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31,700 | The Mathematics College Entrance Examination scores distribution $\xi$ closely follows the normal distribution $N(100, 5^2)$, and $P(\xi < 110) = 0.96$. Find the value of $P(90 < \xi < 100)$. | 0.46 | 67.1875 |
31,701 | The leadership team of a sports event needs to select 4 volunteers from 5 candidates named A, B, C, D, and E to undertake four different tasks: translation, tour guiding, protocol, and driving. If A and B can only undertake the first three tasks, while the other three candidates can undertake all four tasks, determine the number of different selection schemes. | 72 | 8.59375 |
31,702 | Given a cyclic quadrilateral \(ABCD\) with a circumradius of \(200\sqrt{2}\) and sides \(AB = BC = CD = 200\). Find the length of side \(AD\). | 500 | 13.28125 |
31,703 | There are 3012 positive numbers with both their sum and the sum of their reciprocals equal to 3013. Let $x$ be one of these numbers. Find the maximum value of $x + \frac{1}{x}.$ | \frac{12052}{3013} | 0 |
31,704 | Let $a,$ $b,$ $c$ be nonzero real numbers such that $a + b + c = 0,$ and $ab + ac + bc \neq 0.$ Find all possible values of
\[
\frac{a^7 + b^7 + c^7}{abc (ab + ac + bc)}.
\] | -7 | 21.875 |
31,705 | Given that the line $y=x-2$ intersects the hyperbola $C: \frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1 (a>0, b>0)$ at the intersection points $A$ and $B (not coincident)$. The perpendicular bisector of line segment $AB$ passes through the point $(4,0)$. The eccentricity of the hyperbola $C$ is ______. | \frac{2\sqrt{3}}{3} | 37.5 |
31,706 | Adam and Sarah start on bicycle trips from the same point at the same time. Adam travels north at 10 mph and Sarah travels west at 5 mph. After how many hours are they 85 miles apart? | 7.6 | 12.5 |
31,707 | Harriet lives in a large family with 4 sisters and 6 brothers, and she has a cousin Jerry who lives with them. Determine the product of the number of sisters and brothers Jerry has in the house. | 24 | 13.28125 |
31,708 | The diagram shows twenty congruent circles arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the rectangle as shown in the diagram. The ratio of the longer dimension of the rectangle to the shorter dimension can be written as $\frac{1}{2}\left(\sqrt{p}-q\right)$ , where $p$ and $q$ are positive integers. Find $p+q$ . [asy] size(250);real x=sqrt(3); int i; draw(origin--(14,0)--(14,2+2x)--(0,2+2x)--cycle); for(i=0; i<7; i=i+1) { draw(Circle((2*i+1,1), 1)^^Circle((2*i+1,1+2x), 1)); } for(i=0; i<6; i=i+1) { draw(Circle((2*i+2,1+x), 1)); } [/asy] | 154 | 0 |
31,709 | Given four points $O,\ A,\ B,\ C$ on a plane such that $OA=4,\ OB=3,\ OC=2,\ \overrightarrow{OB}\cdot \overrightarrow{OC}=3.$ Find the maximum area of $\triangle{ABC}$ . | 2\sqrt{7} + \frac{3\sqrt{3}}{2} | 1.5625 |
31,710 | Compute the sum of the series:
\[ 5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5)))))))))) \] | 305175780 | 0.78125 |
31,711 | There are six students with unique integer scores in a mathematics exam. The average score is 92.5, the highest score is 99, and the lowest score is 76. What is the minimum score of the student who ranks 3rd from the highest? | 95 | 21.875 |
31,712 | Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a half-dollar. What is the probability that at least 30 cents worth of coins come up heads? | \frac{9}{16} | 1.5625 |
31,713 | Points \( C \) and \( D \) have the same \( y \)-coordinate of 25, but different \( x \)-coordinates. What is the difference between the slope and the \( y \)-intercept of the line containing both points? | -25 | 75 |
31,714 | A company decides to increase the price of a product by 20%. If they aim to increase their total income by 10% despite the decrease in demand, by what proportion must the demand decrease to meet this new income goal? | \frac{1}{12} | 75 |
31,715 | In the coordinate plane, points $D(3, 15)$, $E(15, 0)$, and $F(0, t)$ form a triangle $\triangle DEF$. If the area of $\triangle DEF$ is 50, what is the value of $t$? | \frac{125}{12} | 22.65625 |
31,716 | The faces of a cubical die are marked with the numbers $1$, $2$, $3$, $3$, $4$, and $4$. Another die's faces are marked with $2$, $2$, $5$, $6$, $7$, and $8$. Find the probability that the sum of the top two numbers will be $7$, $9$, or $11$.
