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40.3k
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stringlengths 10
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stringlengths 1
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|---|---|---|---|
28,900 | Let $A$ be the set $\{k^{19}-k: 1<k<20, k\in N\}$ . Let $G$ be the GCD of all elements of $A$ .
Then the value of $G$ is? | 798 | 0 |
28,901 | Simplify: $\frac{{x}^{2}-4}{{x}^{2}-4x+4}+\frac{x}{{x}^{2}-x}\div \frac{x-2}{x-1}$, then choose a value of $x$ from the integers in the range $-1\leqslant x\leqslant 3$ to substitute and evaluate. | -\frac{2}{3} | 1.5625 |
28,902 | Let $p$ , $q$ , $r$ , and $s$ be 4 distinct primes such that $p+q+r+s$ is prime, and the numbers $p^2+qr$ and $p^2+qs$ are both perfect squares. What is the value of $p+q+r+s$ ? | 23 | 39.0625 |
28,903 | let $x,y,z$ be positive reals , such that $x+y+z=1399$ find the
$$ \max( [x]y + [y]z + [z]x ) $$
( $[a]$ is the biggest integer not exceeding $a$ ) | 652400 | 30.46875 |
28,904 | Given $\alpha \in \left(0,\pi \right)$, $sin\alpha+cos\alpha=\frac{\sqrt{3}}{3}$, find $\cos 2\alpha$. | -\frac{\sqrt{5}}{3} | 50.78125 |
28,905 | Given that the function $F(x) = f(x) + x^2$ is an odd function, and $f(2) = 1$, find $f(-2)$. | -9 | 99.21875 |
28,906 | Given vectors $\overrightarrow{a} =(\cos x,\sin x)$, $\overrightarrow{b} =(3,-\sqrt{3} )$, with $x\in[0,\pi]$.
$(1)$ If $\overrightarrow{a} \parallel \overrightarrow{b}$, find the value of $x$;
$(2)$ Let $f(x)=\overrightarrow{a} \cdot \overrightarrow{b}$, find the maximum and minimum values of $f(x)$ and the corresponding values of $x$. | -2 \sqrt {3} | 0 |
28,907 | Let $T$ be a subset of $\{1,2,3,...,40\}$ such that no pair of distinct elements in $T$ has a sum divisible by $5$. What is the maximum number of elements in $T$? | 24 | 0.78125 |
28,908 | A regular octagon is inscribed in a circle and another regular octagon is circumscribed about the same circle. What is the ratio of the area of the larger octagon to the area of the smaller octagon? Express your answer as a common fraction. | \frac{4 - 2\sqrt{2}}{2} | 0 |
28,909 | Four friends — Alex, Betty, Clara, and Dave — participated in a relay race by running in pairs, with one pair sitting out each race. Dave ran in 8 races, which was more than any other friend, and Betty ran in 3 races, which was fewer than any other friend. Determine the total number of races those pairs completed. | 10 | 30.46875 |
28,910 | What is the total number of digits used when the first 2500 positive even integers are written? | 9448 | 28.125 |
28,911 | Given the pattern of positive odd numbers shown below, find the 6th number from the left in the 21st row. | 811 | 13.28125 |
28,912 | Given an ellipse $E$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) whose left focus $F_1$ coincides with the focus of the parabola $y^2 = -4x$, and the eccentricity of ellipse $E$ is $\frac{\sqrt{2}}{2}$. A line $l$ with a non-zero slope passes through point $M(m,0)$ ($m > \frac{3}{4}$) and intersects the ellipse $E$ at points $A$ and $B$. Point $P(\frac{5}{4},0)$ is given, and $\overrightarrow{PA} \cdot \overrightarrow{PB}$ is a constant.
- (Ⅰ) Find the equation of the ellipse $E$.
