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 | 1. Find all four-digit numbers $\overline{a b c d}$, for which $\overline{a b c d}=20 \cdot \overline{a b}+16 \cdot \overline{c d}$. | 1264,1580,1896 | 52 | 14 |
math | Define a sequence $(a_n)$ by $a_0 =0$ and $a_n = 1 +\sin(a_{n-1}-1)$ for $n\geq 1$. Evaluate
$$\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^{n} a_k.$$ | 1 | 75 | 1 |
math | 32. (5) Given an isosceles triangle \(ABC (AB = BC)\). A point \(K\) is chosen on side \(AB\), and a point \(L\) is chosen on side \(BC\) such that \(AK + CL = \frac{1}{2} AB\). Find the geometric locus of the midpoints of segments \(KL\). | M_2N_2 | 78 | 6 |
math | 996. Find all cases when a natural number has 8 different divisors. | p^{7}, | 18 | 4 |
math | Petya and 9 other people are playing a game: each of them rolls a die. A player wins a prize if they roll a number that no one else rolls.
a) What is the probability that Petya will win a prize?
b) What is the probability that at least someone will win a prize? | )(5/6)^{9}\approx0.194;b)\approx0.919 | 66 | 22 |
math | $3 \cdot 27$ When the natural number $n \geqslant 2$ is the smallest, find integers $a_{1}, a_{2}, \cdots, a_{n}$, such that the following equation $a_{1}+a_{2}+\cdots+a_{n}=a_{1} \cdot a_{2} \cdots \cdots a_{n}=1990$ holds. | 5 | 95 | 1 |
math | Booin d.A.
Given two sequences: $2,4,8,16,14,10,2$ and 3, 6, 12. In each of them, each number is obtained from the previous one according to the same rule.
a) Find this rule.
b) Find all natural numbers that transform into themselves (according to this rule).
c) Prove that the number $2^{1991}$ will become a single... | 18 | 104 | 2 |
math | 1. (14 points) Let the line $l: y=k x+m(k, m \in$ Z) intersect the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{12}=1$ at two distinct points $A, B$, and intersect the hyperbola $\frac{x^{2}}{4}-\frac{y^{2}}{12}=1$ at two distinct points $C, D$. Question: Does there exist a line $l$ such that the vector $\overrightarrow{A C}... | 9 | 151 | 1 |
math | 3. Given $0 \leq a_{k} \leq 1(k=1,2, \ldots, 2020)$, let $a_{2021}=a_{1}, a_{2022}=a_{2}$, then the maximum value of $\sum_{k=1}^{2020}\left(a_{k}-\right.$ $\left.a_{k+1} a_{k+2}\right)$ is $\qquad$ | 1010 | 106 | 4 |
math | Given a regular polygon with $n$ sides. It is known that there are $1200$ ways to choose three of the vertices of the polygon such that they form the vertices of a [b]right triangle[/b]. What is the value of $n$? | 50 | 57 | 2 |
math | Consider an $8 \times 8$ chessboard.
How many ways are there to place 6 rooks such that no two rooks are on the same row or column? | 564480 | 37 | 6 |
math | 2. (3 points) One of the heights of a triangle is 2 cm. If the height increases by 6 cm, with the base remaining unchanged, the area increases by 12 square cm. The area of the original triangle is $\qquad$ square cm. | 4 | 57 | 1 |
math | Find all quadruples of real numbers $(a,b,c,d)$ such that the equalities
\[X^2 + a X + b = (X-a)(X-c) \text{ and } X^2 + c X + d = (X-b)(X-d)\]
hold for all real numbers $X$.
[i]Morteza Saghafian, Iran[/i] | (-1, -2, 2, 0), (0, 0, 0, 0) | 81 | 24 |
math | Example 2 (1996 National High School League Question) The number of proper subsets of the set $\left\{x \left\lvert\,-1 \leqslant \log _{\frac{1}{x}} 10<-\frac{1}{2}\right., x \in \mathbf{N}_{+}\right\}$ is $\qquad$ . | 2^{90}-1 | 83 | 6 |
math | Are there positive integers $a; b; c$ and $d$ such that $a^3 + b^3 + c^3 + d^3 =100^{100}$ ?
