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24,800 | Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is \(5:4\). Find the minimum possible value of their common perimeter. | 524 | 0 |
24,801 | Given the function $f(2x+1)=x^{2}-2x$, determine the value of $f(\sqrt{2})$. | \frac{5-4\sqrt{2}}{4} | 0 |
24,802 | 8. Andrey likes all numbers that are not divisible by 3, and Tanya likes all numbers that do not contain digits that are divisible by 3.
a) How many four-digit numbers are liked by both Andrey and Tanya?
b) Find the total sum of the digits of all such four-digit numbers. | 14580 | 0 |
24,803 | How many distinct, positive factors does $1320$ have? | 24 | 0 |
24,804 | Alice and Bob each draw one number from 50 slips of paper numbered from $1$ to $50$ placed in a hat. Alice says, "I can't tell who has the larger number." Bob then says, "I am certain who has the larger number." Understanding Bob's certainty, Alice asks Bob if his number is prime. Bob answers, "Yes." Alice then says, "In that case, when I multiply your number by $100$ and add my number, the result is a perfect square. What could my number possibly be?"
**A)** $24$
**B)** $61$
**C)** $56$
**D)** $89$ | 61 | 0.78125 |
24,805 | Given the function $f(x)=\sin 2x$, its graph intersects with the line $2kx-2y-kπ=0 (k > 0)$ at exactly three common points. The x-coordinates of these points in ascending order are $x_{1}$, $x_{2}$, $x_{3}$. Calculate the value of $(x_{1}-x_{3})\tan (x_{2}-2x_{3})$. | -1 | 9.375 |
24,806 | A rectangular piece of paper, PQRS, has PQ = 20 and QR = 15. The piece of paper is glued flat on the surface of a large cube so that Q and S are at vertices of the cube. The shortest distance from P to R, as measured through the cube, can be calculated using the 3D geometry of the cube. | 18.4 | 0 |
24,807 | Define the sequence $b_1, b_2, b_3, \ldots$ by $b_n = \sum\limits_{k=1}^n \cos{k}$, where $k$ represents radian measure. Find the index of the 100th term for which $b_n < 0$. | 632 | 0 |
24,808 | Let \( a_{1}, a_{2}, \cdots, a_{n} \) be an arithmetic sequence, and it is given that
$$
\sum_{i=1}^{n}\left|a_{i}+j\right|=2028 \text{ for } j=0,1,2,3.
$$
Find the maximum value of the number of terms \( n \). | 52 | 1.5625 |
24,809 | An infinite geometric series has a first term of \( 416 \) and a sum of \( 3120 \). What is its common ratio? | \frac{84}{97} | 0 |
24,810 | The famous German mathematician Dirichlet made significant achievements in the field of mathematics. He was the first person in the history of mathematics to pay attention to concepts and consciously "replace intuition with concepts." The function named after him, $D\left(x\right)=\left\{\begin{array}{l}{1, x \text{ is rational}}\\{0, x \text{ is irrational}}\end{array}\right.$, is called the Dirichlet function. Now, a function similar to the Dirichlet function is defined as $L\left(x\right)=\left\{\begin{array}{l}{x, x \text{ is rational}}\\{0, x \text{ is irrational}}\end{array}\right.$. There are four conclusions about the Dirichlet function and the $L$ function:<br/>$(1)D\left(1\right)=L\left(1\right)$;<br/>$(2)$ The function $L\left(x\right)$ is an even function;<br/>$(3)$ There exist four points $A$, $B$, $C$, $D$ on the graph of the $L$ function such that the quadrilateral $ABCD$ is a rhombus;<br/>$(4)$ There exist three points $A$, $B$, $C$ on the graph of the $L$ function such that $\triangle ABC$ is an equilateral triangle.<br/>The correct numbers of the conclusions are ____. | (1)(4) | 0 |
24,811 | Find $\left(\frac{2}{3}\right)^{6} \cdot \left(\frac{5}{6}\right)^{-4}$. | \frac{82944}{456375} | 0 |
24,812 | Given vectors $a$ and $b$ that satisfy $(a+2b)\cdot(5a-4b)=0$, and $|a|=|b|=1$, find the angle $\theta$ between $a$ and $b$. | \dfrac{\pi}{3} | 99.21875 |
24,813 | In how many ways can 13 bishops be placed on an $8 \times 8$ chessboard such that:
(i) a bishop is placed on the second square in the second row,
(ii) at most one bishop is placed on each square,
(iii) no bishop is placed on the same diagonal as another bishop,
(iv) every diagonal contains a bishop?
