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 | 2. [5] The English alphabet, which has 26 letters, is randomly permuted. Let $p_{1}$ be the probability that $\mathrm{AB}, \mathrm{CD}$, and $\mathrm{EF}$ all appear as contiguous substrings. Let $p_{2}$ be the probability that $\mathrm{ABC}$ and $\mathrm{DEF}$ both appear as contiguous substrings. Compute $\frac{p_{1}... | 23 | 97 | 2 |
math | 2. Given are sets $A, B$, and $C$ such that:
- $A \cup B \cup C=\{1,2,3, \ldots, 100\}$;
- $A$ is the set of all natural numbers not greater than 100 that are divisible by 2;
- $B$ is the set of all natural numbers not greater than 100 that are divisible by 3;
- $B \cap C$ is the set of all natural numbers not greater... | 49 | 175 | 2 |
math | Given a set of n elements, find the largest number of subsets such that no subset is contained in any other | \binom{2k}{k} | 22 | 10 |
math | What are the sides of a right-angled triangle whose perimeter is $40 \mathrm{~cm}$ and the radius of the inscribed circle is $3 \mathrm{~cm}$? | a_{1}=8\mathrm{~},b_{1}=15\mathrm{~};a_{2}=15\mathrm{~},b_{2}=8\mathrm{~},=17\mathrm{~} | 40 | 50 |
math | Before the math competition, Dmytro overheard Olena and Mykola talking about their birthdays.
[b]О[/b]: "The day and month of my birthday are half as large as the day and month of Mykola's birthday."
[b]М[/b]: "Also, the day of Olena's birth and the month of my birth are consecutive positive integers."
[b]О[/b]: "And ... | (11, 6, 22, 12) | 135 | 16 |
math | 21st BMO 1985 Problem 6 Find all non-negative integer solutions to 5 a 7 b + 4 = 3 c . | 5^17^0+4=3^2 | 34 | 12 |
math | If $a$ and $b$ are positive integers such that $a \cdot b = 2400,$ find the least possible value of $a + b.$ | 98 | 36 | 2 |
math | Find the period of the repetend of the fraction $\frac{39}{1428}$ by using [i]binary[/i] numbers, i.e. its binary decimal representation.
(Note: When a proper fraction is expressed as a decimal number (of any base), either the decimal number terminates after finite steps, or it is of the form $0.b_1b_2\cdots b_sa_1a_2... | 24 | 181 | 2 |
math | On a circle of radius 1, a point $O$ is marked, and from it, a notch is made to the right with a radius of $l$. From the resulting point $O_{1}$, another notch is made in the same direction with the same radius, and this is repeated 1968 times. After this, the circle is cut at all 1968 notches, resulting in 1968 arcs. ... | 3 | 107 | 1 |
math | 2. Find the length of the side of a square whose area is numerically equal to its perimeter. | 4 | 21 | 1 |
math | ## Task 23/85
Given the $n$-digit natural number $z_{n}=1985$!. Form the natural number $z_{n-1}$ by removing the units digit of $z_{n}$ and subtracting it from the remaining $(n-1)$-digit number. Continue this process until a single-digit number $z$ is obtained. What is $z$? | 0 | 87 | 1 |
math | A rectangle has an area of $16$ and a perimeter of $18$; determine the length of the diagonal of the rectangle.
[i]2015 CCA Math Bonanza Individual Round #8[/i] | 7 | 47 | 1 |
math | 13.163. When harvesting, 210 centners of wheat were collected from each of two plots. The area of the first plot is 0.5 hectares less than the area of the second plot. How many centners of wheat were collected per hectare on each plot if the wheat yield on the first plot was 1 centner per hectare more than on the secon... | 21 | 83 | 2 |
math | 16. A2 (RUS) The numbers from 1 to $n^{2}$ are randomly arranged in the cells of a $n \times n$ square $(n \geq 2)$. For any pair of numbers situated in the same row or in the same column, the ratio of the greater number to the smaller one is calculated.
