task_type stringclasses 1
value | problem stringlengths 23 3.94k | answer stringlengths 1 231 | problem_tokens int64 8 1.39k | answer_tokens int64 1 200 |
|---|---|---|---|---|
math | Three, (50 points) Try to find the last two non-zero digits of 2011!.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 44 | 49 | 2 |
math | 5.2. A smooth sphere with a radius of 1 cm was dipped in blue paint and launched between two perfectly smooth concentric spheres with radii of 4 cm and 6 cm, respectively (this sphere was outside the smaller sphere but inside the larger one). Upon touching both spheres, the sphere leaves a blue trail. During its moveme... | 60.75 | 133 | 5 |
math | Solve the following equation:
$$
\sqrt[3]{x}+\sqrt[3]{9-x}=3
$$ | x_1=1,x_2=8 | 26 | 10 |
math | 4. A fixed point of a function $f$ is a value of $x$ for which $f(x)=x$. Let $f$ be the quadratic function defined by $f(x)=x^{2}-c x+c$ where $c \in \mathbb{R}$. Find, in interval notation, the set consisting of all values of $c$ for which $f \circ f$ has four distinct fixed points. | (-\infty,-1)\cup(3,+\infty) | 90 | 15 |
math | Example 1 Given $x^{2}+x+1=0$, try to find the value of the rational expression $x^{14}+\frac{1}{x^{14}}$. | -1 | 42 | 2 |
math | Example 6 When the volume of a cylindrical metal beverage can is fixed, how should its height and base radius be selected to minimize the material used? | 2R | 30 | 2 |
math | 120. Find $\lim _{x \rightarrow 3}\left(x^{2}-7 x+4\right)$. | -8 | 28 | 2 |
math | 18. Let $a, b, c$ be positive real numbers, find the value of $k$ such that
$$\left(k+\frac{a}{b+c}\right)\left(k+\frac{b}{c+a}\right)\left(k+\frac{c}{a+b}\right) \geqslant\left(k+\frac{1}{2}\right)^{3}$$
(2009 Vietnam National Team Selection Exam Problem) | \frac{\sqrt{5}-1}{4} | 98 | 11 |
math | ## Task A-2.1.
Determine all pairs of real numbers $(x, y)$ that satisfy the system
$$
\begin{aligned}
& x+y^{2}=y^{3} \\
& y+x^{2}=x^{3}
\end{aligned}
$$ | (x,y)\in{(0,0),(\frac{1+\sqrt{5}}{2},\frac{1+\sqrt{5}}{2}),(\frac{1-\sqrt{5}}{2},\frac{1-\sqrt{5}}{2})} | 59 | 58 |
math | Task A-2.5. (8 points)
Let $x_{1}, x_{2}$ be the distinct solutions of the equation $2 x^{2}-3 x+4=0$. Calculate $\frac{1}{x_{1}^{3}}+\frac{1}{x_{2}^{3}}$. | -\frac{45}{64} | 67 | 9 |
math | (N1 2002) Find the smallest integer $t$ such that there exist strictly positive integers $x_{1}, \ldots, x_{t}$, satisfying $x_{1}^{3}+\ldots+x_{t}^{3}=2002^{2002}$. | 4 | 66 | 1 |
math | Solve the following system of equations in the set of real numbers:
$$
|y-x|-\frac{|x|}{x}+1=0 ;|2 x-y|+|x+y-1|+|x-y|+y-1=0
$$ | 0<x=y\leq0.5 | 58 | 9 |
math | 6. (NET 4) ${ }^{\mathrm{IMO} 3} \mathrm{~A}$ rectangular box can be filled completely with unit cubes. If one places cubes with volume 2 in the box such that their edges are parallel to the edges of the box, one can fill exactly $40 \%$ of the box. Determine all possible (interior) sizes of the box. | (2,3,5) \text{ and } (2,5,6) | 83 | 19 |
math | 7.4. The steamship "Raritet" after leaving the city moves at a constant speed for three hours, then drifts for an hour, moving with the current, then moves for three hours at the same speed, and so on. If the steamship starts its journey from city A and heads to city B, it takes 10 hours. If it starts from city B and h... | 60 | 110 | 2 |