(For the purposes of this problem, consider all diagonals of the chessboard to be diagonals, not just the main diagonals). | 1152 | 0 |
24,814 | In a new configuration, six circles with a radius of 5 units intersect at a single point. What is the number of square units in the area of the shaded region? The region is formed similarly to the original problem where the intersections create smaller sector-like areas. Express your answer in terms of $\pi$. | 75\pi - 25\sqrt{3} | 0 |
24,815 | Fill the numbers 1 to 6 into the six boxes in the image. The smallest result you can get is ______ | 342 | 0 |
24,816 | Given that Bob was instructed to subtract 5 from a certain number and then divide the result by 7, but instead subtracted 7 and then divided by 5, yielding an answer of 47, determine what his answer would have been had he worked the problem correctly. | 33 | 1.5625 |
24,817 | Isosceles right triangle $PQR$ (with $\angle PQR = \angle PRQ = 45^\circ$ and hypotenuse $\overline{PQ}$) encloses a right triangle $ABC$ (hypotenuse $\overline{AB}$) as shown. Given $PC = 5$ and $BP = CQ = 4$, compute $AQ$. | \frac{5}{\sqrt{2}} | 0 |
24,818 | Define
\[
d_k = k + \cfrac{1}{3k + \cfrac{1}{3k + \cfrac{1}{3k + \dotsb}}}.
\]
Calculate $\sum_{k = 1}^{10} (d_k^2 + 2)$. | 405 | 0 |
24,819 | Three cones are placed on a table on their bases, touching each other. The radii of their bases are 1, 12, and 12, and the apex angles of the cones are $-4 \operatorname{arctg} \frac{1}{3}$, $4 \operatorname{arctg} \frac{2}{3}$, and $4 \operatorname{arctg} \frac{2}{3}$ respectively (the apex angle of a cone is the angle between its generatrices in an axial section). A sphere is placed on the table, touching all the cones. Find the radius of the sphere. | 40/21 | 0 |
24,820 | How many of the 256 smallest positive integers written in base 8 use the digit 6 or 7 (or both)? | 10 | 0 |
24,821 | If
\[x + \sqrt{x^2 - 4} + \frac{1}{x - \sqrt{x^2 - 4}} = 10,\]
then find
\[x^2 + \sqrt{x^4 - 4} + \frac{1}{x^2 + \sqrt{x^4 - 4}}.\] | \frac{841}{100} | 0 |
24,822 | Given the function $f(x)=\ln (ax+1)+ \frac {x^{3}}{3}-x^{2}-ax(a∈R)$,
(1) Find the range of values for the real number $a$ such that $y=f(x)$ is an increasing function on $[4,+∞)$;
(2) When $a\geqslant \frac {3 \sqrt {2}}{2}$, let $g(x)=\ln [x^{2}(ax+1)]+ \frac {x^{3}}{3}-3ax-f(x)(x > 0)$ and its two extreme points $x_{1}$, $x_{2}(x_{1} < x_{2})$ are exactly the zeros of $φ(x)=\ln x-cx^{2}-bx$, find the minimum value of $y=(x_{1}-x_{2})φ′( \frac {x_{1}+x_{2}}{2})$. | \ln 2- \frac {2}{3} | 0 |
24,823 | Inside a square of side length 1, four quarter-circle arcs are traced with the edges of the square serving as the radii. These arcs intersect pairwise at four distinct points, forming the vertices of a smaller square. This process is repeated for the smaller square, and continuously for each subsequent smaller square. What is the sum of the areas of all squares formed in this manner? | \frac{2}{1 - (2 - \sqrt{3})} | 0 |
24,824 | Suppose a number $m$ is randomly selected from the set $\{7, 9, 12, 18, 21\}$, and a number $n$ is randomly selected from $\{2005, 2006, 2007, \ldots, 2025\}$. Calculate the probability that the last digit of $m^n$ is $6$. | \frac{8}{105} | 0.78125 |
24,825 | Find the largest positive integer $n>10$ such that the residue of $n$ when divided by each perfect square between $2$ and $\dfrac n2$ is an odd number. | 505 | 90.625 |
24,826 | A youth radio station, to attract listeners' attention, gives away gifts and grand prizes among them. Gifts are given away hourly over sixteen hours (one gift per hour), and grand prizes are given away during four evening hours (one grand prize per hour). The probability that listeners win a prize is 0.3, and a grand prize is 0.02.
