task_type stringclasses 4
values | problem stringlengths 21 5.23k | answer stringlengths 1 8.29k | problem_tokens int64 11 1.16k | answer_tokens int64 1 2.04k |
|---|---|---|---|---|
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 | Task B-1.7. Solve the equation:
$$
\frac{2 x+1}{6 x^{2}-3 x}-\frac{2 x-1}{14 x^{2}+7 x}=\frac{8}{12 x^{2}-3}
$$ | x\neq\frac{1}{2} | 61 | 11 |
math | 10. During a rainstorm, 15 litres of water fell per square metre. By how much did the water level in Michael's outdoor pool rise? | 1.5 | 33 | 3 |
math | Compute the number of integers between $1$ and $100$, inclusive, that have an odd number of factors. Note that $1$ and $4$ are the first two such numbers. | 10 | 41 | 2 |
math | 4. On a circle, 60 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? | 1770 | 54 | 4 |
math | One, (40 points) Integers $a, b, c, d$ satisfy $ad - bc = 1$. Find the minimum value of $a^2 + b^2 + c^2 + d^2 + ab + cd - ac - bd - bc$, and determine all quadruples $(a, b, c, d)$ that achieve this minimum value. | 2 | 79 | 1 |
math | Let $P \in \mathbb{R}[X]$ be a monic polynomial of degree 2020 such that $P(n)=n$ for all $n \in \llbracket 0,2019 \rrbracket$. Calculate $P(2020)$.
Hint: Introduce the polynomial $Q=P-X$. | 2020!+2020 | 76 | 10 |
math | 9. Let $x, y$ be real numbers, the algebraic expression
$$
5 x^{2}+4 y^{2}-8 x y+2 x+4
$$
has a minimum value of | 3 | 46 | 1 |
math | 1. Calculate $\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\cdots+$ $\frac{1}{\sqrt{2003}+\sqrt{2004}}=$ $\qquad$ | 2\sqrt{501}-1 | 74 | 9 |
math | 5. A triangular playground has sides, in metres, measuring 7,24 and 25 . Inside the playground, a lawn is designed so that the distance from each point on the edge of the lawn to the nearest side is 2 metres. What is the area of the lawn? | \frac{28}{3} | 60 | 8 |
math | Given segments $a, b$ and $c$. Using
a compass and a straightedge, construct a segment $x$, such that $x: a=b: c$.
# | OD=x | 38 | 2 |
math | A four-digit number, which is a perfect square of a number, has its first two and last two digits equal; find this number. | 7744 | 28 | 4 |
math | Find all integers $n \geq 3$ such that the following property holds: if we list the divisors of $n$ ! in increasing order as $1=d_{1}<d_{2}<\cdots<d_{k}=n$ !, then we have
$$ d_{2}-d_{1} \leq d_{3}-d_{2} \leq \cdots \leq d_{k}-d_{k-1} $$ | n \in \{3,4\} | 98 | 10 |
math | ## Task 1 - V10921
The Soviet pilot K. Kokkinaki set a new world record with the single-engine turbojet aircraft E 66. He flew $100 \mathrm{~km}$ in $170 \mathrm{~s}$.
a) What was his average speed in $\frac{\mathrm{km}}{\mathrm{h}}$?
