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27,900 | Given $x_1$ and $x_2$ are the two real roots of the quadratic equation in $x$: $x^2 - 2(m+2)x + m^2 = 0$.
(1) When $m=0$, find the roots of the equation;
(2) If $(x_1 - 2)(x_2 - 2) = 41$, find the value of $m$;
(3) Given an isosceles triangle $ABC$ with one side length of 9, if $x_1$ and $x_2$ happen to be the lengths of the other two sides of $\triangle ABC$, find the perimeter of this triangle. | 19 | 37.5 |
27,901 | In $\triangle ABC$ , medians $\overline{AD}$ and $\overline{CE}$ intersect at $P$ , $PE=1.5$ , $PD=2$ , and $DE=2.5$ . What is the area of $AEDC?$
[asy]
unitsize(75);
pathpen = black; pointpen=black;
pair A = MP("A", D((0,0)), dir(200));
pair B = MP("B", D((2,0)), dir(-20));
pair C = MP("C", D((1/2,1)), dir(100));
pair D = MP("D", D(midpoint(B--C)), dir(30));
pair E = MP("E", D(midpoint(A--B)), dir(-90));
pair P = MP("P", D(IP(A--D, C--E)), dir(150)*2.013);
draw(A--B--C--cycle);
draw(A--D--E--C);
[/asy] | 13.5 | 1.5625 |
27,902 | Let $ABCDE$ be a convex pentagon such that: $\angle ABC=90,\angle BCD=135,\angle DEA=60$ and $AB=BC=CD=DE$ . Find angle $\angle DAE$ . | 30 | 18.75 |
27,903 | In response to the national policy of expanding domestic demand, a manufacturer plans to hold a promotional event at the beginning of 2015. After investigation and estimation, the annual sales volume (i.e., the annual production volume $x$ in ten thousand units) and the annual promotional expenses $t$ (where $t > 0$) in ten thousand yuan satisfy $x=4- \frac {k}{t}$ (where $k$ is a constant). If the annual promotional expenses $t$ are 1 ten thousand yuan, the annual sales volume of the product is 1 ten thousand units. It is known that the fixed investment for the product in 2015 is 60 thousand yuan, and an additional investment of 120 thousand yuan is required to produce 1 ten thousand units of the product. The manufacturer sets the selling price of each unit to 1.5 times the average cost of the product (the product cost includes both fixed and additional investments).
- (Ⅰ) Express the profit $y$ (in ten thousand yuan) of the manufacturer for this product in 2015 as a function of the annual promotional expenses $t$ (in ten thousand yuan);
- (Ⅱ) How much should the manufacturer invest in annual promotional expenses in 2015 to maximize profit? | 3 \sqrt {2} | 0 |
27,904 | Let $\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{abcd}$ where $a, b, c, d$ are distinct digits. Find the sum of the elements of $\mathcal{T}.$ | 2520 | 7.8125 |
27,905 | Given a constant function on the interval $(0,1)$, $f(x)$, which satisfies: when $x \notin \mathbf{Q}$, $f(x)=0$; and when $x=\frac{p}{q}$ (with $p, q$ being integers, $(p, q)=1, 0<p<q$), $f(x)=\frac{p+1}{q}$. Determine the maximum value of $f(x)$ on the interval $\left(\frac{7}{8}, \frac{8}{9}\right)$. | $\frac{16}{17}$ | 0 |
27,906 | Out of 1500 people surveyed, $25\%$ do not like television, and out of those who do not like television, $15\%$ also do not like books. How many people surveyed do not like both television and books? | 56 | 96.875 |
27,907 | In the expansion of $(2x +3y)^{20}$ , find the number of coefficients divisible by $144$ .
*Proposed by Hannah Shen* | 15 | 61.71875 |
27,908 | What is the total number of digits used when the first 2500 positive even integers are written? | 9448 | 34.375 |
27,909 | Four people, A, B, C, and D, stand in a line from left to right and are numbered 1, 2, 3, and 4 respectively. They have the following conversation:
A: Both people to my left and my right are taller than me.
B: Both people to my left and my right are shorter than me.
C: I am the tallest.
D: There is no one to my right.
