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 | 2. If the real number $\alpha$ satisfies $\cos \alpha=\tan \alpha$, then $\frac{1}{\sin \alpha}+\cos ^{4} \alpha=$ $\qquad$ | 2 | 43 | 1 |
math | Let $a_{n+1} = \frac{4}{7}a_n + \frac{3}{7}a_{n-1}$ and $a_0 = 1$, $a_1 = 2$. Find $\lim_{n \to \infty} a_n$. | \frac{17}{10} | 63 | 9 |
math | Let $c$ be the length of the hypotenuse of a right angle triangle whose two other sides have lengths $a$ and $b$. Prove that $a+b\le c\sqrt{2}$. When does the equality hold? | a + b \leq c\sqrt{2} | 51 | 13 |
math | 7. In a regular triangular prism $A B C-A_{1} B_{1} C_{1}$, all 9 edges are of equal length, $P$ is the midpoint of $C C_{1}$, and the dihedral angle $B-A_{1} P-B_{1}=\alpha$, then $\sin \alpha=$ $\qquad$ | \frac{\sqrt{10}}{4} | 77 | 11 |
math | 9.2. In triangle $A B C$, side $B C$ is equal to segment $A M$, where $M$ is the point of intersection of the medians. Find the angle $\angle B M C$. | 90 | 47 | 2 |
math | 2. Given that $\triangle A B C$ is inscribed in $\odot O$ with radius 1, and $A B \cdot A C=A B-A C=1$. Try to find the measures of the three interior angles of $\triangle A B C$, and prove your conclusion. | \angle A B C=18^{\circ}, \angle A C B=54^{\circ}, \angle B A C=108^{\circ} \text{ or } \angle A B C=18^{\circ}, \angle A C B=126^{\circ}, \angle B A C=36^{\circ} | 61 | 79 |
math | Example 1.7 Find the number of all permutations of $n$ distinct elements $a_{1}, a_{2}, \cdots, a_{n}$ in which $a_{1}$ and $a_{2}$ are not adjacent. | (n-2)(n-1)! | 51 | 8 |
math | Compute the $\textit{number}$ of ordered quadruples $(w,x,y,z)$ of complex numbers (not necessarily nonreal) such that the following system is satisfied:
\begin{align*}
wxyz &= 1\\
wxy^2 + wx^2z + w^2yz + xyz^2 &=2\\
wx^2y + w^2y^2 + w^2xz + xy^2z + x^2z^2 + ywz^2 &= -3 \\
w^2xy + x^2yz + wy^2z + wxz^2 &= -1\end{align*... | 24 | 136 | 2 |
math | 8. Given $x, y \in[0,+\infty)$, then the minimum value of $x^{3}+y^{3}-5 x y$ is $\qquad$ . | -\frac{125}{27} | 42 | 10 |
math | 1. (2 points) Solve the equation $p^{3}-q^{3}=5 r$, where $p, q, r$ are prime numbers.
Answer: $p=7, q=2, r=67$. | p=7,q=2,r=67 | 49 | 10 |
math | ## Task B-3.4.
While preparing for the competition, Matko discovered a bookstore with good mathematical literature. The bookstore offers 7 different books with problems only in geometry, 4 only in number theory, and 5 only in combinatorics. Furthermore, the store also offers books with problems from exactly two areas.... | 270 | 122 | 3 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow-1} \frac{x^{3}+1}{\sin (x+1)}$ | 3 | 38 | 1 |
math | 2. (6 points) A certain duplicator can print 3600 sheets of paper per hour, so printing 240 sheets of paper requires $\qquad$ minutes. | 4 | 39 | 1 |
math | In a group of $n$ people, there are $k$ individuals who each have exactly two acquaintances among the present. Among the remaining $(n-k)$ members of the group, no two know each other.
What is the maximum number of handshakes that can occur if the strangers introduce themselves to each other? (We consider the acquaint... | S_{\max}=\frac{1}{2}n(n-1)-k | 87 | 18 |
math | Shnol D....
