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 | 8.1. Of the three boys named Anton, Vanya, and Sasha, only one always tells the truth. Anton said: "Vanya does not always tell the truth," Vanya said: "I do not always tell the truth," and Sasha said: "Anton does not always tell the truth." Who among them always tells the truth, given that at least one of them lied?
# | Anton | 82 | 2 |
math | On each wall of a regular octahedron, one of the numbers $1,2,3,4,5,6,7$ and 8 is written, with different numbers on different walls. For each wall, Jarda determined the sum of the number written on it and the numbers of the three adjacent walls. Thus, he obtained eight sums, which he also added together.
What values ... | 144 | 107 | 3 |
math | 394. Solve the equation:
$$
\sqrt[3]{x}+\sqrt[3]{x+19}=5
$$ | 8 | 30 | 1 |
math | 5th Mexico 1991 Problem A2 n is palindromic (so it reads the same backwards as forwards, eg 15651) and n = 2 mod 3, n = 3 mod 4, n = 0 mod 5. Find the smallest such positive integer. Show that there are infinitely many such positive integers. | 515 | 77 | 3 |
math | 13.233. A tourist was returning from vacation on a bicycle. On the first leg of the journey, which was 246 km, he traveled on average 15 km less per day than he did on the last leg of the journey, which was 276 km. He arrived home right on time at the end of his last vacation day. It is also known that it took him one ... | 4 | 128 | 1 |
math | 2. A natural number, not ending in zero, had one of its digits erased. As a result, the number decreased by 6 times. Find all numbers for which this is possible. | 108or12awhen=1,2,3,4 | 39 | 16 |
math | Given a positive integer $n (n>2004)$, we put 1, 2, 3, …,$n^2$ into squares of an $n\times n$ chessboard with one number in a square. A square is called a “good square” if the square satisfies following conditions:
1) There are at least 2004 squares that are in the same row with the square such that any number within t... | n^2 - 2004n | 164 | 11 |
math | By what power should $\frac{32}{243}$ be raised to obtain $\frac{27}{8}$? | -\frac{3}{5} | 27 | 7 |
math | Problem 12.1. Solve in integers the equation
$$
2^{a}+8 b^{2}-3^{c}=283
$$
Oleg Mushkarov, Nikolai Nikolov | =2,b=\6,=2 | 45 | 8 |
math | Three real numbers $x$, $y$, and $z$ are such that $(x+4)/2=(y+9)/(z-3)=(x+5)/(z-5)$. Determine the value of $x/y$. | \frac{1}{2} | 49 | 7 |
math | 8.2. Through the vertices $A$ and $C$ of triangle $ABC$, lines are drawn perpendicular to the bisector of angle $ABC$ and intersect the lines $CB$ and $BA$ at points $K$ and $M$ respectively. Find $AB$, if $BM=10, KC=2$. | 8or12 | 69 | 4 |
math | 3. Find the number of integer solutions $(a ; b ; c)$ of the equation param1, satisfying the condition param2.
| param1 | param2 | |
| :---: | :---: | :---: |
| $150^{a} \cdot\left(\frac{200}{3}\right)^{b} \cdot 2250^{c}=33750$ | $\|a+b+c\| \leq 120$ | |
| $150^{a} \cdot\left(\frac{200}{3}\right)^{b} \cdot 2250^{c}=... | 120,90,250,112 | 288 | 14 |
