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26,800 | In the Empire of Westeros, there were 1000 cities and 2017 roads (each road connects two cities). From each city, it was possible to travel to every other city. One day, an evil wizard cursed $N$ roads, making them impassable. As a result, 7 kingdoms emerged, such that within each kingdom, one can travel between any pair of cities on the roads, but it is impossible to travel from one kingdom to another using the roads. What is the maximum possible value of $N$ for which this is possible? | 2011 | 0 |
26,801 | We color certain squares of an $8 \times 8$ chessboard red. How many squares can we color at most if we want no red trimino? How many squares can we color at least if we want every trimino to have at least one red square? | 32 | 46.875 |
26,802 | Form a four-digit number using the digits 0, 1, 2, 3, 4, 5 without repetition.
(I) How many different four-digit numbers can be formed?
(II) How many of these four-digit numbers have a tens digit that is larger than both the units digit and the hundreds digit?
(III) Arrange the four-digit numbers from part (I) in ascending order. What is the 85th number in this sequence? | 2301 | 65.625 |
26,803 | Given the vertices of a pyramid with a square base, and two vertices connected by an edge are called adjacent vertices, with the rule that adjacent vertices cannot be colored the same color, and there are 4 colors to choose from, calculate the total number of different coloring methods. | 72 | 40.625 |
26,804 | Inside a square with side length 8, four congruent equilateral triangles are drawn such that each triangle shares one side with a side of the square and each has a vertex at one of the square's vertices. What is the side length of the largest square that can be inscribed in the space inside the square and outside of the triangles? | 4\sqrt{3} | 2.34375 |
26,805 | A pyramid is constructed on a $5 \times 12$ rectangular base. Each of the four edges joining the apex to the corners of the rectangular base has a length of $15$. Determine the volume of this pyramid. | 270 | 1.5625 |
26,806 | In a convex 10-gon \(A_{1} A_{2} \ldots A_{10}\), all sides and all diagonals connecting vertices skipping one (i.e., \(A_{1} A_{3}, A_{2} A_{4},\) etc.) are drawn, except for the side \(A_{1} A_{10}\) and the diagonals \(A_{1} A_{9}\), \(A_{2} A_{10}\).
A path from \(A_{1}\) to \(A_{10}\) is defined as a non-self-intersecting broken line (i.e., a line such that no two nonconsecutive segments share a common point) with endpoints \(A_{1}\) and \(A_{10}\), where each segment coincides with one of the drawn sides or diagonals. Determine the number of such paths. | 55 | 7.03125 |
26,807 | Given vectors $\overrightarrow{a}, \overrightarrow{b}$ that satisfy $|\overrightarrow{a}|^2, \overrightarrow{a}\cdot \overrightarrow{b} = \frac{3}{2}, |\overrightarrow{a}+ \overrightarrow{b}| = 2\sqrt{2}$, find $|\overrightarrow{b}| = \_\_\_\_\_\_\_.$ | \sqrt{5} | 19.53125 |
26,808 | A square is divided into three congruent rectangles. The middle rectangle is removed and replaced on the side of the original square to form an octagon as shown.
What is the ratio of the length of the perimeter of the square to the length of the perimeter of the octagon?
A $3: 5$
B $2: 3$
C $5: 8$
D $1: 2$
E $1: 1$ | 3:5 | 14.0625 |
26,809 | Consider $\triangle \natural\flat\sharp$ . Let $\flat\sharp$ , $\sharp\natural$ and $\natural\flat$ be the answers to problems $4$ , $5$ , and $6$ , respectively. If the incircle of $\triangle \natural\flat\sharp$ touches $\natural\flat$ at $\odot$ , find $\flat\odot$ .
