It is desired to replace each $a_i$ by an integer approximation $A_i$ , $1\le i \le 7$ , so that the sum of the $A_i$ 's is also $19$ and so that $M$ , the maximum of the "errors" $\lvert A_i-a_i \rvert$ , is as small as possible. For this minimum $M$ , what is $100M$ ? | 61 | 18.75 |
30,929 |
A regular 100-sided polygon is placed on a table, with the numbers $1, 2, \ldots, 100$ written at its vertices. These numbers are then rewritten in order of their distance from the front edge of the table. If two vertices are at an equal distance from the edge, the left number is listed first, followed by the right number. Form all possible sets of numbers corresponding to different positions of the 100-sided polygon. Calculate the sum of the numbers that occupy the 13th position from the left in these sets. | 10100 | 0 |
30,930 | A person forgot the last digit of a phone number and dialed randomly. Calculate the probability of connecting to the call in no more than 3 attempts. | \dfrac{3}{10} | 21.875 |
30,931 | Alice and Bob are stuck in quarantine, so they decide to play a game. Bob will write down a polynomial $f(x)$ with the following properties:
(a) for any integer $n$ , $f(n)$ is an integer;
(b) the degree of $f(x)$ is less than $187$ .
Alice knows that $f(x)$ satisfies (a) and (b), but she does not know $f(x)$ . In every turn, Alice picks a number $k$ from the set $\{1,2,\ldots,187\}$ , and Bob will tell Alice the value of $f(k)$ . Find the smallest positive integer $N$ so that Alice always knows for sure the parity of $f(0)$ within $N$ turns.
*Proposed by YaWNeeT* | 187 | 19.53125 |
30,932 | Five numbers, $a_1$, $a_2$, $a_3$, $a_4$, are drawn randomly and without replacement from the set $\{1, 2, 3, \dots, 50\}$. Four other numbers, $b_1$, $b_2$, $b_3$, $b_4$, are then drawn randomly and without replacement from the remaining set of 46 numbers. Let $p$ be the probability that, after a suitable rotation, a brick of dimensions $a_1 \times a_2 \times a_3 \times a_4$ can be enclosed in a box of dimensions $b_1 \times b_2 \times b_3 \times b_4$, with the sides of the brick parallel to the sides of the box. Compute $p$ in lowest terms and determine the sum of the numerator and denominator. | 71 | 18.75 |
30,933 | Given that $a$, $b$, $c$, $d$ are the thousands, hundreds, tens, and units digits of a four-digit number, respectively, and the digits in lower positions are not less than those in higher positions. When $|a-b|+|b-c|+|c-d|+|d-a|$ reaches its maximum value, the minimum value of this four-digit number is ____. | 1119 | 7.03125 |
30,934 | Square $BCFE$ is inscribed in right triangle $AGD$, as shown in the diagram which is the same as the previous one. If $AB = 36$ units and $CD = 72$ units, what is the area of square $BCFE$? | 2592 | 9.375 |
30,935 | In the rectangular table shown below, the number $1$ is written in the upper-left hand corner, and every number is the sum of the any numbers directly to its left and above. The table extends infinitely downwards and to the right.
\[
\begin{array}{cccccc}
1 & 1 & 1 & 1 & 1 & \cdots
1 & 2 & 3 & 4 & 5 & \cdots
1 & 3 & 6 & 10 & 15 & \cdots
1 & 4 & 10 & 20 & 35 & \cdots
1 & 5 & 15 & 35 & 70 & \cdots
\vdots & \vdots & \vdots & \vdots & \vdots & \ddots
\end{array}
\]
Wanda the Worm, who is on a diet after a feast two years ago, wants to eat $n$ numbers (not necessarily distinct in value) from the table such that the sum of the numbers is less than one million. However, she cannot eat two numbers in the same row or column (or both). What is the largest possible value of $n$ ?