math | Given a random string of 33 bits (0 or 1), how many (they can overlap) occurrences of two consecutive 0's would you expect? (i.e. "100101" has 1 occurrence, "0001" has 2 occurrences) | 8 | 61 | 1 |
math | 32nd IMO 1991 shortlist Problem 23 f(n) is an integer-valued function defined on the integers which satisfies f(m + f( f(n) ) ) = - f( f(m+1)) - n for all m, n. The polynomial g(n) has integer coefficients and g(n) = g( f(n) ) for all n. Find f(1991) and the most general form for g. | f(1991)=-1992,\(x) | 96 | 16 |
math | 4. Suppose $a_{1}=\frac{1}{6}$ and
$$
a_{n}=a_{n-1}-\frac{1}{n}+\frac{2}{n+1}-\frac{1}{n+2}
$$
for $n>1$. Find $a_{100}$. | \frac{1}{10302} | 71 | 11 |
math | 4. Given positive integers $a, b, c$ satisfying
$$
1<a<b<c, a+b+c=111, b^{2}=a c \text {. }
$$
then $b=$ . $\qquad$ | 36 | 50 | 2 |
math | 1st Eötvös 1894 Problem 3 A triangle has sides length a, a + d, a + 2d and area S. Find its sides and angles in terms of d and S. Give numerical answers for d = 1, S = 6. | 3,4,5,A=\sin^{-1}(3/5),C=90,B=90-A | 60 | 24 |
math | For how many different values of the parameter $p$ does the system of equations
$$
x^{2}-y^{2}=0 \quad x y+p x-p y=p^{2}
$$
have exactly one solution? | 1 | 47 | 1 |
math | Twenty-fi ve of the numbers $1, 2, \cdots , 50$ are chosen. Twenty- five of the numbers$ 51, 52, \cdots, 100$ are also chosen. No two chosen numbers diff er by $0$ or $50$. Find the sum of all $50$ chosen numbers. | 2525 | 81 | 4 |
math | 3. Given a cube $A B C D-$ $A_{1} B_{1} C_{1} D_{1}$ with edge length 1, the distance between the skew lines $A C_{1}$ and $B_{1} C$ is $\qquad$ | \frac{\sqrt{6}}{6} | 59 | 10 |
math | 8. (10 points) Let for positive numbers $x, y, z$ the following system of equations holds:
$$
\left\{\begin{array}{l}
x^{2}+x y+y^{2}=27 \\
y^{2}+y z+z^{2}=9 \\
z^{2}+x z+x^{2}=36
\end{array}\right.
$$
Find the value of the expression $x y+y z+x z$. | 18 | 101 | 2 |
math | 8. The sum of the ages of three people, A, B, and C, represented by $x, y, z$ is 120, and $x, y, z \in (20,60)$. The number of ordered triples $(x, y, z)$ is $\qquad$ . | 1141 | 68 | 4 |
math | For a positive integer $n$, let
\[S_n=\int_0^1 \frac{1-(-x)^n}{1+x}dx,\ \ T_n=\sum_{k=1}^n \frac{(-1)^{k-1}}{k(k+1)}\]
Answer the following questions:
(1) Show the following inequality.
\[\left|S_n-\int_0^1 \frac{1}{1+x}dx\right|\leq \frac{1}{n+1}\]
(2) Express $T_n-2S_n$ in terms of $n$.
(3) Find the limit $\li... | 2 \ln 2 - 1 | 151 | 10 |
math | Exercise 2. Find all quadruplets of integers $(a, b, c, p)$ such that $p$ is a prime number and for which
$$
73 p^{2}+6=9 a^{2}+17 b^{2}+17 c^{2}
$$ | (\1,\1,\4,2)\text{}(\1,\4,\1,2) | 64 | 20 |
math | \section*{Problem 4 - 021134}
Determine all solutions of the equation \(\sin ^{3} x+\cos ^{3} x=1\). | x_{2k}=2k\pi\quadx_{2k+1}=\frac{\pi}{2}+2k\pi\quad(k\in\mathbb{N}) | 42 | 41 |
math | Example 7 Let $S=\{1,2,3,4\}$. An $n$-term sequence: $q_{1}, q_{2}, \cdots, q_{n}$ has the following property: For any non-empty subset $B$ of $S$ (the number of elements in $B$ is denoted by $|B|$), there are adjacent $|B|$ terms in the sequence that exactly form the set $B$. Find the minimum value of $n$.