Let us call the characteristic of the arrangement the smallest of... | \frac{n+1}{n} | 106 | 8 |
math | 8. (5 points) When drawing 2 circles on paper, you can get a maximum of 2 intersection points. When drawing 3 circles, you can get a maximum of 6 intersection points. So, if you draw 10 circles on paper, you can get $\qquad$ intersection points. | 90 | 64 | 2 |
math | 6-19 The set of all positive integers can be divided into two disjoint subsets of positive integers $\{f(1), f(2), \cdots, f(n), \cdots\},\{g(1), g(2), \cdots, g(n), \cdots\}$,
where $\quad f(1)<f(2)<\cdots<f(n)<\cdots$
$$
g(1)<g(2)<\cdots<g(n)<\cdots
$$
and $\quad g(n)=f(f(n))+1 \quad(n \geqslant 1)$
Find: $f(240)$. | 388 | 141 | 3 |
math | 3. In this century, 200 years will be marked since the birth of the famous Russian mathematician, a native of Kaluga Governorate, Pafnuty Lvovich Chebyshev. Among the numbers that record his year of birth, the sum of the digits in the hundreds and thousands place is three times the sum of the digits in the units and te... | 1821 | 129 | 4 |
math | 315. Find the variance and standard deviation of a random variable $X$ that is uniformly distributed in the interval $(a, b)$. | D(X)=\frac{(b-)^2}{12},\sigma(X)=\frac{b-}{2\sqrt{3}} | 30 | 30 |
math | Determine the $gcd$ of all numbers of the form $(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)$ where $a, b, c, d$ range over the integers.
## Solutions | 12 | 48 | 2 |
math | 18.7 Let $K$ be the incenter of $\triangle ABC$, and let points $C_{1}$ and $B_{1}$ be the midpoints of sides $AB$ and $AC$, respectively. The line $AC$ intersects $C_{1}K$ at point $B_{2}$, and the line $AB$ intersects $B_{1}K$ at point $C_{2}$. If the area of $\triangle AB_{2}C_{2}$ is equal to the area of $\triangle... | 60 | 121 | 2 |
math | 13.206. A three-digit number ends with the digit 2. If it is moved to the beginning of the number, the resulting number will be 18 more than the original. Find this number. | 202 | 46 | 3 |
math | 9. Planar vectors $\boldsymbol{a}, \boldsymbol{b}$ have an angle of $\frac{\pi}{3}$ between them. If $|\boldsymbol{a}|,|\boldsymbol{b}|,|\boldsymbol{a}+\boldsymbol{b}|$ form an arithmetic sequence, find $|\boldsymbol{a}|:|\boldsymbol{b}|:|\boldsymbol{a}+\boldsymbol{b}|$. | 3:5:7 | 94 | 5 |
math | 8.201. $\sin 2 x+2 \operatorname{ctg} x=3$.
8.201. $\sin 2 x+2 \cot x=3$. | \frac{\pi}{4}(4n+1),n\inZ | 44 | 16 |
math | ## Problem Statement
Are the vectors $c_{1 \text { and }} c_{2}$, constructed from vectors $a \text{ and } b$, collinear?