Given a triangle $ABC$ and an excircle with center $O$, touching side $BC$ and the extensions of sides $AB$ and $AC$. Point $O_{1}$ is symmetric to point $O$ with respect to line $BC$. Find the measure of angle $A$, if it is known that point $O_{1}$ lies on the circumcircle of triangle $ABC$. | 60 | 86 | 2 |
math | What conditions must the numbers $a, b, c$ satisfy so that the expression $x^{2}+x y+y^{2}+a x+b y+c$ is positive for all pairs of numbers $(x, y)$? | ^{2}-+b^{2}<3 | 49 | 9 |
math | A positive integer $n$ between 10 and 99 , inclusive, is chosen at random. If every such integer is equally likely to be chosen, what is the probability that the sum of the digits of $n$ is a multiple of 7 ? | \frac{2}{15} | 54 | 8 |
math | ## Task 2 - 090522
In an HO clothing store, three customers bought the same fabric. The first bought exactly $3 \mathrm{~m}$, the second exactly $5 \mathrm{~m}$, and the third exactly $9 \mathrm{~m}$. The second customer paid $30 \mathrm{M}$ more than the first.
How many Marks did the three customers have to pay in t... | 255\mathrm{M} | 97 | 8 |
math | 5. Let $Q(x)=a_{2023} x^{2023}+a_{2022} x^{2022}+\cdots+a_{1} x+a_{0}$ be a polynomial with integer coefficients. For every odd prime number $p$, we define the polynomial $Q_{p}(x)=a_{2023}^{p-2} x^{2023}+a_{2022}^{p-2} x^{2022}+\cdots+a_{1}^{p-2} x+a_{0}^{p-2}$. It is known that for infinitely many odd prime numbers $... | \frac{2023^{2024}-1}{2022} | 202 | 20 |
math | $4.77 \operatorname{tg} 9^{\circ}+\operatorname{tg} 15^{\circ}-\operatorname{tg} 27^{\circ}-\operatorname{ctg} 27^{\circ}+\operatorname{ctg} 9^{\circ}+\operatorname{ctg} 15^{\circ}=8$. | 8 | 86 | 1 |
math | 1. Once in a physics and mathematics school, September was declared the month of information technology. Therefore, every Monday in September, there was 1 computer science lesson in each class, every Tuesday - 2 lessons, every Wednesday - 3 lessons, every Thursday - 4 lessons, and every Friday - 5 computer science less... | Wednesday | 140 | 1 |
math | 11.1 From a three-digit number $A$, which does not contain zeros in its notation, a two-digit number $B$ was obtained by writing the sum of the first two digits instead of them (for example, the number 243 turns into 63). Find $A$ if it is known that $A=3 B$. | 135 | 73 | 3 |
math | ## Task A-3.3.
Determine all pairs of natural numbers $(a, b)$ for which $a^{2} b$ divides $b^{2}+3 a$. | (1,1)(1,3) | 39 | 9 |
math | PROBLEM 1. In a mathematics competition, 30 exercises are proposed. For each correctly solved exercise, the student receives 10 points, and for each incorrectly solved exercise, the student is penalized with 10 points (10 points are deducted). Dragoş solved all the exercises and received 80 points.