A) $\frac{1}{18}$
B) $\frac{1}{9}$
C) $\frac{1}{3}$
D) $\frac{4}{9}$
E) $\frac{1}{2}$ | \frac{4}{9} | 71.875 |
31,717 | Given that the angles of a triangle at points \( A, B \), and \( C \) are such that \( \angle ABC = 50^\circ \) and \( \angle ACB = 30^\circ \), calculate the value of \( x \). | 80 | 0 |
31,718 | In $\triangle ABC$, $AB=\sqrt{5}$, $BC=1$, and $AC=2$. $I$ is the incenter of $\triangle ABC$ and the circumcircle of $\triangle IBC$ intersects $AB$ at $P$. Find $BP$. | \sqrt{5} - 2 | 0 |
31,719 | Find all numbers that can be expressed in exactly $2010$ different ways as the sum of powers of two with non-negative exponents, each power appearing as a summand at most three times. A sum can also be made from just one summand. | 2010 | 6.25 |
31,720 | In the Cartesian coordinate system $xOy$, the parametric equations of curve $C_{1}$ are $\left\{{\begin{array}{l}{x=1+t,}\\{y=\sqrt{3}t}\end{array}}\right.$ (where $t$ is the parameter), and the parametric equations of curve $C_{2}$ are $\left\{{\begin{array}{l}{x=\sqrt{2}(cosθ+sinθ),}\\{y=cosθ-sinθ}\end{array}}\right.$ (where $θ$ is the parameter). <br/> $(1)$ Convert the parametric equations of curve $C_{2}$ into the standard form equation. <br/> $(2)$ Given the point $M(1,0)$, curves $C_{1}$ and $C_{2}$ intersect at points $A$ and $B$, find $|{\frac{1}{{|MA|}}-\frac{1}{{|MB|}}}|$. | \frac{1}{3} | 13.28125 |
31,721 | From a group of $3$ orthopedic surgeons, $4$ neurosurgeons, and $5$ internists, a medical disaster relief team of $5$ people is to be formed. How many different ways can the team be selected such that there is at least one person from each specialty? | 590 | 45.3125 |
31,722 | Place the numbers $1, 2, 3, \cdots, 2001$ in a clockwise direction on a circle. First, eliminate the number 2. Then proceed to eliminate every second number in a clockwise direction until only one number remains. What is the last remaining number? | 1955 | 25 |
31,723 | In the positive term geometric sequence $\\{a_n\\}$, $a\_$ and $a\_{48}$ are the two roots of the equation $2x^2 - 7x + 6 = 0$. Find the value of $a\_{1} \cdot a\_{2} \cdot a\_{25} \cdot a\_{48} \cdot a\_{49}$. | 9\sqrt{3} | 12.5 |
31,724 | In triangle $XYZ$, where $\angle X = 90^\circ$, $YZ = 20$, and $\tan Z = 3\cos Y$. What is the length of $XY$? | \frac{40\sqrt{2}}{3} | 62.5 |
31,725 | Four numbers, 2613, 2243, 1503, and 985, when divided by the same positive integer, yield the same remainder (which is not zero). Find the divisor and the remainder. | 23 | 3.125 |
31,726 | Given a rectangular grid measuring 8 by 6, there are $48$ grid points, including those on the edges. Point $P$ is placed at the center of the rectangle. Find the probability that the line $PQ$ is a line of symmetry of the rectangle, given that point $Q$ is randomly selected from the other $47$ points. | \frac{12}{47} | 19.53125 |
31,727 | In the xy-plane with a rectangular coordinate system, let vector $\overrightarrow {a}$ = (cosα, sinα) and vector $\overrightarrow {b}$ = (sin(α + π/6), cos(α + π/6)), where 0 < α < π/2.