[i](4 points)[/i] | a = 10^{66}, b = 2 \cdot 10^{66}, c = 3 \cdot 10^{66}, d = 4 \cdot 10^{66} | 51 | 49 |
math | 8. Find all integers $x$ such that $2 x^{2}+x-6$ is a positive integral power of a prime positive integer. | -3,2,5 | 32 | 6 |
math | 1、The set $\{[x]+[2 x]+[3 x] \mid x \in R\} \mid\{x \mid 1 \leq x \leq 100, x \in Z\}$ has $\qquad$ elements, where $[x]$ represents the greatest integer not exceeding $x$. | 67 | 73 | 2 |
math | For a sequence $(a_{n})_{n\geq 1}$ of real numbers it is known that $a_{n}=a_{n-1}+a_{n+2}$ for $n\geq 2$.
What is the largest number of its consecutive elements that can all be positive? | 5 | 66 | 1 |
math | 9. In a rectangular box $A B C D E F G H$ with edge lengths $A B=A D=6$ and $A E=49$, a plane slices through point $A$ and intersects edges $B F, F G, G H, H D$ at points $P, Q, R, S$ respectively. Given that $A P=A S$ and $P Q=Q R=R S$, find the area of pentagon $A P Q R S$. | \frac{141\sqrt{11}}{2} | 103 | 15 |
math | 3. A sequence of numbers, the first three numbers are $1, 9, 9$, and each subsequent number is the remainder of the sum of the three preceding numbers divided by 3. What is the 1999th number in this sequence? | 0 | 55 | 1 |
math | (7) The function $f(x)=\frac{\sin \left(x+45^{\circ}\right)}{\sin \left(x+60^{\circ}\right)}, x \in\left[0^{\circ}, 90^{\circ}\right]$, then the product of the maximum and minimum values of $f(x)$ is . $\qquad$ | \frac{2\sqrt{3}}{3} | 80 | 12 |
math | The sequence $\left\{a_{n}\right\}$ satisfies
$$
\begin{array}{l}
a_{1}=1, \\
a_{n+1}=\sqrt{a_{n}^{2}-2 a_{n}+3}+c\left(n \in \mathbf{N}_{+}\right),
\end{array}
$$
where $c$ is a constant greater than 0.
(1) If $c=1$, find the general term of the sequence $\left\{a_{n}\right\}$;
(2) If the sequence has an upper bound,... | (0,1) | 139 | 5 |
math | 1. Given that $A D$ and $B E$ are the angle bisectors of $\triangle A B C$, and they intersect at point $I$, with $I D=I E$. If $\angle C A B=80^{\circ}$, then $\angle A B C=$ $\qquad$ | 40 | 66 | 2 |
math | Let $p(x)$ and $q(x)$ be two cubic polynomials such that $p(0)=-24$, $q(0)=30$, and \[p(q(x))=q(p(x))\] for all real numbers $x$. Find the ordered pair $(p(3),q(6))$. | (3, -24) | 70 | 8 |
math | Example 9 The range of the function $y=\sin ^{12} x+\cos ^{12} x$ is $\qquad$ | [\frac{1}{32},1] | 32 | 10 |
math | 5. Let $n=1990$, then
$$
\frac{1}{2^{n}}\left(1-3 C_{n}^{2}+3^{2} C_{4}{ }^{n}-3^{3} C_{n}^{6}+\cdots+3^{994} C_{n}^{1988}-3^{995} C_{n}^{1990}\right)
$$
$=$ | -\frac{1}{2} | 103 | 7 |
math | # Problem 4.