$a=\{1 ;-2 ; 4\}$
$b=\{7 ; 3 ; 5\}$
$c_{1}=6 a-3 b$
$c_{2}=b-2 a$ | c_{1}=-3\cdotc_{2} | 80 | 12 |
math | The edge of a regular tetrahedron \(ABCD\) is \(a\). Points \(E\) and \(F\) are taken on the edges \(AB\) and \(CD\) respectively, such that the sphere circumscribed around the tetrahedron intersects the line passing through \(E\) and \(F\) at points \(M\) and \(N\). Find the length of the segment \(EF\), if \(ME: EF: ... | \frac{2a}{\sqrt{7}} | 104 | 11 |
math | There are $2n$ different numbers in a row. By one move we can interchange any two numbers or interchange any $3$ numbers cyclically (choose $a,b,c$ and place $a$ instead of $b$, $b$ instead of $c$, $c$ instead of $a$). What is the minimal number of moves that is always sufficient to arrange the numbers in increasing or... | n | 83 | 1 |
math | Biot and Gay-Lussac, French natural scientists, in 1804 ascended in a balloon and reached a height of $6825 \mathrm{~m}$. Question: How far did the horizon extend on this occasion and how much $\mathrm{km}^{2}$ of land area could be seen from the balloon? (The radius of the Earth is $6377.4 \mathrm{~km}$. ) | =295\mathrm{~},\quadf=273188\mathrm{~}^{2} | 94 | 27 |
math | 5. The six edges of tetrahedron $ABCD$ have lengths $7, 13, 18, 27, 36, 41$, and it is known that $AB=41$, then $CD=$ | 13 | 54 | 2 |
math | 10.4. How many solutions in integers $x, y$ does the equation $|3 x+2 y|+|2 x+y|=100$ have? | 400 | 38 | 3 |
math | Task A-1.4. (4 points)
Let $a, b$ and $c$ be real numbers such that
$$
a+b+c=3 \quad \text { and } \quad \frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0
$$
What is $a^{2}+b^{2}+c^{2}$? | 9 | 86 | 1 |
math | 4.97 Real number $p \geqslant \frac{1}{4}$. Find all positive real solutions $x$ of the following equation:
$$\log _{\sqrt{2}}^{2} x+2 \log _{\sqrt{2}} x+2 \log _{\sqrt{2}}\left(x^{2}+p\right)+p+\frac{15}{4}=0$$ | x=\frac{1}{2} | 90 | 8 |
math | Let $N$ be the largest positive integer with the following property: reading from left to right, each pair of consecutive digits of $N$ forms a perfect square. What are the leftmost three digits of $N$? | 816 | 46 | 3 |
math | ## Task 15/78
1. Given is an annulus in which $n$ circles are inscribed such that each circle touches the inner and outer boundary of the annulus as well as two adjacent circles. The task is to find the ratio of the sum of the areas of the circles to the area of the annulus.
2. Correspondingly, the task is to solve fo... | \frac{\pi}{4},1,\frac{2}{3} | 132 | 15 |
math | Three. (25 points) Find all positive integers $a$ such that the quadratic equation
$$
a x^{2}+2(2 a-1) x+4(a-3)=0
$$
has at least one integer root. | a=1, 6, 10, 3 | 53 | 13 |
math |
1. Solve the system of equations
$$
\begin{aligned}
& x^{2}-y=z^{2}, \\
& y^{2}-z=x^{2}, \\
& z^{2}-x=y^{2}
\end{aligned}
$$
in the domain of real numbers.
| (0,0,0),(1,0,-1),(0,-1,1),(-1,1,0) | 62 | 26 |
math | ## Problem 2:
a) Study whether there exists a non-monotonic sequence $\left(x_{n}\right)_{n \geqslant 1}$ such that $\lim _{n \rightarrow \infty} x_{n}=\infty$.