(1) If $\overrightarrow {a}$ is parallel to $\overrightarrow {b}$, find the value of α.
(2) If tan2α = -1/7, find the value of the dot product $\overrightarrow {a}$ • $\overrightarrow {b}$. | \frac{\sqrt{6} - 7\sqrt{2}}{20} | 8.59375 |
31,728 | Calculate:<br/>$(1)3-\left(-2\right)$;<br/>$(2)\left(-4\right)\times \left(-3\right)$;<br/>$(3)0\div \left(-3\right)$;<br/>$(4)|-12|+\left(-4\right)$;<br/>$(5)\left(+3\right)-14-\left(-5\right)+\left(-16\right)$;<br/>$(6)(-5)÷(-\frac{1}{5})×(-5)$;<br/>$(7)-24×(-\frac{5}{6}+\frac{3}{8}-\frac{1}{12})$;<br/>$(8)3\times \left(-4\right)+18\div \left(-6\right)-\left(-2\right)$;<br/>$(9)(-99\frac{15}{16})×4$. | -399\frac{3}{4} | 0.78125 |
31,729 | A regular $2018$ -gon is inscribed in a circle. The numbers $1, 2, ..., 2018$ are arranged on the vertices of the $2018$ -gon, with each vertex having one number on it, such that the sum of any $2$ neighboring numbers ( $2$ numbers are neighboring if the vertices they are on lie on a side of the polygon) equals the sum of the $2$ numbers that are on the antipodes of those $2$ vertices (with respect to the given circle). Determine the number of different arrangements of the numbers.
(Two arrangements are identical if you can get from one of them to the other by rotating around the center of the circle). | 2 \times 1008! | 0 |
31,730 | Three concentric circles have radii of 1, 2, and 3 units, respectively. Points are chosen on each of these circles such that they are the vertices of an equilateral triangle. What can be the side length of this equilateral triangle? | \sqrt{7} | 51.5625 |
31,731 | What is the largest quotient that can be formed using two numbers chosen from the set $\{-30, -6, -1, 3, 5, 20\}$, where one of the numbers must be negative? | -0.05 | 0 |
31,732 | Three congruent isosceles triangles $DAO$, $AOB$, and $OBC$ have $AD=AO=OB=BC=12$ and $AB=DO=OC=16$. These triangles are arranged to form trapezoid $ABCD$. Point $P$ is on side $AB$ so that $OP$ is perpendicular to $AB$. Points $X$ and $Y$ are the midpoints of $AD$ and $BC$, respectively. When $X$ and $Y$ are joined, the trapezoid is divided into two smaller trapezoids. Find the ratio of the area of trapezoid $ABYX$ to the area of trapezoid $XYCD$ in simplified form and find $p+q$, where the ratio is $p:q$. | 12 | 8.59375 |
31,733 | Given the linear function y=kx+b, where k and b are constants, and the table of function values, determine the incorrect function value. | 12 | 2.34375 |
31,734 | Emily paid for a $\$2$ sandwich using 50 coins consisting of pennies, nickels, and dimes, and received no change. How many dimes did Emily use? | 10 | 25.78125 |
31,735 | A regular hexagon `LMNOPQ` has sides of length 4. Find the area of triangle `LNP`. Express your answer in simplest radical form. | 8\sqrt{3} | 16.40625 |