math | 11.171. A cube is inscribed in a hemisphere of radius $R$ such that four of its vertices lie on the base of the hemisphere, while the other four vertices are located on its spherical surface. Calculate the volume of the cube. | \frac{2R^{3}\sqrt{6}}{9} | 53 | 15 |
math | Determine all positive integers $n$ such that $4k^2 +n$ is a prime number for all non-negative integer $k$ smaller than $n$. | n = 3, 7 | 34 | 8 |
math | Riquinho distributed $R \$ 1000.00$ reais among his friends: Antônio, Bernardo, and Carlos in the following manner: he gave, successively, 1 real to Antônio, 2 reais to Bernardo, 3 reais to Carlos, 4 reais to Antônio, 5 reais to Bernardo, etc. How much did Bernardo receive? | 345 | 91 | 3 |
math | Given $n$ integers $a_{1}=1, a_{2}, a_{3}, \ldots, a_{n}$, and $a_{i} \leq a_{i+1} \leq 2 a_{i}(i=1,2, \ldots, n-1)$, and the sum of all numbers is even. Can these numbers be divided into two groups such that the sums of the numbers in these groups are equal? | S_{1}=S_{2} | 98 | 8 |
math | Task A-3.4. (4 points)
Determine the minimum value of the expression
$$
\sin (x+3)-\sin (x+1)-2 \cos (x+2)
$$
for $x \in \mathbb{R}$. | 2(\sin1-1) | 58 | 7 |
math | 5. A truck and a bus started towards each other simultaneously from two entrances to the highway. Assume that the truck maintains a constant average speed and covers $27 \mathrm{~km}$ in 18 minutes. Assume that the bus also maintains its own constant average speed, which allows it to cover $864 \mathrm{~m}$ in 28.8 sec... | 544.5\mathrm{~} | 121 | 10 |
math | 7. The equation
$$
3 x^{3}+2 \sqrt{2} x^{2}-(17-9 \sqrt{2}) x-(6-5 \sqrt{2})=0
$$
has solutions $x_{1}=$ $\qquad$ ,$x_{2}=$ $\qquad$ ,$x_{3}=$ $\qquad$ | x_{1}=\frac{\sqrt{2}}{3}, x_{2}=\sqrt{2}-1, x_{3}=1-2 \sqrt{2} | 80 | 37 |
math | Let $x$ and $y$ be two non-zero numbers such that $x^{2} + x y + y^{2} = 0$ ($x$ and $y$ are complex numbers, but that's not too important). Find the value of
$$
\left(\frac{x}{x+y}\right)^{2013} + \left(\frac{y}{x+y}\right)^{2013}
$$ | -2 | 94 | 2 |
math | Let $\mathbb{N}=\{1,2,3, \ldots\}$ be the set of all positive integers. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for any positive integers $a$ and $b$, the following two conditions hold:
(1) $f(a b)=f(a) f(b)$, and
(2) at least two of the numbers $f(a), f(b)$ and $f(a+b)$ are equal.
Proposed... | f(n)=a^{v_{p}(n)} | 172 | 11 |
math | Compute the smallest value $C$ such that the inequality $$x^2(1+y)+y^2(1+x)\le \sqrt{(x^4+4)(y^4+4)}+C$$ holds for all real $x$ and $y$. | 4 | 56 | 1 |
math | 5.5. The vector $\overline{O A}$ forms angles with the axes $O x, O y, O z$ that are respectively equal to $\alpha=\frac{\pi}{3}, \beta=\frac{\pi}{3}, \gamma=\frac{\pi}{4}$; the point $B$ has coordinates $(-2; -2; -2 \sqrt{2})$. Find the angle between the vectors $\overline{O A}$ and $\overline{O B}$. | 180 | 105 | 3 |
math | Let's determine those decimal numbers whose square ends in 76. | 24,26,74,76 | 14 | 11 |
math | 6.014. $\frac{4}{x^{2}+4}+\frac{5}{x^{2}+5}=2$. | 0 | 32 | 1 |
math | 4.53 For a given positive integer $n$, let $p(n)$ denote the product of the non-zero digits of $n$ (if $n$ is a single digit, then $p(n)$ is that digit). If $S = p(1) + p(2) + p(3) + \cdots + p(999)$, what is the largest prime factor of $S$?