If all four of them are honest, what is the 4-digit number formed by the sequence of their numbers? | 2314 | 3.125 |
27,910 | In a convex polygon with 1992 sides, the minimum number of interior angles that are not acute is: | 1989 | 7.8125 |
27,911 | A rigid board with a mass \( m \) and a length \( l = 20 \) meters partially lies on the edge of a horizontal surface, overhanging it by three quarters of its length. To prevent the board from falling, a stone with a mass of \( 2m \) was placed at its very edge. How far from the stone can a person with a mass of \( m / 2 \) walk on the board? Neglect the sizes of the stone and the person compared to the size of the board. | 15 | 7.8125 |
27,912 | Let \( n \) be the product of 3659893456789325678 and 342973489379256. Determine the number of digits of \( n \). | 34 | 98.4375 |
27,913 | In $\triangle ABC$, lines $CF$ and $AD$ are drawn such that $\dfrac{CD}{DB}=\dfrac{2}{3}$ and $\dfrac{AF}{FB}=\dfrac{1}{3}$. Let $s = \dfrac{CQ}{QF}$ where $Q$ is the intersection point of $CF$ and $AD$. Find $s$.
[asy]
size(8cm);
pair A = (0, 0), B = (9, 0), C = (3, 6);
pair D = (6, 4), F = (6, 0);
pair Q = intersectionpoints(A--D, C--F)[0];
draw(A--B--C--cycle);
draw(A--D);
draw(C--F);
label("$A$", A, SW);
label("$B$", B, SE);
label("$C$", C, N);
label("$D$", D, NE);
label("$F$", F, S);
label("$Q$", Q, S);
[/asy] | \frac{3}{5} | 0 |
27,914 | Given the ellipse $x^{2}+4y^{2}=16$, and the line $AB$ passes through point $P(2,-1)$ and intersects the ellipse at points $A$ and $B$. If the slope of line $AB$ is $\frac{1}{2}$, then the value of $|AB|$ is ______. | 2\sqrt{5} | 97.65625 |
27,915 | Consider the set of points that are inside or within two units of a rectangular parallelepiped that measures 2 by 3 by 6 units. Calculate the total volume of this set, expressing your answer in the form $\frac{m+n\pi}{p}$, where $m$, $n$, and $p$ are positive integers with $n$ and $p$ being relatively prime. | \frac{540 + 164\pi}{3} | 5.46875 |
27,916 | There are five unmarked envelopes on a table, each containing a letter for a different person. If the mail is randomly distributed to these five people, with each person getting one letter, what is the probability that exactly three people receive the correct letter? | \frac{1}{12} | 88.28125 |
27,917 | Suppose the graph of \( y=g(x) \) includes the points \( (1,4), (2,6), \) and \( (3,2) \).
Based only on this information, there are two points that must be on the graph of \( y=g(g(x)) \). If we call these points \( (a,b) \) and \( (c,d) \), what is \( ab + cd \)? | 20 | 3.125 |
27,918 | The non-negative difference between two numbers \(a\) and \(b\) is \(a-b\) or \(b-a\), whichever is greater than or equal to 0. For example, the non-negative difference between 24 and 64 is 40. In the sequence \(88, 24, 64, 40, 24, \ldots\), each number after the second is obtained by finding the non-negative difference between the previous 2 numbers. The sum of the first 100 numbers in this sequence is: | 760 | 3.90625 |
27,919 | An iterative average of the numbers 2, 3, 4, 6, and 7 is computed by arranging the numbers in some order. Find the difference between the largest and smallest possible values that can be obtained using this procedure. | \frac{11}{4} | 0.78125 |
27,920 | Given the expansion of $\left(x-\frac{a}{x}\right)^{5}$, find the maximum value among the coefficients in the expansion. | 10 | 71.09375 |
27,921 | Let $S_{n}$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_{n}\}$ with distinct terms, given that $a_{3}a_{8}=3a_{11}$, $S_{3}=9$.