31,736 | In the Cartesian coordinate system $xOy$, it is known that the circle $C: x^{2} + y^{2} + 8x - m + 1 = 0$ intersects with the line $x + \sqrt{2}y + 1 = 0$ at points $A$ and $B$. If $\triangle ABC$ is an equilateral triangle, then the value of the real number $m$ is. | -11 | 58.59375 |
31,737 | Given a moving point $A$ on the curve $y=x^{2}$, let $m$ be the tangent line to the curve at point $A$. Let $n$ be a line passing through point $A$, perpendicular to line $m$, and intersecting the curve at another point $B$. Determine the minimum length of the line segment $AB$. | \frac{3\sqrt{3}}{2} | 33.59375 |
31,738 | There are 99 positive integers, and their sum is 101101. The greatest possible value of the greatest common divisor of these 99 positive integers is: | 101 | 46.09375 |
31,739 | The sum of 100 numbers is 1000. The largest of these numbers was doubled, while another number was decreased by 10. After these actions, the sum of all numbers remained unchanged. Find the smallest of the original numbers. | 10 | 20.3125 |
31,740 | A school club buys 1200 candy bars at a price of four for $3 dollars, and sells all the candy bars at a price of three for $2 dollars, or five for $3 dollars if more than 50 are bought at once. Calculate their total profit in dollars. | -100 | 3.90625 |
31,741 | Given that the function $f(x)$ and its derivative $f'(x)$ have a domain of $R$, and $f(x+2)$ is an odd function, ${f'}(2-x)+{f'}(x)=2$, ${f'}(2)=2$, then $\sum_{i=1}^{50}{{f'}}(i)=$____. | 51 | 11.71875 |
31,742 | Let $(1+2x)^2(1-x)^5 = a + a_1x + a_2x^2 + \ldots + a_7x^7$, then $a_1 - a_2 + a_3 - a_4 + a_5 - a_6 + a_7 =$ ? | -31 | 3.125 |
31,743 | Let $x$ be the number of students in Danny's high school. If Maria's high school has $4$ times as many students as Danny's high school, then Maria's high school has $4x$ students. The difference between the number of students in the two high schools is $1800$, so we have the equation $4x-x=1800$. | 2400 | 4.6875 |
31,744 | Given a sequence \( A = (a_1, a_2, \cdots, a_{10}) \) that satisfies the following four conditions:
1. \( a_1, a_2, \cdots, a_{10} \) is a permutation of \{1, 2, \cdots, 10\};
2. \( a_1 < a_2, a_3 < a_4, a_5 < a_6, a_7 < a_8, a_9 < a_{10} \);
3. \( a_2 > a_3, a_4 > a_5, a_6 > a_7, a_8 > a_9 \);
4. There does not exist \( 1 \leq i < j < k \leq 10 \) such that \( a_i < a_k < a_j \).
Find the number of such sequences \( A \). | 42 | 48.4375 |
31,745 | Find the largest real number $k$ , such that for any positive real numbers $a,b$ , $$ (a+b)(ab+1)(b+1)\geq kab^2 $$ | 27/4 | 0 |
31,746 | Twelve congruent rectangles are placed together to make a rectangle $PQRS$. What is the ratio $PQ: QR$?