Find the probability that over 30 days:
a) Listeners will win three grand prizes;
b) Listeners will win between 130 to 160 prizes. | 0.862 | 0.78125 |
24,827 | Given a triangle \( ABC \) with sides \( AB=13 \), \( BC=20 \), and \( AC=21 \). Point \( K \) is on side \( AB \), point \( L \) is on side \( AC \), and point \( N \) is on side \( BC \). It is known that \( AK=4 \), \( CN=1 \), and \( CL=\frac{20}{21} \). A line through point \( K \) parallel to \( NL \) intersects side \( AC \) at point \( M \). Find the area of the quadrilateral \( NLMK \). | 41 | 0 |
24,828 | If there are 150 seats in a row, calculate the fewest number of seats that must be occupied so the next person to be seated must sit next to someone. | 37 | 0 |
24,829 | Given that the total number of units produced by the workshops A, B, C, and D is 2800, and workshops A and C together contributed 60 units to the sample, determine the total number of units produced by workshops B and D. | 1600 | 8.59375 |
24,830 | In triangle ABC, let the lengths of the sides opposite to angles A, B, and C be a, b, and c respectively, and b = 3, c = 1, A = 2B. Find the value of a. | \sqrt{19} | 0 |
24,831 | Find the minimum value of
\[3x^2 + 3xy + y^2 - 3x + 3y + 9\]
over all real numbers $x$ and $y.$ | \frac{45}{8} | 0 |
24,832 | An eight-sided die is rolled seven times. Find the probability of rolling at least a seven at least six times. | \frac{11}{2048} | 0 |
24,833 | Given the digits $5,$ $6,$ $7,$ and $8,$ used exactly once to form four-digit integers, list these integers from least to greatest. For numbers starting with $7$ or $8,$ reverse the order of the last two digits. What is the $20^{\text{th}}$ integer in the list? | 7865 | 3.90625 |
24,834 | Given $a=1$, $b=2$, $C=\frac{2π}{3}$ in triangle $\triangle ABC$, calculate the value of $c$. | \sqrt{9} | 0 |
24,835 | A pedestrian crossing signal at an intersection alternates between red and green lights, with the red light lasting for $30$ seconds. The probability that Little Ming, upon arriving at the intersection and encountering a red light, will have to wait at least $10$ seconds before the green light appears is _______. | \frac{5}{6} | 0 |
24,836 | If $x+y=10$ and $xy=12$, what is the value of $x^3-y^3$? | 176\sqrt{13} | 38.28125 |
24,837 | If we write $\sqrt{8} + \frac{1}{\sqrt{8}} + \sqrt{9} + \frac{1}{\sqrt{9}}$ in the form $\dfrac{a\sqrt{8} + b\sqrt{9}}{c}$ such that $a$, $b$, and $c$ are positive integers and $c$ is as small as possible, then what is $a+b+c$? | 31 | 0 |
24,838 | A 6x6x6 cube is formed by assembling 216 unit cubes. Ten unit squares are painted on each of the six faces of the cube, leaving some rows and columns unpainted. Specifically, two non-adjacent columns and two non-adjacent rows on each face are left unpainted. How many of the 216 unit cubes have no paint on them? | 168 | 0 |
24,839 | In an organization, there are five leaders and a number of regular members. Each year, the leaders are expelled, followed by each regular member recruiting three new members to become regular members. After this, five new leaders are elected from outside the organization. Initially, the organisation had twenty people total. How many total people will be in the organization six years from now? | 10895 | 0 |
24,840 | Using the vertices of a single rectangular solid (cuboid), how many different pyramids can be formed? | 106 | 0 |
24,841 | Given the vertices of a regular 100-sided polygon \( A_{1}, A_{2}, A_{3}, \ldots, A_{100} \), in how many ways can three vertices be selected such that they form an obtuse triangle? | 117600 | 35.15625 |
24,842 | Compute the expressions
\[ C = 3 \times 4 + 5 \times 6 + 7 \times 8 + \cdots + 43 \times 44 + 45 \]
and
\[ D = 3 + 4 \times 5 + 6 \times 7 + \cdots + 42 \times 43 + 44 \times 45 \]
and find the positive difference between integers $C$ and $D$. | 882 | 0.78125 |
24,843 | For finite sets $A$ and $B$ , call a function $f: A \rightarrow B$ an \emph{antibijection} if there does not exist a set $S \subseteq A \cap B$ such that $S$ has at least two elements and, for all $s \in S$ , there exists exactly one element $s'$ of $S$ such that $f(s')=s$ . Let $N$ be the number of antibijections from $\{1,2,3, \ldots 2018 \}$ to $\{1,2,3, \ldots 2019 \}$ . Suppose $N$ is written as the product of a collection of (not necessarily distinct) prime numbers. Compute the sum of the members of this collection. (For example, if it were true that $N=12=2\times 2\times 3$ , then the answer would be $2+2+3=7$ .)