1. Find the general term formula for the sequence $\{a_{n}\}$.
2. If $b_{n}= \frac {1}{ \sqrt {a_{n}}+ \sqrt {a_{n+1}}}$, and the sum of the first $n$ terms of the sequence $\{b_{n}\}$ is $T_{n}$, find the minimum value of $\frac {a_{n+1}}{T_{n}}$. | 4 \sqrt {2} | 0 |
27,922 | If $3 \in \{a, a^2 - 2a\}$, then the value of the real number $a$ is __________. | -1 | 3.90625 |
27,923 | Consider the ellipse $C\_1$: $\frac{x^{2}}{2}+y^{2}=1$ and the ellipse $C\_2$: $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ ($a > b > 0$). One focus of ellipse $C\_2$ has the coordinates $(\sqrt{5},0)$. The line $l$ with slope $1$ intersects ellipse $C\_2$ at points $A$ and $B$, and the midpoint $H$ of the line segment $AB$ has the coordinates $(2,-1)$.
1. Find the equation of ellipse $C\_2$.
2. Let $P$ be a point on the ellipse $C\_2$, and let $M$ and $N$ be points on the ellipse $C\_1$ such that $\overrightarrow{OP}=\overrightarrow{OM}+2\overrightarrow{ON}$. Determine whether the product of the slopes of lines $OM$ and $ON$ is constant. If it is, find the constant value; if not, explain the reason. | -\frac{1}{2} | 60.15625 |
27,924 | Let \( x \) and \( y \) be real numbers, \( y > x > 0 \), such that
\[ \frac{x}{y} + \frac{y}{x} = 4. \]
Find the value of
\[ \frac{x + y}{x - y}. \] | \sqrt{3} | 7.8125 |
27,925 | Among the following four propositions:
(1) The domain of the function $y=\tan (x+ \frac {π}{4})$ is $\{x|x\neq \frac {π}{4}+kπ,k\in Z\}$;
(2) Given $\sin α= \frac {1}{2}$, and $α\in[0,2π]$, the set of values for $α$ is $\{\frac {π}{6}\}$;
(3) The graph of the function $f(x)=\sin 2x+a\cos 2x$ is symmetric about the line $x=- \frac {π}{8}$, then the value of $a$ equals $(-1)$;
(4) The minimum value of the function $y=\cos ^{2}x+\sin x$ is $(-1)$.
Fill in the sequence number of the propositions you believe are correct on the line ___. | (1)(3)(4) | 0 |
27,926 | Given the digits $1$ through $7$ , one can form $7!=5040$ numbers by forming different permutations of the $7$ digits (for example, $1234567$ and $6321475$ are two such permutations). If the $5040$ numbers are then placed in ascending order, what is the $2013^{\text{th}}$ number? | 3546127 | 0 |
27,927 | Six chairs sit in a row. Six people randomly seat themselves in the chairs. Each person randomly chooses either to set their feet on the floor, to cross their legs to the right, or to cross their legs to the left. There is only a problem if two people sitting next to each other have the person on the right crossing their legs to the left and the person on the left crossing their legs to the right. The probability that this will **not** happen is given by $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$ . | 1106 | 1.5625 |
27,928 | The spinner shown is divided into 6 sections of equal size. Determine the probability of landing on a section that contains the letter Q using this spinner. | \frac{2}{6} | 0 |
27,929 | 1. Calculate the value of the following expression:
$(1)(2\frac{7}{9})^{\frac{1}{2}} - (2\sqrt{3} - \pi)^{0} - (2\frac{10}{27})^{-\frac{2}{3}} + 0.25^{-\frac{3}{2}}$
2. Given that $x + x^{-1} = 4 (0 < x < 1)$, find the value of ${x^{\frac{1}{2}}} + {x^{-\frac{1}{2}}}$. | \frac{389}{48} | 0 |
27,930 | Using the digits $0$, $1$, $2$, $3$, $4$ to form a four-digit number without repeating any digit, determine the total number of four-digit numbers less than $2340$. | 40 | 10.9375 |
27,931 | Please fold a long rope in half, then fold it in half again along the middle of the folded rope, and continue to fold it in half 5 times in total. Finally, cut the rope along the middle after it has been folded 5 times. At this point, the rope will be cut into ___ segments. | 33 | 3.90625 |
27,932 | Let \( ABC \) be a triangle such that \( AB = 2 \), \( CA = 3 \), and \( BC = 4 \). A semicircle with its diameter on \(\overline{BC}\) is tangent to \(\overline{AB}\) and \(\overline{AC}\). Compute the area of the semicircle. | \frac{27 \pi}{40} | 3.125 |
27,933 |
Consider all possible quadratic polynomials $x^2 + px + q$ with a positive discriminant, where the coefficients $p$ and $q$ are integers divisible by 5. Find the largest natural number $n$ such that for any polynomial with the described properties, the sum of the hundredth powers of the roots is an integer divisible by $5^n$. | 50 | 4.6875 |