The integer part $[x]$ of a real number $x$ is defined as the greatest integer $M$ such that $M \leq x$. For example, $[\sqrt{2}]=1,[2]=2,[\pi]=3$. Find all positive real numbers $x$ for which
$$
x[x[x[x]]]<2018
$$ | 0<x<7 | 83 | 4 |
math | Problem 3. If we divide the numbers 701 and 592 by the same natural number, we get remainders of 8 and 7, respectively. By which number did we divide the given numbers? | 9 | 47 | 1 |
math | A college math class has $N$ teaching assistants. It takes the teaching assistants $5$ hours to grade homework assignments. One day, another teaching assistant joins them in grading and all homework assignments take only $4$ hours to grade. Assuming everyone did the same amount of work, compute the number of hours it w... | 20 | 78 | 2 |
math | Let $a, b>0$. Find all functions $f: \mathbb{R}_{+} \rightarrow \mathbb{R}$ such that for all positive real numbers $x, y$:
$$
f(x) f(y)=y^{a} f\left(\frac{x}{2}\right)+x^{b} f\left(\frac{y}{2}\right)
$$ | f(x)=^{} | 83 | 5 |
math | ## Zadatak B-2.5.
Jednadžbe $x^{2}-45 x+4 a+4=0$ i $x^{2}-47 x+5 a-4=0, a \in \mathbb{R}, a<20$ imaju jedno zajedničko realno rješenje. Koliko iznosi umnožak preostalih dvaju rješenja tih jednadžbi?
| 2024 | 106 | 4 |
math | # Problem 4. (Kuyanov 9.)
Find all quadruples of natural numbers $a, b, c, d$ for which the following equations are satisfied:
\[
\left\{
\begin{array}{l}
a+b=c d \\
c+d=a b
\end{array}
\right.
\] | (2,2,2,2),(1,2,3,5),(2,1,3,5),(1,2,5,3),(2,1,5,3),(3,5,1,2),(5,3,1,2),(3,5,2,1),(5,3,2,1) | 70 | 73 |
math | 4. There is a simple pendulum, which has a period of $T=1$ second in summer. In winter, the pendulum length shortens by 0.01 cm. In winter, this pendulum is approximately faster by $\qquad$ seconds in 24 hours (rounded to 1 second).
Note: The formula for the period of a simple pendulum is $T=2 \pi \sqrt{\frac{l}{g}}$,... | 17 | 135 | 2 |
math | Example 4 Let the set $A=\{1,2,3,4,5,6\}$, and the mapping $f: A \rightarrow A$, such that its third composite mapping $f \cdot f \cdot f$ is the identity mapping. How many such $f$ are there?
(1996. Japan Mathematical Olympiad Preliminary) | 81 | 77 | 2 |
math | Example 2.44. Investigate the conditional and absolute convergence of the improper integral
$$
I=\int_{0}^{2}\left[2 x \sin \left(\frac{\pi}{x^{2}}\right)-\frac{2 \pi}{x} \cos \left(\frac{\pi}{x^{2}}\right)\right] d x
$$ | 2\sqrt{2} | 81 | 6 |
math | 1077*. Find all natural numbers $k$ and $n$ for which the number
$$
\frac{7 k+15 n-1}{3 k+4 n}
$$
is a natural number. | k=3a-2,n=2a-1(\inN) | 48 | 16 |
math | [Help me] Determine the smallest value of the sum M =xy-yz-zx where x; y; z are real numbers satisfying the following condition $x^2+2y^2+5z^2 = 22$. | \frac{-55 - 11\sqrt{5}}{10} | 50 | 18 |
math | Find all functions $f : [0,\infty) \to [0,\infty)$ such that $f(f(x)) +f(x) = 12x$, for all $x \ge 0$.
| f(x) = 3x | 47 | 8 |
math | One, (40 points) Find all integer triples $(a, b, c)$ such that
$$
N=\frac{(a-b)(b-c)(c-a)}{2}+2
$$
is a positive integer of the form $1729^{m}(m \in \mathbf{N})$. | (,b,)=(k,k+1,k+2) | 69 | 13 |
math | ## Task 4 - 100614
A total of 25 answer cards marked "very well solved" were sent by the editorial team to 15 participants in the mathematics student magazine "alpha" competition, and each of these participants received at least one such answer card.