b) Consider the sequence $\left(x_{n}\right)_{n \geq 1}$ for which $x_{n}^{2}-2 n x_{n}+n^{2}-1 \leq 0, \forall n \geq 1$. Calculate $\lim _{n \rightarrow \i... | 1 | 142 | 1 |
math | Example 5 Let $a_{1}, a_{2}, \cdots, a_{n}$ be given non-zero real numbers. If the inequality
$$\begin{array}{l}
r_{1}\left(x_{1}-a_{1}\right)+r_{2}\left(x_{2}-a_{2}\right)+\cdots+r_{n}\left(x_{n}-a_{n}\right) \leqslant \\
\sqrt[m]{x_{1}^{m}+x_{2}^{m}+\cdots+x_{n}^{m}}-\sqrt[m]{a_{1}^{m}+a_{2}^{m}+\cdots+a_{n}^{m}}
\en... | r_{i}=\left(\frac{a_{i}}{\sqrt[m]{a_{1}^{m}+a_{2}^{m}+\cdots+a_{n}^{m}}}\right)^{m-1}(i=1,2, \cdots, n) | 244 | 62 |
math | 3. Let the function be
$$
f(x)=\sqrt{2 x^{2}+2 x+41}-\sqrt{2 x^{2}+4 x+4}(x \in \mathbf{R}) \text {. }
$$
Then the maximum value of $f(x)$ is $\qquad$ | 5 | 70 | 1 |
math | 16. Given that $\alpha, \beta$ are real numbers, for any real numbers $x, y, z$, we have
$$
\alpha(x y+y z+z x) \leqslant M \leqslant \beta\left(x^{2}+y^{2}+z^{2}\right),
$$
where, $M=\sum \sqrt{x^{2}+x y+y^{2}} \sqrt{y^{2}+y z+z^{2}}$, and $\sum$ denotes the cyclic sum. Find the maximum value of $\alpha$ and the mini... | \alpha=3,\beta=3 | 132 | 8 |
math | 52. Find the angle between the vectors: a) $\vec{a}=(4 ; 0)$ and $\vec{b}=(2 ;-2)$; b) $\vec{a}=(5 ;-3)$ and $\vec{b}=(3 ; 5)$. | \frac{\pi}{4} | 60 | 7 |
math | ## Problem Statement
Find the second-order derivative $y_{x x}^{\prime \prime}$ of the function given parametrically.
$$
\left\{\begin{array}{l}
x=2(t-\sin t) \\
y=4(2+\cos t)
\end{array}\right.
$$ | \frac{1}{(1-\cos())^2} | 67 | 13 |
math | 2.204. $\sqrt{2+\sqrt{3}} \cdot \sqrt{2+\sqrt{2+\sqrt{3}}} \cdot \sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}} \cdot \sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}$. | 1 | 71 | 1 |
math | 4. If the inequality about $x$
$$
|x+a|<|x|+|x+1|
$$
has the solution set $\mathbf{R}$, then the range of values for $a$ is | 0<a<1 | 47 | 4 |
math | 7.2 In the set $\left\{-3,-\frac{5}{4},-\frac{1}{2}, 0, \frac{1}{3}, 1, \frac{4}{5}, 2\right\}$, two numbers are drawn without replacement. Find the probability that the two numbers are the slopes of a pair of perpendicular lines. | \frac{3}{28} | 77 | 8 |
math | Compute the number of subsets $S$ of $\{0,1,\dots,14\}$ with the property that for each $n=0,1,\dots,
6$, either $n$ is in $S$ or both of $2n+1$ and $2n+2$ are in $S$.
[i]Proposed by Evan Chen[/i] | 2306 | 79 | 4 |
math | 10. (10 points) A stick of length $L$, is divided into 8, 12, and 18 equal segments by red, blue, and black lines respectively. The stick is then cut at each division line. How many segments can be obtained in total? What is the length of the shortest segment? | 28 | 69 | 2 |
math | Find all functions $f,g : N \to N$ such that for all $m ,n \in N$ the following relation holds: $$f(m ) - f(n) = (m - n)(g(m) + g(n))$$.
Note: $N = \{0,1,2,...\}$ | (f(n), g(n)) = (an^2 + 2bn + c, an + b) | 69 | 24 |
math | Solve the following equation:
$$
\frac{1}{x-a}+\frac{1}{x-b}+\frac{1}{x-c}=0
$$ | \frac{+b+\\sqrt{(+b+)^{2}-3(++)}}{3} | 35 | 23 |
math | 5-2. In a sports tournament, a team of 10 people participates. The regulations stipulate that 8 players from the team are always on the field, changing from time to time. The duration of the match is 45 minutes, and all 10 participants on the team must play an equal number of minutes. How many minutes will each player ... | 36 | 83 | 2 |
math | (1) (This question is worth 35 points) Let the set $A=\{1,2,3, \cdots, 366\}$. If a binary subset $B=\{a, b\}$ of $A$ satisfies 17|(a $+b$), then $B$ is said to have property $P$.