*Proposed by Evan Chen* | 2.5 | 0.78125 |
26,810 | From the numbers $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, four different numbers are selected, denoted as $a$, $b$, $c$, $d$ respectively. If the parity of $a+b$ is the same as the parity of $c+d$, then the total number of ways to select $a$, $b$, $c$, $d$ is ______ (provide the answer in numerical form). | 912 | 0.78125 |
26,811 | Determine the number of ways to arrange the letters of the word PERSEVERANCE. | 19,958,400 | 0 |
26,812 | Kelvin the Frog lives in the 2-D plane. Each day, he picks a uniformly random direction (i.e. a uniformly random bearing $\theta\in [0,2\pi]$ ) and jumps a mile in that direction. Let $D$ be the number of miles Kelvin is away from his starting point after ten days. Determine the expected value of $D^4$ . | 200 | 14.84375 |
26,813 | Using the digits 0 to 9, how many three-digit even numbers can be formed without repeating any digits? | 288 | 3.90625 |
26,814 | Given a 50-term sequence $(b_1, b_2, \dots, b_{50})$, the Cesaro sum is 500. What is the Cesaro sum of the 51-term sequence $(2, b_1, b_2, \dots, b_{50})$? | 492 | 10.9375 |
26,815 | Medians $\overline{DP}$ and $\overline{EQ}$ of $\triangle DEF$ are perpendicular. An altitude $\overline{ER}$ from $E$ to line $DF$ is such that $ER = 15$. Given $DP= 27$ and $EQ = 36$, determine the length of ${DF}$. | 45 | 6.25 |
26,816 | How many of the numbers from the set $\{1, 2, 3, \ldots, 100\}$ have a perfect square factor other than one? | 48 | 0 |
26,817 | Given the sequence $\{a_n\}$ such that the sum of the first $n$ terms is $S_n$, $S_1=6$, $S_2=4$, $S_n>0$ and $S_{2n}$, $S_{2n-1}$, $S_{2n+2}$ form a geometric progression, while $S_{2n-1}$, $S_{2n+2}$, $S_{2n+1}$ form an arithmetic progression. Determine the value of $a_{2016}$.
Options:
A) $-1009$
B) $-1008$
C) $-1007$
D) $-1006$ | -1009 | 2.34375 |
26,818 | In a regular tetrahedron with edge length $2\sqrt{6}$, the total length of the intersection between the sphere with center $O$ and radius $\sqrt{3}$ and the surface of the tetrahedron is ______. | 8\sqrt{2}\pi | 5.46875 |
26,819 | A square and a regular octagon have equal perimeters. If the area of the square is 16, what is the area of the octagon?
A) $8 + 4\sqrt{2}$
B) $4 + 2\sqrt{2}$
C) $16 + 8\sqrt{2}$
D) $4\sqrt{2}$
E) $8\sqrt{2}$ | 8 + 4\sqrt{2} | 15.625 |
26,820 | Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a
running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7. Play continues until either player wins, or else inde nitely. If Nathaniel goes fi rst, determine the probability that he ends up winning. | 5/11 | 3.90625 |
26,821 | Five packages are delivered to five houses, one to each house. If these packages are randomly delivered, what is the probability that exactly three of them are delivered to the correct houses? | \frac{1}{6} | 0.78125 |
26,822 | A permutation of a finite set $S$ is a one-to-one function from $S$ to $S$ . A permutation $P$ of the set $\{ 1, 2, 3, 4, 5 \}$ is called a W-permutation if $P(1) > P(2) < P(3) > P(4) < P(5)$ . A permutation of the set $\{1, 2, 3, 4, 5 \}$ is selected at random. Compute the probability that it is a W-permutation. | 2/15 | 93.75 |
26,823 | A hemisphere-shaped bowl with radius 1 foot is filled full with chocolate. All of the chocolate is then evenly distributed between 36 cylindrical molds, each having a height equal to their diameter. What is the diameter of each cylinder? | \frac{2^{1/3}}{3} | 0 |
26,824 | In a right-angled triangle $ABC$, where $AB = AC = 1$, an ellipse is constructed with point $C$ as one of its foci. The other focus of the ellipse lies on side $AB$, and the ellipse passes through points $A$ and $B$. Determine the focal length of the ellipse. | \frac{\sqrt{5}}{2} | 3.90625 |