A) $2: 3$
B) $3: 4$
C) $5: 6$
D) $7: 8$
E) $8: 9$ | 8 : 9 | 3.125 |
31,747 | If the function $f(x) = x^2 + a|x - 1|$ is monotonically increasing on the interval $[-1, +\infty)$, then the set of values for the real number $a$ is ______. | \{-2\} | 35.15625 |
31,748 | Three distinct numbers are selected simultaneously and at random from the set $\{1, 2, 3, 4, 6\}$. What is the probability that at least one number divides another among the selected numbers? Express your answer as a common fraction. | \frac{9}{10} | 32.03125 |
31,749 | The clock shows $00:00$, with both the hour and minute hands coinciding. Considering this coincidence as number 0, determine after what time interval (in minutes) they will coincide for the 21st time. If the answer is not an integer, round the result to the nearest hundredth. | 1374.55 | 19.53125 |
31,750 | The average of the numbers 47 and $x$ is 53. Besides finding the positive difference between 47 and $x$, also determine their sum. | 106 | 93.75 |
31,751 | Given \( n \in \mathbf{N}, n > 4 \), and the set \( A = \{1, 2, \cdots, n\} \). Suppose there exists a positive integer \( m \) and sets \( A_1, A_2, \cdots, A_m \) with the following properties:
1. \( \bigcup_{i=1}^{m} A_i = A \);
2. \( |A_i| = 4 \) for \( i=1, 2, \cdots, m \);
3. Let \( X_1, X_2, \cdots, X_{\mathrm{C}_n^2} \) be all the 2-element subsets of \( A \). For every \( X_k \) \((k=1, 2, \cdots, \mathrm{C}_n^2)\), there exists a unique \( j_k \in\{1, 2, \cdots, m\} \) such that \( X_k \subseteq A_{j_k} \).
Find the smallest value of \( n \). | 13 | 5.46875 |
31,752 | If lines $l_{1}$: $ax+2y+6=0$ and $l_{2}$: $x+(a-1)y+3=0$ are parallel, find the value of $a$. | -1 | 21.09375 |
31,753 | Let the functions $f(\alpha,x)$ and $g(\alpha)$ be defined as \[f(\alpha,x)=\dfrac{(\frac{x}{2})^\alpha}{x-1}\qquad\qquad\qquad g(\alpha)=\,\dfrac{d^4f}{dx^4}|_{x=2}\] Then $g(\alpha)$ is a polynomial is $\alpha$ . Find the leading coefficient of $g(\alpha)$ . | 1/16 | 64.84375 |
31,754 | Given two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ with an acute angle between them, and satisfying $|\overrightarrow{a}|= \frac{8}{\sqrt{15}}$, $|\overrightarrow{b}|= \frac{4}{\sqrt{15}}$. If for any $(x,y)\in\{(x,y)| |x \overrightarrow{a}+y \overrightarrow{b}|=1, xy > 0\}$, it holds that $|x+y|\leqslant 1$, then the minimum value of $\overrightarrow{a} \cdot \overrightarrow{b}$ is \_\_\_\_\_\_. | \frac{8}{15} | 38.28125 |
31,755 | Observe the following equations:
1=1
1-4=-（1+2）=-3
1-4+9=1+2+3=6
1-4+9-16=-（1+2+3+4）=-10
Then, the 5th equation is
The value of the 20th equation is
These equations reflect a certain pattern among integers. Let $n$ represent a positive integer, try to express the pattern you discovered using an equation related to $n$. | -210 | 67.96875 |
31,756 | Given that $40\%$ of the birds were geese, $20\%$ were swans, $15\%$ were herons, and $25\%$ were ducks, calculate the percentage of the birds that were not ducks and were geese. | 53.33\% | 31.25 |
31,757 | The graphs \( y = 2 \cos 3x + 1 \) and \( y = - \cos 2x \) intersect at many points. Two of these points, \( P \) and \( Q \), have \( x \)-coordinates between \(\frac{17 \pi}{4}\) and \(\frac{21 \pi}{4}\). The line through \( P \) and \( Q \) intersects the \( x \)-axis at \( B \) and the \( y \)-axis at \( A \). If \( O \) is the origin, what is the area of \( \triangle BOA \)? | \frac{361\pi}{8} | 0 |
31,758 | If $e^{i \theta} = \frac{1 + i \sqrt{2}}{2}$, then find $\sin 3 \theta.$ | \frac{\sqrt{2}}{8} | 0.78125 |
31,759 | Given the line $y=-x+1$ and the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$, they intersect at points $A$ and $B$. $OA \perp OB$, where $O$ is the origin. If the eccentricity of the ellipse $e \in [\frac{1}{2}, \frac{\sqrt{3}}{2}]$, find the maximum value of $a$. | \frac{\sqrt{10}}{2} | 6.25 |
31,760 | Let $(b_1,b_2,b_3,\ldots,b_{10})$ be a permutation of $(1,2,3,\ldots,10)$ for which
$b_1>b_2>b_3>b_4 \mathrm{\ and \ } b_4<b_5<b_6<b_7<b_8<b_9<b_{10}.$
Find the number of such permutations. | 84 | 2.34375 |
31,761 | Find \( g\left(\frac{1}{1996}\right) + g\left(\frac{2}{1996}\right) + g\left(\frac{3}{1996}\right) + \cdots + g\left(\frac{1995}{1996}\right) \) where \( g(x) = \frac{9x}{3 + 9x} \). | 997 | 2.34375 |
31,762 | Given the function $f(x)=x^{2}+x-2$, determine the characteristics of the function $f(x)$ in the interval $[-1,1)$. | -\frac{9}{4} | 13.28125 |
31,763 | In a box, there are red and black socks. If two socks are randomly taken from the box, the probability that both of them are red is $1/2$.