In addition, it is known about these 15 participants that at least ... | 11 | 141 | 2 |
math | Example 4. Find the indefinite solution of the equation
$$
x+y+z=20
$$ | 231 | 22 | 3 |
math | 8 Arrange fifteen students numbered $1,2,3, \ldots, 15$ in a circle facing inward, in numerical order. The first time, the student numbered 1 turns around. The second time, the students numbered 2 and 3 turn around. The third time, the students numbered $4,5,6$ turn around, .... The 15th time, all students turn around.... | 12 | 105 | 2 |
math | 9. (10 points) A three-digit number whose sum of the tens digit and the units digit equals the hundreds digit is called a "good number". How many good numbers are there?
The above text is translated into English, preserving the original text's line breaks and format. | 54 | 57 | 2 |
math | 752. Output the formula
$$
I_{i}^{*}=\frac{b-a}{n} \sum_{i=1}^{n}\left[\varphi\left(x_{i}\right)-g\left(x_{i}\right)\right]+\int_{a}^{b} g(x) \mathrm{d} x
$$
where $x_{i}=a+(b-a) r_{i}, g(x) \simeq \varphi(x)$, for estimating the integral
$$
I=\int_{a}^{b} \varphi(x) \mathrm{d} x
$$ | I_{i}^{*}=\frac{b-}{n}\sum_{i=1}^{n}[\varphi(x_{i})-(x_{i})]+\int_{}^{b}(x) | 134 | 45 |
math | For each positive integer $k$ greater than $1$, find the largest real number $t$ such that the following hold:
Given $n$ distinct points $a^{(1)}=(a^{(1)}_1,\ldots, a^{(1)}_k)$, $\ldots$, $a^{(n)}=(a^{(n)}_1,\ldots, a^{(n)}_k)$ in $\mathbb{R}^k$, we define the score of the tuple $a^{(i)}$ as
\[\prod_{j=1}^{k}\#\{1\leq ... | t = \frac{1}{k-1} | 349 | 12 |
math | Example 3 If $x, y, z$ are all positive real numbers, find the maximum value of $\frac{x y z}{(1+5 x)(4 x+3 y)(5 y+6 z)(z+18)}$. (2003 Singapore Mathematical Olympiad) | \frac{1}{5120} | 62 | 10 |
math | Example 9 For positive real numbers $a, b, c$ satisfying $abc=1$, find the maximum value of
$$\left(a-1+\frac{1}{b}\right)\left(b-1+\frac{1}{c}\right)\left(c-1+\frac{1}{a}\right)$$ | 1 | 68 | 1 |
math | 3. Let $[x]$ denote the greatest integer not exceeding the real number $x$. Set $A=\left[\frac{7}{8}\right]+\left[\frac{7^{2}}{8}\right]+\cdots+\left[\frac{7^{2016}}{8}\right]$. Then the remainder when $A$ is divided by 50 is $\qquad$ . | 42 | 85 | 2 |
math | Think about Question 2 Given $x_{1}=1, x_{2}=6$, and $x_{n+1}=6 x_{n}-9 x_{n-1}+3^{n}(n=2,3, \cdots)$, find the general term formula of the sequence $\left\{x_{n}\right\}$. | x_{n}=\frac{3^{n-1}}{2}(n^{2}-n+2) | 75 | 24 |
math | 9.3. Find five different numbers if all possible sums of triples of these numbers are equal to $3,4,6$, $7,9,10,11,14,15$ and 17. The numbers do not have to be integers. | -3,2,4,5,8 | 58 | 10 |
math | 1.5.13 $\star \star$ Find the largest real number $k$, such that for any positive real numbers $a, b, c$, we have
$$
\begin{aligned}
& \frac{(b-c)^{2}(b+c)}{a}+\frac{(c-a)^{2}(c+a)}{b}+\frac{(a-b)^{2}(a+b)}{c} \\
\geqslant & k\left(a^{2}+b^{2}+c^{2}-a b-b c-c a\right)
\end{aligned}
$$ | 2 | 128 | 1 |
math | 31. Given that 2 is both the square root of $x-2$ and the cube root of $2 x-y+1$, then the arithmetic square root of $x^{2}-4 y$ is | 4 | 45 | 1 |
math | 1. Let $\mathbf{P}$ be the set of all prime numbers. Find all functions $f: \mathbf{P} \rightarrow \mathbf{P}$ such that for any $p, q \in \mathbf{P}$, we have
$$
(f(p))^{f(q)} + q^{p} = (f(q))^{f(p)} + p^{q} .