*Proposed by Evan Chen* | 19 | 14.0625 |
30,936 | In quadrilateral \(ABCD\), we have \(AB=5\), \(BC=6\), \(CD=5\), \(DA=4\), and \(\angle ABC=90^\circ\). Let \(AC\) and \(BD\) meet at \(E\). Compute \(\frac{BE}{ED}\). | \sqrt{3} | 0 |
30,937 | How many such five-digit Shenma numbers exist, where the middle digit is the smallest, the digits increase as they move away from the middle, and all the digits are different? | 1512 | 0 |
30,938 | Given two lines $l_1: ax+2y+6=0$ and $l_2: x+(a-1)y+a^2-1=0$. When $a$ \_\_\_\_\_\_, $l_1$ intersects $l_2$; when $a$ \_\_\_\_\_\_, $l_1$ is perpendicular to $l_2$; when $a$ \_\_\_\_\_\_, $l_1$ coincides with $l_2$; when $a$ \_\_\_\_\_\_, $l_1$ is parallel to $l_2$. | -1 | 47.65625 |
30,939 | Four pairs of socks in different colors are randomly selected from a wardrobe, and it is known that two of them are from the same pair. Calculate the probability that the other two are not from the same pair. | \frac{8}{9} | 0.78125 |
30,940 | Given functions $f(x)=-2x$ for $x<0$ and $g(x)=\frac{x}{\ln x}+x-2$. If $f(x_{1})=g(x_{2})$, find the minimum value of $x_{2}-2x_{1}$. | 4\sqrt{e}-2 | 23.4375 |
30,941 | Find the maximum of
\[
\sqrt{x + 31} + \sqrt{17 - x} + \sqrt{x}
\]
for $0 \le x \le 17$. | 12 | 3.125 |
30,942 | We define $N$ as the set of natural numbers $n<10^6$ with the following property:
There exists an integer exponent $k$ with $1\le k \le 43$ , such that $2012|n^k-1$ .
Find $|N|$ . | 1988 | 10.15625 |
30,943 | Let $\mathcal S$ be the sphere with center $(0,0,1)$ and radius $1$ in $\mathbb R^3$ . A plane $\mathcal P$ is tangent to $\mathcal S$ at the point $(x_0,y_0,z_0)$ , where $x_0$ , $y_0$ , and $z_0$ are all positive. Suppose the intersection of plane $\mathcal P$ with the $xy$ -plane is the line with equation $2x+y=10$ in $xy$ -space. What is $z_0$ ? | 40/21 | 44.53125 |
30,944 | How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots? | 100 | 14.0625 |
30,945 |
Given positive real numbers \( a, b, c, d \) that satisfy the equalities
\[ a^{2}+d^{2}-ad = b^{2}+c^{2}+bc \quad \text{and} \quad a^{2}+b^{2} = c^{2}+d^{2}, \]
find all possible values of the expression \( \frac{ab+cd}{ad+bc} \). | \frac{\sqrt{3}}{2} | 23.4375 |
30,946 | Let \( n = 2^{40}5^{15} \). How many positive integer divisors of \( n^2 \) are less than \( n \) but do not divide \( n \)? | 599 | 83.59375 |
30,947 | Given points $A(-2,0)$ and $P(1, \frac{3}{2})$ on the ellipse $M: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a>b>0)$, and two lines with slopes $k$ and $-k (k>0)$ passing through point $P$ intersect ellipse $M$ at points $B$ and $C$.
(I) Find the equation of ellipse $M$ and its eccentricity.