(1) Find the number of all binary subsets of $A$ that have property $P$;
(2) Find the number of all pairwise disjoint binary subsets of $A$ ... | 3928 | 121 | 4 |
math | 215. Given the area of a right-angled triangle and its hypotenuse, find the legs.
Problems by Leonardo Fibonacci from "Liber Abaci".
Problem "De duobus hominibus" (About two men). | \begin{aligned}&\frac{\sqrt{^{2}+4\Delta}+\sqrt{^{2}-4\Delta}}{2}\\&\frac{\sqrt{^{2}+4\Delta}-\sqrt{^{2}-4\Delta}}{2}\end{aligned} | 50 | 62 |
math | 1. Evaluate
$$
\sin \left(1998^{\circ}+237^{\circ}\right) \sin \left(1998^{\circ}-1653^{\circ}\right)
$$ | -\frac{1}{4} | 53 | 7 |
math | Call a k-digit positive integer a [i]hyperprime[/i] if all its segments consisting of $ 1, 2, ..., k$ consecutive digits are prime. Find all hyperprimes. | 2, 3, 5, 7, 23, 37, 53, 73, 373 | 42 | 31 |
math | 5. Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=1, a_{2}=2, a_{n+1}-3 a_{n}+2 a_{n-1}=1\left(n \geqslant 2, n \in \mathbf{N}^{*}\right)$, then the general term formula of $\left\{a_{n}\right\}$ is $\qquad$ . | 2^{n}-n | 99 | 5 |
math | For what values of the parameter $a$ does the equation $\frac{\log _{a} x}{\log _{a} 2}+\frac{\log _{x}(2 a-x)}{\log _{x} 2}=\frac{1}{\log _{\left(a^{2}-1\right)} 2}$ have:
(1) solutions? (2) exactly one solution? | 2 | 87 | 1 |
math | 3. Let the sequence $\left(x_{n}\right)_{n \geq 1}, x_{1}=\frac{1}{2014}, x_{n+1}=x_{n}\left(1+x_{1}+x_{1}^{2}+\ldots+x_{1}^{n}\right)$, for any $n \geq 1$. We denote $S=\frac{x_{1}}{x_{2}}+\frac{x_{2}}{x_{3}}+\ldots+\frac{x_{2014}}{x_{2015}}$. Find $[S]$. | 2013 | 136 | 4 |
math | For any positive integer $n$, let $D_n$ denote the set of all positive divisors of $n$, and let $f_i(n)$ denote the size of the set
$$F_i(n) = \{a \in D_n | a \equiv i \pmod{4} \}$$
where $i = 1, 2$.
Determine the smallest positive integer $m$ such that $2f_1(m) - f_2(m) = 2017$. | 2 \cdot 5^{2016} | 108 | 13 |
math | 2A. Let the complex number $i$ be given. Determine all complex numbers $z$ such that the number $a=\frac{u-\bar{u} z}{1-z}$ is a real number. | z\neq1 | 45 | 5 |
math | Task 1. Represent in the form of an irreducible fraction:
$$
\frac{12+15}{18}+\frac{21+24}{27}+\ldots+\frac{48+51}{54}
$$ | \frac{171}{20} | 56 | 10 |
math | 6. Given 95 numbers $a_{1}, a_{2}, \cdots, a_{95}$, each of which can only take one of the two values +1 or -1, then the sum of their pairwise products
$$
a_{1} a_{2}+a_{1} a_{3}+\cdots+a_{94} a_{95}
$$
is the smallest positive value . $\qquad$ | 13 | 95 | 2 |
math | 10. $f(x)$ is a function defined on $(-\infty,+\infty)$ that is odd, and $f(1+x)+f(1-x)=f(1), f(x)$ is a decreasing function on $[0,1]$, then $-f\left(\frac{10}{3}\right) 、 f\left(-2+\frac{\sqrt{2}}{2}\right) 、 f\left(\frac{9}{2}\right)$ arranged from smallest to largest is $\qquad$ - | f(-2+\frac{\sqrt{2}}{2})<-f(\frac{10}{3})<f(\frac{9}{2}) | 117 | 32 |
math | 1495. A randomly selected phone number consists of 5 digits. What is the probability that in it: 1) all digits are different; 2) all digits are odd? | 0.3024 | 40 | 6 |
math | 6. Ivan, Josip, and Tomislav together have 12000 kuna. Ivan divides half of his money into two equal parts and gives them to Josip and Tomislav, and keeps the other half for himself. Josip does the same, and then Tomislav, after which all three friends have the same amount of money. How much money did each boy have at ... | Ivanhad2000kn,Josip3500kn,Tomislav6500kn | 86 | 25 |
math | Example 18. Solve the inequality
$$