b) What possible error is associated with this value if the distance measurement w... | 2118\mathrm{~}\cdot\mathrm{}^{-1} | 123 | 16 |
math | 4. Determine all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for every $n \in \mathbb{N}$,
$$
2 n+2001 \leqslant f(f(n))+f(n) \leqslant 2 n+2002
$$
(Romania) | f(n)=n+667 | 78 | 8 |
math | Let $x,y,$ and $z$ be real numbers satisfying the system
\begin{align*} \log_2(xyz-3+\log_5 x)&=5,\\ \log_3(xyz-3+\log_5 y)&=4,\\ \log_4(xyz-3+\log_5 z)&=4.\\ \end{align*}
Find the value of $|\log_5 x|+|\log_5 y|+|\log_5 z|$. | 265 | 108 | 3 |
math | 1. Determine all values of the parameter $m$ such that for every $x$ the following holds
$$
\left(4 m^{2}-5 m+1\right) x^{2}+2(m+1) x+1>0
$$ | \in(-\infty,0)\cup(\frac{7}{3},+\infty) | 56 | 21 |
math | 7. (10 points) A natural number that reads the same from left to right as from right to left is called a "palindrome number", for example:
909. Then the average of all three-digit palindrome numbers is. | 550 | 50 | 3 |
math | Determine value of real parameter $\lambda$ such that equation $$\frac{1}{\sin{x}} + \frac{1}{\cos{x}} = \lambda $$ has root in interval $\left(0,\frac{\pi}{2}\right)$ | \lambda \geq 2\sqrt{2} | 53 | 13 |
math | 14. (FRG 1) How many words with $n$ digits can be formed from the alphabet $\{0,1,2,3,4\}$, if neighboring digits must differ by exactly one? | x_{2n}=8\cdot3^{n-1},\quadx_{2n+1}=14\cdot3^{n-1} | 46 | 33 |
math | 14. (25 points) Given real numbers $a, b, c$ satisfy
$$
a^{2}+b^{2}+c^{2}=1 \text {. }
$$
Find the maximum and minimum values of $M=a^{2} b c+a b^{2} c+a b c^{2}$. | \frac{1}{3} | 71 | 7 |
math | 3. Find all natural numbers $n$ for which $2^{n}+n^{2016}$ is a prime number. | 1 | 29 | 1 |
math | Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
f\left(x^{3}+y^{3}+x y\right)=x^{2} f(x)+y^{2} f(y)+f(x y)
$$
for all $x, y \in \mathbb{R}$. | f(x)=x f(1) | 79 | 8 |
math | A regular heptagon $ABCDEFG$ is given. The lines $AB$ and $CE$ intersect at $ P$. Find the measure of the angle $\angle PDG$. | \frac{\pi}{2} | 36 | 7 |
math | 16. If a positive integer cannot be written as the difference of two square numbers, then the integer is called a "cute" integer. For example, 1,2 and 4 are the first three "cute" integers. Find the $2010^{\text {th }}$ "cute" integer.
(Note: A square number is the square of a positive integer. As an illustration, 1,4,... | 8030 | 104 | 4 |
math | 1. (5 points) Calculate: $0.15 \div 2.1 \times 56=$ | 4 | 25 | 1 |
math | 1. Does there exist a positive integer divisible by 2020, in whose representation the digits $0,1, \cdots, 9$ appear the same number of times? | 12123434565679798080 | 40 | 20 |
math | 4.1. The sea includes a bay with more saline water. The salinity of the water in the sea is 120 per mille, in the bay 240 per mille, in the part of the sea not including the bay - 110 per mille. How many times is the volume of water in the sea larger than the volume of water in the bay? The volume of water is considere... | 13 | 128 | 2 |
math | 11. $[\mathbf{8}]$ For positive integers $m, n$, let $\operatorname{gcd}(m, n)$ denote the largest positive integer that is a factor of both $m$ and $n$. Compute
$$
\sum_{n=1}^{91} \operatorname{gcd}(n, 91)
$$ | 325 | 76 | 3 |
math | (5) The maximum value of the function $y=2 x-5+\sqrt{11-3 x}$ is $\qquad$ . | \frac{65}{24} | 31 | 9 |
math | Example 2 Given $x, y, z \in \mathbf{R}$, and satisfying $x+y+z=2$, find the minimum value of $T=x^{2}-2 x+2 y^{2}+3 z^{2}$. | -\frac{5}{11} | 54 | 8 |
math | 9. In the sequence $\left\{a_{n}\right\}$, $a_{1}=\sqrt{2}, a_{n}=\sqrt{2+a_{n-1}}$, then $a_{n}=$ | 2\cos\frac{\pi}{2^{n+1}} | 49 | 14 |
math | (treatise)
For what condition on $y \in \mathbb{N}$ is $y^{2}+5 y+12$ a square? | 3 | 34 | 1 |
math | 6. The brakes of a car allow it to stand on an inclined asphalt surface with an angle at the base of no more than $30^{\circ}$. Determine the minimum braking distance of this car when moving at a speed of $30 \, \text{m/s}$ on a flat horizontal road with the same surface. The acceleration due to gravity $g=10 \, \text{... | 78\, | 127 | 4 |
math | Auto: Kovamidjei A.K.