27,934 | A motorist left point A for point D, covering a distance of 100 km. The road from A to D passes through points B and C. At point B, the GPS indicated that 30 minutes of travel time remained, and the motorist immediately reduced speed by 10 km/h. At point C, the GPS indicated that 20 km of travel distance remained, and the motorist immediately reduced speed by another 10 km/h. (The GPS determines the remaining time based on the current speed of travel.) Determine the initial speed of the car if it is known that the journey from B to C took 5 minutes longer than the journey from C to D. | 100 | 1.5625 |
27,935 | Let $ABC$ be a right triangle with $m(\widehat{A})=90^\circ$ . Let $APQR$ be a square with area $9$ such that $P\in [AC]$ , $Q\in [BC]$ , $R\in [AB]$ . Let $KLMN$ be a square with area $8$ such that $N,K\in [BC]$ , $M\in [AB]$ , and $L\in [AC]$ . What is $|AB|+|AC|$ ? | 12 | 13.28125 |
27,936 | Calculate $(2.1)(50.5 + 0.15)$ after increasing $50.5$ by $5\%$. What is the product closest to? | 112 | 10.15625 |
27,937 | Find the number of ordered quintuples $(a,b,c,d,e)$ of nonnegative real numbers such that:
\begin{align*}
a^2 + b^2 + c^2 + d^2 + e^2 &= 5, \\
(a + b + c + d + e)(a^3 + b^3 + c^3 + d^3 + e^3) &= 25.
\end{align*} | 31 | 0 |
27,938 | Find the largest integer $n$ for which $2^n$ divides \[ \binom 21 \binom 42 \binom 63 \dots \binom {128}{64}. \]*Proposed by Evan Chen* | 193 | 82.03125 |
27,939 | In rectangle \(ABCD\), \(\overline{AB}=30\) and \(\overline{BC}=15\). Let \(E\) be a point on \(\overline{CD}\) such that \(\angle CBE=45^\circ\) and \(\triangle ABE\) is isosceles. Find \(\overline{AE}.\) | 15 | 12.5 |
27,940 | What is the total number of digits used when the first 2500 positive even integers are written? | 9449 | 4.6875 |
27,941 | Given an ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, where the upper vertex of $C$ is $A$, and the two foci are $F_{1}$ and $F_{2}$, with an eccentricity of $\frac{1}{2}$. A line passing through $F_{1}$ and perpendicular to $AF_{2}$ intersects $C$ at points $D$ and $E$, where $|DE| = 6$. Find the perimeter of $\triangle ADE$. | 13 | 14.84375 |
27,942 | What is the smallest positive integer with exactly 16 positive divisors? | 120 | 90.625 |
27,943 | What is the least positive integer with exactly $12$ positive factors? | 72 | 0 |
27,944 | Students are in classroom with $n$ rows. In each row there are $m$ tables. It's given that $m,n \geq 3$ . At each table there is exactly one student. We call neighbours of the student students sitting one place right, left to him, in front of him and behind him. Each student shook hands with his neighbours. In the end there were $252$ handshakes. How many students were in the classroom? | 72 | 3.125 |
27,945 | In $\triangle ABC$, $\angle A = 60^{\circ}$ and $AB > AC$. Point $O$ is the circumcenter, and the altitudes $BE$ and $CF$ intersect at point $H$. Points $M$ and $N$ lie on segments $BH$ and $HF$, respectively, such that $BM = CN$. Find the value of $\frac{MH + NH}{OH}$. | \sqrt{3} | 0.78125 |
27,946 | A circle is inscribed in quadrilateral $EFGH$, tangent to $\overline{EF}$ at $R$ and to $\overline{GH}$ at $S$. Given that $ER=25$, $RF=35$, $GS=40$, and $SH=20$, and that the circle is also tangent to $\overline{EH}$ at $T$ with $ET=45$, find the square of the radius of the circle. | 3600 | 0 |
27,947 | There is a beach soccer tournament with 17 teams, where each team plays against every other team exactly once. A team earns 3 points for a win in regular time, 2 points for a win in extra time, and 1 point for a win in a penalty shootout. The losing team earns no points. What is the maximum number of teams that can each earn exactly 5 points? | 11 | 3.90625 |
27,948 | Given a structure formed by joining eight unit cubes where one cube is at the center, and each face of the central cube is shared with one additional cube, calculate the ratio of the volume to the surface area in cubic units to square units. | \frac{4}{15} | 8.59375 |
27,949 | Given a right rectangular prism $B$ with edge lengths $1,$ $3,$ and $4$, and the set $S(r)$ of points within a distance $r$ of some point in $B$, express the volume of $S(r)$ as $ar^{3} + br^{2} + cr + d$ and determine the ratio $\frac{bc}{ad}$. | 19 | 8.59375 |
27,950 | Each row of a $24 \times 8$ table contains some permutation of the numbers $1, 2, \cdots , 8.$ In each column the numbers are multiplied. What is the minimum possible sum of all the products?