*Proposed by Ankit Bisain* | 1363641 | 0 |
24,844 | What is the area of the quadrilateral formed by the points of intersection of the circle \(x^2 + y^2 = 16\) and the ellipse \((x-3)^2 + 4y^2 = 36\). | 14 | 0 |
24,845 | Using a $4 \times 4$ grid where points are spaced equally at 1 unit apart both horizontally and vertically, how many rectangles are there whose four vertices are points on this grid? | 101 | 0 |
24,846 | Nina and Tadashi play the following game. Initially, a triple $(a, b, c)$ of nonnegative integers with $a+b+c=2021$ is written on a blackboard. Nina and Tadashi then take moves in turn, with Nina first. A player making a move chooses a positive integer $k$ and one of the three entries on the board; then the player increases the chosen entry by $k$ and decreases the other two entries by $k$ . A player loses if, on their turn, some entry on the board becomes negative.
Find the number of initial triples $(a, b, c)$ for which Tadashi has a winning strategy. | 6561 | 0 |
24,847 | Rectangular prism P Q R S W T U V has a square base P Q R S. Point X is on the face T U V W so that P X = 12, Q X = 10, and R X = 8. Determine the maximum possible area of rectangle P Q U T. | 67.82 | 0 |
24,848 | Given that the ratio of bananas to yogurt to honey is 3:2:1, and that Linda has 10 bananas, 9 cups of yogurt, and 4 tablespoons of honey, determine the maximum number of servings of smoothies Linda can make. | 13 | 0 |
24,849 | John is planning to fence a rectangular garden such that the area is at least 150 sq. ft. The length of the garden should be 20 ft longer than its width. Additionally, the total perimeter of the garden must not exceed 70 ft. What should the width, in feet, be? | -10 + 5\sqrt{10} | 4.6875 |
24,850 | Determine the volume of the right rectangular parallelepiped whose edges are formed by the distances from the orthocenter to the vertices of a triangle, where the radius of the circumcircle $r = 2.35$ and the angles are: $\alpha = 63^{\circ} 18^{\prime} 13^{\prime \prime}, \beta = 51^{\circ} 42^{\prime} 19^{\prime \prime}$. | 12.2 | 0 |
24,851 | Given that $\frac{x}{2} = y^2$ and $\frac{x}{5} = 3y$, solve for $x$. | 112.5 | 7.03125 |
24,852 | Let \( n \) be a two-digit number such that the square of the sum of the digits of \( n \) is equal to the sum of the digits of \( n^2 \). Find the sum of all possible values of \( n \). | 139 | 0 |
24,853 | For any positive integers \( m \) and \( n \), define \( r(m, n) \) as the remainder of \( m \div n \) (for example, \( r(8,3) \) represents the remainder of \( 8 \div 3 \), so \( r(8,3)=2 \)). What is the smallest positive integer solution satisfying the equation \( r(m, 1) + r(m, 2) + r(m, 3) + \cdots + r(m, 10) = 4 \)? | 120 | 96.09375 |
24,854 | In triangle \(ABC\), angle \(B\) is \(120^\circ\), and \(AB = 2BC\). The perpendicular bisector of side \(AB\) intersects \(AC\) at point \(D\). Find the ratio \(AD:DC\). | 3/2 | 0 |
24,855 | The equation $y = -16t^2 + 34t + 25$ describes the height (in feet) of a ball thrown upwards at $34$ feet per second from $25$ feet above the ground. Determine the time (in seconds) when the ball will hit the ground. | \frac{25}{8} | 0 |
24,856 | In the isosceles triangle \( ABC \) (\( AB = BC \)), medians \( AD \) and \( EC \) intersect at point \( O \). The ratio of the radius of the circle inscribed in triangle \( AOC \) to the radius of the circle inscribed in quadrilateral \( ODBE \) is \(\frac{2}{3}\). Find the ratio \(\frac{AC}{BC}\). | 20/17 | 0 |
24,857 | An ellipse \( \frac{x^2}{9} + \frac{y^2}{16} = 1 \) has chords passing through the point \( C = (2, 2) \). If \( t \) is defined as
\[
t = \frac{1}{AC} + \frac{1}{BC}
