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 | 1. If $2 x^{2}-6 y^{2}+x y+k x+6$ can be factored into the product of two linear factors, then the value of $k$ is | \7 | 42 | 2 |
math | The quartic (4th-degree) polynomial P(x) satisfies $P(1)=0$ and attains its maximum value of $3$ at both $x=2$ and $x=3$. Compute $P(5)$. | -24 | 53 | 3 |
math | 16. Two bullets are placed in two consecutive chambers of a 6-chamber pistol. The cylinder is then spun. The pistol is fired but the first shot is a blank. Let $p$ denote the probability that the second shot is also a blank if the cylinder is spun after the first shot and let $\mathrm{q}$ denote the probability that th... | 89 | 118 | 2 |
math | ## Task 1 - 040831
If the digits of a two-digit number $n$ are reversed, the resulting number is $\frac{8}{3}$ times as large as $n$. The number $n$ is to be determined. | 27 | 55 | 2 |
math | Find all positive integers $n$ such that $6^n+1$ it has all the same digits when it is writen in decimal representation. | n = 1 \text{ and } n = 5 | 30 | 13 |
math | 1A. Determine the largest term of the sequence $a_{n}=\frac{2016^{n}}{n!}, n \in \mathbb{N}$. | a_{2015} | 39 | 7 |
math | A finite set of positive integers is called [i]isolated [/i]if the sum of the numbers in any given proper subset is co-prime with the sum of the elements of the set.
a) Prove that the set $A=\{4,9,16,25,36,49\}$ is isolated;
b) Determine the composite numbers $n$ for which there exist the positive integers $a,b$... | n = 6 | 130 | 5 |
math | Question 49, if the three interior angles $A, B, C$ of $\triangle ABC$ satisfy $\cos A = \sin B = 2 \tan \frac{C}{2}$, then the value of $\sin A + \cos A + 2 \tan A$ is $\qquad$ - | 2 | 67 | 1 |
math | Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for $x, y \in \mathbb{R}$, we have:
$$
f(\lfloor x\rfloor y)=f(x)\lfloor f(y)\rfloor
$$ | f=with=0or\in[1,2[ | 63 | 13 |
math | 1.1. The lengths of the sides of a right-angled triangle form an arithmetic progression with a common difference of 1 cm. Find the length of the hypotenuse. | 5 | 37 | 1 |
math | Fibonacci sequences is defined as $f_1=1$,$f_2=2$, $f_{n+1}=f_{n}+f_{n-1}$ for $n \ge 2$.
a) Prove that every positive integer can be represented as sum of several distinct Fibonacci number.
b) A positive integer is called [i]Fib-unique[/i] if the way to represent it as sum of several distinct Fibonacci number is uniqu... | n = 1 | 141 | 5 |
math | 2. To walk 4 km, ride 6 km on a bicycle, and drive 40 km by car, Uncle Vanya needs 2 hours and 12 minutes. If he needs to walk 5 km, ride 8 km on a bicycle, and drive 30 km by car, it will take him 2 hours and 24 minutes. How much time will Uncle Vanya need to walk 8 km, ride 10 km on a bicycle, and drive 160 km by car... | 5.8 | 110 | 3 |
math | 5. (2000 World Inter-City League) A bus, starting a 100-kilometer journey at 12:20 PM, has a computer that at 1:00 PM, 2:00 PM, 3:00 PM, 4:00 PM, 5:00 PM, and 6:00 PM, says: “If the average speed from now on is the same as the average speed so far, then it will take one more hour to reach the destination.” Is this poss... | 85 | 136 | 2 |
math | For some positive integers $p$, there is a quadrilateral $ABCD$ with positive integer side lengths, perimeter $p$, right angles at $B$ and $C$, $AB=2$, and $CD=AD$. How many different values of $p<2015$ are possible?
$\textbf{(A) }30\qquad\textbf{(B) }31\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$ | 31 | 122 | 2 |
math | [ Percentage and Ratio Problems ]
Karlson ate $40 \%$ of the cake for breakfast, and Little One ate 150 g. For lunch, Fräulein Bock ate $30 \%$ of the remainder and another 120 g, and Matilda licked the remaining 90 g of crumbs from the cake. What was the initial weight of the cake?