(1997, Shan... | 8 | 118 | 1 |
math | Four points $A, O, B, O^{\prime}$ are aligned in this order on a line. Let $C$ be the circle with center $O$ and radius 2015, and $C^{\prime}$ be the circle with center $O^{\prime}$ and radius 2016. Suppose that $A$ and $B$ are the intersections of two common tangents to the two circles. Calculate $A B$ given that $A B... | 8124480 | 163 | 7 |
math | Try to use a function of $n$ to represent the product
$$
9 \times 99 \times 9999 \times \cdots \times\left(10^{2^{n}}-1\right)
$$
the sum of the digits in decimal notation. | 9\times2^{n} | 62 | 7 |
math | Find all bases of logarithms in which a real positive number can be equal to its logarithm or prove that none exist. | n = a^{1/a} | 25 | 8 |
math | 11. We define: $a @ b=a \times(a+1) \times \ldots \times(a+b-1)$. Given that $x @ y @ 2=420$, then $y @ x=$ ( Exam point: Solving equations and defining new operations | 20or120 | 61 | 6 |
math | 4. Solve the equation $6(x-1)(x+2)-4(x-3)(x+4)=2(x-5)(x-6)$. | 1 | 34 | 1 |
math | Solve the following equation:
$$
a \cdot b^{x} \cdot c^{2 x}=\sqrt[3 x]{d} \cdot \sqrt[4 x]{e}
$$
Numerical examples: the values of $a, b, c, d, e$ are respectively: I. 2, 3, 5, 7, 11; $\quad$ II. 5, 3, 2, 1/7, $1 / 11$. | x_{1}\approx-0.624,x_{2}\approx0.464 | 107 | 21 |
math | 1. [20] For how many positive integers $n \leq 1000$ does the equation in real numbers
$$
x^{\lfloor x\rfloor}=n
$$
have a positive solution for $x$ ? (For a real number $x,\lfloor x\rfloor$ denotes the largest integer that is not greater than $x$.) | 412 | 82 | 3 |
math | (11) There are 6 different books, including one math book, two English books, and three music books. Arrange them in a row so that the English books are not adjacent, and the music books are also not adjacent. The number of different arrangements is $\qquad$ . | 120 | 59 | 3 |
math | 11. Given that $a$ and $b$ are real numbers, satisfying:
$$
\sqrt[3]{a}-\sqrt[3]{b}=12, \quad a b=\left(\frac{a+b+8}{6}\right)^{3} \text {. }
$$
Then $a-b=$ $\qquad$ (Proposed by Thailand) | 468 | 79 | 3 |
math | 15. Random Vectors. There are $n$ random vectors of the form $\left(y_{1}, y_{2}, y_{3}\right)$, where exactly one random coordinate is equal to 1, and the others are 0. They are added together. The resulting random vector is $\vec{a}$ with coordinates $\left(Y_{1}, Y_{2}, Y_{3}\right)$.
a) (from 9th grade. 2 points).... | \frac{2n+n^{2}}{3} | 151 | 12 |
math | 4. (20 points) The ballroom in the palace of the thirtieth kingdom is a region on the plane, the coordinates of which satisfy the conditions $|x| \leqslant 4,|y| \leqslant 6$. How many identical parquet tiles, having the shape of a rectangle with sides 1.5 and 2, are needed to tile the floor of the room? Tiling is cons... | 32 | 113 | 2 |
math | 10. (20 points) Given that circles $C_{1}$ and $C_{2}$ intersect at two points, one of which has coordinates $(9,6)$, and the product of the radii of the two circles is 68. If the x-axis and the line $y=m x (m>0)$ are both tangent to the two circles, find the value of $m$.
| \frac{12\sqrt{221}}{49} | 86 | 16 |
math | . In the decimal writing of $A$, the digits appear in (strictly) increasing order from left to right. What is the sum of the digits of $9 A$? | 9 | 37 | 1 |
math | (10 Given $\lg x_{1}, \lg x_{2}, \lg x_{3}, \lg x_{4}, \lg x_{5}$ are consecutive positive integers (in ascending or descending order), and $\left(\lg x_{4}\right)^{2}<\lg x_{1} \cdot \lg x_{5}$, then the minimum value of $x_{1}$ is $\qquad$ . | 100000 | 90 | 6 |
math | In the acute triangle $ABC$, the sum of the distances from the vertices $B$ and $C$ to of the orthocenter $H$ is equal to $4r,$ where $r$ is the radius of the circle inscribed in this triangle. Find the perimeter of triangle $ABC$ if it is known that $BC=a$.
(Gryhoriy Filippovskyi) | 3a | 82 | 4 |
math | # Problem 5. (3 points)
$A B C D$ is a cyclic quadrilateral. The extension of side $A B$ beyond point $B$ and the extension of side $C D$ beyond point $C$ intersect at point $P$. The extension of side $A D$ beyond point $D$ and the extension of side $B C$ beyond point $C$ intersect at point $Q$. It turns out that angl... | 1040 | 161 | 4 |
math | Berezin V.N.