\]
where \( AC \) and \( BC \) are distances from \( A \) and \( B \) (endpoints of the chords) to \( C \). Find the constant \( t \). | \frac{4\sqrt{5}}{5} | 0 |
24,858 | Let $T$ be a positive integer whose only digits are 0s and 1s. If $X = T \div 18$ and $X$ is an integer, what is the smallest possible value of $X$? | 6172839500 | 0 |
24,859 | At time $0$ , an ant is at $(1,0)$ and a spider is at $(-1,0)$ . The ant starts walking counterclockwise around the unit circle, and the spider starts creeping to the right along the $x$ -axis. It so happens that the ant's horizontal speed is always half the spider's. What will the shortest distance ever between the ant and the spider be? | \frac{\sqrt{14}}{4} | 1.5625 |
24,860 | Given an ellipse \( C: \frac{x^{2}}{2}+y^{2}=1 \) with left and right foci \( F_{1}, F_{2} \) respectively, let \( P \) be a point on the ellipse \( C \) in the first quadrant. The extended lines \( P F_{1}, P F_{2} \) intersect the ellipse \( C \) at points \( Q_{1}, Q_{2} \) respectively. Find the maximum value of the difference of areas of triangles \( \triangle P F_{1} Q_{2} \) and \( \triangle P F_{2} Q_{1} \). | \frac{2\sqrt{2}}{3} | 2.34375 |
24,861 | A square is inscribed in the ellipse whose equation is $x^2 + 3y^2 = 3$. One vertex of the square is at $(0, 1)$, and one diagonal of the square lies along the y-axis. Determine the square of the length of each side of the square. | \frac{5}{3} - 2\sqrt{\frac{2}{3}} | 0 |
24,862 | Consider a parallelogram where each vertex has integer coordinates and is located at $(0,0)$, $(4,5)$, $(11,5)$, and $(7,0)$. Calculate the sum of the perimeter and the area of this parallelogram. | 9\sqrt{41} | 0 |
24,863 | 1. There are 5 different books, and we need to choose 3 books to give to 3 students, one book per student. There are a total of different ways to do this.
2. There are 5 different books, and we want to buy 3 books to give to 3 students, one book per student. There are a total of different ways to do this. | 125 | 0.78125 |
24,864 | Given that the cosine value of the vertex angle of an isosceles triangle equals $\dfrac{4}{5}$, calculate the sine value of the base angle of this triangle. | \dfrac{2\sqrt{3}}{5} | 0 |
24,865 | Given the inequality $x\ln x - kx > 3$, which holds true for any $x > 1$, determine the maximum value of the integer $k$. | -3 | 46.875 |
24,866 | The slopes of lines $l_1$ and $l_2$ are the two roots of the equation $6x^2+x-1=0$, respectively. The angle between lines $l_1$ and $l_2$ is __________. | \frac{\pi}{4} | 53.125 |
24,867 | Find the square root of $\dfrac{9!}{210}$. | 216\sqrt{3} | 0 |
24,868 | Given the equations $3x + 2y = 6$ and $2x + 3y = 7$, find $14x^2 + 25xy + 14y^2$. | 85 | 2.34375 |
24,869 | Initially, a prime number is displayed on the computer screen. Every second, the number on the screen is replaced with a number obtained by adding the last digit of the previous number, increased by 1. After how much time, at the latest, will a composite number appear on the screen? | 996 | 0 |
24,870 | In a regular quadrilateral frustum with lateral edges \(A A_{1}, B B_{1}, C C_{1}, D D_{1}\), the side length of the upper base \(A_{1} B_{1} C_{1} D_{1}\) is 1, and the side length of the lower base is 7. A plane passing through the edge \(B_{1} C_{1}\) perpendicular to the plane \(A D_{1} C\) divides the frustum into two equal-volume parts. Find the volume of the frustum. | \frac{38\sqrt{5}}{5} | 0 |
24,871 | In America, temperature is measured in degrees Fahrenheit. This is a linear scale where the freezing point of water is $32^{\circ} \mathrm{F}$ and the boiling point is $212^{\circ} \mathrm{F}$.