Is the number $4^{9}+6^{10}+3^{20}$ prime? | (2^{9}+3^{10})^{2} | 32 | 14 |
math | 3-4. Solve the equation:
\[
\sqrt{x+3-4 \sqrt{x-1}}+\sqrt{x+8-6 \sqrt{x-1}}=1
\] | 5\leqslantx\leqslant10 | 41 | 14 |
math | Find all $f: \mathbb{Q}_{+} \rightarrow \mathbb{R}$ such that \[ f(x)+f(y)+f(z)=1 \] holds for every positive rationals $x, y, z$ satisfying $x+y+z+1=4xyz$. | f(x) = a \times \frac{1}{2x+1} + (1-a) \frac{1}{3} | 61 | 30 |
math | N1. Find all pairs $(k, n)$ of positive integers for which $7^{k}-3^{n}$ divides $k^{4}+n^{2}$. | (2,4) | 37 | 5 |
math | Let $f$ be a monic cubic polynomial such that the sum of the coefficients of $f$ is $5$ and such that the sum of the roots of $f$ is $1$. Find the absolute value of the sum of the cubes of the roots of $f$. | 14 | 58 | 2 |
math | B1. A number of students participated in a test that could be scored up to 100 points. Everyone scored at least 60 points. Exactly five students scored 100 points. The average score of the participating students was 76 points.
How many students participated in this test at a minimum? | 13 | 66 | 2 |
math | 7. $\triangle A B C$ is equilateral with side length 4. $D$ is a point on $B C$ such that $B D=1$. If $r$ and $s$ are the radii of the inscribed circles of $\triangle A D B$ and $\triangle A D C$ respectively, find $r s$.
(1 mark) $A B C$ 是ç‰é‚Šä¸‰è§’å½¢, 邊長爲 $4 \circ D$ 是 $B C$ 上的一點, 使得 $B D=1 \circ$ è‹¥ $r$ å’Œ $\mathrm{s}$ 分別是 $... | 4-\sqrt{13} | 160 | 7 |
math | To each positive integer $ n$ it is assigned a non-negative integer $f(n)$ such that the following conditions are satisfied:
(1) $ f(rs) \equal{} f(r)\plus{}f(s)$
(2) $ f(n) \equal{} 0$, if the first digit (from right to left) of $ n$ is 3.
(3) $ f(10) \equal{} 0$.
Find $f(1985)$. Justify your answer. | 0 | 107 | 1 |
math | 6. If three angles $x, y, z$ form an arithmetic sequence with a common difference of $\frac{\pi}{3}$, then $\tan x \cdot \tan y + \tan y \cdot \tan z + \tan z \cdot \tan x$ $=$ . $\qquad$ | -3 | 64 | 2 |
math | Find such numbers $A, B, C, a, b, c$, so that the identity
$$
(4 x-2) /\left(x^{3}-x\right)=A /(x-a)+B /(x-b)+C /(x-c)
$$
holds. | A=2,B=1,C=-3,=0,b=1,=-1 | 58 | 18 |
math | ## Task B-1.2.