(12th American Invitational Mathematics Examination, 1994) | 103 | 105 | 3 |
math | 21.3.9 $\star \star$ Let $a_{1}, a_{2}, \cdots, a_{n}$ be a permutation of $1,2, \cdots, n$, satisfying $a_{i} \neq i(i=1$, $2, \cdots, n)$, there are $D_{n}$ such groups of $a_{1}, a_{2}, \cdots, a_{n}$, find $D_{n}$. | D_{n}=n!\cdot\sum_{k=0}^{n}\frac{(-1)^{k}}{k!} | 103 | 29 |
math | ## 5. Middle Number
Arrange in ascending order all three-digit numbers less than 550 whose hundreds digit is equal to the product of the other two digits. Among these arranged numbers, which number is in the middle?
## Result: $\quad 331$ | 331 | 57 | 3 |
math | You roll three fair six-sided dice. Given that the highest number you rolled is a $5$, the expected value of the sum of the three dice can be written as $\tfrac ab$ in simplest form. Find $a+b$. | 706 | 50 | 3 |
math | Compute the sum $S=\sum_{i=0}^{101} \frac{x_{i}^{3}}{1-3 x_{i}+3 x_{i}^{2}}$ for $x_{i}=\frac{i}{101}$. | 51 | 59 | 2 |
math | 5. Solve the system of equations:
$$
\left\{\begin{array}{l}
\frac{1}{\sqrt{1+2 x^{2}}}+\frac{1}{\sqrt{1+2 y^{2}}}=\frac{2}{\sqrt{1+2 x y}}, \\
\sqrt{x(1-2 x)}+\sqrt{y(1-2 y)}=\frac{2}{9}
\end{array} .\right.
$$ | \frac{1}{4}\\frac{\sqrt{73}}{36} | 101 | 18 |
math | 2. The equations $x^{2}-a=0$ and $3 x^{4}-48=0$ have the same real solutions. What is the value of $a$ ? | 4 | 40 | 1 |
math | 1. Find all natural numbers $n$ such that the number
$$
1!+2!+3!+\ldots+n!
$$
is a perfect square. (Here: $k!=1 \cdot 2 \cdot \ldots \cdot k, 1!=1$) | 1,3 | 62 | 3 |
math | ## Problem Statement
Write the equation of the plane passing through point $A$ and perpendicular to vector $\overrightarrow{B C}$.
$A(-3 ; 6 ; 4)$
$B(8 ;-3 ; 5)$
$C(10 ;-3 ; 7)$ | x+z-1=0 | 62 | 6 |
math | 9. Let $A B C$ be a triangle, and let $B C D E, C A F G, A B H I$ be squares that do not overlap the triangle with centers $X, Y, Z$ respectively. Given that $A X=6, B Y=7$, and $C Z=8$, find the area of triangle $X Y Z$. | \frac{21\sqrt{15}}{4} | 79 | 14 |
math | 20. Find the sum of the fourth powers of the real roots of the equation
$$
x^{4}-1000 x^{2}+2017=0
$$ | 1991932 | 41 | 7 |
math | 6. The sum of $n$ terms of the geometric progression $\left\{b_{n}\right\}$ is 6 times less than the sum of their reciprocals. Find the product $b_{1} b_{n}$. | \frac{1}{6} | 51 | 7 |
math | G3.2 There are $R$ zeros at the end of $\underbrace{99 \ldots 9}_{2009 \text { of }} \times \underbrace{99 \ldots 9}_{2009 \text { of } 9^{\prime} s}+1 \underbrace{99 \ldots 9}_{2009 \text { of } 9^{\prime} s}$, find the value of $R$. | 4018 | 105 | 4 |
math | 2. Let $A$ be the set of all two-digit positive integers $n$ for which the number obtained by erasing its last digit is a divisor of $n$. How many elements does $A$ have? | 32 | 45 | 2 |
math | 6.344 Find the coefficients $a$ and $b$ of the equation $x^{4}+x^{3}-18 x^{2}+a x+b=0$, if among its roots there are three equal integer roots. | =-52,b=-40 | 52 | 7 |
math | 2. The sum of the following 7 numbers is exactly 19:
$$
\begin{array}{l}
a_{1}=2.56, a_{2}=2.61, a_{3}=2.65, a_{4}=2.71, a_{5}=2.79, a_{6}= \\
2.82, a_{7}=2.86 .
\end{array}
$$
We want to use integers $A_{i}$ to approximate $a_{i}$ $(1 \leqslant i \leqslant 7)$, such that the sum of $A_{i}$ is still 19, and the maximu... | 61 | 190 | 2 |
math | ## Zadatak B-3.4.