- (Ⅱ) Find the maximum area of $\triangle OAB$. | \frac{\sqrt{2}}{2} | 10.15625 |
28,913 | Jonah’s five cousins are visiting and there are four identical rooms for them to stay in. If any number of cousins can occupy any room, how many different ways can the cousins be arranged among the rooms? | 51 | 46.875 |
28,914 | In a regular octagon, there are two types of diagonals - one that connects alternate vertices (shorter) and another that skips two vertices between ends (longer). What is the ratio of the shorter length to the longer length? Express your answer as a common fraction in simplest form. | \frac{\sqrt{2}}{2} | 13.28125 |
28,915 | A certain department store sells a batch of shirts. The cost price of each shirt is $80. On average, 30 shirts can be sold per day, with a profit of $50 per shirt. In order to increase sales and profits, the store decides to take appropriate price reduction measures. After investigation, it is found that if the price of each shirt is reduced by $1, the store can sell an additional 2 shirts per day on average. If the store makes an average daily profit of $2000, what should be the selling price of each shirt? | 120 | 13.28125 |
28,916 | Function $f(x, y): \mathbb N \times \mathbb N \to \mathbb Q$ satisfies the conditions:
(i) $f(1, 1) =1$ ,
(ii) $f(p + 1, q) + f(p, q + 1) = f(p, q)$ for all $p, q \in \mathbb N$ , and
(iii) $qf(p + 1, q) = pf(p, q + 1)$ for all $p, q \in \mathbb N$ .
Find $f(1990, 31).$ | \frac{30! \cdot 1989!}{2020!} | 0 |
28,917 | How many positive integers divide $5n^{11}-2n^5-3n$ for all positive integers $n$. | 12 | 39.84375 |
28,918 | A cone with a base radius of $15$ cm and a height of $30$ cm has a sphere inscribed within it. The radius of the sphere can be expressed in the form $b\sqrt{d} - b$ cm. Determine the values of $b$ and $d$. | 12.5 | 0 |
28,919 | If two 4'' by 4'' squares are added at each successive stage, what will be the area of the rectangle at Stage 4, in square inches? | 128 | 17.1875 |
28,920 | A point \((x, y)\) is a distance of 14 units from the \(x\)-axis. It is a distance of 8 units from the point \((1, 8)\). Given that \(x > 1\), what is the distance \(n\) from this point to the origin? | 15 | 11.71875 |
28,921 | On a street, there are 10 lamps numbered 1, 2, 3, …, 10. Now, we need to turn off four of them, but we cannot turn off two or three adjacent lamps, nor can we turn off the two lamps at both ends. How many ways are there to turn off the lamps under these conditions? | 20 | 0 |
28,922 | If non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}| = |\overrightarrow{b}|$ and $(\sqrt{3}\overrightarrow{a} - 2\overrightarrow{b}) \cdot \overrightarrow{a} = 0$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is __________. | \frac{\pi}{6} | 84.375 |
28,923 | If $a$ and $b$ are positive integers and the equation \( ab - 8a + 7b = 395 \) holds true, what is the minimal possible value of \( |a - b| \)? | 15 | 29.6875 |
28,924 | For some positive integer $n$, the number $120n^3$ has $120$ positive integer divisors, including $1$ and the number $120n^3$. How many positive integer divisors does the number $64n^4$ have? | 375 | 0.78125 |
28,925 | Suppose in a right triangle where angle \( Q \) is at the origin and \( \cos Q = 0.5 \). If the length of \( PQ \) is \( 10 \), what is \( QR \)? | 20 | 26.5625 |
28,926 | What is the smallest positive integer with exactly 20 positive divisors? | 144 | 0 |
28,927 | Three not necessarily integer numbers are given. If each of these numbers is increased by 1, the product of the numbers also increases by 1. If each of these numbers is increased by 2, their product also increases by 2. Find these numbers. | -1 | 0.78125 |
28,928 | Let $\Omega$ be a unit circle and $A$ be a point on $\Omega$ . An angle $0 < \theta < 180^\circ$ is chosen uniformly at random, and $\Omega$ is rotated $\theta$ degrees clockwise about $A$ . What is the expected area swept by this rotation? | 2\pi | 0.78125 |
28,929 | Mike and Alain play a game in which each player is equally likely to win. The first player to win three games becomes the champion, and no further games are played. If Mike has won the first game, what is the probability that Mike becomes the champion? | $\frac{11}{16}$ | 0 |
28,930 | Given that point $P$ is any point on the curve $(x-1)^2+(y-2)^2=9$ with $y \geq 2$, find the minimum value of $x+ \sqrt {3}y$. | 2\sqrt{3} - 2 | 14.84375 |
28,931 | An unpainted cone has radius \( 3 \mathrm{~cm} \) and slant height \( 5 \mathrm{~cm} \). The cone is placed in a container of paint. With the cone's circular base resting flat on the bottom of the container, the depth of the paint in the container is \( 2 \mathrm{~cm} \). When the cone is removed, its circular base and the lower portion of its lateral surface are covered in paint. The fraction of the total surface area of the cone that is covered in paint can be written as \( \frac{p}{q} \) where \( p \) and \( q \) are positive integers with no common divisor larger than 1. What is the value of \( p+q \)?