Determine the set of all integers $n$ for which the inequality
$$
\left(1+\frac{1}{3}\right)\left(1+\frac{1}{8}\right)\left(1+\frac{1}{15}\right) \cdots\left(1+\frac{1}{2019^{2}-1}\right)<\frac{2019}{n^{2}}
$$
holds. | {-31,-30,\ldots,-1,1,2,\ldots,30,31} | 102 | 25 |
math | 11.2. Find all triples of real numbers such that each of these numbers is equal to the square of the difference of the other two. | (0,0,0),(1,1,0),(1,0,1),(0,1,1) | 30 | 25 |
math | 21. You roll a fair 12 -sided die repeatedly. The probability that all the primes show up at least once before seeing any of the other numbers can be expressed as a fraction $p / q$ in lowest terms. What is $p+q$ ? | 793 | 57 | 3 |
math | 15. $A, B, C$ are all positive integers. It is known that $A$ has 7 divisors, $B$ has 6 divisors, $C$ has 3 divisors, $A \times B$ has 24 divisors, $B \times C$ has 10 divisors. Then the minimum value of $A+B+C$ is $\qquad$ _ | 91 | 88 | 2 |
math | 1. Boris distributes 8 white and 8 black balls into two boxes. Nastya randomly chooses a box and then takes a ball from it without looking. Can Boris distribute the balls in such a way that the probability of drawing a white ball is greater than $\frac{2}{3}$? | \frac{22}{30}>\frac{2}{3} | 61 | 16 |
math | 8.4. On the Island of Liars and Knights, a circular arrangement is called correct if each person standing in the circle can say that among their two neighbors, there is a representative of their tribe. Once, 2019 natives formed a correct circular arrangement. A liar approached them and said: "Now we can also form a cor... | 1346 | 87 | 4 |
math | [Theorem of the length of a tangent and a secant; the product of the entire secant and its external part [ Sine Theorem The midline of a triangle
A circle with radius 3 passes through vertex $B$, the midpoints of sides $A B$ and $B C$, and is tangent to side $A C$ of triangle $A B C$. Angle $B A C$ is acute, and $\s... | 16\sqrt{2} | 118 | 7 |
math | 1. Given positive integers $a, b, c$ satisfy
$$
10 a^{2}-3 a b+7 c^{2}=0 \text {. }
$$
Find the minimum value of $(a, b)(b, c)(c, a)$. | 3 | 57 | 1 |
math | 7. (6 points) Xiaopang needs 90 seconds to go from the first floor to the third floor. At this speed, it would take him $\qquad$ seconds to go from the second floor to the seventh floor. | 225 | 49 | 3 |
math | 16. Let $\alpha, \beta$ be real numbers. If for any real numbers $x, y, z$, we have $\alpha(x y+y z+z x) \leqslant M \leqslant \beta\left(x^{2}+y^{2}+z^{2}\right)$ always holds, where $M=\sqrt{x^{2}+x y+y^{2}} \cdot \sqrt{y^{2}+y z+z^{2}}+\sqrt{y^{2}+y z+z^{2}} \cdot \sqrt{z^{2}+z x+x^{2}}+\sqrt{z^{2}+z x+x^{2}} \cdot ... | 3 | 179 | 1 |
math | Find the least real number $k$ with the following property: if the real numbers $x$, $y$, and $z$ are not all positive, then \[k(x^{2}-x+1)(y^{2}-y+1)(z^{2}-z+1)\geq (xyz)^{2}-xyz+1.\] | \frac{16}{9} | 73 | 9 |
math | Let $\mathbb{R}_{>0}$ be the set of positive real numbers. Let $a \in \mathbb{R}_{>0}$ be given. Find all functions $f: \mathbb{R}_{>0} \rightarrow \mathbb{R}$ such that $f(a)=1$ and
$$
\forall x, y \in \mathbb{R}_{>0}: f(x) f(y)+f\left(\frac{a}{x}\right) f\left(\frac{a}{y}\right)=2 f(x y)
$$ | f: \mathbb{R}_{>0} \rightarrow \mathbb{R}: x \mapsto 1 | 122 | 25 |
math | 9. A chemistry student conducted an experiment: from a tank filled with syrup solution, he poured out several liters of liquid, refilled the tank with water, then poured out twice as much liquid and refilled the tank with water again. As a result, the amount of syrup in the tank decreased by $\frac{8}{3}$ times. Determ... | 250 | 98 | 3 |
math | 13.353. When multiplying two positive numbers, one of which is 75 more than the other, the product was mistakenly obtained as 1000 less than the true product. As a result, when dividing (during verification) the incorrect product by the smaller of the multipliers, a quotient of 227 and a remainder of 113 were obtained.... | 159234 | 85 | 6 |
math | 1. a) Perform: $[0,(6)+3,(63)+0,2(37)]:\left(\frac{2}{3}+\frac{40}{11}+\frac{47}{198}\right)$.