5^{\frac{1}{4} \log _{5}^{2} x} \geqslant 5 x^{\frac{1}{5} \log _{5} x}
$$ | (0;5^{-2\sqrt{5}}]\cup[5^{2}\sqrt{5};+\infty) | 56 | 26 |
math | 1. $[\mathbf{3}]$ Suppose that $p(x)$ is a polynomial and that $p(x)-p^{\prime}(x)=x^{2}+2 x+1$. Compute $p(5)$. | 50 | 49 | 2 |
math | 8. $A B$ is the common perpendicular segment of skew lines $a, b$, $A$ is on line $a$, $B$ is on line $b$, $A B=2$, the skew lines $a, b$ form a $30^{\circ}$ angle, and on line $a$ take $A P=4$, then the distance from point $P$ to line $b$ is | 2\sqrt{2} | 89 | 6 |
math | 7. There are $n$ ellipses centered at the origin with the coordinate axes as axes of symmetry, and the directrices of all ellipses are $x=1$. If the eccentricity of the $k$-th $(k=1,2, \cdots, n)$ ellipse is $e_{k}=2^{-k}$, then the sum of the major axes of these $n$ ellipses is $\qquad$ | 2-2^{1-n} | 94 | 7 |
math | 4. In the tetrahedron $O A B C$, $O A \perp$ plane $A B C, A B \perp$ $A C$, point $P$ satisfies $O P=l O A+m O B+n O C$, where $l, m, n$ are positive numbers and $l+m+n=1$. If the line $O P$ is formed by points that are equidistant from the planes $O B C$, $O C A$, and $O A B$, then the cosine value of the dihedral an... | \frac{n}{l} | 140 | 6 |
math | 1. When $x=\frac{\sqrt{21}-5}{2}$, the value of the algebraic expression
$$
x(x+1)(x+2)(x+3)(x+4)(x+5)
$$
is $\qquad$ . | -15 | 57 | 3 |
math | Example 5 Let $n(\geqslant 3)$ be a positive integer, and $M$ be an $n$-element set. Find the maximum positive integer $k$ such that there exists a family $\psi$ of $k$ three-element subsets of $M$ with the property that the intersection of any two elements (note: elements of $\psi$ are three-element sets) is non-empty... | C_{n-1}^{2} | 87 | 9 |
math | 8.362. $\sin ^{6} x+\cos ^{6} x=\frac{7}{16}$.
8.362. $\sin ^{6} x+\cos ^{6} x=\frac{7}{16}$. | \frac{\pi}{6}(3k\1),k\inZ | 58 | 16 |
math | Let $S_{n}$ denote the sum of the digits of the number $n$ in the decimal system. Determine all numbers $n$ for which $n=S^{2}(n)-S(n)+1$ | 1,13,43,91,157 | 44 | 14 |
math | What is the largest positive integer that is not the sum of a positive integral multiple of 42 and a positive composite integer? | 215 | 26 | 3 |
math | 12.413 The lengths of the four arcs into which the entire circumference of radius $R$ is divided form a geometric progression with a common ratio of 3. The points of division serve as the vertices of a quadrilateral inscribed in this circle. Find its area. | \frac{R^{2}\sqrt{2}}{4} | 58 | 14 |
math | 2. Masha, Dasha, and Sasha are tasked with harvesting all the currants from the bushes on the garden plot. Masha and Dasha together can collect all the berries in 7 hours 30 minutes, Masha and Sasha - in 6 hours, and Dasha and Sasha - in 5 hours. How many hours will it take the children to collect all the berries if th... | 4 | 85 | 1 |
math | 1. $[\mathbf{3}]$ Compute:
$$
\lim _{x \rightarrow 0} \frac{x^{2}}{1-\cos (x)}
$$ | 2 | 38 | 1 |
math | 4. Given $P(1,4,5)$ is a fixed point in the rectangular coordinate system $O-x y z$, a plane is drawn through $P$ intersecting the positive half-axes of the three coordinate axes at points $A$, $B$, and $C$ respectively. Then the minimum value of the volume $V$ of all such tetrahedrons $O-A B C$ is $\qquad$ | 90 | 89 | 2 |
math | 2. Solve the system of equations
$$
\left\{\begin{array}{l}
x_{1}+x_{2}+\cdots+x_{n}=n \\
x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=n \\
\cdots \cdots \\
x_{1}^{n}+x_{2}^{n}+\cdots+x_{n}^{n}=n
\end{array}\right.