$$ | f(p)=p | 87 | 4 |
math | Example 2 From the numbers $1,2, \cdots, 14$, select $a_{1}, a_{2}, a_{3}$ in ascending order, and $a_{2}-a_{1} \geqslant 3, a_{3}-a_{2} \geqslant 3$. How many different ways of selection are there that meet the conditions? | 120 | 84 | 3 |
math | Example 10 We notice that $6!=8 \cdot 9 \cdot 10$. Try to find the largest positive integer $n$ such that $n!$ can be expressed as the product of $n-3$ consecutive natural numbers. | 23 | 53 | 2 |
math | 1. (16 points) Does there exist a real number $a$, such that the line $y=a x+1$ intersects the hyperbola $3 x^{2}-y^{2}=1$ at two points $A$ and $B$, and the circle with diameter $AB$ passes exactly through the origin of the coordinate system? | a= \pm 1 | 72 | 6 |
math | 23. (2004 National College Entrance Examination, Fujian Province) Two people, A and B, participate in an English test. It is known that among the 10 questions available, A can answer 6 of them, and B can answer 8 of them. It is stipulated that each test will randomly select 3 questions from the available questions for ... | \frac{44}{45} | 154 | 9 |
math | 8. A coin collector has 100 coins that look the same. The collector knows that 30 of them are genuine, and 70 are fake, and that all genuine coins weigh the same, while all fake coins weigh different and are heavier than the genuine ones. The collector has a balance scale that can be used to compare the weight of two g... | 70 | 100 | 2 |
math | $$
\begin{array}{l}
\text { Example } 4 \text { Given } \frac{x-a-b}{c}+\frac{x-b-c}{a}+ \\
\frac{x-c-a}{b}=3 \text {, and } \frac{1}{a}+\frac{1}{b}+\frac{1}{c} \neq 0 \text {. Then } x-a- \\
b-c=
\end{array}
$$
Example 4 Given $\frac{x-a-b}{c}+\frac{x-b-c}{a}+$ $\frac{x-c-a}{b}=3$, and $\frac{1}{a}+\frac{1}{b}+\frac{... | x-a-b-c=0 | 183 | 6 |
math | 8. (3 points) For a certain product, if the purchase price decreases by 10%, and the selling price remains unchanged, then the gross profit margin (Gross Profit Margin = $\frac{\text{Selling Price - Purchase Price}}{\text{Purchase Price}} \times 100 \%$) can increase by $12 \%$, then the original gross profit margin fo... | 8 | 90 | 1 |
math | II. (This question is worth 25 points) Arrange all positive integers that are coprime with 105 in ascending order. Find the 1000th term of this sequence. | 2186 | 43 | 4 |
math | 75. Let positive real numbers $a, b, c$ satisfy the condition $(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)=13$. Find the maximum and minimum values of the following expression.
$$P=\frac{a^{3}+b^{3}+c^{3}}{a b c} \text { (Pham Kim Hung) }$$ | 11 \pm 2\sqrt{3} | 95 | 11 |
math | 56. Determine the largest real number $C$, for all real numbers $x, y, x \neq y$, and $xy=2$, such that the inequality $\frac{\left[(x+y)^{2}-6\right]\left[(x-y)^{2}+8\right]}{(x-y)^{2}} \geqslant C$ holds. (2002 Austrian Mathematical Olympiad Problem) | 18 | 91 | 2 |
math | The first term of a sequence is $a_{1}=1$ and for each subsequent term,
$$
a_{n+1}=1+\frac{n}{a_{n}}, \quad n=1,2,3 \ldots
$$
Does the following limit exist? If it does, determine it.
$$
\lim _{n \rightarrow \infty}\left(a_{n}-\sqrt{n}\right)
$$ | \frac{1}{2} | 90 | 7 |
math | 68. In the USSR Cup football games, after the qualifying matches, the football teams are divided by lot into two equal groups. In each group, the winner of that group is determined separately, and the two winners meet in the cup final. Let the total number of teams that have passed the qualifying matches be 20.