26,825 | To obtain the graph of the function $y=\sin\left(2x+\frac{\pi}{3}\right)$, find the transformation required to obtain the graph of the function $y=\cos\left(2x-\frac{\pi}{3}\right)$. | \left(\frac{\pi}{12}\right) | 0 |
26,826 | Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $2500$ are not factorial tails? | 500 | 7.8125 |
26,827 | The number $5\,41G\,507\,2H6$ is divisible by $40.$ Determine the sum of all distinct possible values of the product $GH.$ | 225 | 9.375 |
26,828 | Form a four-digit number without repeating digits using the numbers 0, 1, 2, 3, 4, 5, 6, where the sum of the digits in the units, tens, and hundreds places is even. How many such four-digit numbers are there? (Answer with a number) | 324 | 53.125 |
26,829 | Taylor is tiling his 12 feet by 16 feet living room floor. He plans to place 1 foot by 1 foot tiles along the edges to form a border, and then use 2 feet by 2 feet tiles to fill the remaining floor area, so find the total number of tiles he will use. | 87 | 67.96875 |
26,830 | What is the total number of digits used when the first 2500 positive even integers are written? | 9449 | 3.90625 |
26,831 | Convert $645_{10}$ to base 5. | 10400_5 | 0 |
26,832 | Given $(a+i)i=b+ai$, solve for $|a+bi|$. | \sqrt{2} | 3.125 |
26,833 | Alice cycled 240 miles in 4 hours, 30 minutes. Then, she cycled another 300 miles in 5 hours, 15 minutes. What was Alice's average speed in miles per hour for her entire journey? | 55.38 | 90.625 |
26,834 | The ratio of a geometric sequence is an integer. We know that there is a term in the sequence which is equal to the sum of some other terms of the sequence. What can the ratio of the sequence be? | -1 | 6.25 |
26,835 | A regular tetrahedron is formed by joining the centers of four non-adjacent faces of a cube. Determine the ratio of the volume of the tetrahedron to the volume of the cube. | \frac{1}{3} | 6.25 |
26,836 | Given vectors $\overrightarrow {m}=(\sin x，-1)$ and $\overrightarrow {n}=( \sqrt {3}\cos x，- \frac {1}{2})$, and the function $f(x)= \overrightarrow {m}^{2}+ \overrightarrow {m}\cdot \overrightarrow {n}-2$.
(I) Find the maximum value of $f(x)$ and the set of values of $x$ at which the maximum is attained.
(II) Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ of triangle $ABC$, respectively, and that they form a geometric sequence. Also, angle $B$ is acute, and $f(B)=1$. Find the value of $\frac{1}{\tan A} + \frac{1}{\tan C}$. | \frac{2\sqrt{3}}{3} | 78.125 |
26,837 | A student travels from his university to his hometown, a distance of 150 miles, in a sedan that averages 25 miles per gallon. For the return trip, he drives his friend's truck, which averages only 15 miles per gallon. Additionally, they make a detour of 50 miles at an average of 10 miles per gallon in the truck. Calculate the average fuel efficiency for the entire trip. | 16.67 | 83.59375 |
26,838 | If the line $x+ay+6=0$ is parallel to the line $(a-2)x+3y+2a=0$, determine the value of $a$. | -1 | 14.84375 |
26,839 | The Gropkas of Papua New Guinea have ten letters in their alphabet: A, E, G, I, K, O, R, U, and V. Suppose license plates of four letters use only the letters in the Gropka alphabet. How many possible license plates are there of four letters that begin with either A or E, end with V, cannot contain P, and have no letters that repeat? | 84 | 67.1875 |
26,840 | Using the six digits $0$, $1$, $2$, $3$, $4$, $5$, form integers without repeating any digit. Determine how many such integers satisfy the following conditions:
$(1)$ How many four-digit even numbers can be formed?
$(2)$ How many five-digit numbers that are multiples of $5$ and have no repeated digits can be formed?