(The lateral surface of a cone is its external surface not including the circular base. A cone with radius \( r \), height \( h \), and slant height \( s \) has lateral surface area equal to \( \pi r s \).) | 59 | 7.8125 |
28,932 | Define a function $f$ from nonnegative integers to real numbers, with $f(1) = 1$ and the functional equation:
\[ f(m+n) + f(m-n) = 3(f(m) + f(n)) \]
for all nonnegative integers $m \ge n$. Determine $f(10)$. | 100 | 12.5 |
28,933 | The equation
\[(x - \sqrt[3]{7})(x - \sqrt[3]{29})(x - \sqrt[3]{61}) = \frac{1}{5}\]
has three distinct solutions $u,$ $v,$ and $w.$ Calculate the value of $u^3 + v^3 + w^3.$ | 97.6 | 0 |
28,934 | Two concentric squares share the same center $O$, each with sides of length 2. The length of the line segment $\overline{AB}$ is $\frac{1}{3}$, and the area of the octagon $ABCDEFGH$ formed by the intersection of lines drawn from the vertices of the inner square to the midpoints of sides of the outer square is sought. Express this area as a fraction $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
[asy] //code taken from thread for problem
real alpha = 15;
pair W=dir(225), X=dir(315), Y=dir(45), Z=dir(135), O=origin;
pair w=dir(alpha)*W, x=dir(alpha)*X, y=dir(alpha)*Y, z=dir(alpha)*Z;
draw(W--X--Y--Z--cycle^^w--x--y--z--cycle);
pair A=intersectionpoint(Y--Z, y--z), C=intersectionpoint(Y--X, y--x), E=intersectionpoint(W--X, w--x), G=intersectionpoint(W--Z, w--z), B=intersectionpoint(Y--Z, y--x), D=intersectionpoint(Y--X, w--x), F=intersectionpoint(W--X, w--z), H=intersectionpoint(W--Z, y--z);
dot(O);
label("$O$", O, SE);
label("$A$", A, dir(O--A));
label("$B$", B, dir(O--B));
label("$C$", C, dir(O--C));
label("$D$", D, dir(O--D));
label("$E$", E, dir(O--E));
label("$F$", F, dir(O--F));
label("$G$", G, dir(O--G));
label("$H$", H, dir(O--H));[/asy] | 11 | 3.90625 |
28,935 | Given the parabola \(\Gamma: y^{2}=8 x\) with focus \(F\), a line \(l\) passing through \(F\) intersects parabola \(\Gamma\) at points \(A\) and \(B\). Tangents to parabola \(\Gamma\) at \(A\) and \(B\) intersect the \(y\)-axis at points \(P\) and \(Q\) respectively. Find the minimum area of the quadrilateral \(APQB\). | 12 | 0 |
28,936 | Hari is obsessed with cubics. He comes up with a cubic with leading coefficient 1, rational coefficients and real roots $0 < a < b < c < 1$ . He knows the following three facts: $P(0) = -\frac{1}{8}$ , the roots form a geometric progression in the order $a,b,c$ , and \[ \sum_{k=1}^{\infty} (a^k + b^k + c^k) = \dfrac{9}{2} \] The value $a + b + c$ can be expressed as $\frac{m}{n}$ , where $m,n$ are relatively prime positive integers. Find $m + n$ .
*Proposed by Akshar Yeccherla (TopNotchMath)* | 19 | 9.375 |
28,937 | In the rectangular coordinate system $(xOy)$, the curve $C\_1$: $ \begin{cases} x=a\cos φ \ y=b\sin φ\end{cases}(φ)$ is a parameter, where $(a > b > 0)$, and in the polar coordinate system with $O$ as the pole and the positive semi-axis of $x$ as the polar axis, the curve $C\_2$: $ρ=2\cos θ$, the ray $l$: $θ=α(ρ≥0)$, intersects the curve $C\_1$ at point $P$, and when $α=0$, the ray $l$ intersects the curve $C\_2$ at points $O$ and $Q$, $(|PQ|=1)$; when $α= \dfrac {π}{2}$, the ray $l$ intersects the curve $C\_2$ at point $O$, $(|OP|= \sqrt {3})$.