a) Wha... | )\frac{10}{19};b)\frac{28}{323}\approx0.08,(\frac{28}{323})^2\approx0.0064;)\frac{135}{323}\approx0.42 | 150 | 63 |
math | 4. (5 points) If $\frac{1}{2}+\frac{1}{4}+\cdots+\frac{1}{2^{n}}>\frac{315}{412}$ ( $n$ is a natural number greater than 0), then the smallest value of $n$ that satisfies the condition is . $\qquad$ | 3 | 75 | 1 |
math | 4. 2. 35 * Given Given that the equation $x^{3} \sin \theta-(\sin \theta+2) x^{2}+6 x-4=0$ has 3 positive real roots, find
$$
u-\frac{9 \sin ^{2} \theta-4 \sin \theta+3}{(1-\cos \theta)(2 \cos \theta-6 \sin \theta-3 \sin 2 \theta+2)}
$$
the minimum value. | \frac{621}{8} | 111 | 9 |
math | 13. Let $S=\{1,2, \cdots, n\}, A$ be an arithmetic sequence with at least two terms, a positive common difference, and all terms in $S$. Additionally, adding any other element from $S$ to $A$ does not form an arithmetic sequence with the same common difference as $A$. Find the number of such $A$ (here, sequences with o... | [\frac{n^{2}}{4}] | 94 | 9 |
math | Example 10 Find all functions $f: \mathbf{Q} \rightarrow \mathbf{Q}$, satisfying the condition $f[x+f(y)]=f(x) \cdot f(y)(x, y \in \mathbf{Q})$.
| f(x)=0orf(x)=1 | 56 | 8 |
math | 3. Solve the inequality
$$
(1+\sqrt{3}) \sin 2 x+2 \cos ^{2} x \geq 2\left(1+\sqrt{3} \cos ^{2} x\right)
$$ | [\frac{\pi}{4}+k\pi,\frac{\pi}{3}+k\pi] | 54 | 23 |
math | 5. Let triangle $A B C$ be such that $A B=A C=22$ and $B C=11$. Point $D$ is chosen in the interior of the triangle such that $A D=19$ and $\angle A B D+\angle A C D=90^{\circ}$. The value of $B D^{2}+C D^{2}$ can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$... | 36104 | 118 | 5 |
math | 2. Let $n$ be a given positive integer, find the smallest positive integer $m$, such that
$$2^{m} \equiv 1\left(\bmod 5^{n}\right)$$ | 4 \times 5^{n-1} | 45 | 10 |
math | 2. Find all real numbers $x$ for which
$$
\left(x^{2}-7 x+11\right)^{x^{2}+5 x-6}=1
$$ | 1,2,3,4,5,-6 | 42 | 11 |
math | Cátia leaves school every day at the same time and returns home by bicycle. When she cycles at $20 \mathrm{~km} / \mathrm{h}$, she arrives home at 4:30 PM. If she cycles at $10 \mathrm{~km} / \mathrm{h}$, she arrives home at 5:15 PM. At what speed should she cycle to arrive home at 5:00 PM? | 12\mathrm{~}/\mathrm{} | 96 | 10 |
math | 10. Find all values of parameters $a$ and $b$, for which the polynomial
$$
f(x)=x^{5}-3 x^{4}+a x^{3}+b x^{2}-5 x-5
$$
is divisible by $x^{2}-1$ without a remainder. | =4,b=8 | 67 | 5 |
math | Example 5 Calculate $\sec \frac{2 \pi}{9}+\sec \frac{4 \pi}{9}+\sec \frac{6 \pi}{9}+\sec \frac{8 \pi}{9}$. | 4 | 49 | 1 |
math | \section*{Exercise 3 - 011213}
How many different three-digit numbers can be formed using the digits
a) 1 and 2,
b) 1, 2, and 3,
c) 1, 2, 3, and 4,
where the digits can be used multiple times? Try to find a pattern!
1) What solution do you get for four-digit numbers?