(II) If quadrilateral $PABC$ is a parallelogram, find the value of $k$. | \frac{3}{2} | 4.6875 |
30,948 | Given the planar vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, with $|\overrightarrow{a}| = 1$, $|\overrightarrow{b}| = \sqrt{2}$, and $\overrightarrow{a} \cdot \overrightarrow{b} = 1$, find the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{\pi}{4} | 98.4375 |
30,949 | Let $S$ be a set of $2020$ distinct points in the plane. Let
\[M=\{P:P\text{ is the midpoint of }XY\text{ for some distinct points }X,Y\text{ in }S\}.\]
Find the least possible value of the number of points in $M$ . | 4037 | 0 |
30,950 | Let $n$ be a positive integer. Each of the numbers $1,2,3,\ldots,100$ is painted with one of $n$ colors in such a way that two distinct numbers with a sum divisible by $4$ are painted with different colors. Determine the smallest value of $n$ for which such a situation is possible. | 25 | 10.9375 |
30,951 | In rectangle $ABCD$, $AB = 4$ and $BC = 8$. The rectangle is folded so that points $B$ and $D$ coincide, forming the pentagon $ABEFC$. What is the length of segment $EF$? Express your answer in simplest radical form. | \sqrt{10} | 0 |
30,952 | The function $g$ is defined on the set of integers and satisfies \[g(n)= \begin{cases} n-4 & \mbox{if }n\ge 1010 \\ g(g(n+7)) & \mbox{if }n<1010. \end{cases}\] Find $g(77)$. | 1011 | 0 |
30,953 | In a WeChat group, there are 5 individuals: A, B, C, D, and E, playing a game involving grabbing red envelopes. There are 4 red envelopes, each person may grab at most one, and all red envelopes must be grabbed. Among the 4 red envelopes, there are two 2-yuan envelopes, one 3-yuan envelope, and one 4-yuan envelope (envelopes with the same amount are considered the same). How many situations are there where both A and B grab a red envelope? (Answer with a numeral). | 36 | 45.3125 |
30,954 | Find the integer $n$, $-90 \le n \le 90$, such that $\sin n^\circ = \sin 782^\circ$. | -62 | 12.5 |
30,955 | A manager schedules an informal review at a café with two of his team leads. He forgets to communicate a specific time, resulting in all parties arriving randomly between 2:00 and 4:30 p.m. The manager will wait for both team leads, but only if at least one has arrived before him or arrives within 30 minutes after him. Each team lead will wait for up to one hour if the other isn’t present, but not past 5:00 p.m. What is the probability that the review meeting successfully occurs? | \frac{1}{2} | 2.34375 |
30,956 | The sides of triangle $DEF$ are in the ratio $3:4:5$. Segment $EG$ is the angle bisector drawn to the shortest side, dividing it into segments $DG$ and $GE$. If the length of side $DE$ (the base) is 12 inches, what is the length, in inches, of the longer segment of side $EF$ once the bisector is drawn from $E$ to $EF$? | \frac{80}{7} | 57.03125 |
30,957 | Let $f(x)$ be a function defined on $R$ such that $f(x+3) + f(x+1) = f(2) = 1$. Find $\sum_{k=1}^{2023} f(k) =$ ____. | 1012 | 55.46875 |
30,958 | How many ways are there to distribute 6 distinguishable balls into 2 indistinguishable boxes if no box can hold more than 4 balls? | 25 | 55.46875 |
30,959 | The product of two consecutive even negative integers is 2496. What is the sum of these two integers? | -102 | 64.84375 |
30,960 | The ship decided to determine the depth of the ocean at its location. The signal sent by the echo sounder was received on the ship 8 seconds later. The speed of sound in water is 1.5 km/s. Determine the depth of the ocean. | 6000 | 6.25 |
30,961 | A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths 5, 7, and 8. What is the area of the triangle and the radius of the circle? | \frac{10}{\pi} | 3.90625 |
30,962 | Using the digits 0, 2, 3, 5, 7, how many four-digit numbers divisible by 5 can be formed if:
(1) Digits do not repeat;
(2) Digits can repeat. | 200 | 55.46875 |
30,963 | The real number \( a \) makes the equation \( 4^{x} - 4^{-x} = 2 \cos(ax) \) have exactly 2015 solutions. For this \( a \), how many solutions does the equation \( 4^{x} + 4^{-x} = 2 \cos(ax) + 4 \) have? | 4030 | 24.21875 |
30,964 | Given an ellipse $E$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity of $\frac{\sqrt{3}}{2}$ and a minor axis length of $2$.