a) What is the minimum number of socks that can be in the box?
b) What is the minimum number of socks that can be in the box, given that the number of black socks is even? | 21 | 0 |
31,764 | Let the set $\mathbf{A}=\{1, 2, 3, 4, 5, 6\}$ and a bijection $f: \mathbf{A} \rightarrow \mathbf{A}$ satisfy the condition: for any $x \in \mathbf{A}$, $f(f(f(x)))=x$. Then the number of bijections $f$ satisfying the above condition is: | 81 | 19.53125 |
31,765 | Grace is trying to solve the following equation by completing the square: $$36x^2-60x+25 = 0.$$ She successfully rewrites the equation in the form: $$(ax + b)^2 = c,$$ where $a$, $b$, and $c$ are integers and $a > 0$. What is the value of $a + b + c$? | 26 | 0 |
31,766 | The greatest common divisor of two integers is $(x+3)$ and their least common multiple is $x(x+3)$, where $x$ is a positive integer. If one of the integers is 36, what is the smallest possible value of the other one? | 108 | 1.5625 |
31,767 | Evaluate the following expression:
$$\frac { \sqrt {3}tan12 ° -3}{sin12 ° (4cos ^{2}12 ° -2)}$$ | -4 \sqrt {3} | 0 |
31,768 | In a right triangle, side AB is 8 units, and side BC is 15 units. Calculate $\cos C$ and $\sin C$. | \frac{8}{15} | 0 |
31,769 | How many multiples of 15 are there between 40 and 240? | 14 | 97.65625 |
31,770 | Given that the complex number $z$ satisfies the equation $\frac{1-z}{1+z}={i}^{2018}+{i}^{2019}$ (where $i$ is the imaginary unit), find the value of $|2+z|$. | \frac{5\sqrt{2}}{2} | 11.71875 |
31,771 | Fill in the blanks with appropriate numbers.
4 liters 25 milliliters = ___ milliliters
6.09 cubic decimeters = ___ cubic centimeters
4.9 cubic decimeters = ___ liters ___ milliliters
2.03 cubic meters = ___ cubic meters ___ cubic decimeters. | 30 | 71.875 |
31,772 | Definition: If the line $l$ is tangent to the graphs of the functions $y=f(x)$ and $y=g(x)$, then the line $l$ is called the common tangent line of the functions $y=f(x)$ and $y=g(x)$. If the functions $f(x)=a\ln x (a>0)$ and $g(x)=x^{2}$ have exactly one common tangent line, then the value of the real number $a$ is ______. | 2e | 54.6875 |
31,773 | Chester traveled from Hualien to Lukang in Changhua to participate in the Hua Luogeng Gold Cup Math Competition. Before leaving, his father checked the car’s odometer, which displayed a palindromic number of 69,696 kilometers (a palindromic number reads the same forward and backward). After driving for 5 hours, they arrived at the destination with the odometer showing another palindromic number. During the journey, the father's driving speed never exceeded 85 kilometers per hour. What is the maximum possible average speed (in kilometers per hour) that Chester's father could have driven? | 82.2 | 8.59375 |
31,774 | In the triangle $ABC$ it is known that $\angle A = 75^o, \angle C = 45^o$ . On the ray $BC$ beyond the point $C$ the point $T$ is taken so that $BC = CT$ . Let $M$ be the midpoint of the segment $AT$ . Find the measure of the $\angle BMC$ .