Riješite jednadžbu
$$
\frac{\sin \left(x-\frac{2015 \pi}{2}\right)-\cos ^{3} x}{\sin (2 x+2015 \pi)}=\frac{1}{2}
$$
| nosolution | 72 | 2 |
math | Three. (20 points) Given a sequence $\left\{a_{n}\right\}$ with all terms no less than 1, satisfying: $a_{1}=1, a_{2}=1+\frac{\sqrt{2}}{2},\left(\frac{a_{n}}{a_{n+1}-1}\right)^{2}+$ $\left(\frac{a_{n}-1}{a_{n-1}}\right)^{2}=2$. Try to find:
(1) The general term formula of the sequence $\left\{a_{n}\right\}$;
(2) The va... | \frac{2}{3} | 157 | 7 |
math | Let $a, b, c, d$ be an increasing arithmetic sequence of positive real numbers with common difference $\sqrt2$. Given that the product $abcd = 2021$, $d$ can be written as $\frac{m+\sqrt{n}}{\sqrt{p}}$ , where $m, n,$ and $p$ are positive integers not divisible by the square of any prime. Find $m + n + p$. | 100 | 91 | 3 |
math | 9. For the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length 1, the distance between the lines $A_{1} C_{1}$ and $B D_{1}$ is $\qquad$ | \frac{\sqrt{6}}{6} | 58 | 10 |
math | 6. The solution to the equation $\arcsin x+\arcsin 2 x=\arccos x+\arccos 2 x$ is | \frac{\sqrt{5}}{5} | 33 | 10 |
math | 2. Solve the equation $\left(\frac{7}{4}-3 \cos 2 x\right) \cdot|1+2 \cos 2 x|=\sin x(\sin x+\sin 5 x)$. | \\frac{\pi}{6}+\frac{k\pi}{2},k\inZ | 48 | 19 |
math | 4. Let $k$ be the number of real roots of the equation $|x|^{\frac{1}{4}}+|x|^{\frac{1}{2}}-\cos x=0$, then $k=$ | 2 | 47 | 1 |
math | 4. (4th "Hope Cup" Invitational Competition Question) Rectangle $A B C D$ is congruent to rectangle $A B E F$, $D-A B-E$ is a right dihedral angle, $M$ is the midpoint of $A B$, $F M$ and $B D$ form an angle $\theta, \sin \theta=\frac{\sqrt{78}}{9}$, find $\left.\frac{A B \mid}{\mid B C} \right\rvert\,$. | \frac{\sqrt{2}}{2} | 111 | 10 |
math | 1. A bag contains 8 white balls and 2 red balls. Each time a ball is randomly drawn from the bag, and then a white ball is put back. What is the probability that all red balls are exactly drawn by the fourth draw? | 0.0434 | 51 | 6 |
math | 2. Let positive real numbers $a, b, c$ satisfy $a^{2}+b^{2}+c^{2}=1$. Then the maximum value of $s=a^{2} b c+a b^{2} c+a b c^{2}$ is $\qquad$ | \frac{1}{3} | 61 | 7 |
math | 7. Let the function $f(x)=x^{2}-x+1$. Define $f^{(n)}(x)$ as follows:
$$
f^{(1)}(x)=f(x), f^{(n)}(x)=f\left(f^{(n-1)}(x)\right) \text {. }
$$
Let $r_{n}$ be the arithmetic mean of all the roots of $f^{(n)}(x)=0$. Then $r_{2015}=$ $\qquad$ . | \frac{1}{2} | 113 | 7 |
math | ## Task 2 - 330822
Susann asks Xaver, Yvonne, and Zacharias to each say a natural number. She then tells them the sum of these three numbers. Each of them multiplies the sum with the number they originally said. As a result, Xaver gets 240, Yvonne gets 270, and Zacharias gets 390.
Determine whether the three original... | 8,9,13 | 110 | 6 |
math | 6. A stork, a cormorant, a sparrow, and a pigeon decided to weigh themselves. The weight of each of them turned out to be an integer number of parrots, and the total weight of all four was 32 parrots. Moreover,
- the sparrow is lighter than the pigeon;
- the sparrow and the pigeon together are lighter than the cormora... | A=13,B=4,\Gamma=5,V=10 | 146 | 15 |
math | 12.10*. On a circle with diameter $A B$, points $C$ and $D$ are taken. The line $C D$ and the tangent to the circle at point $B$ intersect at point $X$. Express $B X$ in terms of the radius of the circle $R$ and the angles $\varphi=\angle B A C$ and $\psi=\angle B A D$.