Find the sums
a) $1 \cdot n+2(n-1)+3(n-2)+\ldots+n \cdot 1$.
b) $S_{n, k}=(1 \cdot 2 \cdot \ldots \cdot k) \cdot(n(n-1) \ldots(n-k+1))+(2 \cdot 3 \cdot \ldots \cdot(k+1)) \cdot((n-1)(n-2) \ldots(n-k))+\ldots+((n-k+1)(n-k+$
2) $\ldots \cdot n) \cdot(k(k-1) \cdot \ldots \cdot 1)$ | (k!)^2\cdotC_{n+k+1}^{2k+1} | 151 | 19 |
math | Example 3 So far, apart from calculating specific values, we have no method to solve the indeterminate equation
$$x^{2}-11 y^{2}=1, \quad x>0, y>0$$ | x=10, y=3 ; \quad x=199, y=60 ; \quad x=3970, y=1197 | 46 | 37 |
math | 6. Among the first 999 positive integers, the numbers that are neither divisible by 5 nor by 7 are
$\qquad$ in number. | 686 | 34 | 3 |
math | ## Zadatak B-3.1.
Odredite umnožak svih rješenja jednadžbe
$$
\sqrt{2021} x^{\log _{2021} x}=x^{2}
$$
| 2021^2 | 58 | 6 |
math | Mr. Ambulando is at the intersection of $5^{\text{th}}$ and $\text{A St}$, and needs to walk to the intersection of $1^{\text{st}}$ and $\text{F St}$. There's an accident at the intersection of $4^{\text{th}}$ and $\text{B St}$, which he'd like to avoid.
[center]<see attached>[/center]
Given that Mr. Ambulando wants ... | 56 | 118 | 2 |
math | Problem 1. Find the last digit of the number $n=2^{0}+2^{1}+2^{2}+\ldots+2^{2014}$. | 7 | 40 | 1 |
math | 5. (10 points) A small railway wagon with a jet engine is standing on the tracks. The tracks are laid in the form of a circle with a radius of $R=4$ m. The wagon starts from rest, with the jet force having a constant value. What is the maximum speed the wagon will reach after one full circle, if its acceleration over t... | 2.8\mathrm{~}/\mathrm{} | 99 | 11 |
math | Example 13 On the coordinate plane, points with both integer horizontal and vertical coordinates are called integer points. For any natural number $n$, connect the origin $O$ with $A_{n}(n, n+3)$, and let $f(n)$ denote the number of integer points on the line segment $O A_{n}$, excluding the endpoints. Then
$$
f(1)+f(2... | 1334 | 121 | 4 |
math | 2. For a natural number $n$, we will say it is superprime if the difference between any two of its consecutive positive divisors is a prime number. Determine all superprime numbers. | 3 | 39 | 1 |
math | 17.5 Given that the area of $\triangle A B C$ is $S$, points $D$, $E$, and $F$ are the midpoints of $B C$, $C A$, and $A B$ respectively, $I_{1}$, $I_{2}$, and $I_{3}$ are the incenters of $\triangle A E F$, $\triangle B F D$, and $\triangle C D E$ respectively, and the area of $\triangle I_{1} I_{2} I_{3}$ is $S^{\pri... | \frac{1}{4} | 133 | 7 |
math | 1. Given $\tan \alpha=m, \frac{3 \sin \alpha+\sin 3 \alpha}{3 \cos \alpha+\cos 3 \alpha}=$ | \frac{}{2}(^{2}+3) | 37 | 12 |
math | 4. (3 points) A swimming pool has three water inlet pipes, A, B, and C. If only pipe A is opened, it takes 20 hours to fill the pool; if pipes A and B are opened together, it takes 8 hours to fill the pool; if pipes B and C are opened together, it takes 6 hours to fill the pool. Then, if only pipe C is opened, it will ... | 10\frac{10}{11} | 100 | 11 |
math | 1. How many non-empty subsets of the set $\{0,1, \ldots, 9\}$ have the sum of their elements divisible by three?