$(3)$ How many four-digit numbers greater than $1325$ and with no repeated digits can be formed? | 270 | 10.15625 |
26,841 | Three distinct integers, $x$, $y$, and $z$, are randomly chosen from the set $\{1,2,3,4,5,6,7,8,9,10,11,12\}$. What is the probability that $xyz-xy-xz-yz$ is even? | \frac{10}{11} | 19.53125 |
26,842 | Let \( f(n) = \sum_{k=2}^{\infty} \frac{1}{k^n \cdot k!} \). Calculate \( \sum_{n=2}^{\infty} f(n) \). | 3 - e | 0.78125 |
26,843 | If $z$ is a complex number such that
\[
z + z^{-1} = 2\sqrt{2},
\]
what is the value of
\[
z^{100} + z^{-100} \, ?
\] | -2 | 64.0625 |
26,844 | If a line is perpendicular to a plane, then this line and the plane form a "perpendicular line-plane pair". In a cube, the number of "perpendicular line-plane pairs" formed by a line determined by two vertices and a plane containing four vertices is _________. | 36 | 9.375 |
26,845 | Let $a$, $b$, and $c$ be three positive real numbers whose sum is 2. If no one of these numbers is more than three times any other, find the minimum value of the product $abc$. | \frac{2}{9} | 0.78125 |
26,846 | 20 shareholders are seated around a round table. What is the minimum total number of their shares if it is known that:
a) any three of them together have more than 1000 shares,
b) any three consecutive shareholders together have more than 1000 shares? | 6674 | 0.78125 |
26,847 | What is the smallest positive integer with exactly 12 positive integer divisors? | 60 | 91.40625 |
26,848 | Let $\mathrm {Q}$ be the product of the roots of $z^8+z^6+z^4+z^3+z+1=0$ that have a positive imaginary part, and suppose that $\mathrm {Q}=s(\cos{\phi^{\circ}}+i\sin{\phi^{\circ}})$, where $0<s$ and $0\leq \phi <360$. Find $\phi$. | 180 | 11.71875 |
26,849 | There are $2024$ people, which are knights and liars and some of them are friends. Every person is asked for the number of their friends and the answers were $0,1, \ldots, 2023$ . Every knight answered truthfully, while every liar changed the real answer by exactly $1$ . What is the minimal number of liars? | 1012 | 41.40625 |
26,850 | Point P moves on the parabola $y^2=4x$, and point Q moves on the line $x-y+5=0$. Find the minimum value of the sum of the distance $d$ from point P to the directrix of the parabola and the distance $|PQ|$ between points P and Q. | 3\sqrt{2} | 3.90625 |
26,851 | Given Farmer Euclid has a field in the shape of a right triangle with legs of lengths $5$ units and $12$ units, and he leaves a small unplanted square of side length S in the corner where the legs meet at a right angle, where $S$ is $3$ units from the hypotenuse, calculate the fraction of the field that is planted. | \frac{431}{480} | 3.125 |
26,852 | What is the ratio of the area of a regular hexagon inscribed in an equilateral triangle with side length $s$ to the area of a regular hexagon inscribed in a circle with radius $r$? Assume the height of the equilateral triangle equals the diameter of the circle, thus $s = r \sqrt{3}$. | \dfrac{9}{16} | 3.90625 |
26,853 | How many three-digit whole numbers have at least one 5 or consecutively have the digit 1 followed by the digit 2? | 270 | 3.90625 |
26,854 | The function $f(x)=(m^{2}-m-1)x^{m^{2}+m-1}$ is a power function, and it is decreasing on $(0,+\infty)$. Find the real number $m$. | -1 | 26.5625 |
26,855 | Let $ABCD$ and $AEFG$ be two faces of a cube with edge $AB=10$. A beam of light emanates from vertex $A$ and reflects off face $AEFG$ at point $Q$, which is 3 units from $\overline{EF}$ and 6 units from $\overline{AG}$. The beam continues to be reflected off the faces of the cube. The length of the light path from the time it leaves point $A$ until it next reaches a vertex of the cube is given by $p\sqrt{q}$, where $p$ and $q$ are integers and $q$ is not divisible by the square of any prime. Find $p+q$. | 155 | 4.6875 |
26,856 | Consider three squares: $PQRS$, $TUVW$, and $WXYZ$, where each side of the squares has length $s=1$. $S$ is the midpoint of $WY$, and $R$ is the midpoint of $WU$. Calculate the ratio of the area of the shaded quadrilateral $PQSR$ to the sum of the areas of the three squares.