(Anton Trygub) | 45 | 14.0625 |
31,775 | Given the expression $\frac{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \cdot 9}{1+2+3+4+5+6+7+8+9+10}$, evaluate the expression. | 6608 | 0 |
31,776 | In triangle \(ABC\), the angle at vertex \(A\) is \(60^{\circ}\). A circle is drawn through points \(B\), \(C\), and point \(D\), which lies on side \(AB\), intersecting side \(AC\) at point \(E\). Find \(AE\) if \(AD = 3\), \(BD = 1\), and \(EC = 4\). Find the radius of the circle. | \sqrt{7} | 2.34375 |
31,777 | Given a non-right triangle $\triangle ABC$, where the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $c=1$, also $$C= \frac {\pi}{3}$$, if $\sin C + \sin(A-B) = 3\sin 2B$, then the area of $\triangle ABC$ is \_\_\_\_\_\_. | \frac {3 \sqrt {3}}{28} | 0 |
31,778 | Let $p(x)$ be a monic polynomial of degree 7 such that $p(0) = 0,$ $p(1) = 1,$ $p(2) = 2,$ $p(3) = 3,$ $p(4) = 4,$ $p(5) = 5,$ $p(6) = 6,$ and $p(7) = 7.$ Find $p(8).$ | 40328 | 69.53125 |
31,779 | Let the function \( f(x) = x^3 + a x^2 + b x + c \), where \( a \), \( b \), and \( c \) are non-zero integers. If \( f(a) = a^3 \) and \( f(b) = b^3 \), then the value of \( c \) is: | 16 | 17.96875 |
31,780 | Let $ABCD$ be a unit square. $E$ and $F$ trisect $AB$ such that $AE<AF. G$ and $H$ trisect $BC$ such that $BG<BH. I$ and $J$ bisect $CD$ and $DA,$ respectively. Let $HJ$ and $EI$ meet at $K,$ and let $GJ$ and $FI$ meet at $L.$ Compute the length $KL.$ | \frac{6\sqrt{2}}{35} | 24.21875 |
31,781 | The sum of four prime numbers $P,$ $Q,$ $P-Q,$ and $P+Q$ must be expressed in terms of a single letter indicating the property of the sum. | 17 | 15.625 |
31,782 | Assuming that the new demand is given by \( \frac{1}{1+ep} \), where \( p = 0.20 \) and \( e = 1.5 \), calculate the proportionate decrease in demand. | 0.23077 | 0 |
31,783 | Determine the minimum possible value of the sum
\[
\frac{a}{3b} + \frac{b}{6c} + \frac{c}{9a},
\]
where \(a, b, c\) are positive real numbers. | 3 \cdot \frac{1}{\sqrt[3]{162}} | 0 |
31,784 | Jun Jun is looking at an incorrect single-digit multiplication equation \( A \times B = \overline{CD} \), where the digits represented by \( A \), \( B \), \( C \), and \( D \) are all different from each other. Clever Jun Jun finds that if only one digit is changed, there are 3 ways to correct it, and if only the order of \( A \), \( B \), \( C \), and \( D \) is changed, the equation can also be corrected. What is \( A + B + C + D = \) ? | 17 | 0 |
31,785 | Given the vectors $\overrightarrow{a} = (\cos 25^\circ, \sin 25^\circ)$, $\overrightarrow{b} = (\sin 20^\circ, \cos 20^\circ)$, and $\overrightarrow{u} = \overrightarrow{a} + t\overrightarrow{b}$, where $t\in\mathbb{R}$, find the minimum value of $|\overrightarrow{u}|$. | \frac{\sqrt{2}}{2} | 92.96875 |
31,786 | In 2023, a special international mathematical conference is held. Let $A$, $B$, and $C$ be distinct positive integers such that the product $A \cdot B \cdot C = 2023$. What is the largest possible value of the sum $A+B+C$? | 297 | 62.5 |
31,787 | The teacher gave each of her $37$ students $36$ pencils in different colors. It turned out that each pair of students received exactly one pencil of the same color. Determine the smallest possible number of different colors of pencils distributed. | 666 | 14.0625 |