a) How many exercis... | 19,11,2,300 | 134 | 11 |
math | Test $\mathbf{G}$ Given $a, b, c, d \in \mathbf{R}$, and satisfying $a+b+c+d=3, a^{2}+2 b^{2}+$ $3 c^{2}+6 d^{2}=5$, try to find the maximum and minimum values of $a$. | 1\leqslant\leqslant2 | 72 | 12 |
math | Determine the number of all positive integers which cannot be written in the form $80k + 3m$, where $k,m \in N = \{0,1,2,...,\}$ | 79 | 42 | 2 |
math | 10. Given that $f(x)$ is a function defined on $\mathbf{R}$, $f(1)=1$ and for any $x \in \mathbf{R}$, $f(x+5) \geqslant f(x)+5, f(x+1) \leqslant f(x)+1$. If $g(x)=f(x)+1-x$, then $g(2002)$ $=$ . $\qquad$ | 1 | 100 | 1 |
math | A rectangle has area $1100$. If the length is increased by ten percent and the width is
decreased by ten percent, what is the area of the new rectangle? | 1089 | 39 | 4 |
math | Example 11 Suppose $1995 x^{3}=1996 y^{3}=1997 z^{3}$, $x y z>0$, and $\sqrt[3]{1995 x^{2}+1996 y^{2}+1997 z^{2}}=$ $\sqrt[3]{1995}+\sqrt[3]{1996}+\sqrt[3]{1997}$. Then $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=$
(1996, National Junior High School Mathematics League) | 1 | 139 | 1 |
math | Example 5. Solve the equation:
a) $\log _{1 / 5} \frac{2+x}{10}=\log _{1 / 5} \frac{2}{x+1}$;
b) $\log _{3}\left(x^{2}-4 x+3\right)=\log _{3}(3 x+21)$;
c) $\log _{1 / 10} \frac{2 x^{2}-54}{x+3}=\log _{1 / 10}(x-4)$;
d) $\log _{(5+x) / 3} 3=\log _{-1 /(x+1)} 3$. | 3,-2,9,6,-4 | 150 | 9 |
math | Example: Find the value of $\arcsin \frac{1}{\sqrt{10}}+\arccos \frac{5}{\sqrt{26}}$ $+\operatorname{arctg} \frac{1}{7}+\arccos \frac{8}{\sqrt{65}}$. | \frac{\pi}{4} | 69 | 7 |
math | For how many integers $x$ is the expression $\frac{\sqrt{75-x}}{\sqrt{x-25}}$ equal to an integer? | 5 | 32 | 1 |
math | 6. Solve the equation $\sqrt{\frac{x-3}{2 x+1}}+2=3 \sqrt{\frac{2 x+1}{x-3}}$. | -4 | 37 | 2 |
math | ## Task 5 - 020525
On a straight line, consecutively mark the segments $A B=3 \mathrm{~cm}, B C=5 \mathrm{~cm}$ and $C D=4 \mathrm{~cm}$! How large is the distance between the midpoints of the segments $A B$ and $C D$? Justify your answer by calculation! | 8.5 | 87 | 3 |
math | ## Task B-4.4.
A sequence is defined by the recursive formula
$$
x_{1}=1, x_{n+1}=x_{n}+2 n+1, n \geqslant 1
$$
Determine $x_{2018}$. | 2018^2 | 63 | 6 |
math | 4. Find all pairs $(p, q)$ of real numbers such that the polynomial $x^{2}+p x+q$ is a divisor of the polynomial $x^{4}+p x^{2}+q$.
---
Note: The translation preserves the original formatting and structure of the text. | (0,0),(0,-1),(-1,0),(1,1),(-2,1) | 64 | 23 |
math | 14. Let \( f(x) = A(x^2 - 2x)e^x - e^x + 1 \). For any \( x \leq 0 \), \( f(x) \geq 0 \) holds. Determine the range of the real number \( A \). | [-\frac{1}{2},+\infty) | 64 | 12 |
math | 5. Find the largest positive integer $x$ such that $x$ is divisible by all the positive integers $\leq \sqrt[3]{x}$. | 420 | 33 | 3 |
math | We denote with $m_a$, $m_g$ the arithmetic mean and geometrical mean, respectively, of the positive numbers $x,y$.