(Eliška Macáková) | 351 | 43 | 3 |
math | Determine all positive integers $n$ such that:
$$
5^{n-1}+3^{n-1} \mid 5^{n}+3^{n}
$$ | 1 | 39 | 1 |
math | Compute the sum of all positive integers $n$ such that $n^n$ has 325 positive integer divisors. (For example, $4^4=256$ has 9 positive integer divisors: 1, 2, 4, 8, 16, 32, 64, 128, 256.) | 93 | 81 | 2 |
math | 10. For the equation $x^{2}+m x+1+2 \mathrm{i}=0$ with respect to $x$, if it has real roots, then the minimum value of the modulus of the complex number $m$ is | \sqrt{2+2\sqrt{5}} | 51 | 11 |
math | 8. (10 points) In the expression $(x+y+z)^{2032}+(x-y-z)^{2032}$, the parentheses were expanded and like terms were combined. How many monomials $x^{a} y^{b} z^{c}$ with a non-zero coefficient were obtained? | 1034289 | 69 | 7 |
math | 5. Given the function
$$
f(x)=\log _{a}\left(a x^{2}-x+\frac{1}{2}\right)
$$
is always positive in the interval $[1,2]$. Then the range of the real number $a$ is $\qquad$ . | (\frac{1}{2},\frac{5}{8})\cup(\frac{3}{2},+\infty) | 64 | 27 |
math | 10.131. A circle with an area of $Q$ is inscribed in a rhombus with an acute angle of $30^{\circ}$. Find the area of the rhombus. | \frac{8Q}{\pi} | 46 | 9 |
math | 8. Arrange the numbers $1,2, \cdots, 6$ in any order. If the number $k$ appears exactly in the $k$-th position, it is called a match. The expected number of matches is $\qquad$ | 1 | 54 | 1 |
math | 11.1. A unified control test in mathematics was conducted among all eleventh-graders in the school. As a result, 5/8 of the students received fives, 11/20 of the number of excellent students received fours, and the remaining three eleventh-graders did not come to the test due to illness. How many eleventh-graders are t... | 96 | 82 | 2 |
math | ## Task B-4.5.
Determine all prime numbers $p$ and $q$ for which the equation
$$
5 p^{3}-8 q+5 p-10=0
$$
holds. | p=2,q=5 | 47 | 6 |
math | ## Task 3A - 301233A
Determine all seventeen-digit natural numbers $n$ whose 17 digits $x_{1}, x_{2}, \ldots, x_{17}$ satisfy the following conditions (1) and (2):
(1) It holds that: $x_{1} \geq x_{2} \geq \ldots \geq x_{17}$.
(2) For the sum $s=x_{1}+x_{2}+\ldots+x_{17}$ and the product $p=x_{1} \cdot x_{2} \cdot \... | 9311111111111111,6221111111111111,55111111111111111 | 206 | 51 |
math | 12.44 Find the natural number solutions \(x, y\) for the equation \(x^{3}-y^{3}=x y+61\).
(15th All-Soviet Union Mathematical Olympiad, 1981) | x=6,y=5 | 53 | 6 |
math | Determine all functions $f:\mathbb R\to\mathbb R$ such that equality
$$f(x + y + yf(x)) = f(x) + f(y) + xf(y)$$
holds for all real numbers $x$, $y$.
Proposed by Athanasios Kontogeorgis | f(x) \equiv x | 67 | 6 |
math | 4.007. When the ninth term of an arithmetic progression is divided by the second term, the quotient is 5, and when the thirteenth term is divided by the sixth term, the quotient is 2 and the remainder is 5. Find the first term and the common difference of the progression. | 3;4 | 65 | 3 |
math | $4 \cdot 102$ Solve the system of equations
$$
\left\{\begin{array}{l}
x(x+1)(3 x+5 y)=144 \\
x^{2}+4 x+5 y=24
\end{array}\right.
$$ | (-4,\frac{24}{5}),(3,\frac{3}{5}) | 64 | 19 |
math | Solve the following equation:
$$
\log (7 x-9)^{2}+\log (3 x-4)^{2}=2
$$ | x_{1}=2,\quadx_{2}=\frac{13}{21} | 33 | 20 |
math | Find all integers $n \in \mathbb{N}$ such that $n \mid\left(n^{2}+2 n+27\right)$. | 1,3,9,27 | 35 | 8 |
math | 1. Given $A_{1}, A_{2}, \cdots, A_{n}$ are 11 stations sequentially on a straight highway, and
$$
\begin{array}{l}
A_{i} A_{i+2} \leqslant 12(i=1,2, \cdots, 9), \\
A_{i} A_{i+3} \geqslant 17(i=1,2, \cdots, 8) .