b) Determine the value of the natural number $a$ knowing that the number $B=\frac{2}{a}+\frac{1}{3}$ is a natural number. | 3 | 93 | 1 |
math | Joana wrote the numbers from 1 to 10000 on the blackboard and then erased all multiples of 7 and 11. What number remained in the 2008th position? | 2577 | 45 | 4 |
math | Task B-2.4. The glider had to return to a place it had passed to refuel at the port. It was delayed for 12 minutes. This delay was compensated for on the remaining 60 km of the journey by increasing the speed by 10 km/h. What was the glider's speed before docking at the port? | 50\mathrm{~}/\mathrm{} | 74 | 10 |
math | $1 \cdot 33$ The integers $1,2, \cdots, n$ are arranged in a permutation such that: each number is either greater than all the numbers before it, or less than all the numbers before it. How many such permutations are there? | 2^{n-1} | 57 | 6 |
math | Three. (50 points) Given sets $A_{1}, A_{2}, \cdots, A_{n}$ are different subsets of the set $\{1,2, \cdots, n\}$, satisfying the following conditions:
(i) $i \notin A_{i}$ and $\operatorname{Card}\left(A_{i}\right) \geqslant 3, i=1,2, \cdots, n$;
(ii) $i \in A_{j}$ if and only if $j \notin A_{i}(i \neq j$, $i, j=1,2, ... | 7 | 187 | 1 |
math | 12. Given two quadratic functions $y_{1}$ and $y_{2}$, when $x=\alpha(\alpha>0)$, $y_{1}$ reaches its maximum value of 5, and $y_{2}=25$. Also, the minimum value of $y_{2}$ is $-2$, and $y_{1}+y_{2}=x^{2}+16 x+13$. Find the value of $\alpha$ and the analytical expressions of the quadratic functions $y_{1}, y_{2}$. | y_{1}=-2 x^{2}+4 x+3, y_{2}=3 x^{2}+12 x+10, \alpha=1 | 116 | 37 |
math | Are there positive integers $a, b$ with $b \ge 2$ such that $2^a + 1$ is divisible by $2^b - 1$? | b = 2 | 39 | 5 |
math | The quadrilateral $ABCD$ is an isosceles trapezoid with $AB = CD = 1$, $BC = 2$, and $DA = 1+ \sqrt{3}$. What is the measure of $\angle ACD$ in degrees? | 90^\circ | 59 | 4 |
math | [b]Q12.[/b] Find all positive integers $P$ such that the sum and product of all its divisors are $2P$ and $P^2$, respectively. | 6 | 40 | 1 |
math | In a rectangular plot of land, a man walks in a very peculiar fashion. Labeling the corners $ABCD$, he starts at $A$ and walks to $C$. Then, he walks to the midpoint of side $AD$, say $A_1$. Then, he walks to the midpoint of side $CD$ say $C_1$, and then the midpoint of $A_1D$ which is $A_2$. He continues in this fashi... | 793 | 143 | 3 |
math | ## Zadatak B-3.4.
Odredite neki period funkcije $f(x)=2 \sin \left(\frac{3}{4} x\right)+\cos \left(\frac{4}{5} x-\frac{\pi}{3}\right)$.
| 40\pi | 60 | 4 |
math | ## Task 15/66
For the calculation of the square root of a number $z=p^{2}+a$ with $0 \leq a \leq 2 p+1$, the approximation formula is
$$
\sqrt{z}=\sqrt{p^{2}+a} \approx p+\frac{a}{2 p+1}
$$
How large is the maximum error of this approximation in dependence on $a$? How does this change in dependence on $p$? | \frac{1}{4(2p+1)} | 108 | 12 |
math | Determine all composite positive integers $n$ with the following property: If $1 = d_1 < d_2 < \cdots < d_k = n$ are all the positive divisors of $n$, then
$$(d_2 - d_1) : (d_3 - d_2) : \cdots : (d_k - d_{k-1}) = 1:2: \cdots :(k-1)$$
(Walther Janous) | n = 4 | 104 | 5 |
math | 4. Let $f(x)=(x+a)(x+b)$ (where $a, b$ are given positive real numbers), and $n \geqslant 2$ be a given integer. For non-negative real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying
$$
x_{1}+x_{2}+\cdots+x_{n}=1
$$
find the maximum value of
$$
F=\sum_{1 \leq i<j \leq n} \min \left\{f\left(x_{i}\right), f\left(x_{j}\... | \frac{n-1}{2}\left(\frac{1}{n}+a+b+n a b\right) | 145 | 25 |
math | ## Problem Statement
Find the derivative.