Someone provides the temperature rounded to whole degrees Fahrenheit, which we then convert to Celsius and afterwards round to whole degrees. What is the maximum possible deviation of the obtained value from the original temperature in Celsius degrees? | 13/18 | 0 |
24,872 | Given a circle and $2006$ points lying on this circle. Albatross colors these $2006$ points in $17$ colors. After that, Frankinfueter joins some of the points by chords such that the endpoints of each chord have the same color and two different chords have no common points (not even a common endpoint). Hereby, Frankinfueter intends to draw as many chords as possible, while Albatross is trying to hinder him as much as he can. What is the maximal number of chords Frankinfueter will always be able to draw? | 117 | 1.5625 |
24,873 | Given that $x$ is the geometric mean of $4$ and $16$, find the value of $x$. | -8 | 0 |
24,874 | A circle inscribed in triangle \( ABC \) divides median \( BM \) into three equal parts. Find the ratio \( BC: CA: AB \). | 5:10:13 | 0 |
24,875 | The sum of all roots of the equation $2x^2-3x-5=0$ and the equation $x^2-6x+2=0$ is equal to $\boxed{\text{blank}}$, and the product of all roots is equal to $\boxed{\text{blank}}$. | 7\frac{1}{2} | 0 |
24,876 | Let \( A = (-4, 0) \), \( B = (-1, 2) \), \( C = (1, 2) \), and \( D = (4, 0) \). Suppose that point \( P \) satisfies
\[ PA + PD = 10 \quad \text{and} \quad PB + PC = 10. \]
Find the \( y \)-coordinate of \( P \), when simplified, which can be expressed in the form \( \frac{-a + b \sqrt{c}}{d} \), where \( a, b, c, d \) are positive integers. Find \( a + b + c + d \). | 35 | 0.78125 |
24,877 | Given that circle $C$ passes through the point $(0,2)$ with a radius of $2$, if there exist two points on circle $C$ that are symmetric with respect to the line $2x-ky-k=0$, find the maximum value of $k$. | \frac{4\sqrt{5}}{5} | 9.375 |
24,878 | Find the maximum number $E$ such that the following holds: there is an edge-colored graph with 60 vertices and $E$ edges, with each edge colored either red or blue, such that in that coloring, there is no monochromatic cycles of length 3 and no monochromatic cycles of length 5. | 1350 | 3.125 |
24,879 | Two out of three independently operating components of a computing device have failed. Find the probability that the first and second components have failed, given that the failure probabilities of the first, second, and third components are 0.2, 0.4, and 0.3, respectively. | 0.3 | 0.78125 |
24,880 | The four zeros of the polynomial $x^4 + jx^2 + kx + 256$ are distinct real numbers in arithmetic progression. Compute the value of $j$. | -40 | 0 |
24,881 | Antal and Béla start from home on their motorcycles heading towards Cegléd. After traveling one-fifth of the way, Antal for some reason turns back. As a result, he accelerates and manages to increase his speed by one quarter. He immediately sets off again from home. Béla, continuing alone, decreases his speed by one quarter. They travel the final section of the journey together at $48$ km/h and arrive 10 minutes later than planned. What can we calculate from all this? | 40 | 0.78125 |
24,882 | With the same amount of a monoatomic ideal gas, two cyclic processes $1-2-3-4-1$ and $1-3-4-1$ are carried out. Find the ratio of their efficiencies. | 18/13 | 0 |
24,883 | In an isosceles right triangle $ABC$, with $AB = AC$, a circle is inscribed touching $AB$ at point $D$, $AC$ at point $E$, and $BC$ at point $F$. If line $DF$ is extended to meet $AC$ at point $G$ and $\angle BAC = 90^\circ$, and the length of segment $DF$ equals $6$ cm, calculate the length of $AG$.