(I) Find the general equation of the curve $C\_1$;
(II) If the line $l′$: $ \begin{cases} x=-t \ y= \sqrt {3}t\end{cases}(t)$ is a parameter, $t≠0$, intersects the curve $C\_2$ at point $R$, and $α= \dfrac {π}{3}$, find the area of $△OPR$. | \dfrac {3 \sqrt {30}}{20} | 0 |
28,938 | Emma's telephone number is $548-1983$ and her apartment number contains different digits. The sum of the digits in her four-digit apartment number is the same as the sum of the digits in her phone number. What is the lowest possible value for Emma’s apartment number? | 9876 | 2.34375 |
28,939 | In an enterprise, no two employees have jobs of the same difficulty and no two of them take the same salary. Every employee gave the following two claims:
(i) Less than $12$ employees have a more difficult work;
(ii) At least $30$ employees take a higher salary.
Assuming that an employee either always lies or always tells the truth, find how many employees are there in the enterprise. | 42 | 24.21875 |
28,940 | Altitudes $\overline{AP}$ and $\overline{BQ}$ of an acute triangle $\triangle ABC$ intersect at point $H$. If $HP=8$ and $HQ=3$, then calculate $(BP)(PC)-(AQ)(QC)$. | 55 | 17.96875 |
28,941 | In three sugar bowls, there is an equal number of sugar cubes, and the cups are empty. If each cup receives $\frac{1}{18}$ of the contents of each sugar bowl, then each sugar bowl will have 12 more sugar cubes than each cup.
How many sugar cubes were originally in each sugar bowl? | 36 | 7.03125 |
28,942 | Given that $| \overrightarrow{a}|=1$, $| \overrightarrow{b}|= \sqrt {2}$, and $\overrightarrow{a} \perp ( \overrightarrow{a}- \overrightarrow{b})$, find the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac {\pi}{4} | 96.875 |
28,943 | Let \( p(x) \) be a monic quartic polynomial such that \( p(1) = 1, p(2) = 9, p(3) = 28, \) and \( p(4) = 65. \) Find \( p(5) \). | 126 | 22.65625 |
28,944 | What is the largest number, with all different digits, whose digits add up to 19? | 982 | 67.1875 |
28,945 | Two circles are externally tangent. Lines $\overline{PAB}$ and $\overline{PA'B'}$ are common tangents with points $A$, $A'$ on the smaller circle and $B$, $B'$ on the larger circle. If $PA=AB=5$ and the radius of the larger circle is 3 times the radius of the smaller circle, find the area of the smaller circle. | 5\pi | 0.78125 |
28,946 | A palindrome between $10000$ and $100000$ is chosen at random. What is the probability that it is divisible by $11$? | \frac{41}{450} | 18.75 |
28,947 | A cardboard box in the shape of a rectangular parallelopiped is to be enclosed in a cylindrical container with a hemispherical lid. If the total height of the container from the base to the top of the lid is $60$ centimetres and its base has radius $30$ centimetres, find the volume of the largest box that can be completely enclosed inside the container with the lid on. | 108000 | 43.75 |
28,948 | Let $G$ be the centroid of quadrilateral $ABCD$. If $GA^2 + GB^2 + GC^2 + GD^2 = 116$, find the sum $AB^2 + AC^2 + AD^2 + BC^2 + BD^2 + CD^2$. | 464 | 51.5625 |
28,949 | The centers of the three circles A, B, and C are collinear with the center of circle B lying between the centers of circles A and C. Circles A and C are both externally tangent to circle B, and the three circles share a common tangent line. Given that circle A has radius $12$ and circle B has radius $42,$ find the radius of circle C. | 147 | 4.6875 |
28,950 | My frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score. The game ends after one of the two teams scores three points (total, not necessarily consecutive). If every possible sequence of scores is equally likely, what is the expected score of the losing team? | 3/2 | 4.6875 |
28,951 | What is the sum of all positive integers less than 500 that are fourth powers of even perfect squares? | 272 | 35.9375 |
28,952 | A function $f$ is defined for all real numbers and satisfies the conditions $f(3+x) = f(3-x)$ and $f(8+x) = f(8-x)$ for all $x$. If $f(0) = 0$, determine the minimum number of roots that $f(x) = 0$ must have in the interval $-500 \leq x \leq 500$. | 201 | 10.15625 |
28,953 | Given that $| \overrightarrow{a}|=5$, $| \overrightarrow{b}|=4$, and $\overrightarrow{a} \cdot \overrightarrow{b}=-10$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ (denoted as $\langle \overrightarrow{a}, \overrightarrow{b} \rangle$). | \frac{2\pi}{3} | 91.40625 |
28,954 | In December 2022, $550$ cm of snow fell in Winterpark, Canada. What was the average snowfall in cm per minute during that month?