A) $\frac{1}{12}$
B) $\frac{1}{6}$
C) $\frac{1}{8}$
D) $\frac{1}{3}$
E) $\frac{1}{24}$ | \frac{1}{12} | 11.71875 |
26,857 | Let $x$ be inversely proportional to $y$. If $x = 4$ when $y = 2$, find the value of $x$ when $y = -3$ and when $y = 6$. | \frac{4}{3} | 96.09375 |
26,858 | For positive real numbers $a,$ $b,$ and $c,$ compute the maximum value of:
\[\frac{abc(a + b + c)}{(a + b)^3 (b + c)^3}.\] | \frac{1}{8} | 0 |
26,859 | Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|=1$ and $(\overrightarrow{a} + \overrightarrow{b}) \perp \overrightarrow{b}$, calculate the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{2\pi}{3} | 90.625 |
26,860 | On December 8, 2022, the Joint Prevention and Control Mechanism of the State Council held a press conference to introduce further optimization of the implementation of epidemic prevention and control measures. It was emphasized that individuals are responsible for their own health. Xiao Hua prepared some medicines. There are three types of fever-reducing medicines and five types of cough-suppressing medicines to choose from. Xiao Hua randomly selects two types. Let event $A$ represent the selection of at least one fever-reducing medicine, and event $B$ represent the selection of exactly one cough-suppressing medicine. Find $P(A)=$____ and $P(B|A)=$____. | \frac{5}{6} | 50.78125 |
26,861 | If $a=2 \int_{-3}^{3} (x+|x|) \, dx$, determine the total number of terms in the expansion of $(\sqrt{x} - \frac{1}{\sqrt[3]{x}})^a$ where the power of $x$ is not an integer. | 14 | 2.34375 |
26,862 | Among the natural numbers not exceeding 10,000, calculate the number of odd numbers with distinct digits. | 2605 | 39.0625 |
26,863 | Find the volume of the region in space defined by
\[|x - y + z| + |x - y - z| \le 10\]and $x, y, z \ge 0$. | 125 | 19.53125 |
26,864 | The hyperbola $C:\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ $(a > 0,b > 0)$ has an asymptote perpendicular to the line $x+2y+1=0$. Let $F_1$ and $F_2$ be the foci of $C$, and let $A$ be a point on the hyperbola. If $|F_1A|=2|F_2A|$, then $\cos \angle AF_2F_1=$ __________. | \dfrac{\sqrt{5}}{5} | 46.09375 |
26,865 | Given that $f(\alpha) = \cos\alpha \sqrt{\frac{\cot\alpha - \cos\alpha}{\cot\alpha + \cos\alpha}} + \sin\alpha \sqrt{\frac{\tan\alpha - \sin\alpha}{\tan\alpha + \sin\alpha}}$, and $\alpha$ is an angle in the second quadrant.
(1) Simplify $f(\alpha)$.