# | 750 | 83 | 3 |
math | 6. Solve the congruence $111 x \equiv 75(\bmod 321)$. | x\equiv99,206,313(\bmod321) | 26 | 20 |
math | 7. Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$, such that $f(0) \neq 0$, and for all real numbers $x, y$, we have
$$
\begin{array}{l}
f^{2}(x+y) \\
=2 f(x) f(y)+\max \left\{f\left(x^{2}\right)+f\left(y^{2}\right), f\left(x^{2}+y^{2}\right)\right\} .
\end{array}
$$ | f(x)=-1orf(x)=x-1 | 122 | 11 |
math | ## Task 1 - 090621
Klaus participated in a track race as a member of the cycling section of a company sports club. After the race, his brother Reiner asked Klaus about the outcome of the race. Klaus said:
"When I crossed the finish line, no rider was next to me; exactly one third of the participating riders had alrea... | 3 | 111 | 1 |
math | Suppose that $2^{2n+1}+ 2^{n}+1=x^{k}$, where $k\geq2$ and $n$ are positive integers. Find all possible values of $n$. | n = 4 | 48 | 5 |
math | Bogdanov I.I.
In a certain country, there are 100 cities (consider the cities as points on a plane). In a directory, for each pair of cities, there is a record of the distance between them (a total of 4950 records).
a) One record has been erased. Can it always be uniquely restored from the remaining ones?
b) Suppose... | 96 | 123 | 2 |
math | Solve the following system of equations:
$$
\begin{aligned}
& y^{2}-z x=a(x+y+z)^{2} \\
& x^{2}-y z=b(x+y+z)^{2} \\
& z^{2}-x y=c(x+y+z)^{2}
\end{aligned}
$$ | ^{2}+b^{2}+^{2}-(++ca)-(+b+)=0 | 68 | 21 |
math | 12.240. The ratio of the volume of a sphere inscribed in a cone to the volume of a circumscribed sphere is $k$. Find the angle between the slant height of the cone and the plane of its base and the permissible values of $k$. | \arccos\frac{1\\sqrt{1-2\sqrt[3]{k}}}{2},0<k\leq\frac{1}{8} | 58 | 36 |
math | 3. How many triples $(A, B, C)$ of positive integers (positive integers are the numbers $1,2,3,4, \ldots$ ) are there such that $A+B+C=10$, where order does not matter (for instance the triples $(2,3,5)$ and $(3,2,5)$ are considered to be the same triple) and where two of the integers in a triple could be the same (for... | 8 | 108 | 1 |
math | 5. Place 7 goldfish of different colors into 3 glass fish tanks numbered $1, 2, 3$. If the number of fish in each tank must be no less than its number, then the number of different ways to place the fish is $\qquad$ kinds. | 455 | 59 | 3 |
math | Find all $a, b, c \in \mathbb{N}$ such that $2^{a} \cdot 3^{b}=7^{c}-1$. | (1,1,1)(4,1,2) | 36 | 13 |
math | We want to make a square-based, open-top box from a square piece of paper with a side length of $20 \mathrm{~cm}$ by cutting out congruent squares from each of the four corners of the paper and folding up the resulting "flaps". What is the maximum possible volume of the box? | \frac{1}{4}\cdot(\frac{40}{3})^{3}\mathrm{~}^{3} | 65 | 26 |
math | 9. $A$ and $B$ are two points on a circle with center $O$, and $C$ lies outside the circle, on ray $A B$. Given that $A B=24, B C=28, O A=15$, find $O C$. | 41 | 61 | 2 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 2 \pi}(\cos x)^{\frac{\operatorname{ctg} 2 x}{\sin 3 x}}$ | e^{-\frac{1}{12}} | 47 | 10 |
math | 2. Determine all positive integers that are equal to 300 times the sum of their digits. | 2700 | 21 | 4 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty} \frac{(n+1)^{3}+(n-1)^{3}}{n^{3}+1}$ | 2 | 49 | 1 |
math | Let $n$ be a natural number divisible by $4$. Determine the number of bijections $f$ on the set $\{1,2,...,n\}$ such that $f (j )+f^{-1}(j ) = n+1$ for $j = 1,..., n.$ | \frac{(\frac{n}{2})!}{(\frac{n}{4})!} | 64 | 19 |
math | 1. Let $f(x)=\mathrm{e}^{2 x}-1, g(x)=\ln (x+1)$. Then the solution set of the inequality $f(g(x))-g(f(x)) \leqslant 1$ is $\qquad$ | (-1,1] | 58 | 5 |
math | An $m \times n$ chessboard where $m \le n$ has several black squares such that no two rows have the same pattern. Determine the largest integer $k$ such that we can always color $k$ columns red while still no two rows have the same pattern. | n - m + 1 | 58 | 7 |
math | A collection of 8 cubes consists of one cube with edge-length $k$ for each integer $k,\thinspace 1 \le k \le 8.$ A tower is to be built using all 8 cubes according to the rules:
$\bullet$ Any cube may be the bottom cube in the tower.