$$
y=e^{\sin x}\left(x-\frac{1}{\cos x}\right)
$$ | e^{\sinx}\cdot(x\cdot\cosx-\frac{\sinx}{\cos^2x}) | 31 | 25 |
math | 11. Let $S=\{1,2,3, \cdots \cdots n\}$, and $A$ be a subset of $S$. Arrange the elements of $A$ in descending order, then alternately subtract or add the subsequent numbers starting from the largest number to get the alternating sum of $A$. For example, if $A=\{1,4,9,6,2\}$, rearranging it gives $\{9,6,4,2,1\}$, and it... | n\cdot2^{n-1} | 134 | 9 |
math | 942. A wire of length $20 m$ is required to fence a flower bed, which should have the shape of a circular sector. What radius of the circle should be taken so that the area of the flower bed is maximized? | 5\mathrm{~} | 51 | 6 |
math | 2. Given a natural number $n \geqslant 5$, try to find:
(1) In the $n$-element set $\left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$, how many different numbers are generated by $a_{i}+a_{j}(1 \leqslant i<j \leqslant n)$ at least?
(2) Determine all $n$-element sets that achieve the above minimum value. | 2n-3 | 108 | 4 |
math | There are $13$ positive integers greater than $\sqrt{15}$ and less than $\sqrt[3]{B}$. What is the smallest integer value of $B$? | 4097 | 38 | 4 |
math | Let $\mathbb{R}^*$ be the set of all real numbers, except $1$. Find all functions $f:\mathbb{R}^* \rightarrow \mathbb{R}$ that satisfy the functional equation $$x+f(x)+2f\left(\frac{x+2009}{x-1}\right)=2010$$. | f(x) = \frac{1}{3}\left(x + 2010 - 2\frac{x+2009}{x-1}\right) | 77 | 38 |
math | ## Problem Statement
Find the point of intersection of the line and the plane.
$\frac{x-7}{3}=\frac{y-3}{1}=\frac{z+1}{-2}$
$2 x+y+7 z-3=0$ | (10;4;-3) | 56 | 8 |
math | If a right triangle is drawn in a semicircle of radius $1 / 2$ with one leg (not the hypotenuse) along the diameter, what is the triangle's maximum possible area? | \frac{3\sqrt{3}}{32} | 42 | 13 |
math | 11.1. Solve the equation $\sin ^{2} x+1=\cos (\sqrt{2} x)$. | 0 | 27 | 1 |
math | 8. In $\triangle A B C$, $A C=3, \sin C=k \sin A(k \geq 2)$, then the maximum value of the area of $\triangle A B C$ is $\qquad$ | 3 | 49 | 1 |
math | ## Task A-1.7.
How many five-digit natural numbers are there whose product of digits is equal to 900? | 210 | 28 | 3 |
math | 5. If from the numbers $1,2, \cdots, 14$, we select $a_{1}, a_{2}, a_{3}$ in increasing order, such that
$$
a_{2}-a_{1} \geqslant 3 \text { and } a_{3}-a_{2} \geqslant 3 \text {, }
$$
then the total number of different ways to select them is $\qquad$ kinds. | 120 | 102 | 3 |
math | 7. Given two non-zero complex numbers $x, y$ whose sum of cubes is 0, then $\left(\frac{x}{x-y}\right)^{2017}+\left(\frac{y}{x-y}\right)^{2017}=$ | 0or\\sqrt{3}i | 58 | 8 |
math | 1. In the Cartesian coordinate system $x O y$, points $A$ and $B$ lie on the parabola $y^{2}=2 x$, satisfying $\overrightarrow{O A} \cdot \overrightarrow{O B}=-1, F$ is the focus of the parabola. Then the minimum value of $S_{\triangle O F A}+S_{\triangle O F B}$ is $\qquad$ | \frac{\sqrt{2}}{2} | 93 | 10 |
math | 8. Let $a, b$ be positive integers, and $a-b \sqrt{3}=(2-\sqrt{3})^{100}$, then the units digit of $a b$ is | 2 | 44 | 1 |
math | Task 1 - 150821 The weighing of a container filled with water resulted in a total mass (container and water mass) of 2000 g. If 20% of the water is poured out, the weighed total mass decreases to 88%.