1. Find the equation of the ellipse $E$;
2. A line $l$ is tangent to a circle $C$: $x^{2}+y^{2}=r^{2}(0 < r < b)$ at any point and intersects the ellipse $E$ at points $A$ and $B$, with $OA \perp OB$ ($O$ is the origin of the coordinate system), find the value of $r$. | \frac{2\sqrt{5}}{5} | 46.09375 |
30,965 | Let \( A, B, C \) be positive integers such that the number \( 1212017ABC \) is divisible by 45. Find the difference between the largest and the smallest possible values of the two-digit number \( AB \). | 85 | 16.40625 |
30,966 | The tangent value of the angle between the slant height and the base is when the lateral area of the cone with volume $\frac{\pi}{6}$ is minimum. | \sqrt{2} | 18.75 |
30,967 | Calculate the value of $\cos \frac{\pi}{7} \cos \frac{2\pi}{7} \cos \frac{4\pi}{7} = \_\_\_\_\_\_$. | -\frac{1}{8} | 10.15625 |
30,968 | After folding the long rope in half 6 times, calculate the number of segments the rope will be cut into. | 65 | 0 |
30,969 | Let $\triangle PQR$ be a right triangle with $Q$ as a right angle. A circle with diameter $QR$ intersects side $PR$ at point $S$. If $PS = 3$ and $QS = 9$, find the length of $RS$. | 27 | 64.0625 |
30,970 | Consider the infinite series $1 - \frac{1}{3} - \frac{1}{9} + \frac{1}{27} - \frac{1}{81} - \frac{1}{243} + \frac{1}{729} - \cdots$. Let \( T \) be the sum of this series. Find \( T \). | \frac{15}{26} | 28.90625 |
30,971 | Alan, Beth, Carla, and Dave weigh themselves in pairs. Together, Alan and Beth weigh 280 pounds, Beth and Carla weigh 230 pounds, Carla and Dave weigh 250 pounds, and Alan and Dave weigh 300 pounds. How many pounds do Alan and Carla weigh together? | 250 | 3.125 |
30,972 |
Absent-Minded Scientist had a sore knee. The doctor prescribed him 10 pills for his knee: taking one pill daily. These pills are effective in $90\%$ of cases, and in $2\%$ of cases, there is a side effect - absence of absent-mindedness, if it existed.
Another doctor prescribed the Scientist pills for absent-mindedness - also one per day for 10 consecutive days. These pills cure absent-mindedness in $80\%$ of cases, but in $5\%$ of cases, there is a side effect - knees stop hurting.
The bottles with pills look similar, and when the Scientist left for a ten-day business trip, he took one bottle with him, but paid no attention to which one. He took one pill a day for ten days and returned completely healthy. Both absent-mindedness and knee pain were gone. Find the probability that the Scientist took the pills for absent-mindedness. | 0.69 | 0 |
30,973 | In a new diagram showing the miles traveled by bikers Alberto, Bjorn, and Carlos over a period of 6 hours. The straight lines represent their paths on a coordinate plot where the y-axis represents miles and x-axis represents hours. Alberto's line passes through the points (0,0) and (6,90), Bjorn's line passes through (0,0) and (6,72), and Carlos’ line passes through (0,0) and (6,60). Determine how many more miles Alberto has traveled compared to Bjorn and Carlos individually after six hours. | 30 | 61.71875 |
30,974 | How many of the integers between 1 and 1500, inclusive, can be expressed as the difference of the squares of two positive integers? | 1124 | 2.34375 |
30,975 | Analogical reasoning is an important method of reasoning. Based on the similarity of two things in some characteristics, conclusions can be drawn that they may be similar in other characteristics. Reading perception: In addition and subtraction of fractions with different denominators, it is often necessary to first convert them into the same denominator, and then add or subtract the numerators. For example, $\frac{1}{2}-\frac{1}{3}=\frac{3}{{2×3}}-\frac{2}{{3×2}}=\frac{{3-2}}{6}=\frac{1}{6}$. Reversing the above calculation process, we get $\frac{1}{6}=\frac{1}{{2×3}}=\frac{1}{2}-\frac{1}{3}$. This equivalent transformation process in mathematics is called splitting terms. Similarly, for $\frac{1}{{4×6}}$, it can be transformed using the method of splitting terms as: $\frac{1}{{4×6}}=\frac{1}{2}({\frac{1}{4}-\frac{1}{6}})$. Analogous to the above method, solve the following problems.<br/>【Analogical Inquiry】(1) Guess and write: $\frac{1}{{n×({n+1})}}=$______；<br/>【Understanding Application】(2) Analogous to the method of splitting terms, calculate: $\frac{1}{{1×2}}+\frac{1}{{2×3}}+\frac{1}{{3×4}}+⋯+\frac{1}{{99×100}}$；<br/>【Transfer Application】(3) Investigate and calculate: $\frac{1}{{-1×3}}+\frac{1}{{-3×5}}+\frac{1}{{-5×7}}+\frac{1}{{-7×9}}+⋯+\frac{1}{{-2021×2023}}$. | -\frac{1011}{2023} | 89.84375 |
30,976 | Arthur, Bob, and Carla each choose a three-digit number. They each multiply the digits of their own numbers. Arthur gets 64, Bob gets 35, and Carla gets 81. Then, they add corresponding digits of their numbers together. The total of the hundreds place is 24, that of the tens place is 12, and that of the ones place is 6. What is the difference between the largest and smallest of the three original numbers?