$\bullet$ The cube immediately on top of a cube with edge-length $k$ must have ed... | 458 | 127 | 3 |
math | Example 3 Arrange the positive integers that are coprime with 105 in ascending order, and find the 1000th term of this sequence. | 2186 | 35 | 4 |
math | ## Problem Statement
Find the second-order derivative $y_{x x}^{\prime \prime}$ of the function given parametrically.
$\left\{\begin{array}{l}x=e^{t} \\ y=\arcsin t\end{array}\right.$ | \frac{^{2}+-1}{e^{2}\cdot\sqrt{(1-^{2})^{3}}} | 58 | 25 |
math | ## Task 23/87
There are prime numbers $p_{i}$ that have the following properties:
1. They are (in decimal notation) truly four-digit.
2. Their digit sum is $Q\left(p_{i}\right)=25$.
3. If you add 4 to them, the result is a "mirror number".
A "mirror number" is a number whose sequence of digits is symmetric with resp... | 1987,3769 | 108 | 9 |
math | Example 4. Express $x$ in the following equation using the inverse cosine function: $\cos x=\frac{3}{5} \quad x \in(3,6.1)$. | x=3+\arccos \left(\frac{3}{5} \cos 3 - \frac{4}{5} \sin 3\right) | 41 | 35 |
math | 1. If the equation $\log _{2}\left[k\left(x-\frac{1}{2}\right)\right]=2 \log _{2}\left(x+\frac{1}{2}\right)$ has only one real root, then the maximum value of $k$ that satisfies the condition is
$\qquad$ . | 4 | 70 | 1 |
math | 4B. Three cards are given. The number 19 is written on one, the number 97 on another, and a two-digit number on the third. If we add all the six-digit numbers obtained by arranging the cards in a row, we get the number 3232320. What number is written on the third card? | 44 | 74 | 2 |
math | Mr. Tik and Mr. Tak sold alarm clocks in the stores Before the Corner and Behind the Corner. Mr. Tik claimed that Before the Corner sold 30 more alarm clocks than Behind the Corner, while Mr. Tak claimed that Before the Corner sold three times as many alarm clocks as Behind the Corner. In the end, it turned out that bo... | 60 | 97 | 2 |
math | 1. Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$, such that the equation
$$
f([x] y)=f(x)[f(y)]
$$
holds for all $x, y \in \mathbf{R}$ (where $[z]$ denotes the greatest integer less than or equal to the real number $z$).
(French contribution) | f(x)=c \text{ (constant), where, } c=0 \text{ or } 1 \leqslant c<2 | 85 | 31 |
math | A natural number was sequentially multiplied by each of its digits. The result was 1995. Find the original number.
# | 57 | 27 | 2 |
math | 13.274. According to the schedule, the train covers a 120 km section at a constant speed. Yesterday, the train covered half of the section at this speed and was forced to stop for 5 minutes. To arrive on time at the final point of the section, the driver had to increase the train's speed by 10 km/h on the second half o... | 100 | 140 | 3 |
math | 30. A round cake is cut into $n$ pieces with 3 cuts. Find the product of all possible values of $n$. | 840 | 29 | 3 |
math | 1. (8 points) Calculate: $6 \times\left(\frac{1}{2}-\frac{1}{3}\right)+12 \times\left(\frac{1}{3}+\frac{1}{4}\right)+19-33+21-7+22=$ $\qquad$ | 30 | 70 | 2 |
math | ## Task Condition
Find the differential $d y$.