A) $3\sqrt{2}$ cm
B) $4\sqrt{2}$ cm
C) $6\sqrt{2}$ cm
D) $5\sqrt{2}$ cm | 3\sqrt{2} | 7.03125 |
24,884 | What is the 7th term of an arithmetic sequence of 15 terms where the first term is 3 and the last term is 72? | 33 | 0 |
24,885 | It takes 42 seconds for a clock to strike 7 times. How many seconds does it take for it to strike 10 times? | 60 | 0 |
24,886 | There is a graph with 30 vertices. If any of 26 of its vertices with their outgoiing edges are deleted, then the remained graph is a connected graph with 4 vertices.
What is the smallest number of the edges in the initial graph with 30 vertices? | 405 | 0.78125 |
24,887 | In the given $5 \times 5$ grid, there are 6 letters. Divide the grid along the lines to form 6 small rectangles (including squares) of different areas, so that each rectangle contains exactly one letter, and each letter is located in a corner square of its respective rectangle. If each of these six letters is equal to the area of the rectangle it is in, what is the five-digit number $\overline{\mathrm{ABCDE}}$? | 34216 | 0 |
24,888 | A furniture store received a batch of office chairs that were identical except for their colors: 15 chairs were black and 18 were brown. The chairs were in demand and were being bought in a random order. At some point, a customer on the store's website discovered that only two chairs were left for sale. What is the probability that these two remaining chairs are of the same color? | 0.489 | 0 |
24,889 | Given the number
\[e^{11\pi i/40} + e^{21\pi i/40} + e^{31 \pi i/40} + e^{41\pi i /40} + e^{51 \pi i /40},\]
express it in the form $r e^{i \theta}$, where $0 \le \theta < 2\pi$. Find $\theta$. | \frac{11\pi}{20} | 3.125 |
24,890 | Calculate the roundness of 1,728,000. | 19 | 0 |
24,891 | A triangle has vertices $A=(4,3)$, $B=(-4,-1)$, and $C=(9,-7)$. Calculate the equation of the bisector of $\angle A$ in the form $3x - by + c = 0$. Determine the value of $b+c$. | -6 | 0 |
24,892 | What is the sum of all positive integers $\nu$ for which $\mathop{\text{lcm}}[\nu, 18] = 72$? | 60 | 0 |
24,893 | A cube with side length $2$ is inscribed in a sphere. A second cube, with faces parallel to the first, is inscribed between the sphere and one face of the first cube. What is the length of a side of the smaller cube? | \frac{2}{3} | 0.78125 |
24,894 | Each vertex of this parallelogram has integer coordinates. The perimeter of the parallelogram is \( p \) units, and the area is \( a \) square units. If the parallelogram is defined by vertices \((2, 3)\), \((7, 3)\), \((x, y)\), and \((x-5, y)\), where \(x\) and \(y\) are integers, find the value of \(p + a\). | 38 | 1.5625 |
24,895 | If $\frac{x^2}{2^2} + \frac{y^2}{\sqrt{2}^2} = 1$, what is the largest possible value of $|x| + |y|$? | 2\sqrt{3} | 1.5625 |
24,896 | Let $T$ be a subset of $\{1,2,3,\ldots,2021\}$ such that no two members of $T$ differ by $5$ or $8$. What is the largest number of elements $T$ can have? | 918 | 0 |
24,897 | On side \(BC\) of square \(ABCD\), point \(E\) is chosen such that it divides the segment into \(BE = 2\) and \(EC = 3\). The circumscribed circle of triangle \(ABE\) intersects the diagonal \(BD\) a second time at point \(G\). Find the area of triangle \(AGE\). | 43.25 | 0 |
24,898 | Given that \( P \) is the vertex of a right circular cone with an isosceles right triangle as its axial cross-section, \( PA \) is a generatrix of the cone, and \( B \) is a point on the base of the cone. Find the maximum value of \(\frac{PA + AB}{PB}\). | \sqrt{4 + 2\sqrt{2}} | 0 |
24,899 | Compute the value of \[M = 50^2 + 48^2 - 46^2 + 44^2 + 42^2 - 40^2 + \cdots + 4^2 + 2^2 - 0^2,\] where the additions and subtractions alternate in triplets. | 2600 | 0 |
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