$$ | x_{1}=x_{2}=\cdots=x_{n}=1 | 107 | 16 |
math | 10th Chinese 1995 Problem A2 N + is the set of positive integers. f: N + → N + satisfies f(1) = 1, f(2n) < 6 f(n), and 3 f(n) f(2n+1) = f(2n) + 3 f(n) f(2n) for all n. Find all m, n such that f(m) + f(n) = 293. | (47,5),(45,7),(39,13),(37,15) | 102 | 23 |
math |
A1. A set $A$ is endowed with a binary operation $*$ satisfying the following four conditions:
(1) If $a, b, c$ are elements of $A$, then $a *(b * c)=(a * b) * c$;
(2) If $a, b, c$ are elements of $A$ such that $a * c=b * c$, then $a=b$;
(3) There exists an element $e$ of $A$ such that $a * e=a$ for all $a$ in $A$;... | 3 | 223 | 1 |
math | 10. Person A and Person B start from points $A$ and $B$ respectively (Person A starts from $A$), walking towards each other and continuously moving back and forth between the two points. Person A's speed is 4 times that of Person B. It is known that the distance between $A$ and $B$ is $S$ kilometers, where $S$ is a pos... | 105 | 284 | 3 |
math | 8. If 4 lines in a plane intersect each other pairwise and no three lines are concurrent, then there are $\qquad$ pairs of consecutive interior angles. | 24 | 33 | 2 |
math | 2. Find all values of the parameter $a$ for which the equation $2 x^{2}-a x+2 a=0$ has two distinct integer roots. | -2;18 | 35 | 5 |
math | 7. (10 points) $\left[x-\frac{1}{2}\right]=3 x-5$, here $[x]$ represents the greatest integer not exceeding $x$, then $x=$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 2 | 68 | 1 |
math | 9. (12 points) Three people, A, B, and C, depart from location $A$ to location $B$. A departs at 8:00, B at 8:20, and C at 8:30. They all travel at the same speed. 10 minutes after C departs, the distance from A to $B$ is exactly half the distance from B to $B$. At this moment, C is 2015 meters away from $B$. Therefore... | 2418 | 127 | 4 |
math | Find all real values of the parameter $a$ for which the system of equations
\[x^4 = yz - x^2 + a,\]
\[y^4 = zx - y^2 + a,\]
\[z^4 = xy - z^2 + a,\]
has at most one real solution. | a \leq 0 | 67 | 6 |
math | 15.8 (League 2001) $D, E$ are points on side $BC$ of $\triangle ABC$, $F$ is a point on the extension of $BA$, $\angle DAE = \angle CAF$.
(1) Determine the positional relationship between the circumcircle of $\triangle ABD$ and the circumcircle of $\triangle AEC$, and prove your conclusion.