*(C. Wu)* | 8 * (8!)^3 | 0 |
27,951 | Santa Claus has 36 identical gifts divided into 8 bags. The number of gifts in each of the 8 bags is at least 1 and all are different. You need to select some of these bags to evenly distribute all their gifts to 8 children, such that all gifts are distributed completely (each child receives at least one gift). How many different selections are there? | 31 | 25 |
27,952 | Five persons wearing badges with numbers $1, 2, 3, 4, 5$ are seated on $5$ chairs around a circular table. In how many ways can they be seated so that no two persons whose badges have consecutive numbers are seated next to each other? (Two arrangements obtained by rotation around the table are considered different) | 10 | 72.65625 |
27,953 | What is the minimum number of shots required in the game "Battleship" on a 7x7 board to definitely hit a four-cell battleship (which consists of four consecutive cells in a single row)? | 12 | 4.6875 |
27,954 | In triangle $XYZ$, $XY = 15$, $XZ = 17$, and $YZ = 24$. The medians $XM$, $YN$, and $ZL$ of triangle $XYZ$ intersect at the centroid $G$. Let $Q$ be the foot of the altitude from $G$ to $YZ$. Find $GQ$. | 3.5 | 0 |
27,955 | Let $\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{abcd}$ where $a, b, c, d$ are distinct digits. Find the sum of the elements of $\mathcal{T}.$ | 2520 | 9.375 |
27,956 | Find the smallest positive integer $k$ such that, for any subset $A$ of $S=\{1,2,\ldots,2012\}$ with $|A|=k$ , there exist three elements $x,y,z$ in $A$ such that $x=a+b$ , $y=b+c$ , $z=c+a$ , where $a,b,c$ are in $S$ and are distinct integers.
*Proposed by Huawei Zhu* | 1008 | 8.59375 |
27,957 | Five squares and two right-angled triangles are positioned as shown. The areas of three squares are \(3 \, \mathrm{m}^{2}, 7 \, \mathrm{m}^{2}\), and \(22 \, \mathrm{m}^{2}\). What is the area, in \(\mathrm{m}^{2}\), of the square with the question mark?
A) 18
B) 19
C) 20
D) 21
E) 22 | 18 | 10.15625 |
27,958 | In the diagram, triangle \(ABC\) is isosceles, with \(AB = AC\). If \(\angle ABC = 50^\circ\) and \(\angle DAC = 60^\circ\), the value of \(x\) is: | 70 | 10.9375 |
27,959 | How many of the smallest 2401 positive integers written in base 7 include the digits 4, 5, or 6? | 2146 | 4.6875 |
27,960 | The 31st World University Summer Games will be held in Chengdu, Sichuan from July 28th to August 8th, 2023. A company decided to evaluate a certain product under its umbrella for bidding for related endorsement activities. The original selling price of the product was $25 per unit, with an annual sales volume of 80,000 units.
$(1)$ According to market research, if the price is increased by $1, the sales volume will decrease by 2,000 units. To ensure that the total revenue from sales is not less than the original revenue, what is the maximum price per unit that the product can be priced at?