\end{array}
$$
If $A_{1} A_{11}=56 \mathrm{~km}$, then $A_{2} A_{7}=$ $\qquad$ $\mathrm{km}$. | 29 | 154 | 2 |
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 | 4. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
f(x+y f(x))=f(f(x))+x f(y)
$$ | f(x)=x | 43 | 4 |
math | Example 2. Calculate the area bounded by the curve $y=6 x-x^{2}-5$ and the $O x$ axis. | \frac{32}{3} | 30 | 8 |
math | ## Problem Statement
Find the second-order derivative $y_{x x}^{\prime \prime}$ of the function given parametrically.
$$
\left\{\begin{array}{l}
x=\frac{\cos t}{1+2 \cos t} \\
y=\frac{\sin t}{1+2 \cos t}
\end{array}\right.
$$ | -\frac{(1+2\cos)^{3}}{\sin^{3}} | 78 | 17 |
math | Let $p,q$ and $s{}$ be prime numbers such that $2^sq =p^y-1$ where $y > 1.$ Find all possible values of $p.$ | 3 \text{ and } 5 | 41 | 8 |
math | Find the smallest positive value of $36^k - 5^m$, where $k$ and $m$ are positive integers. | 11 | 29 | 2 |
math | Determine all positive integers $n$ for which there exist positive integers $a_1,a_2, ..., a_n$
with $a_1 + 2a_2 + 3a_3 +... + na_n = 6n$ and $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+ ... +\frac{n}{a_n}= 2 + \frac1n$ | n = 3 | 100 | 5 |
math | 4. How many nonempty subsets of $\{1,2, \ldots, 10\}$ have the property that the sum of its largest element and its smallest element is 11 ? | 341 | 42 | 3 |
math | 8. Given $2014+$ Ying $=2015+$ Xin $=2016+$ Nian, and Ying $\times$ Xin $\times$ Nian $=504$, then Ying $\times$ Xin + Xin $\times$ Nian
$\qquad$ | 128 | 63 | 3 |
math | 19. (MEX 1) Let $f(n)$ be a function defined on the set of all positive integers and having its values in the same set. Suppose that $f(f(m)+f(n))=m+n$ for all positive integers $n, m$. Find all possible values for $f(1988)$. | 1988 | 71 | 4 |
math | 4. (7 points) The numbers $a, b, c, d$ belong to the interval $[-7 ; 7]$. Find the maximum value of the expression $a+2 b+c+2 d-a b-b c-c d-d a$. | 210 | 54 | 3 |
math | Example 3 Let $n$ be a positive integer,
$$
\begin{aligned}
S= & \{(x, y, z) \mid x, y, z \in\{0,1, \cdots, n\}, \\
& x+y+z>0\}
\end{aligned}
$$
is a set of $(n+1)^{3}-1$ points in three-dimensional space. Try to find the minimum number of planes whose union contains $S$ but does not contain $(0,0,0)$. | 3n | 114 | 2 |
math | 5. Given a regular triangular prism $A B C-A_{1} B_{1} C_{1}$ with a height of 2 and a base edge length of 1, the center of the upper base $\triangle A_{1} B_{1} C_{1}$ is $P$. A plane $B C D \perp A P$ is made through the lower base edge $B C$, intersecting the edge $A A_{1}$ at $D$. Then the area of the section $B C ... | \frac{\sqrt{13}}{8} | 115 | 11 |
math | ## Task Condition
Find the derivative.