A) $\frac{550}{31\times 24 \times 60}$
B) $\frac{550 \times 31}{24 \times 60}$
C) $\frac{550 \times 24 \times 60}{31}$
D) $\frac{31 \times 24 \times 60}{550}$
E) $550 \times 31 \times 24 \times 60$ | \frac{550}{31\times 24 \times 60} | 86.71875 |
28,955 | Let \( m \) be the smallest positive integer that is a multiple of \( 100 \) and has exactly \( 100 \) positive integral divisors, including 1 and itself. Find \( \frac{m}{100} \). | 4050 | 2.34375 |
28,956 | A triangle has sides of length $48$ , $55$ , and $73$ . A square is inscribed in the triangle such that one side of the square lies on the longest side of the triangle, and the two vertices not on that side of the square touch the other two sides of the triangle. If $c$ and $d$ are relatively prime positive integers such that $c/d$ is the length of a side of the square, find the value of $c+d$ . | 200689 | 6.25 |
28,957 | In triangle $XYZ$, $XY=25$ and $XZ=14$. The angle bisector of $\angle X$ intersects $YZ$ at point $E$, and point $N$ is the midpoint of $XE$. Let $Q$ be the point of the intersection of $XZ$ and $YN$. The ratio of $ZQ$ to $QX$ can be expressed in the form $\dfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. | 39 | 18.75 |
28,958 | Compute the number of ordered triples of integers $(a,b,c)$ between $1$ and $12$ , inclusive, such that, if $$ q=a+\frac{1}{b}-\frac{1}{b+\frac{1}{c}}, $$ then $q$ is a positive rational number and, when $q$ is written in lowest terms, the numerator is divisible by $13$ .
*Proposed by Ankit Bisain* | 132 | 54.6875 |
28,959 | Vertex E of equilateral triangle ∆ABE is inside square ABCD. F is the intersection point of diagonal BD and line segment AE. If AB has length √(1+√3), calculate the area of ∆ABF. | \frac{\sqrt{3}}{2} | 40.625 |
28,960 | $ABCDEFGH$ is a cube. Find $\cos \angle GAC$. | \frac{\sqrt{3}}{3} | 4.6875 |
28,961 | Six congruent copies of the parabola $y = x^2$ are arranged in the plane so that each vertex is tangent to a circle, and each parabola is tangent to its two neighbors. Assume that each parabola is tangent to a line that forms a $45^\circ$ angle with the x-axis. Find the radius of the circle. | \frac{1}{4} | 1.5625 |
28,962 | Dima calculated the factorials of all natural numbers from 80 to 99, found the reciprocals of them, and printed the resulting decimal fractions on 20 endless ribbons (for example, the last ribbon had the number \(\frac{1}{99!}=0. \underbrace{00\ldots00}_{155 \text{ zeros}} 10715 \ldots \) printed on it). Sasha wants to cut a piece from one ribbon that contains \(N\) consecutive digits without any decimal points. For what largest \(N\) can he do this so that Dima cannot determine from this piece which ribbon Sasha spoiled? | 155 | 25 |
28,963 | Calculate the distance between the foci of the ellipse defined by the equation
\[\frac{x^2}{36} + \frac{y^2}{9} = 9.\] | 2\sqrt{3} | 0.78125 |
28,964 | Given the function $f(x) = ax^7 + bx - 2$, if $f(2008) = 10$, then the value of $f(-2008)$ is. | -12 | 0 |
28,965 | The perimeter of a rectangle exceeds the perimeter of an equilateral triangle by 1950 cm. The length of each side of the rectangle exceeds the length of each side of the triangle by \( d \ \text{cm} \). All sides of the triangle are equal, and the rectangle is such that its length is triple that of its width. The triangle has a perimeter greater than 0. Determine how many positive integers are NOT possible values for \( d \). | 650 | 10.9375 |
28,966 | A box of chocolates in the shape of a cuboid was full of chocolates arranged in rows and columns. Míša ate some of them, and the remaining chocolates were rearranged to fill three entire rows completely, except for one space. Míša ate the remaining chocolates from another incomplete row. Then he rearranged the remaining chocolates and filled five columns completely, except for one space. He again ate the chocolates from the incomplete column. In the end, one-third of the original number of chocolates remained in the box. Determine:
a) How many chocolates were there in the entire box originally?