$(2)$ To seize this opportunity, expand the influence of the product, and increase the annual sales volume, the company decided to immediately carry out a comprehensive technological innovation and marketing strategy reform on the product, and increase the price to $x per unit. The company plans to invest $\frac{1}{6}(x^{2}-600)$ million as a technological innovation cost, $50$ million as fixed advertising costs, and $\frac{x}{5}$ million as variable advertising costs. What is the minimum sales volume $a$ that the product should reach after the reform in order to ensure that the sales revenue after the reform is not less than the original revenue plus the total investment? Also, determine the price per unit of the product at this point. | 30 | 6.25 |
27,961 | In tetrahedron \( SABC \), the circumradius of triangles \( \triangle SAB \), \( \triangle SBC \), and \( \triangle SCA \) are all 108. The center of the inscribed sphere is \( I \) and \( SI = 125 \). Let \( R \) be the circumradius of \( \triangle ABC \). If \( R \) can be expressed as \( \sqrt{\frac{m}{n}} \) (where \( m \) and \( n \) are positive integers and \(\gcd(m, n) = 1\)), what is \( m+n \)? | 11665 | 1.5625 |
27,962 | Consider a square pyramid $S-ABCD$ with a height of $h$. The base $ABCD$ is a square with side length 1. Points $S$, $A$, $B$, $C$, and $D$ all lie on the surface of a sphere with radius 1. The task is to find the distance between the center of the base $ABCD$ and the vertex $S$. | \frac{\sqrt{2}}{2} | 27.34375 |
27,963 | In the diagram below, $ABCD$ is a trapezoid such that $\overline{AB}\parallel \overline{CD}$ and $\overline{AC}\perp\overline{CD}$. If $CD = 15$, $\tan C = 2$, and $\tan B = \frac{3}{2}$, then what is $BD$? | 10\sqrt{13} | 10.15625 |
27,964 | Given that the discrete random variable \\(\xi\\) follows a normal distribution \\(N \sim (2,1)\\), and \\(P(\xi < 3) = 0.968\\), then \\(P(1 < \xi < 3) =\\) \_\_\_\_\_\_. | 0.936 | 40.625 |
27,965 | Juan rolls a fair regular decagonal die marked with numbers from 1 to 10. Then Amal rolls a fair eight-sided die marked with numbers from 1 to 8. What is the probability that the product of the two rolls is a multiple of 4? | \frac{19}{40} | 8.59375 |
27,966 | What is the number of square units in the area of trapezoid EFGH with vertices E(0,0), F(0,3), G(5,3), and H(3,0)? | 7.5 | 41.40625 |
27,967 | In a tetrahedron \(ABCD\), \(AD = \sqrt{2}\) and all other edge lengths are 1. Find the shortest path distance from the midpoint \(M\) of edge \(AB\) to the midpoint \(N\) of edge \(CD\) along the surface of the tetrahedron. | \frac{\sqrt{3}}{2} | 20.3125 |
27,968 | Find the probability \( P(B^c | A^c) \) given the probabilities:
\[ P(A \cap B) = 0.72, \quad P(A \cap \bar{B}) = 0.18 \] | 0.90 | 0 |
27,969 | How many of the natural numbers from 1 to 800, inclusive, contain the digit 7 at least once? | 152 | 0 |
27,970 | How many of the 512 smallest positive integers written in base 8 use 5 or 6 (or both) as a digit? | 296 | 0.78125 |
27,971 | A community organization begins with twenty members, among which five are leaders. The leaders are replaced annually. Each remaining member persuades three new members to join the organization every year. Additionally, five new leaders are elected from outside the community each year. Determine the total number of members in the community five years later. | 15365 | 10.15625 |
27,972 | The function $f(x)=1+x- \frac {x^{2}}{2}+ \frac {x^{3}}{3}$, $g(x)=1-x+ \frac {x^{2}}{2}- \frac {x^{3}}{3}$, if the function $F(x)=f(x+3)g(x-4)$, and the zeros of the function $F(x)$ are all within $[a,b]$ $(a < b$, $a$, $b\in\mathbb{Z})$, then the minimum value of $b-a$ is \_\_\_\_\_\_. | 10 | 17.1875 |