$y=\frac{e^{x^{3}}}{1+x^{3}}$ | \frac{3x^{5}\cdote^{x^{3}}}{(1+x^{3})^{2}} | 26 | 25 |
math | Example 3 Find the largest integer $n$, such that all non-zero solutions of the equation $(z+1)^{n}=z^{n}+1$ lie on the unit circle. | 7 | 40 | 1 |
math | 10. Let $\square A B C D$ be a trapezoid with parallel sides $A B$ and $C D$ of lengths 6 units and 8 units, respectively. Let $E$ be the point of intersection of the extensions of the nonparallel sides of the trapezoid. If the area of $\triangle B E A$ is 60 square units, what is the area of $\triangle B A D$ ? | 20 | 94 | 2 |
math | ## 13. The Scout and the Drummer
On the occasion of the village festival, a procession was organized that stretched for 250 m; the scouts led the procession, and the musicians brought up the rear. Soon after the march began, the youngest scout remembered that he had not tied his neckerchief, which was left with his fr... | 3 | 134 | 1 |
math | A3. We call a positive integer alternating if the digits of the number alternate between even and odd. Thus, 2381 and 3218 are alternating, but 2318 is not. We call a number completely alternating if the number itself is alternating and double the number is also alternating. Thus, 505 is completely alternating, because... | 70 | 116 | 2 |
math | (EGMO 2012)(M-D) Find all functions $f$ from $\mathbb{R}$ to $\mathbb{R}$ such that, for all real numbers $x$ and $y$, we have:
$$
f(y f(x+y)+f(x))=4 x+2 y f(x+y)
$$ | f(x)=2x | 71 | 5 |
math | ## Task Condition
Find the differential $d y$.
$$
y=\operatorname{arctg} \frac{x^{2}-1}{x}
$$ | \frac{x^{2}+1}{x^{4}-x^{2}+1}\cdot | 34 | 21 |
math | 4. (8 points) Jia, Yi, Bing, and Ding won the top 4 places in the school (no ties), and they said:
Jia: “I am neither first nor second”; Yi said: “I am neither second nor third”;
Bing: “My rank is adjacent to Yi's”; Ding: “My rank is adjacent to Bing's”.
Now it is known that Jia, Yi, Bing, and Ding respectively obtaine... | 4123 | 142 | 4 |
math | 3. The range of the function $y=\sqrt{7-x}+\sqrt{9+x}$ is the interval | [4,4 \sqrt{2}] | 24 | 9 |
math | 11. (5 points) The Yirong ferry has a speed of 40 kilometers per hour. On odd days, it travels downstream from $A$ to $B$, and on even days, it travels upstream from $B$ to $A$. (The water speed is 24 kilometers per hour.) On an odd day, the ferry lost power while traveling to a point $C$ and could only drift to $B$. T... | 192 | 205 | 3 |
math | Find the maximum value of $M$ for which for all positive real numbers $a, b, c$ we have
\[ a^3+b^3+c^3-3abc \geq M(ab^2+bc^2+ca^2-3abc) \] | \frac{3}{4^{\frac{1}{3}}} | 59 | 15 |
math | Someones age is equal to the sum of the digits of his year of birth. How old is he and when was he born, if it is known that he is older than $11$.
P.s. the current year in the problem is $2010$. | 24 | 57 | 4 |
math | 21. In a family, there are 4 children aged 5, 8, 13, and 15, and their names are Tanya, Yura, Svetlana, and Lena. How old is each of them, if one girl attends a kindergarten, Tanya is older than Yura, and the sum of Tanya's and Svetlana's ages is divisible by 3? | Tanya:13,Yura:8,Svetlana:5,Lena:15 | 88 | 20 |
math | Four, $n^{2}(n \geqslant 4)$ positive numbers are arranged in $n$ rows and $n$ columns,
\begin{tabular}{llllll}
$a_{11}$ & $a_{12}$ & $a_{13}$ & $a_{14}$ & $\cdots$ & $a_{1 n}$ \\
$a_{21}$ & $a_{22}$ & $a_{23}$ & $a_{24}$ & $\cdots$ & $a_{2 n}$ \\
$a_{31}$ & $a_{32}$ & $a_{33}$ & $a_{34}$ & $\cdots$ & $a_{3 n}$ \\
$a_{... | 2-\frac{1}{2^{n-1}}-\frac{n}{2^{n}} | 359 | 20 |
math | $4 \cdot 100$ Solve the system of equations $\quad\left\{\begin{array}{l}x+y+z=0, \\ x^{3}+y^{3}+z^{3}=-18 .\end{array}\right.$ for integer solutions. | \begin{pmatrix}{\begin{array}{}{x=-3,}\{y=1,}\{z=2;}\end{pmatrix}\quad{\begin{array}{}{x=-3,}\{y=2,}\{z=1;}\end{pmatrix}\quad{} | 62 | 67 |
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