a) If $m_a+m_g=y-x$, determine the value of $\dfrac xy$;
b) Prove that there exists exactly one pair of different positive integers $(x,y)$ for which $m_a+m_g=40$. | (5, 45) | 80 | 7 |
math | 1. The quadratic trinomial $f(x)=a x^{2}+b x+c$ has exactly one root, the quadratic trinomial $2 f(2 x-3)-f(3 x+1)$ also has exactly one root. Find the root of the trinomial $f(x)$. | -11 | 66 | 3 |
math | 7. Given sets of real numbers $A, B$, define the operation $A \otimes B=\{x \mid x=a b+a+b, a \in A, b \in B\}$. Let $A=\{0,2,4, \cdots, 18\}, B=\{98,99,100\}$, then the sum of all elements in $A \otimes B$ is | 29970 | 94 | 5 |
math | Example 3 Given the real-coefficient equation $x^{3}+2(k-1) x^{2}+9 x+5(k-1)=0$ has a complex root with modulus $\sqrt{5}$, find the value of $k$, and solve this equation. | k=-1ork=3 | 59 | 6 |
math | 14. [9] Compute the sum of all positive integers $n$ for which
$$
9 \sqrt{n}+4 \sqrt{n+2}-3 \sqrt{n+16}
$$
is an integer. | 18 | 48 | 2 |
math | The proportion of potassium nitrate, sulfur, and charcoal in the formulation of ancient Chinese "black powder" is $15: 2: 3$. Given 50 kilograms of charcoal, to prepare "black powder" weighing 1000 kilograms, how many more kilograms of charcoal are needed? | 100 | 63 | 3 |
math | Three, (50 points) Given non-negative real numbers $a, b, c, d$ satisfying $a+b+c+d=4$. Find the minimum value of $\sum \frac{b+3}{a^{2}+4}$, where “$\sum$” denotes the cyclic sum. | 3 | 63 | 1 |
math | 1. If the line $x \cos \theta+y \sin \theta=\cos ^{2} \theta-\sin ^{2} \theta$ $(0<\theta<\pi)$ intersects the circle $x^{2}+y^{2}=\frac{1}{4}$, then the range of values for $\theta$ is $\qquad$ | \frac{\pi}{6} \leqslant \theta \leqslant \frac{\pi}{3} \text{ or } \frac{2 \pi}{3} \leqslant \theta \leqslant \frac{5 \pi}{6} | 77 | 61 |
math | 2. (17 points) A tourist travels from point $A$ to point $B$ in 2 hours and 14 minutes. The route from $A$ to $B$ goes uphill first, then on flat terrain, and finally downhill. What is the length of the uphill road if the tourist's speed downhill is 6 km/h, uphill is 4 km/h, and on flat terrain is 5 km/h, and the total... | 6 | 123 | 1 |
math | 4. Given the function $y=x^{3}$, the tangent line at $x=a_{k}$ intersects the $x$-axis at point $a_{k+1}$. If $a_{1}=1, S_{n}=\sum_{i=1}^{n} a_{i}$, then $\lim _{n \rightarrow \infty} S_{n}$ $=$ . $\qquad$ | 3 | 89 | 1 |
math | A square pyramid with base $ABCD$ and vertex $E$ has eight edges of length 4. A plane passes through the midpoints of $\overline{AE}$, $\overline{BC}$, and $\overline{CD}$. The plane's intersection with the pyramid has an area that can be expressed as $\sqrt{p}$. Find $p$. | p = 80 | 80 | 6 |
math | Four, (15 points) Given a positive integer $n$ that satisfies the following condition: among any $n$ integers greater than 1 and not exceeding 2009 that are pairwise coprime, at least one is a prime number. Find the minimum value of $n$.
| 15 | 62 | 2 |
math | ## Task B-4.4.
Determine all polynomials $p$ with real coefficients for which the equality
$$
x \cdot p(x-1)=(x-2021) \cdot p(x)
$$
is satisfied for all real numbers $x$. | p(x)=\cdotx(x-1)(x-2)\cdots(x-2020),\in\mathbb{R} | 57 | 31 |
math | 4.94. Solve the Cauchy problem: $y^{\prime \prime}-8 y^{3}=0 ; y(0)=-1 ; y^{\prime}(0)=2$. | -\frac{1}{1+2x} | 43 | 10 |
math | 8,9
Calculate the volume of a regular tetrahedron if the radius of the circle circumscribed around its face is $R$.
# | \frac{R^3\sqrt{6}}{4} | 32 | 14 |
math | An infinite sequence of positive real numbers is defined by $a_0=1$ and $a_{n+2}=6a_n-a_{n+1}$ for $n=0,1,2,\cdots$. Find the possible value(s) of $a_{2007}$. | 2^{2007} | 64 | 7 |
math | Example 10 Let $x, y, z$ be positive real numbers, and satisfy $x y z + x + z = y$, find
$$
p=\frac{2}{x^{2}+1}-\frac{2}{y^{2}+1}+\frac{3}{z^{2}+1}
$$
the maximum value. | \frac{10}{3} | 77 | 8 |
math | 24. Find the number of permutations $a_{1} a_{2} a_{3} a_{4} a_{5} a_{6}$ of the six integers from 1 to 6 such that for all $i$ from 1 to $5, a_{i+1}$ does not exceed $a_{i}$ by 1 . | 309 | 75 | 3 |
math | ## Problem 1
A palindrome is a positive integers which is unchanged if you reverse the order of its digits. For example, 23432. If all palindromes are written in increasing order, what possible prime values can the difference between successive palindromes take?