$$
y=\ln \left|2 x+2 \sqrt{x^{2}+x}+1\right|
$$ | \frac{}{\sqrt{x^{2}+x}} | 39 | 12 |
math | 1. Calculate: $3752 \div(39 \times 2)+5030 \div(39 \times 10)=$ | 61 | 35 | 2 |
math | 4. (10 points) Two classes are planting trees. Class one plants 3 trees per person, and class two plants 5 trees per person, together planting a total of 115 trees. The maximum sum of the number of people in the two classes is $\qquad$ . | 37 | 61 | 2 |
math | Find the maximal value of
$$ S=\sqrt[3]{\frac{a}{b+7}}+\sqrt[3]{\frac{b}{c+7}}+\sqrt[3]{\frac{c}{d+7}}+\sqrt[3]{\frac{d}{a+7}} $$
where $a, b, c, d$ are nonnegative real numbers which satisfy $a+b+c+d=100$. (Taiwan) Answer: $\frac{8}{\sqrt[3]{7}}$, reached when $(a, b, c, d)$ is a cyclic permutation of $(1,49,1,4... | \frac{8}{\sqrt[3]{7}} | 142 | 12 |
math | ## Task 24/88
The following system of equations is to be solved. Here, $n_{1}$ and $n_{2}$ are natural numbers, and $p_{i} (i=1,2, \ldots, 6)$ are prime numbers.
$$
\begin{array}{r}
n_{1}=p_{1}^{2} p_{2} p_{3} \\
n_{2}=n_{1}+1=p_{4}^{2} p_{5} p_{6} \\
p_{2}=p_{3}-p_{1}^{6} \\
p_{3}=p_{1} p_{4}^{3}+p_{6} \\
p_{5}=p_{2... | n_{1}=1988,\quadn_{2}=1989=n_{1}+1 | 201 | 24 |
math | 1. Given the sequence $\left\{a_{n}\right\}$ satisfies
$$
\begin{array}{l}
a_{1}=\mathrm{e}, a_{2}=\mathrm{e}^{3}, \\
\mathrm{e}^{1-k} a_{n}^{k+2}=a_{n+1} a_{n-1}^{2 k}\left(n \geqslant 2, n \in \mathbf{Z}_{+}, k \in \mathbf{R}_{+}\right) . \\
\text {Find } \prod_{i=1}^{2017} a_{i} . \quad \text { (Hao Hongbin) }
\end{... | \mathrm{e}^{2^{2018}-2019} | 166 | 18 |
math | 2. For the function $y=f(x)(x \in D)$, if for any $x_{1} \in D$, there exists a unique $x_{2} \in D$ such that
$$
\sqrt{f\left(x_{1}\right) f\left(x_{2}\right)}=M \text {, }
$$
then the function $f(x)$ is said to have a geometric mean of $M$ on $D$.
Given $f(x)=x^{3}-x^{2}+1(x \in[1,2])$. Then the geometric mean of th... | \sqrt{5} | 160 | 5 |
math | The integers $1, 2, \dots, n$ are written in order on a long slip of paper. The slip is then cut into five pieces, so that each piece consists of some (nonempty) consecutive set of integers. The averages of the numbers on the five slips are $1234$, $345$, $128$, $19$, and $9.5$ in some order. Compute $n$.