b) How many chocolates did Míša eat before the first rearrangement? | 25 | 0 |
28,967 | It is given polygon with $2013$ sides $A_{1}A_{2}...A_{2013}$ . His vertices are marked with numbers such that sum of numbers marked by any $9$ consecutive vertices is constant and its value is $300$ . If we know that $A_{13}$ is marked with $13$ and $A_{20}$ is marked with $20$ , determine with which number is marked $A_{2013}$ | 67 | 0 |
28,968 | Let the rational number $p/q$ be closest to but not equal to $22/7$ among all rational numbers with denominator $< 100$ . What is the value of $p - 3q$ ? | 14 | 98.4375 |
28,969 | In a right triangle, instead of having one $90^{\circ}$ angle and two small angles sum to $90^{\circ}$, consider now the acute angles are $x^{\circ}$, $y^{\circ}$, and a smaller angle $z^{\circ}$ where $x$, $y$, and $z$ are all prime numbers, and $x^{\circ} + y^{\circ} + z^{\circ} = 90^{\circ}$. Determine the largest possible value of $y$ if $y < x$ and $y > z$. | 47 | 25 |
28,970 | Given Professor Chen has ten different language books, including two Arabic, four German, and four Spanish books, arranged on the shelf so that the Arabic books are together, the Spanish books are together, and two of the German books are treated as indistinguishable, calculate the number of ways to arrange the ten books. | 576 | 2.34375 |
28,971 | Let $k$ be a positive integer. Marco and Vera play a game on an infinite grid of square cells. At the beginning, only one cell is black and the rest are white.
A turn in this game consists of the following. Marco moves first, and for every move he must choose a cell which is black and which has more than two white neighbors. (Two cells are neighbors if they share an edge, so every cell has exactly four neighbors.) His move consists of making the chosen black cell white and turning all of its neighbors black if they are not already. Vera then performs the following action exactly $k$ times: she chooses two cells that are neighbors to each other and swaps their colors (she is allowed to swap the colors of two white or of two black cells, though doing so has no effect). This, in totality, is a single turn. If Vera leaves the board so that Marco cannot choose a cell that is black and has more than two white neighbors, then Vera wins; otherwise, another turn occurs.
Let $m$ be the minimal $k$ value such that Vera can guarantee that she wins no matter what Marco does. For $k=m$ , let $t$ be the smallest positive integer such that Vera can guarantee, no matter what Marco does, that she wins after at most $t$ turns. Compute $100m + t$ .
*Proposed by Ashwin Sah* | 203 | 6.25 |
28,972 | There is a certain regularity in the operation between rational numbers and irrational numbers. For example, if $a$ and $b$ are rational numbers, and $a(\pi +3)+b=0$, then $a=0$, $b=0$. Given that $m$ and $n$ are rational numbers:<br/>$(1)$ If $(m-3)×\sqrt{6}+n-3=0$, then the square root of $mn$ is ______;<br/>$(2)$ If $(2+\sqrt{3})m-(1-\sqrt{3})n=5$, where $m$ and $n$ are square roots of $x$, then the value of $x$ is ______. | \frac{25}{9} | 24.21875 |
28,973 | Determine the number of six-digit palindromes. | 9000 | 7.8125 |
28,974 |
The extensions of sides \(AD\) and \(BC\) of a convex quadrilateral \(ABCD\) intersect at point \(M\), and the extensions of sides \(AB\) and \(CD\) intersect at point \(O\). Segment \(MO\) is perpendicular to the angle bisector of \(\angle AOD\). Find the ratio of the areas of triangle \(AOD\) and quadrilateral \(ABCD\), given that \(OA = 12\), \(OD = 8\), and \(CD = 2\). | 2:1 | 0 |
28,975 | Given an ellipse with $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), its left focal point is $F_1(-1, 0)$, and vertex P on the ellipse satisfies $\angle PF_1O = 45^\circ$ (where O is the origin).