(2) If $f(-\alpha) = \frac{1}{5}$, find the value of $\frac{1}{\tan\alpha} - \frac{1}{\cot\alpha}$. | -\frac{7}{12} | 29.6875 |
26,866 | Determine if there are other composite numbers smaller than 20 that are also abundant besides number 12. If so, list them. If not, confirm 12 remains the smallest abundant number. | 12 | 35.15625 |
26,867 | Find the maximum real number $\lambda$ such that for the real coefficient polynomial $f(x)=x^{3}+a x^{2}+b x+c$ having all non-negative real roots, it holds that $f(x) \geqslant \lambda(x-a)^{3}$ for all $x \geqslant 0$. Additionally, determine when equality holds in the given expression. | -1/27 | 0 |
26,868 | Find the sum: \( S = 19 \cdot 20 \cdot 21 + 20 \cdot 21 \cdot 22 + \cdots + 1999 \cdot 2000 \cdot 2001 \). | 6 \left( \binom{2002}{4} - \binom{21}{4} \right) | 0 |
26,869 | A \(3\times 5\) rectangle and a \(4\times 6\) rectangle need to be contained within a square without any overlapping at their interior points, and the square's sides are parallel to the sides of the given rectangles. Determine the smallest possible area of this square. | 81 | 12.5 |
26,870 | How many of the natural numbers from 1 to 800, inclusive, contain the digit 7 at least once? | 152 | 0 |
26,871 | Jeremy wakes up at 6:00 a.m., catches the school bus at 7:00 a.m., has 7 classes that last 45 minutes each, enjoys 45 minutes for lunch, and spends an additional 2.25 hours (which includes 15 minutes for miscellaneous activities) at school. He takes the bus home and arrives at 5:00 p.m. Calculate the total number of minutes he spends on the bus. | 105 | 24.21875 |
26,872 | Let a three-digit number \( n = \overline{a b c} \). If the digits \( a, b, c \) can form an isosceles (including equilateral) triangle, calculate how many such three-digit numbers \( n \) are there. | 165 | 84.375 |
26,873 | How many distinct products can you obtain by multiplying two or more distinct elements from the set $\{1, 2, 3, 5, 7, 11\}$? | 26 | 0 |
26,874 | Given acute angles $ \alpha $ and $ \beta $ satisfy $ \sin \alpha =\frac{\sqrt{5}}{5},\sin (\alpha -\beta )=-\frac{\sqrt{10}}{10} $, then $ \beta $ equals \_\_\_\_\_\_\_\_\_\_\_\_. | \frac{\pi}{4} | 85.9375 |
26,875 | Triangle $ABC$ has a right angle at $C$ , and $D$ is the foot of the altitude from $C$ to $AB$ . Points $L, M,$ and $N$ are the midpoints of segments $AD, DC,$ and $CA,$ respectively. If $CL = 7$ and $BM = 12,$ compute $BN^2$ . | 193 | 18.75 |
26,876 | Given the digits 0, 1, 2, 3, 4, 5, how many unique six-digit numbers greater than 300,000 can be formed where the digit in the thousand's place is less than 3? | 216 | 27.34375 |
26,877 | A number is randomly selected from the interval $[-π, π]$. Calculate the probability that the value of the function $y = \cos x$ falls within the range $[-\frac{\sqrt{3}}{2}, \frac{\sqrt{3}}{2}]$. | \frac{2}{3} | 52.34375 |
26,878 | Class A and Class B each send 2 students to participate in the grade math competition. The probability of each participating student passing the competition is 0.6, and the performance of the participating students does not affect each other. Find:
(1) The probability that there is exactly one student from each of Class A and Class B who passes the competition;
(2) The probability that at least one student from Class A and Class B passes the competition. | 0.9744 | 77.34375 |
26,879 | To rebuild homes after an earthquake for disaster relief, in order to repair a road damaged during the earthquake, if Team A alone takes 3 months to complete the work, costing $12,000 per month; if Team B alone takes 6 months to complete the work, costing $5,000 per month. How many months will it take for Teams A and B to cooperate to complete the construction? How much will it cost in total? | 34,000 | 0 |
26,880 | Calculate the integer nearest to $500\sum_{n=4}^{10005}\frac{1}{n^2-9}$. | 174 | 17.96875 |
26,881 | If I roll 7 standard 6-sided dice and multiply the number on the face of each die, what is the probability that the result is a composite number and the sum of the numbers rolled is divisible by 3? | \frac{1}{3} | 10.15625 |
26,882 | Determine $\sqrt[7]{218618940381251}$ without a calculator. | 102 | 3.90625 |
26,883 | What is the area of the polygon with vertices at $(2, 1)$, $(4, 3)$, $(6, 1)$, $(5, -2)$, and $(3, -2)$? | 13 | 10.9375 |
26,884 | Given the ellipse $E$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$, its minor axis length is $2$, and the eccentricity is $\frac{\sqrt{6}}{3}$. The line $l$ passes through the point $(-1,0)$ and intersects the ellipse $E$ at points $A$ and $B$. $O$ is the coordinate origin.