(2) If the circumradius of $... | \frac{8}{3} | 984 | 7 |
math | Calculate $x^{m}+x^{m+1}+x^{m+2}+\ldots+x^{n-2}+x^{n-1}+x^{n}$, with $x \neq 1$ and $m<n$. | \frac{x^{n+1}-x^{}}{x-1} | 56 | 16 |
math | 13. (12 points) Cars A and B start from point A to point B at the same time. At the start, car A's speed is 2.5 kilometers per hour faster than car B's. After 10 minutes, car A slows down; 5 minutes later, car B also slows down, at which point car B is 0.5 kilometers per hour slower than car A. Another 25 minutes later... | 10 | 120 | 2 |
math | $9 \cdot 7$ Let $A$ be the sum of the digits of the decimal number $4444^{4444}$, and let $B$ be the sum of the digits of $A$, find the sum of the digits of $B$ (all numbers here are in decimal).
（17th International Mathematical Olympiad, 1975） | 7 | 81 | 1 |
math | 11.1. Mikhail leaves from Berdsk to Cherapanovo at $8:00$ AM; On the same day, at the same time, and on the same road, Khariton and Nikolay leave from Cherapanovo to Berdsk. At 9:30 AM, Khariton was exactly halfway between Mikhail and Nikolay; at 10:00 AM, Mikhail was exactly halfway between Khariton and Nikolay. Deter... | MikhailKharitonmetat9:48AM,MikhailNikolay-at10:15AM | 127 | 25 |
math | 7. Find all positive real numbers $a$ such that the equation
$$
a^{2} x^{2}+a x+1-13 a^{2}=0
$$
has two integer roots. $\qquad$ | 1,\frac{1}{3},\frac{1}{4} | 51 | 15 |
math | 3. (6 points) $(1234+2341+3412+4123) \div 11=$ | 1010 | 33 | 4 |
math | ## Problem Statement
Find the point of intersection of the line and the plane.
$\frac{x-1}{2}=\frac{y+2}{-5}=\frac{z-3}{-2}$
$x+2 y-5 z+16=0$ | (3,-7,1) | 58 | 7 |
math | ## 196. Math Puzzle $9 / 81$
In a house, there are three clocks. On January 1st, they all showed the exact time. However, only the first clock was correct, the second clock lost one minute per day, and the third clock gained one minute per day.
After how much time will all three clocks, if they continue to run this w... | 720 | 88 | 3 |
math | Example 3 (2003 Hungarian Mathematical Competition) Given the circumradius of $\triangle A B C$ is $R$, if $\frac{a \cos \alpha+b \cos \beta+c \cos \gamma}{a \sin \beta+b \sin \gamma+c \sin \alpha}=$ $\frac{a+b+c}{9 R}$, where $a, b, c$, are the lengths of the three sides of $\triangle A B C$; $\alpha, \beta, \gamma$ a... | \alpha=\beta=\gamma=60 | 135 | 9 |
math | ## Task $7 / 89$
All integers $x$ are to be determined for which $f(x)=x^{5}-x+5$ is a prime number! | x_{1}=-1,x_{2}=0,x_{3}=1 | 37 | 16 |
math | 9. How many six-digit numbers exist where each digit in an even position is one greater than the digit to its left (digits are numbered from left to right)? | 648 | 33 | 3 |
math | 7. In the corners $C$ and $B$ of triangle $ABC$, circles with centers $O_{1}$ and $O_{2}$ of equal radius are inscribed, and point $O$ is the center of the circle inscribed in triangle $ABC$. These circles touch side $BC$ at points $K_{1}, K_{2}$, and $K$ respectively, with $C K_{1}=3, B K_{2}=7$, and $B C=16$.
a) Fin... | )CK=\frac{24}{5},b)\angleACB=2\arcsin\frac{3}{5}=\arccos\frac{7}{25} | 180 | 40 |
math | B2. Find all pairs of real numbers $x$ and $y$ that solve the system of equations
$$
\begin{gathered}
\sqrt{x}-\sqrt{y}=1 \\
\sqrt{8 x+7}-\sqrt{8 y+7}=2
\end{gathered}
$$
## Problems for 3rd Year
Time for solving: 45 minutes. | \frac{9}{4},\frac{1}{4} | 85 | 14 |
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