(1) Determine the values of $a$ and $b$;
(2) Given that line $l_1: y = kx + m_1$ intersects the ellipse at points A and B, and line $l_2: y = kx + m_2$ ($m_1 \neq m_2$) intersects the ellipse at points C and D, and $|AB| = |CD|$:
① Find the value of $m_1 + m_2$;
② Determine the maximum value of the area S of quadrilateral ABCD. | 2\sqrt{2} | 17.96875 |
28,976 | Let $(a_1,a_2,a_3,\ldots,a_{14})$ be a permutation of $(1,2,3,\ldots,14)$ where $a_1 > a_2 > a_3 > a_4 > a_5 > a_6 > a_7$ and $a_7 < a_8 < a_9 < a_{10} < a_{11} < a_{12} < a_{13} < a_{14}$. An example of such a permutation is $(7,6,5,4,3,2,1,8,9,10,11,12,13,14)$. Determine the number of such permutations. | 1716 | 1.5625 |
28,977 | On some cells of a 10x10 board, there is a flea. Every minute, the fleas jump simultaneously, each one to a neighboring cell (adjacent by side). Each flea jumps strictly in one of the four directions parallel to the board's sides and maintains this direction as long as possible; otherwise, it changes to the opposite direction. Barbos the dog observed the fleas for an hour and never saw two fleas on the same cell. What is the maximum number of fleas that could have been jumping on the board? | 40 | 0.78125 |
28,978 | 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}{12} | 84.375 |
28,979 | Rhombus $PQRS$ has sides of length $4$ and $\angle Q = 150^\circ$. Region $T$ is defined as the area inside the rhombus that is closer to vertex $Q$ than to any of the other vertices $P$, $R$, or $S$. Calculate the area of region $T$.
A) $\frac{2\sqrt{3}}{3}$
B) $\frac{4\sqrt{3}}{3}$
C) $\frac{6\sqrt{3}}{3}$
D) $\frac{8\sqrt{3}}{9}$
E) $\frac{10\sqrt{3}}{3}$ | \frac{8\sqrt{3}}{9} | 2.34375 |
28,980 | Two spheres are inscribed in a dihedral angle such that they touch each other. The radius of one sphere is 4 times that of the other, and the line connecting the centers of the spheres forms an angle of \(60^\circ\) with the edge of the dihedral angle. Find the measure of the dihedral angle. Provide the cosine of this angle, rounded to two decimal places if necessary. | 0.04 | 0 |
28,981 | Let $A$, $B$, $C$, and $D$ be vertices of a regular tetrahedron where each edge is 1 meter. A bug starts at vertex $A$ and at each vertex chooses randomly among the three incident edges to move along. Compute the probability $p$ that the bug returns to vertex $A$ after exactly 10 meters, where $p = \frac{n}{59049}$. | 4921 | 0 |
28,982 | Compute
\[\sin^2 6^\circ + \sin^2 12^\circ + \sin^2 18^\circ + \dots + \sin^2 174^\circ.\] | \frac{31}{2} | 0 |
28,983 | In the Cartesian coordinate system, the coordinates of the two foci of an ellipse are $F_{1}(-2\sqrt{2},0)$ and $F_{2}(2\sqrt{2},0)$. The minimum distance from a point on ellipse $C$ to the right focus is $3-2\sqrt{2}$.
$(1)$ Find the equation of ellipse $C$;
$(2)$ Suppose a line with a slope of $-2$ intersects curve $C$ at points $E$ and $F$. Find the equation of the trajectory of the midpoint $N$ of segment $EF$;
$(3)$ Suppose a line passing through point $F_{1}(-2\sqrt{2},0)$ intersects curve $C$ and forms a chord segment $PQ$. Find the maximum area of $\triangle PQO$ ($O$ is the origin). | \dfrac{3}{2} | 5.46875 |
28,984 | $-2^{3}+|2-3|-2\times \left(-1\right)^{2023}$. | -5 | 95.3125 |
28,985 | The graph of the function $f(x)=\sqrt{3}\cos 2x-\sin 2x$ can be obtained by translating the graph of the function $f(x)=2\sin 2x$ by an unspecified distance. | \dfrac{\pi}{6} | 16.40625 |
28,986 | In a certain football invitational tournament, 16 cities participate, with each city sending two teams, Team A and Team B. According to the competition rules, after several days of matches, it was found that aside from Team A from city $A$, the number of matches already played by each of the other teams was different. Find the number of matches already played by Team B from city $A$. | 15 | 69.53125 |
28,987 | Divide 6 volunteers into 4 groups, with two groups having 2 people each and the other two groups having 1 person each, to serve at four different pavilions of the World Expo. How many different allocation schemes are there? (Answer with a number). | 1080 | 85.15625 |
28,988 | Suppose in a right triangle LMN, where angle M is the right angle, $\cos N = \frac{4}{5}$ and length LM is given by 20 units. What is the length of LN? | 25 | 54.6875 |
28,989 | In a particular year, the price of a commodity increased by $30\%$ in January, decreased by $10\%$ in February, increased by $20\%$ in March, decreased by $y\%$ in April, and finally increased by $15\%$ in May. Given that the price of the commodity at the end of May was the same as it had been at the beginning of January, determine the value of $y$. | 38 | 10.15625 |
28,990 | Let $g(x)$ be the function defined on $-2 \leq x \leq 2$ by the formula $$g(x) = 2 - \sqrt{4-x^2}.$$ This is a vertically stretched version of the previously given function. If a graph of $x=g(y)$ is overlaid on the graph of $y=g(x)$, then one fully enclosed region is formed by the two graphs. What is the area of that region, rounded to the nearest hundredth? | 2.28 | 14.0625 |
28,991 | Given the function $f(x)=\ln x-xe^{x}+ax$ where $a\in \mathbb{R}$.