27,973 | Let ellipse $C$:$\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\left(a \gt b \gt 0\right)$ have foci $F_{1}(-c,0)$ and $F_{2}(c,0)$. Point $P$ is the intersection point of $C$ and the circle $x^{2}+y^{2}=c^{2}$. The bisector of $\angle PF_{1}F_{2}$ intersects $PF_{2}$ at $Q$. If $|PQ|=\frac{1}{2}|QF_{2}|$, then find the eccentricity of ellipse $C$. | \sqrt{3}-1 | 0 |
27,974 | In $\triangle ABC,$ $AB=20$, $AC=24$, and $BC=18$. Points $D,E,$ and $F$ are on sides $\overline{AB},$ $\overline{BC},$ and $\overline{AC},$ respectively. Angle $\angle BAC = 60^\circ$ and $\overline{DE}$ and $\overline{EF}$ are parallel to $\overline{AC}$ and $\overline{AB},$ respectively. What is the perimeter of parallelogram $ADEF$? | 44 | 18.75 |
27,975 | What is the largest four-digit negative integer congruent to $2 \pmod{25}$? | -1023 | 20.3125 |
27,976 | The least common multiple of $a$ and $b$ is $18$, and the least common multiple of $b$ and $c$ is $20$. Find the least possible value of the least common multiple of $a$ and $c$. | 90 | 27.34375 |
27,977 | Given an ellipse $\dfrac {x^{2}}{a^{2}}+ \dfrac {y^{2}}{b^{2}}=1$ with its left and right foci being $F_{1}$ and $F_{2}$ respectively, a perpendicular line to the x-axis through the left focus $F_{1}(-2,0)$ intersects the ellipse at points $P$ and $Q$. The line $PF_{2}$ intersects the y-axis at $E(0, \dfrac {3}{2})$. $A$ and $B$ are points on the ellipse located on either side of $PQ$.
- (I) Find the eccentricity $e$ and the standard equation of the ellipse;
- (II) When $\angle APQ=\angle BPQ$, is the slope $K_{AB}$ of line $AB$ a constant value? If so, find this constant value; if not, explain why. | - \dfrac {1}{2} | 0 |
27,978 | Bryan has some stamps of 3 cents, 4 cents, and 6 cents. What is the least number of stamps he can combine so the value of the stamps is 50 cents? | 10 | 53.90625 |
27,979 | At 17:00, the speed of a racing car was 30 km/h. Every subsequent 5 minutes, the speed increased by 6 km/h. Determine the distance traveled by the car from 17:00 to 20:00 on the same day. | 425.5 | 0 |
27,980 | Given that the plane vectors $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ satisfy $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=2$, and for all $t\in \mathbb{R}$, $|\overrightarrow{b}+t\overrightarrow{a}| \geq |\overrightarrow{b}-\overrightarrow{a}|$ always holds, determine the angle between $2\overrightarrow{a}-\overrightarrow{b}$ and $\overrightarrow{b}$. | \frac{2\pi}{3} | 89.0625 |
27,981 | Given vectors $\overrightarrow{a}=(1,3)$ and $\overrightarrow{b}=(-2,4)$, calculate the projection of $\overrightarrow{a}$ in the direction of $\overrightarrow{b}$. | \sqrt{5} | 0 |
27,982 | For a nonnegative integer $n$, let $r_7(n)$ represent the remainder when $n$ is divided by $7$. Determine the $15^{\text{th}}$ entry in an ordered list of all nonnegative integers $n$ that satisfy $$r_7(3n)\le 3.$$ | 22 | 21.875 |
27,983 | Use the Horner's method to compute the value of the polynomial $f(x)=0.5x^{5}+4x^{4}-3x^{2}+x-1$ when $x=3$, and determine the first operation to perform. | 5.5 | 0.78125 |
27,984 | Given two squares $ABCD$ and $DCFE$ with side lengths of $1$, where the planes they reside in are perpendicular to each other. Points $P$ and $Q$ are moving points on line segments $BC$ and $DE$ (including endpoints), with $PQ = \sqrt{2}$. Let the trajectory of the midpoint of line segment $PQ$ be curve $\mathcal{A}$. Determine the length of $\mathcal{A}$. | \frac{\pi}{4} | 4.6875 |
27,985 | Let \( f : \mathbb{C} \to \mathbb{C} \) be defined by \( f(z) = z^2 - 2iz + 2 \). Determine how many complex numbers \( z \) exist such that \( \text{Im}(z) > 0 \) and both the real and the imaginary parts of \( f(z) \) are integers within \( |a|, |b| \leq 5 \). | 110 | 0 |
27,986 | What is the sum of the squares of the lengths of the medians of a triangle whose side lengths are $13, 13,$ and $10$? | 432 | 1.5625 |