Calculate the mass of the empty container! | 800\mathrm{~} | 68 | 8 |
math | 3. Given the set $\Omega=\left\{(x, y) \mid x^{2}+y^{2} \leqslant 2008\right\}$, if points $P(x, y)$ and $P^{\prime}\left(x^{\prime}, y^{\prime}\right)$ satisfy $x \leqslant x^{\prime}$ and $y \geqslant y^{\prime}$, then point $P$ is said to dominate $P^{\prime}$. If a point $Q$ in set $\Omega$ satisfies: there does no... | {(x,y)\midx^{2}+y^{2}=2008,x\leqslant0 | 156 | 25 |
math | A square board of three rows by three columns contains nine cells. In how many different ways can we write the three letters A, B, and $\mathbf{C}$ in three different cells, so that in each row exactly one of these three letters is written? | 162 | 54 | 3 |
math | 11. If $\cot \alpha+\cot \beta+\cot \gamma=-\frac{4}{5}, \tan \alpha+\tan \beta+\tan \gamma=\frac{17}{6}$ and $\cot \alpha \cot \beta+\cot \beta \cot \gamma+\cot \gamma \cot \alpha=-\frac{17}{5}$, find the value of $\tan (\alpha+\beta+\gamma)$. | 11 | 93 | 2 |
math | $6 \cdot 140$ Find all functions $f: Q \rightarrow$ $Q$ (where $Q$ is the set of rational numbers) that satisfy $f(1)=2$ and
$f(x y) \equiv f(x) f(y)-f(x+y)+1, x, y \in Q$. | f(x)=x+1 | 70 | 6 |
math | Five lighthouses are located, in order, at points $A, B, C, D$, and $E$ along the shore of a circular lake with a diameter of $10$ miles. Segments $AD$ and $BE$ are diameters of the circle. At night, when sitting at $A$, the lights from $B, C, D$, and $E$ appear to be equally spaced along the horizon. The perimeter in ... | 95 | 129 | 2 |
math | 1. (8 points) Calculate: $2013 \div(25 \times 52-46 \times 15) \times 10=$ | 33 | 38 | 2 |
math | # Task â„– 6.1
## Condition:
Yasha and Grisha are playing a game: first, they take turns naming a number from 1 to 105 (Grisha names the number first, the numbers must be different). Then each counts the number of different rectangles with integer sides, the perimeter of which is equal to the named number. The one with... | 104 | 128 | 3 |
math | Example 1. Find the minimum value of $|x-1|+|x-3|+|x-5|$. | 4 | 28 | 1 |
math | Solve in the set of real numbers, the system:
$$x(3y^2+1)=y(y^2+3)$$
$$y(3z^2+1)=z(z^2+3)$$
$$z(3x^2+1)=x(x^2+3)$$ | (1, 1, 1), (-1, -1, -1), (0, 0, 0) | 68 | 27 |
math | 5. Let $p_{1}, p_{2}, \ldots, p_{k}$ be distinct prime numbers. Determine the number of natural numbers of the form $p_{1}^{\alpha_{1}} p_{2}^{\alpha_{2}} \ldots p_{k}^{\alpha_{k}}, \alpha_{i} \in \mathbb{N}$ for which
$$
\alpha_{1} \alpha_{2} \ldots \alpha_{k}=p_{1} p_{2} \ldots p_{k}
$$ | k^{k} | 120 | 4 |
math | 8.120. $\cos ^{2} \frac{x}{2}+\cos ^{2} \frac{3 x}{2}-\sin ^{2} 2 x-\sin ^{2} 4 x=0$. | x_{1}=\frac{\pi}{4}(2k+1),x_{2}=\frac{\pi}{7}(2n+1),x_{3}=\frac{\pi}{5}(2+1)\quadk,n,\inZ | 53 | 53 |
math | 6. Compare the numbers $\frac{100}{101} \times \frac{102}{103} \times \ldots \times \frac{1020}{1021} \times \frac{1022}{1023}$ and $\frac{5}{16}$ | A<\frac{5}{16} | 73 | 10 |
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