[i]Propose... | 2014 | 106 | 4 |
math | Task 11. Ten different books are randomly arranged on one bookshelf. Find the probability that two specific books will be placed next to each other (event $A$). | 0.2 | 36 | 3 |
math | 5. The maximum value of the orthogonal projection area of a regular tetrahedron with edge length 1 on a horizontal plane is $\qquad$ | \frac{1}{2} | 31 | 7 |
math | 2. Let
$$
\begin{array}{l}
f(x)=x^{2}-53 x+196+\left|x^{2}-53 x+196\right| \\
\text { then } f(1)+f(2)+\cdots+f(50)=
\end{array}
$$ | 660 | 72 | 3 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow \pi} \frac{e^{\pi}-e^{x}}{\sin 5 x-\sin 3 x}$ | \frac{e^{\pi}}{2} | 44 | 11 |
math | Example 2 Given $f(x)=\left\{\begin{array}{cc}x^{2}-1, & 0 \leqslant x \leqslant 1, \\ x^{2}, & -1 \leqslant x<0 .\end{array}\right.$ Find $f^{-1}(x)$. | f^{-1}(x)={\begin{pmatrix}\sqrt{x+1},(-1\leqslantx\leqslant0),\\-\sqrt{x},(0<x\leqslant1)0\end{pmatrix}.} | 74 | 56 |
math | 4. A farmer has a flock of $n$ sheep, where $2000 \leq n \leq 2100$. The farmer puts some number of the sheep into one barn and the rest of the sheep into a second barn. The farmer realizes that if she were to select two different sheep at random from her flock, the probability that they are in different barns is exact... | 2025 | 98 | 4 |
math | 2B. Determine all solutions to the equation
$$
3^{x}+3^{y}+3^{z}=21897
$$
such that $x, y, z \in \mathbb{N}$ and $x<y<z$. | 3,7,9 | 57 | 5 |
math | 11. (20 points) Given the sequence of positive integers $\left\{a_{n}\right\}$ satisfies:
$$
\begin{array}{l}
a_{1}=a, a_{2}=b, \\
a_{n+2}=\frac{a_{n}+2018}{a_{n+1}+1}(n \geqslant 1) .
\end{array}
$$
Find all possible values of $a+b$. | 1011or2019 | 103 | 9 |
math | Example 2. Solve the equation $y^{\prime}=\frac{1}{x \cos y+\sin 2 y}$. | Ce^{\siny}-2(1+\siny) | 29 | 13 |
math | Find all pairs $ (p, q)$ of primes such that $ {p}^{p}\plus{}{q}^{q}\plus{}1$ is divisible by $ pq$. | (5, 2) | 38 | 6 |
math | 8. If a positive integer $n$ makes the equation $x^{3}+y^{3}=z^{n}$ have positive integer solutions $(x, y, z)$, then $n$ is called a "good number". Then, the number of good numbers not exceeding 2,019 is $\qquad$ . | 1346 | 70 | 4 |
math | 10. The difference between $180^{\circ}$ and each interior angle of a quadrilateral is called the angle defect of that angle. The sum of the angle defects of all interior angles of a quadrilateral is
$\qquad$ degrees. | 360 | 53 | 3 |
math | In a convex $n$-sided polygon, the difference between any two adjacent interior angles is $18^{\circ}$. Try to find the maximum value of $n$.
The difference between any two adjacent interior angles of a convex $n$-sided polygon is $18^{\circ}$. Try to find the maximum value of $n$. | 38 | 76 | 2 |
math | 419. Calculate $\sin \left(-\frac{5 \pi}{3}\right)+\cos \left(-\frac{5 \pi}{4}\right)+\operatorname{tg}\left(-\frac{11 \pi}{6}\right)+$ $+\operatorname{ctg}\left(-\frac{4 \pi}{3}\right)$. | \frac{\sqrt{3}-\sqrt{2}}{2} | 80 | 15 |
math | Two circles have radius $2$ and $3$, and the distance between their centers is $10$. Let $E$ be the intersection of their two common external tangents, and $I$ be the intersection of their two common internal tangents. Compute $EI$.
(A [i]common external tangent[/i] is a tangent line to two circles such that the circl... | 24 | 128 | 2 |
math | Let $P$ be a point inside a square $ABCD$ such that $PA:PB:PC$ is $1:2:3$. Determine the angle $\angle BPA$. | 135^\circ | 39 | 5 |
math | 9.3. The road from Kimovsk to Moscow consists of three sections: Kimovsk - Novomoskovsk (35 km), Novomoskovsk - Tula (60 km), and Tula - Moscow (200 km). A bus, whose speed nowhere exceeded 60 km/h, traveled from
Kimovsk to Tula in 2 hours, and from Novomoskovsk to Moscow in 5 hours. How long could the bus have been ... | 5\frac{7}{12} | 107 | 9 |
math | [Inequality problems. Case analysis]
In a certain school, a council of 5 students was elected in each of the 20 classes. Petya turned out to be the only boy elected to the council along with four girls. He noticed that in 15 other classes, more girls were elected than boys, although in total, the number of boys and gi... | 19 | 103 | 2 |
math | While taking the SAT, you become distracted by your own answer sheet. Because you are not bound to the College Board's limiting rules, you realize that there are actually $32$ ways to mark your answer for each question, because you could fight the system and bubble in multiple letters at once: for example, you could m... | 2013 | 241 | 4 |
math | [b]M[/b]ary has a sequence $m_2,m_3,m_4,...$ , such that for each $b \ge 2$, $m_b$ is the least positive integer m for
which none of the base-$b$ logarithms $log_b(m),log_b(m+1),...,log_b(m+2017)$ are integers. Find the largest number in her sequence.