(1) Find the equation of the ellipse $E$;
(2) Find the maximum area of $\triangle OAB$. | \frac{\sqrt{6}}{3} | 3.90625 |
26,885 | Given an ellipse with its foci on the $y$-axis, an eccentricity of $\frac{2\sqrt{2}}{3}$, and one focus at $(0, 2\sqrt{2})$.
(1) Find the standard equation of the ellipse;
(2) A moving line $l$ passes through point $P(-1,0)$, intersecting a circle $O$ centered at the origin with a radius of $2$ at points $A$ and $B$. $C$ is a point on the ellipse such that $\overrightarrow{AB} \cdot \overrightarrow{CP} = 0$. Find the length of chord $AB$ when $|\overrightarrow{CP}|$ is at its maximum. | \frac{\sqrt{62}}{2} | 0 |
26,886 | Let $a$ , $b$ , $c$ be positive integers with $a \le 10$ . Suppose the parabola $y = ax^2 + bx + c$ meets the $x$ -axis at two distinct points $A$ and $B$ . Given that the length of $\overline{AB}$ is irrational, determine, with proof, the smallest possible value of this length, across all such choices of $(a, b, c)$ . | \frac{\sqrt{13}}{9} | 38.28125 |
26,887 | Right $ \triangle ABC$ has $ AB \equal{} 3$ , $ BC \equal{} 4$ , and $ AC \equal{} 5$ . Square $ XYZW$ is inscribed in $ \triangle ABC$ with $ X$ and $ Y$ on $ \overline{AC}$ , $ W$ on $ \overline{AB}$ , and $ Z$ on $ \overline{BC}$ . What is the side length of the square?
[asy]size(200);defaultpen(fontsize(10pt)+linewidth(.8pt));
real s = (60/37);
pair A = (0,0);
pair C = (5,0);
pair B = dir(60)*3;
pair W = waypoint(B--A,(1/3));
pair X = foot(W,A,C);
pair Y = (X.x + s, X.y);
pair Z = (W.x + s, W.y);
label(" $A$ ",A,SW);
label(" $B$ ",B,NW);
label(" $C$ ",C,SE);
label(" $W$ ",W,NW);
label(" $X$ ",X,S);
label(" $Y$ ",Y,S);
label(" $Z$ ",Z,NE);
draw(A--B--C--cycle);
draw(X--W--Z--Y);[/asy] | \frac {60}{37} | 24.21875 |
26,888 | A point P is taken on the circle x²+y²=4. A vertical line segment PD is drawn from point P to the x-axis, with D being the foot of the perpendicular. As point P moves along the circle, what is the trajectory of the midpoint M of line segment PD? Also, find the focus and eccentricity of this trajectory. | \frac{\sqrt{3}}{2} | 44.53125 |
26,889 | Given vectors $\overrightarrow{a} = (\cos \frac{3x}{2}, \sin \frac{3x}{2})$ and $\overrightarrow{b} = (\cos \frac{x}{2}, -\sin \frac{x}{2})$, with $x \in \left[-\frac{\pi}{3}, \frac{\pi}{4}\right]$,
(Ⅰ) Find $\overrightarrow{a} \cdot \overrightarrow{b}$ and $|\overrightarrow{a} + \overrightarrow{b}|$.