## §2. The Law of Cosines | BX=2R\sin\varphi\sin\psi/\sin|\varphi\\psi| | 96 | 21 |
math | A5. Consider the six-digit multiples of three with at least one of each of the digits 0 , 1 and 2 , and no other digits. What is the difference between the largest and the smallest of these numbers? | 122208 | 47 | 6 |
math | Solve the following system of equations:
$$
\begin{aligned}
& x^{2}+y \sqrt{x y}=336 \\
& y^{2}+x \sqrt{x y}=112
\end{aligned}
$$ | 18,2 | 53 | 4 |
math | 9. (5 points) A and B start running in opposite directions on a circular track at the same time and place. It is known that A's speed is $180 \mathrm{~m}$ per minute, and B's speed is $240 \mathrm{~m}$ per minute. Within 30 minutes, they meet 24 times. What is the maximum length of the track in meters? | 525 | 88 | 3 |
math | Lines intersect at point $P$, draw perpendiculars to the tangents through points $A$ and $B$, and their intersection is point $Q$. Find the expression for the coordinates of point $Q$ in terms of the coordinates of point $P$, and answer: (1) When the y-coordinate of point $P$ remains constant and the x-coordinate chang... | y_{Q}=2y_{P}x_{Q}-2y_{P}(2y_{P}^{2}+1) | 115 | 29 |
math | 9.5 $n$ is the smallest integer with the following property: it is a multiple of 15, and each of its digits is 0 or 8. Find $\frac{n}{15}$.
(2nd American Invitational Mathematics Examination, 1984) | 592 | 60 | 3 |
math | One, (20 points) For the four-digit number $\overline{a b c d}$, the sum of its digits $a + b + c + d$ is a perfect square. Reversing the digits forms the four-digit number $\overline{d c b a}$, which is 4995 greater than the original number. Find all such four-digit numbers. | 2007, 1116, 1996, 2887, 3778, 4669 | 81 | 34 |
math | 12. (2004 Czech and Slovak Mathematical Olympiad) Find the positive integer $n$, such that $\frac{n}{1!}+\frac{n}{2!}+\cdots+\frac{n}{n!}$ is an integer. | n\in{1,2,3} | 52 | 10 |
math | 119. Extracting Roots. Once, in a conversation with Professor Simon Greathead, a man of rather eccentric mind, I mentioned the extraction of cube roots.
- Amazing, - said the professor, - what ignorance people show in such a simple matter! It seems that in the extraction of roots, since the only roots were those extra... | 5832,17576,19683 | 292 | 16 |
math | 16. (3 points) Use the ten different digits $0-9$ to form a ten-digit number that can be divided by 11, the largest number is | 9876524130 | 36 | 10 |
math | ## Task A-1.2.
Tea mixed a dough from three ingredients: flour, water, and eggs. The mass of flour in the dough to the mass of water is in the ratio of $7: 2$, while the mass of water to the mass of eggs is in the ratio of 5 : 2. The total mass of the dough is 1470 grams. Determine the mass of each ingredient. | 1050 | 89 | 4 |
math | C1. A train travelling at constant speed takes five seconds to pass completely through a tunnel which is $85 \mathrm{~m}$ long, and eight seconds to pass completely through a second tunnel which is $160 \mathrm{~m}$ long. What is the speed of the train? | 25\mathrm{~}/\mathrm{} | 63 | 10 |
math | ## Task 5
Which number would you write in the box? Justify!