27,987 | Anya, Vanya, Danya, and Tanya collected apples. Each of them collected a whole number percentage from the total number of apples, and all these numbers are distinct and more than zero. Then Tanya, who collected the most apples, ate her apples. After that, each of the remaining kids still had a whole percentage of the remaining apples. What is the minimum number of apples that could have been collected? | 20 | 0 |
27,988 | A triangle has vertices \( P=(-7,4) \), \( Q=(-14,-20) \), and \( R=(2,-8) \). The equation of the bisector of \( \angle P \) can be written in the form \( ax+2y+c=0 \). Find \( a+c \). | 40 | 0 |
27,989 | The increasing sequence consists of all those positive integers which are either powers of 2, powers of 3, or sums of distinct powers of 2 and 3. Find the $50^{\rm th}$ term of this sequence. | 57 | 7.8125 |
27,990 | Find the number of six-digit palindromes. | 900 | 91.40625 |
27,991 | \(ABCD\) is a square with sides \(8 \, \text{cm}\). \(M\) is a point on \(CB\) such that \(CM = 2 \, \text{cm}\). If \(N\) is a variable point on the diagonal \(DB\), find the least value of \(CN + MN\). | 10 | 17.1875 |
27,992 | There is a positive integer s such that there are s solutions to the equation $64sin^2(2x)+tan^2(x)+cot^2(x)=46$ in the interval $(0,\frac{\pi}{2})$ all of the form $\frac{m_k}{n_k}\pi$ where $m_k$ and $n_k$ are relatively prime positive integers, for $k = 1, 2, 3, . . . , s$ . Find $(m_1 + n_1) + (m_2 + n_2) + (m_3 + n_3) + · · · + (m_s + n_s)$ . | 100 | 0 |
27,993 | The first number in the following sequence is $1$ . It is followed by two $1$ 's and two $2$ 's. This is followed by three $1$ 's, three $2$ 's, and three $3$ 's. The sequence continues in this fashion.
\[1,1,1,2,2,1,1,1,2,2,2,3,3,3,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,\dots.\]
Find the $2014$ th number in this sequence. | 13 | 0.78125 |
27,994 | In triangle $ABC$, the altitude and the median from vertex $C$ each divide the angle $ACB$ into three equal parts. Determine the ratio of the sides of the triangle. | 2 : \sqrt{3} : 1 | 1.5625 |
27,995 | Given that vectors $\overrightarrow{OA}$ and $\overrightarrow{OB}$ are perpendicular and $|\overrightarrow{OA}| = |\overrightarrow{OB}| = 24$. If $t \in [0,1]$, the minimum value of $|t \overrightarrow{AB} - \overrightarrow{AO}| + \left|\frac{5}{12} \overrightarrow{BO} - (1-t) \overrightarrow{BA}\right|$ is: | 26 | 12.5 |
27,996 | From a \(6 \times 6\) square grid, gray triangles were cut out. What is the area of the remaining shape? The length of each side of the cells is 1 cm. Provide your answer in square centimeters. | 27 | 46.875 |
27,997 | Given a cube, calculate the total number of pairs of diagonals on its six faces, where the angle formed by each pair is $60^{\circ}$. | 48 | 5.46875 |
27,998 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. The vectors $m=(\cos (A-B),\sin (A-B))$, $n=(\cos B,-\sin B)$, and $m\cdot n=-\frac{3}{5}$.
(1) Find the value of $\sin A$.
(2) If $a=4\sqrt{2}$, $b=5$, find the measure of angle $B$ and the projection of vector $\overrightarrow{BA}$ onto the direction of $\overrightarrow{BC}$. | \frac{\sqrt{2}}{2} | 3.125 |
27,999 | An ellipse $\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1 (a>b>0)$ has its two foci and the endpoints of its minor axis all lying on the circle $x^{2}+y^{2}=1$. A line $l$ (not perpendicular to the x-axis) passing through the right focus intersects the ellipse at points A and B. The perpendicular bisector of segment AB intersects the x-axis at point P.
(1) Find the equation of the ellipse;
(2) Investigate whether the ratio $\frac {|AB|}{|PF|}$ is a constant value. If it is, find this constant value. If not, explain why. | 2 \sqrt {2} | 0 |
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