(Ⅱ) Let $f(x) = \overrightarrow{a} \cdot \overrightarrow{b} - |\overrightarrow{a} + \overrightarrow{b}|$, find the maximum and minimum values of $f(x)$. | -\frac{3}{2} | 55.46875 |
26,890 | What are the rightmost three digits of $3^{1987}$? | 187 | 0 |
26,891 | What is the least positive integer with exactly $12$ positive factors? | 72 | 0 |
26,892 | Define the sequence \(\{a_n\}\) where \(a_n = n^3 + 4\) for \(n \in \mathbf{N}_+\). Let \(d_n = \gcd(a_n, a_{n+1})\), which is the greatest common divisor of \(a_n\) and \(a_{n+1}\). Find the maximum value of \(d_n\). | 433 | 53.125 |
26,893 | $(1)\sqrt{5}-27+|2-\sqrt{5}|-\sqrt{9}+(\frac{1}{2})^{2}$；<br/>$(2)2\sqrt{40}-5\sqrt{\frac{1}{10}}-\sqrt{10}$；<br/>$(3)(3\sqrt{12}-2\sqrt{\frac{1}{3}}-\sqrt{48})÷4\sqrt{3}-{(\sqrt{2}-1)^0}$；<br/>$(4)(-\sqrt{5}+\sqrt{3})(\sqrt{5}+\sqrt{3})+(-\sqrt{3}-1)^{2}$. | 2+2\sqrt{3} | 48.4375 |
26,894 | 9 pairs of table tennis players participate in a doubles match, their jersey numbers are 1, 2, …, 18. The referee is surprised to find that the sum of the jersey numbers of each pair of players is exactly a perfect square. The player paired with player number 1 is . | 15 | 57.03125 |
26,895 | A projectile is launched with an initial velocity of $u$ at an angle of $\alpha$ from the ground. The trajectory can be modeled by the parametric equations:
\[
x = ut \cos \alpha, \quad y = ut \sin \alpha - \frac{1}{2} kt^2,
\]
where $t$ denotes time and $k$ denotes a constant acceleration, forming a parabolic arch.
Suppose $u$ is constant, but $\alpha$ varies over $0^\circ \le \alpha \le 90^\circ$. The highest points of each parabolic arch are plotted. Determine the area enclosed by the curve traced by these highest points, and express it in the form:
\[
d \cdot \frac{u^4}{k^2}.
\] | \frac{\pi}{8} | 0 |
26,896 | Let the sample space be $\Omega =\{1,2,3,4,5,6,7,8\}$ with equally likely sample points, and events $A=\{1,2,3,4\}$, $B=\{1,2,3,5\}$, $C=\{1,m,n,8\}$, such that $p\left(ABC\right)=p\left(A\right)p\left(B\right)p\left(C\right)$, and satisfying that events $A$, $B$, and $C$ are not pairwise independent. Find $m+n$. | 13 | 34.375 |
26,897 | A uniform solid semi-circular disk of radius $R$ and negligible thickness rests on its diameter as shown. It is then tipped over by some angle $\gamma$ with respect to the table. At what minimum angle $\gamma$ will the disk lose balance and tumble over? Express your answer in degrees, rounded to the nearest integer.
[asy]
draw(arc((2,0), 1, 0,180));
draw((0,0)--(4,0));
draw((0,-2.5)--(4,-2.5));
draw(arc((3-sqrt(2)/2, -4+sqrt(2)/2+1.5), 1, -45, 135));
draw((3-sqrt(2), -4+sqrt(2)+1.5)--(3, -4+1.5));
draw(anglemark((3-sqrt(2), -4+sqrt(2)+1.5), (3, -4+1.5), (0, -4+1.5)));
label(" $\gamma$ ", (2.8, -3.9+1.5), WNW, fontsize(8));
[/asy]
*Problem proposed by Ahaan Rungta* | 23 | 1.5625 |
26,898 | Given angles $\alpha$ and $\beta$ have their vertices at the origin of coordinates, and their initial sides coincide with the positive half-axis of the x-axis, $\alpha, \beta \in (0, \pi)$. The terminal side of angle $\beta$ intersects the unit circle at a point whose x-coordinate is $-\frac{5}{13}$, and the terminal side of angle $\alpha + \beta$ intersects the unit circle at a point whose y-coordinate is $\frac{3}{5}$. Then, find the value of $\cos\alpha$. | \frac{56}{65} | 13.28125 |
26,899 | A polynomial $g(x)=x^4+px^3+qx^2+rx+s$ has real coefficients, and it satisfies $g(3i)=g(3+i)=0$. | 49 | 21.09375 |
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