$$
2,4,6, \square, 10,12
$$ | 8 | 36 | 1 |
math | 14. In a $3 \times 3$ grid filled with the numbers $1 \sim 9$, the largest number in each row is colored red, and the smallest number in each row is colored green. Let $M$ be the smallest number among the three red squares, and $m$ be the largest number among the three green squares. Then $M-m$ can have $\qquad$ differ... | 8 | 87 | 1 |
math | 4. On a circle, 40 red points and one blue point are marked. All possible polygons with vertices at the marked points are considered. Which type of polygons is more numerous, and by how many: those with a blue vertex, or those without it? | 780 | 54 | 3 |
math | 1. Given $x=\frac{1}{2-\sqrt{5}}$, then $x^{3}+3 x^{2}-5 x+1=$ | 0 | 34 | 1 |
math | 4. Let the circle $O: x^{2}+y^{2}=5$ intersect the parabola $C: y^{2}=2 p x(p>0)$ at point $A\left(x_{0}, 2\right)$, and let $AB$ be the diameter of circle $O$. A line passing through $B$ intersects $C$ at two distinct points $D, E$. Then the product of the slopes of lines $AD$ and $AE$ is $\qquad$. | 2 | 107 | 1 |
math | Question 234: In the interval $[1,1000]$, take $\mathrm{n}$ different numbers $\mathrm{a}_{1}, \mathrm{a}_{2}, \ldots, \mathrm{a}_{\mathrm{n}}$. There always exist two numbers $\mathrm{a}_{\mathrm{i}},$ $a_{j}$, such that $0<\left|a_{i}-a_{j}\right|<1+3 \sqrt[3]{a_{i} a_{j}}$. Find the minimum possible value of $n$. | 11 | 120 | 2 |
math | 5. (10 points) The sum of all natural numbers less than 200 and coprime with 200 is $\qquad$ | 8000 | 33 | 4 |
math | 1. It is known that for three consecutive natural values of the argument, the quadratic function $f(x)$ takes the values $-9, -9$, and $-15$ respectively. Find the greatest possible value of $f(x)$. | -\frac{33}{4} | 51 | 8 |
math | Solve the following equation:
$$
5 x-7 x^{2}+8 \sqrt{7 x^{2}-5 x+1}=8
$$ | x_{1}=0,\quadx_{2}=\frac{5}{7},\quadx_{3}=3,\quadx_{4}=-\frac{16}{7} | 34 | 40 |
math | Example 4 Let $x, y \in \mathbf{R}$. Given
$$
\left(\sqrt{x^{2}+1}-x+1\right)\left(\sqrt{y^{2}+1}-y+1\right)=2 \text{. }
$$
Find the value of $xy$. | xy=1 | 70 | 3 |
math | Find all functions $f ,g,h : R\to R$ such that $f(x+y^3)+g(x^3+y) = h(xy)$ for all $x,y \in R$
| f(x) = c | 43 | 6 |
math | Point $P$ lies on the diagonal $AC$ of square $ABCD$ with $AP>CP$. Let $O_1$ and $O_2$ be the circumcenters of triangles $ABP$ and $CDP$ respectively. Given that $AB=12$ and $\angle O_1 P O_2 = 120^\circ$, then $AP=\sqrt{a}+\sqrt{b}$ where $a$ and $b$ are positive integers. Find $a+b$. | 96 | 108 | 2 |
math | $1 \cdot 71$ A circle has $n$ points $(n>1)$, connect these $n$ points, sequentially denoted as $P_{1}, P_{2}$,
$\cdots, P_{n}$, such that the polyline $P_{1} P_{2} \cdots P_{n}$ does not intersect itself. How many such connection methods are there? | n\cdot2^{n-2} | 84 | 9 |
math | 1. Answer: $2011^{2011}+2009^{2009}>2011^{2009}+2009^{2011}$. | 2011^{2011}+2009^{2009}>2011^{2009}+2009^{2011} | 48 | 42 |
math | ## Task Condition
Find the derivative.
$$
y=\frac{1}{a\left(1+a^{2}\right)}\left(\operatorname{arctg}(a \cos x)+a \ln \left(\operatorname{tg} \frac{x}{2}\right)\right)
$$ | \frac{\cosx\cdot\operatorname{ctg}x}{1+^{2}\cdot\cos^{2}x} | 64 | 29 |
math | Example 36 (1978 Kyiv Mathematical Olympiad) Find the smallest natural numbers $a$ and $b (b>1)$, such that
$$
\sqrt{a \sqrt{a \sqrt{a}}}=b .
$$ | =256,b=128 | 54 | 9 |
math | Place a circle of the largest possible radius inside a cube.