2) What can be conjectured for four-digit number... | n^4 | 117 | 3 |
math | 2. (India 2004) $S$ is the set of all ordered tuples $(a, b, c, d, e, f)$ where $a, b, c, d, e, f$ are integers, and $a^{2}+b^{2}+c^{2}+d^{2}+e^{2}=f^{2}$. Find the largest $k$ such that for all elements of $S$, $k$ divides $a b c d e f$.
| 24 | 109 | 2 |
math | 3. In $\triangle A B C$, it is known that $\overrightarrow{A B} \cdot \overrightarrow{A C}+2 \overrightarrow{B A} \cdot \overrightarrow{B C}=3 \overrightarrow{C A} \cdot \overrightarrow{C B}$, then the minimum value of $\cos C$ is | \frac{\sqrt{2}}{3} | 75 | 10 |
math | Example. Find the integral curves of the differential equation
$$
y^{\prime}=\frac{x^{2}+2 x y-5 y^{2}}{2 x^{2}-6 x y}
$$ | 2\operatorname{arctg}\frac{y}{x}-\ln\frac{(x^{2}+y^{2})^{3}}{|x|^{5}}=C | 45 | 40 |
math | A 7-divisible number $K$ in binary form is as follows:
$$
K=10110101010101 x y z 110
$$
Determine the digits $x, y, z$. | 0,1,0 | 55 | 5 |
math | # 7. Clone 1
In the expression OL $*$ IM $* P *$ IA * DA, it is required to replace the asterisks with two plus signs and two minus signs, and to replace the letters with digits according to the rules of a cryptarithm (identical letters with identical digits, and different letters with different digits). What is the m... | 263 | 103 | 3 |
math | 14. If it is said to you: divide 100 square cubits into 2 unknown parts and take $\frac{3}{4}$ of the side of one as the side of the other, give me each of the unknown parts (i.e., divide an area of 100 sq. cubits into 2 squares, the sides of which are in the ratio $\left.1: \frac{3}{4}\right)$.
## Problems from the R... | 8,6 | 104 | 3 |
math | In a tetrahedron, at each vertex, edges of lengths $5$, $\sqrt{41}$, and $\sqrt{34}$ meet. What are the lengths of the segments connecting the midpoints of the opposite edges of the tetrahedron? | 3,4,5 | 55 | 5 |
math | 1. Find the largest real number $\theta(\theta<\pi)$ such that
$$
\prod_{k=0}^{10} \cos 2^{k} \theta \neq 0 \text {, and } \prod_{k=0}^{10}\left(1+\frac{1}{\cos 2^{k} \theta}\right)=1
$$
(2015, Harvard-MIT Mathematics Tournament) | \frac{2046 \pi}{2047} | 98 | 15 |
math | Let $D$ be the footpoint of the altitude from $B$ in the triangle $A B C$, where $A B=1$. The incentre of triangle $B C D$ coincides with the centroid of triangle $A B C$. Find the lengths of $A C$ and $B C$. | A C=B C=\sqrt{\frac{5}{2}} | 65 | 13 |
math | Example 9. On five identical cards, letters are written: on two cards $l$, on the other three $i$. These cards are randomly laid out in a row. What is the probability that the word "lilii" will be formed? | 0.1 | 52 | 3 |
math | ## Problem 4
Find $x>1$ for which $\frac{1}{[x]}+\frac{1}{\{x\}}=2014 x$, where $[x]$ denotes the integer part of $x$, and $\{x\}$ represents the fractional part of $x$.
## Mathematical Gazette
## Note.
All problems are mandatory.
Each problem is worth 7 points.
Working time 3 hours.