*Proposed by Jacob Weiner* | 182 | 6.25 |
30,977 | Let (b_1, b_2, ... b_7) be a list of the first 7 odd positive integers such that for each 2 ≤ i ≤ 7, either b_i + 2 or b_i - 2 (or both) must appear before b_i in the list. How many such lists are there? | 64 | 53.90625 |
30,978 | Given that the vertex of angle $\alpha$ coincides with the origin $O$, its initial side coincides with the non-negative semi-axis of the $x$-axis, and its terminal side passes through point $P(-\frac{3}{5}, -\frac{4}{5})$.
(1) Find the value of $\sin(\alpha + \pi)$;
(2) If angle $\beta$ satisfies $\sin(\alpha + \beta) = \frac{5}{13}$, find the value of $\cos(\beta)$. | \frac{16}{65} | 4.6875 |
30,979 | From a 12 × 12 grid, a 4 × 4 square has been cut out, located at the intersection of horizontals from the fourth to the seventh and the same verticals. What is the maximum number of rooks that can be placed on this board such that no two rooks attack each other, given that rooks do not attack across the cut-out cells? | 15 | 0 |
30,980 | A square sheet of paper has an area of $12 \text{ cm}^2$. The front is white and the back is black. When the paper is folded so that point $A$ rests on the diagonal and the visible black area is equal to the visible white area, how far is point A from its original position? Give your answer in simplest radical form. | 2\sqrt{6} | 0.78125 |
30,981 | We have 21 pieces of type $\Gamma$ (each formed by three small squares). We are allowed to place them on an $8 \times 8$ chessboard (without overlapping, so that each piece covers exactly three squares). An arrangement is said to be maximal if no additional piece can be added while following this rule. What is the smallest $k$ such that there exists a maximal arrangement of $k$ pieces of type $\Gamma$? | 16 | 7.8125 |
30,982 | When 5 integers are arranged in ascending order, the median is 6. If the only mode of this data set is 8, determine the possible maximum sum of these 5 integers. | 31 | 68.75 |
30,983 | Let \(x, y \in \mathbf{R}\). Define \( M \) as the maximum value among \( x^2 + xy + y^2 \), \( x^2 + x(y-1) + (y-1)^2 \), \( (x-1)^2 + (x-1)y + y^2 \), and \( (x-1)^2 + (x-1)(y-1) + (y-1)^2 \). Determine the minimum value of \( M \). | \frac{3}{4} | 50 |
30,984 | Given \( x_{i}=\frac{i}{101} \), find the value of \( S=\sum_{i=1}^{101} \frac{x_{i}^{3}}{3 x_{i}^{2}-3 x_{i}+1} \). | 51 | 80.46875 |
30,985 | Xiao Xiao did an addition problem, but he mistook the second addend 420 for 240, and the result he got was 390. The correct result is ______. | 570 | 96.875 |
30,986 | Given complex numbers \( z \) and \( \omega \) satisfying the following two conditions:
1. \( z + \omega + 3 = 0 \);
2. \( |z|, 2, |\omega| \) form an arithmetic sequence.