# | \frac{\sqrt{6}}{4} | 13 | 10 |
math | Find the number of ordered triples of sets $(T_1, T_2, T_3)$ such that
1. each of $T_1, T_2$, and $T_3$ is a subset of $\{1, 2, 3, 4\}$,
2. $T_1 \subseteq T_2 \cup T_3$,
3. $T_2 \subseteq T_1 \cup T_3$, and
4. $T_3\subseteq T_1 \cup T_2$. | 625 | 116 | 3 |
math | Find the sum of all prime numbers $p$ which satisfy \[p = a^4 + b^4 + c^4 - 3\] for some primes (not necessarily distinct) $a$, $b$ and $c$. | 719 | 50 | 3 |
math | 2. $42 N$ is the set of all positive integers. For a subset $S$ of $N$ and $n \in N$, define
$$S \oplus\{n\}=\{s+n \mid s \in S\}$$
Additionally, define the subset $S_{k}$ as follows:
$$S_{1}=\{1\}, S_{k}=\left\{S_{k-1} \oplus\{k\}\right\} \cup\{2 k-1\}, k=2,3,4 \cdots .$$
(1) Find $N-\bigcup_{k=1}^{\infty} S_{k}$.
(2... | 500 | 178 | 3 |
math | Example 14 Let the three sides of $\triangle ABC$ be $a, b, c$ with corresponding altitudes $h_{a}$, $h_{b}$, $h_{c}$, and the radius of the incircle of $\triangle ABC$ be $r=2$. If $h_{a}+h_{b}+h_{c}=18$, find the area of $\triangle ABC$. | 12\sqrt{3} | 88 | 7 |
math | What is the maximum number of elements of a subset of $\{1,2, \ldots, 100\}$ such that none of its elements is a multiple of any other? | 50 | 40 | 2 |
math | G1.4 Let $r$ and $s$ be the two distinct real roots of the equation $2\left(x^{2}+\frac{1}{x^{2}}\right)-3\left(x+\frac{1}{x}\right)=1$. If $d=r+s$, find the value of $d$. | \frac{5}{2} | 69 | 7 |
math | 12.313. The height of the cone is $H$, and the angle between the generatrix and the base plane is $\alpha$. The total surface area of this cone is divided in half by a plane perpendicular to its height. Find the distance from this plane to the base of the cone. | 2H\sin^{2}\frac{\alpha}{4} | 63 | 13 |
math | 5. A bag contains five white balls, four red balls, and three yellow balls. If four balls are drawn at random, the probability that balls of all colors are included is $\qquad$ . | \frac{6}{11} | 41 | 8 |
math | We have 30 padlocks and for each one, we have a key that does not open any of the other padlocks. Someone randomly drops the keys into the closed padlocks, one into each. We break open two padlocks. What is the probability that we can open the rest without breaking any more padlocks? | \frac{1}{15} | 66 | 8 |
math | 9. Divisible by 11 with a remainder of 7, divisible by 7 with a remainder of 5, and not greater than 200, the sum of all such natural numbers is $\qquad$ | 351 | 47 | 3 |
math | A racing turtle moves in a straight line as follows. In the first segment of the path, which measures $\frac{1}{2} \mathrm{~m}$, she runs at a speed of $3 \mathrm{~m} / \mathrm{s}$. In the second segment, which measures $\frac{1}{3} \mathrm{~m}$, she runs at a speed of $4 \mathrm{~m} / \mathrm{s}$. In the third segment... | \frac{1}{2}-\frac{1}{2015} | 211 | 17 |
math | 3.2.3 ** Let $g(n)$ denote the greatest odd divisor of the positive integer $n$, for example, $g(3)=3, g(14)=7$, etc. Find the value of $g(1)+g(2)+g(3)+\cdots+g\left(2^{n}\right)$. | \frac{1}{3}(4^{n}+2) | 74 | 14 |
math | Let $A B C$ be a triangle with circumradius $R$, perimeter $P$ and area $K$. Determine the maximum value of $K P / R^{3}$. | \frac{27}{4} | 38 | 8 |
math | 19. Let the three sides of $\triangle ABC$ be $a, b, c$, and $a+b+c=3$. Find the minimum value of $f(a, b, c)=a^{2}+$ $b^{2}+c^{2}+\frac{4}{3} a b c$. (2007 Northern Mathematical Olympiad Problem) | \frac{13}{3} | 78 | 8 |
math | Problem 9.4. In city $\mathrm{N}$, there are exactly three monuments. One day, a group of 42 tourists arrived in this city. Each of them took no more than one photo of each of the three monuments. It turned out that any two tourists together had photos of all three monuments. What is the minimum number of photos that a... | 123 | 82 | 3 |
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