| 2188 | 88 | 4 |
math | \section*{Problem 5 - 331035}
Determine all pairs \((m ; n)\) of positive integers \(m\) and \(n\) for which
\[
\frac{m^{2}}{m+1}+\frac{n^{2}}{n+1}
\]
is an integer. | (1;1) | 71 | 5 |
math | 6.28 Given the value of $\sin \alpha$. Find: (a) $\sin \frac{\alpha}{2}$, (b) $\sin \frac{\alpha}{3}$, respectively, how many different values can they have at most? | 4,3 | 53 | 3 |
math | 10. There is a sequence of numbers +1 and -1 of length $n$. It is known that the sum of every 10 neighbouring numbers in the sequence is 0 and that the sum of every 12 neighbouring numbers in the sequence is not zero.
What is the maximal value of $n$ ? | 15 | 66 | 2 |
math | 29.37. Calculate the length of the astroid given by the equation $x^{2 / 3}+y^{2 / 3}=a^{2 / 3}$. | 6a | 41 | 2 |
math | Solve for $a, b, c$ given that $a \le b \le c$, and
$a+b+c=-1$
$ab+bc+ac=-4$
$abc=-2$ | a = -1 - \sqrt{3}, \, b = -1 + \sqrt{3}, \, c = 1 | 43 | 28 |
math | Find all values of the parameter $a$ such that the equation
$$
a x^{2}-(a+3) x+2=0
$$
admits two real roots of opposite signs.
Proposition 3.36 (Viète's Relations in the General Case). Let $P(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0} \in \mathbb{K}[X]$ with $a_{n} \neq 0$. If $\alpha_{1}, \ldots, \alpha_{n... | <0 | 762 | 2 |
math | 25. $[\mathbf{1 3}]$ Evaluate the sum
$$
\cos \left(\frac{2 \pi}{18}\right)+\cos \left(\frac{4 \pi}{18}\right)+\cdots+\cos \left(\frac{34 \pi}{18}\right) .
$$ | -1 | 73 | 2 |
math | 17. [10] How many ways are there to color every integer either red or blue such that $n$ and $n+7$ are the same color for all integers $n$, and there does not exist an integer $k$ such that $k, k+1$, and $2 k$ are all the same color? | 6 | 71 | 1 |
math | 59. Let $a, b$ be positive constants. $x_{1}, x_{2}, \cdots, x_{n}$ are positive real numbers. $n \geqslant 2$ is a positive integer. Find the maximum value of $y=$ $\frac{x_{1} x_{2} \cdots x_{n}}{\left(a+x_{1}\right)\left(x_{1}+x_{2}\right) \cdots\left(x_{n-1}+x_{n}\right)\left(x_{n}+b\right)}$. (1999 Polish Mathemat... | \frac{1}{(\sqrt[n+1]{a}+\sqrt[n+1]{b})^{n+1}} | 136 | 26 |
math | Problem 10.7. At one meal, Karlson can eat no more than 5 kg of jam. If he opens a new jar of jam, he must eat it completely during this meal. (Karlson will not open a new jar if he has to eat more than 5 kg of jam together with what he has just eaten.)