Proposers: Prof. Hecser Enik... | n+\frac{1}{2014n}, | 159 | 12 |
math | Let $ABC$ be a triangle, let the $A$-altitude meet $BC$ at $D$, let the $B$-altitude meet $AC$ at $E$, and let $T\neq A$ be the point on the circumcircle of $ABC$ such that $AT || BC$. Given that $D,E,T$ are collinear, if $BD=3$ and $AD=4$, then the area of $ABC$ can be written as $a+\sqrt{b}$, where $a$ and $b$ are po... | 112 | 145 | 3 |
math | Example 1 Try to determine all triples $(p, q, n)$ that simultaneously satisfy
$$
\begin{array}{l}
q^{n+2} \equiv 3^{n+2}\left(\bmod p^{n}\right), \\
p^{n+2} \equiv 3^{n+2}\left(\bmod q^{n}\right)
\end{array}
$$
where $p, q$ are odd primes, and $n$ is an integer greater than 1. ${ }^{[1]}$
(2008, China Mathematical Ol... | (3,3, n)(n=2,3, \cdots) | 126 | 17 |
math | 10,11
The bases of three equal cones are located in the same plane and touch each other. The axial section of each cone is an equilateral triangle with side $a$. Find the radius of the sphere that touches the lateral surface of each cone and the plane in which their bases are located. | \frac{(2-\sqrt{3})}{2} | 63 | 12 |
math | 8. Given $0<x<\frac{\pi}{2}, \sin x-\cos x=\frac{\pi}{4}$. If $\tan x+\frac{1}{\tan x}$ can be expressed in the form $\frac{a}{b-\pi^{c}}$ ($a$, $b$, $c$ are positive integers), then $a+b+c=$ $\qquad$ | 50 | 82 | 2 |
math | Let $x = \left( 1 + \frac{1}{n}\right)^n$ and $y = \left( 1 + \frac{1}{n}\right)^{n+1}$ where $n \in \mathbb{N}$. Which one of the numbers $x^y$, $y^x$ is bigger ? | x^y = y^x | 77 | 8 |
math | Find the integer $n$ such that
\[n + \left\lfloor\sqrt{n}\right\rfloor + \left\lfloor\sqrt{\sqrt{n}}\right\rfloor = 2017.\] Here, as usual, $\lfloor\cdot\rfloor$ denotes the floor function. | 1967 | 69 | 4 |
math | Example 8. Determine the area bounded by the arc of the cosine curve from $x=-\frac{\pi}{2}$ to $x=\frac{\pi}{2}$ and the $O x$ axis. | 2 | 44 | 1 |
math | 4. Let $x, y, z$ be positive numbers, and $x^{2}+y^{2}+z^{2}=1$. Try to find the minimum value of the following expression:
$$
S=\frac{x y}{z}+\frac{y z}{x}+\frac{z x}{y} .
$$
(22nd All-Soviet Union Olympiad) | \sqrt{3} | 84 | 5 |
math | In $\triangle{ABC}, AB=10, \angle{A}=30^\circ$ , and $\angle{C=45^\circ}$. Let $H, D,$ and $M$ be points on the line $BC$ such that $AH\perp{BC}$, $\angle{BAD}=\angle{CAD}$, and $BM=CM$. Point $N$ is the midpoint of the segment $HM$, and point $P$ is on ray $AD$ such that $PN\perp{BC}$. Then $AP^2=\dfrac{m}{n}$, where ... | 77 | 148 | 2 |
math | We say that some positive integer $m$ covers the number $1998$, if $1,9,9,8$ appear in this order as digits of $m$. (For instance $1998$ is covered by $2\textbf{1}59\textbf{9}36\textbf{98}$ but not by $213326798$.) Let $k(n)$ be the number of positive integers that cover $1998$ and have exactly $n$ digits ($n\ge 5$), a... | 1 | 146 | 1 |
math | Example 5 Given that $x, y$ are real numbers, and $x^{2}+x y+$ $y^{2}-2=0$. Then the range of values for $x^{2}-x y+y^{2}$ is $\qquad$
(1996, Huanggang Region, Hubei Province Junior High School Mathematics Competition) | \frac{2}{3} \leqslant x^{2}-x y+y^{2} \leqslant 6 | 75 | 29 |
math | 3. Let $p>3$ be a prime number. For an arbitrary set $S \subseteq \mathbb{Z}$ and $a \in \mathbb{Z}$, let
$$
S_{a}=\left\{x \in\{0,1, \ldots, p-1\} \mid(\exists s \in S) \quad x \equiv_{p} a \cdot s\right\}
$$
(a) How many sets $S \subseteq\{1,2, \ldots, p-1\}$ are there such that the sequence $S_{1}, S_{2}, \ldots, ... | 2 | 241 | 1 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.