(Ⅰ) If the function $f(x)$ is monotonically decreasing on $\left[1,+\infty \right)$, find the range of real number $a$.
(Ⅱ) If $a=1$, find the maximum value of $f(x)$. | -1 | 4.6875 |
28,992 | A certain unit decides to invest $3200$ yuan to build a warehouse (in the shape of a rectangular prism) with a constant height. The back wall will be built reusing the old wall at no cost, the front will be made of iron grilles at a cost of $40$ yuan per meter in length, and the two side walls will be built with bricks at a cost of $45$ yuan per meter in length. The top will have a cost of $20$ yuan per square meter. Let the length of the iron grilles be $x$ meters and the length of one brick wall be $y$ meters. Find:<br/>$(1)$ Write down the relationship between $x$ and $y$;<br/>$(2)$ Determine the maximum allowable value of the warehouse area $S$. In order to maximize $S$ without exceeding the budget, how long should the front iron grille be designed? | 15 | 62.5 |
28,993 | Given that $\cos (α-β)= \frac{3}{5}$, $\sin β=- \frac{5}{13}$, and $α∈\left( \left. 0, \frac{π}{2} \right. \right)$, $β∈\left( \left. - \frac{π}{2},0 \right. \right)$, find the value of $\sin α$. | \frac{33}{65} | 32.8125 |
28,994 | Find the largest value of the expression $\frac{p}{R}\left( 1- \frac{r}{3R}\right)$ , where $p,R, r$ is, respectively, the perimeter, the radius of the circumscribed circle and the radius of the inscribed circle of a triangle. | \frac{5\sqrt{3}}{2} | 91.40625 |
28,995 | Given the set $A={3,3^{2},3^{3},…,3^{n}}$ $(n\geqslant 3)$, choose three different numbers from it and arrange them in a certain order to form a geometric sequence. Denote the number of geometric sequences that satisfy this condition as $f(n)$.
(I) Find $f(5)=$ _______ ;
(II) If $f(n)=220$, find $n=$ _______ . | 22 | 15.625 |
28,996 | Line $l_1$ has equation $4x - 3y = 2$ and passes through point $D = (-2, -3)$. Line $l_2$ has equation $y = 2$ and intersects line $l_1$ at point $E$. Line $l_3$ has a positive slope, passes through point $D$, and meets $l_2$ at point $F$. The area of $\triangle DEF$ is $6$. What is the slope of $l_3$? | \frac{25}{32} | 21.875 |
28,997 | In an isosceles triangle \(ABC\) with base \(AC\), point \(D\) divides side \(BC\) in the ratio \(2:1\) from vertex \(B\), and point \(E\) is the midpoint of side \(AB\). It is known that the median \(CQ\) of triangle \(CED\) is equal to \(\frac{\sqrt{23}}{2}\), and \(DE = \frac{\sqrt{23}}{2}\). Find the radius of the circumcircle of triangle \(ABC\). | 12/5 | 0 |
28,998 | Alice and Ali each select a positive integer less than 250. Alice's number is a multiple of 25, and Ali's number is a multiple of 30. What is the probability that they selected the same number? Express your answer as a common fraction. | \frac{1}{80} | 0 |
28,999 | The average age of 8 people in Room C is 35. The average age of 6 people in Room D is 30. Calculate the average age of all people when the two groups are combined. | \frac{460}{14} | 0 |
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