Little Boy has several jars of raspberry jam weighing a total of ... | 12 | 120 | 2 |
math | How many ways are there to remove an $11\times11$ square from a $2011\times2011$ square so that the remaining part can be tiled with dominoes ($1\times 2$ rectangles)? | 2,002,001 | 53 | 9 |
math | 2. It is known that the numbers $x, y, z$ form an arithmetic progression in the given order with a common difference $\alpha=\arccos \left(-\frac{1}{3}\right)$, and the numbers $\frac{1}{\cos x}, \frac{3}{\cos y}, \frac{1}{\cos z}$ also form an arithmetic progression in the given order. Find $\cos ^{2} y$. | \frac{4}{5} | 95 | 7 |
math | The Bell Zoo has the same number of rhinoceroses as the Carlton Zoo has lions. The Bell Zoo has three more elephants than the Carlton Zoo has lions. The Bell Zoo has the same number of elephants as the Carlton Zoo has rhinoceroses. The Carlton Zoo has two more elephants than rhinoceroses. The Carlton Zoo has twice as m... | 57 | 195 | 2 |
math | Three, (This question is worth 16 points) How many ordered pairs of positive integers $(x, y)$ have the following properties:
$y<x \leqslant 100$, and $\frac{x}{y}$ and $\frac{x+1}{y+1}$ are both integers? | 85 | 64 | 2 |
math | 11. How many strikes will a clock make in a day if it strikes the whole number of hours and also marks the midpoint of each hour with one strike? | 180 | 33 | 3 |
math | 5. Given the function $f(x)$
$=\log _{a}\left(a x^{2}-x+\frac{1}{2}\right)$ is always positive on the interval $[1,2]$, then the range of the real number $a$ is $\qquad$ | (\frac{1}{2},\frac{5}{8})\cup(\frac{3}{2},+\infty) | 61 | 27 |
math | Sorrelkova N.P.
Under one of the cells of an $8 \times 8$ board, a treasure is buried. Under each of the other cells, there is a sign indicating the minimum number of steps required to reach the treasure from that cell (one step allows moving from a cell to an adjacent cell by side). What is the minimum number of cell... | 3 | 88 | 1 |
math | Find all real numbers with the following property: the difference of its cube and
its square is equal to the square of the difference of its square and the number itself. | 0, 1, 2 | 34 | 7 |
math | Example 1. Find the Lagrange interpolation polynomial that takes the values $y_{0}=-5, y_{1}=-11, y_{2}=10$ at the points $x_{0}=-3, x_{1}=-1, x_{2}=2$. | 2x^{2}+5x-8 | 61 | 10 |
math | 2. $f(n)$ is defined on the set of positive integers, and: (1) for any positive integer $n, f[f(n)]=4 n+9$; (2) for any non-negative integer $k, f\left(2^{k}\right)=2^{k+1}+3$. Determine $f(1789)$.
(1989 Australian Olympiad Problem) | 3581 | 88 | 4 |
math | We roll a die until a six comes up. What is the probability that we do not roll a five in the meantime? | \frac{1}{2} | 25 | 7 |
math | 9. Let $0<\theta<\pi$, find the maximum value of $y=\sin \frac{\theta}{2}(1+\cos \theta)$ | \frac{4\sqrt{3}}{9} | 34 | 12 |
math | ## Task 16/86
The angle bisectors of a parallelogram determine another parallelogram. Determine the ratio of the areas of this and the original parallelogram. | \frac{(-b)^{2}}{} | 39 | 10 |
math | [ equations in integers $]$ [GCD and LCM. Mutual simplicity ]
Find all pairs of integers $(x, y)$, satisfying the equation $3 \cdot 2^{x}+1=y^{2}$. | (0,\2),(3,\5),(4,\7) | 46 | 13 |
math | 2. What is the remainder when the number
$1+1 \cdot 2+1 \cdot 2 \cdot 3+1 \cdot 2 \cdot 3 \cdot 4+1 \cdot 2 \cdot 3 \cdot 4 \cdot 5+\ldots+1 \cdot 2 \cdot 3 \cdot 4 \cdot \ldots \cdot 14+1 \cdot 2 \cdot 3 \cdot 4 \cdot \ldots \cdot 14 \cdot 15$
is divided by 72? | 9 | 124 | 1 |
math | In a taxi, a passenger can sit in the front and three passengers can sit in the back. In how many ways can four passengers sit in a taxi if one of these passengers wants to sit by the window? | 18 | 43 | 2 |
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