31,788 | Graphistan has $2011$ cities and Graph Air (GA) is running one-way flights between all pairs of these cities. Determine the maximum possible value of the integer $k$ such that no matter how these flights are arranged it is possible to travel between any two cities in Graphistan riding only GA flights as long as the absolute values of the difference between the number of flights originating and terminating at any city is not more than $k.$ | 1005 | 41.40625 |
31,789 | A right-angled triangle has sides of lengths 6, 8, and 10. A circle is drawn so that the area inside the circle but outside this triangle equals the area inside the triangle but outside the circle. The radius of the circle is closest to: | 2.8 | 0.78125 |
31,790 | Given that a certain middle school has 3500 high school students and 1500 junior high school students, if 70 students are drawn from the high school students, calculate the total sample size $n$. | 100 | 90.625 |
31,791 | Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a half dollar. What is the probability that at least 25 cents worth of coins come up heads? | \dfrac{13}{16} | 0.78125 |
31,792 | $ABCD$ is a square where each side measures 4 units. $P$ and $Q$ are the midpoints of $\overline{BC}$ and $\overline{CD},$ respectively. Find $\sin \phi$ where $\phi$ is the angle $\angle APQ$.
 | \frac{3}{5} | 93.75 |
31,793 | If $x$ and $y$ are positive integers less than $30$ for which $x + y + xy = 104$, what is the value of $x + y$? | 20 | 45.3125 |
31,794 | 14 students attend the IMO training camp. Every student has at least $k$ favourite numbers. The organisers want to give each student a shirt with one of the student's favourite numbers on the back. Determine the least $k$ , such that this is always possible if: $a)$ The students can be arranged in a circle such that every two students sitting next to one another have different numbers. $b)$ $7$ of the students are boys, the rest are girls, and there isn't a boy and a girl with the same number. | k = 2 | 43.75 |
31,795 | Find $x$ such that $\lceil x \rceil \cdot x = 210$. Express $x$ as a decimal. | 14.0 | 32.8125 |
31,796 | For $n > 1$ , let $a_n$ be the number of zeroes that $n!$ ends with when written in base $n$ . Find the maximum value of $\frac{a_n}{n}$ . | 1/2 | 77.34375 |
31,797 | What is the smallest positive value of $x$ such that $x + 8765$ results in a palindrome? | 13 | 0.78125 |
31,798 | 1. The converse of the proposition "If $x > 1$, then ${x}^{2} > 1$" is ________.
2. Let $P$ be a point on the parabola ${{y}^{2}=4x}$ such that the distance from $P$ to the line $x+2=0$ is $6$. The distance from $P$ to the focus $F$ of the parabola is ________.
3. In a geometric sequence $\\{a\_{n}\\}$, if $a\_{3}$ and $a\_{15}$ are roots of the equation $x^{2}-6x+8=0$, then $\frac{{a}\_{1}{a}\_{17}}{{a}\_{9}} =$ ________.
4. Let $F$ be the left focus of the hyperbola $C$: $\frac{{x}^{2}}{4}-\frac{{y}^{2}}{12} =1$. Let $A(1,4)$ and $P$ be a point on the right branch of $C$. When the perimeter of $\triangle APF$ is minimum, the distance from $F$ to the line $AP$ is ________. | \frac{32}{5} | 3.125 |
31,799 | The absolute value of -1.2 is ____, and its reciprocal is ____. | -\frac{5}{6} | 13.28125 |
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