Is there a maximum value for \( \cos(\arg z - \arg \omega) \)? If so, find it. | \frac{1}{8} | 0.78125 |
30,987 | Given $a \in \mathbb{R}$, the function $f(x) = ax^3 - 3x^2$, and $x = 2$ is an extreme point of the function $y = f(x)$.
1. Find the value of $a$.
2. Find the extreme values of the function $f(x)$ in the interval $[-1, 5]$. | 50 | 45.3125 |
30,988 | In triangle $ABC$, the sides opposite angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and it is given that $3a\cos A = \sqrt{6}(b\cos C + c\cos B)$.
(1) Calculate the value of $\tan 2A$.
(2) If $\sin\left(\frac{\pi}{2} + B\right) = \frac{1}{3}$ and $c = 2\sqrt{2}$, find the area of triangle $ABC$. | \frac{8}{5}\sqrt{2} | 0 |
30,989 | For two lines $ax+2y+1=0$ and $3x+(a-1)y+1=0$ to be parallel, determine the value of $a$ that satisfies this condition. | -2 | 24.21875 |
30,990 | In a convex pentagon \( P Q R S T \), the angle \( P R T \) is half of the angle \( Q R S \), and all sides are equal. Find the angle \( P R T \). | 30 | 0.78125 |
30,991 | In triangle $XYZ$, $XY = 12$, $YZ = 16$, and $XZ = 20$, with $ZD$ as the angle bisector. Find the length of $ZD$. | \frac{16\sqrt{10}}{3} | 8.59375 |
30,992 | Given the ellipse C: $$\frac {x^{2}}{25}+ \frac {y^{2}}{9}=1$$, F is the right focus, and l is a line passing through point F (not parallel to the y-axis), intersecting the ellipse at points A and B. l′ is the perpendicular bisector of AB, intersecting the major axis of the ellipse at point D. Then the value of $$\frac {DF}{AB}$$ is __________. | \frac {2}{5} | 42.96875 |
30,993 | Given an ellipse and a hyperbola,\[\frac{x^2}{16} - \frac{y^2}{25} = 1\]and \[\frac{x^2}{K} + \frac{y^2}{25} = 1\], have the same asymptotes. Find the value of $K$. | 16 | 94.53125 |
30,994 | The sum of the coefficients of all terms in the expanded form of $(C_4^1x + C_4^2x^2 + C_4^3x^3 + C_4^4x^4)^2$ is 256. | 256 | 22.65625 |
30,995 | Suppose two distinct numbers are chosen from between 6 and 20, inclusive. What is the probability that their product is even, or exactly one of the numbers is a prime? | \frac{94}{105} | 11.71875 |
30,996 | Calculate the value of the following expression and find angle $\theta$ if the number can be expressed as $r e^{i \theta}$, where $0 \le \theta < 2\pi$:
\[ e^{11\pi i/60} + e^{21\pi i/60} + e^{31 \pi i/60} + e^{41\pi i /60} + e^{51 \pi i /60} \] | \frac{31\pi}{60} | 0 |
30,997 | Compute $1-2+3-4+\dots+100-101$. | 51 | 7.03125 |
30,998 | Solve the equations:
(1) $2x^2-3x-2=0$;
(2) $2x^2-3x-1=0$ (using the method of completing the square). | \frac{3-\sqrt{17}}{4} | 2.34375 |
30,999 | For a finite sequence $B=(b_1,b_2,\dots,b_{50})$ of numbers, the Cesaro sum of $B$ is defined to be
\[\frac{T_1 + \cdots + T_{50}}{50},\] where $T_k = b_1 + \cdots + b_k$ and $1 \leq k \leq 50$.
If the Cesaro sum of the 50-term sequence $(b_1,\dots,b_{50})$ is 500, what is the Cesaro sum of the 51-term sequence $(2, b_1,\dots,b_{50})$? | 492 | 6.25 |
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