| 2,11 | 60 | 4 |
math | 11. (20 points) Let $n (n \geqslant 2)$ be a given positive integer. Find the value of $\prod_{k=1}^{n-1} \sin \frac{n(k-1)+k}{n(n+1)} \pi$.
| \frac{1}{2^{n-1}} | 63 | 11 |
math | 1. Find the largest three-digit number from which, after erasing any digit, we get a prime number. | 731 | 23 | 3 |
math | 9. For the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length 1, the distance between the lines $A_{1} C_{1}$ and $B D_{1}$ is $\qquad$ | \frac{\sqrt{6}}{6} | 58 | 10 |
math | 13.435 The desired number is greater than 400 and less than 500. Find it, if the sum of its digits is 9 and it is equal to 47/36 of the number represented by the same digits but written in reverse order. | 423 | 61 | 3 |
math | There are real numbers $a, b, c,$ and $d$ such that $-20$ is a root of $x^3 + ax + b$ and $-21$ is a root of $x^3 + cx^2 + d.$ These two polynomials share a complex root $m + \sqrt{n} \cdot i,$ where $m$ and $n$ are positive integers and $i = \sqrt{-1}.$ Find $m+n.$ | 330 | 101 | 3 |
math | 881*. Solve the equation in natural numbers
$$
x!+y!=z!
$$ | (1;1;2) | 21 | 7 |
math | 8. A coin collector has 100 coins that look the same. The collector knows that 30 of them are genuine, 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 group... | 70 | 99 | 2 |
math | II. (50 points)
Given that $x, y, z$ are all positive real numbers, find the minimum value of $\iota=\frac{(x+y-z)^{2}}{(x+y)^{2}+z^{2}}+\frac{(z+x-y)^{2}}{(z+x)^{2}+y^{2}}+\frac{(y+z-x)^{2}}{(y+z)^{2}+x^{2}}$. | \frac{3}{5} | 97 | 7 |
math | 1. In tetrahedron $ABCD$, $AD \perp$ plane $BCD$, $\angle ABD = \angle BDC = \theta < 45^{\circ}$. It is known that $E$ is a point on $BD$ such that $CE \perp BD$, and $BE = AD = 1$.
(1) Prove: $\angle BAC = \theta$;
(2) If the distance from point $D$ to plane $ABC$ is $\frac{4}{13}$, find the value of $\cos \theta$.
| \cos \theta = \frac{4}{5} | 127 | 12 |
math | 1. The digit at the 2007th position after the decimal point of the irrational number $0.2342343423434342343434342 \cdots$ is $\qquad$ . | 3 | 59 | 1 |
math | 4. Given positive integers $a, b, c$ satisfy
$$
1<a<b<c, a+b+c=111, b^{2}=a c \text {. }
$$
then $b=$ $\qquad$ | 36 | 49 | 2 |
math | Let's play heads or tails in the following way: We toss the coin four times and then as many times as there were heads in the first four tosses. What is the probability that we will get at least 5 heads in all our tosses? | \frac{47}{256} | 52 | 10 |
math | 41. Under what condition do three lines, given by the equations: $A_{1} x+B_{1} y+C_{1}=0 ; \quad A_{2} x+B_{2} y+C_{2}=0 \quad$ and $\quad A_{3} x+B_{3} y+C_{3}=0$, pass through the same point? | |\begin{pmatrix}A_{1}&B_{1}&C_{1}\\A_{2}&B_{2}&C_{2}\\A_{3}&B_{3}&C_{3}\end{pmatrix}|=0 | 77 | 49 |
math | ## Task 1 - 200711
On the occasion of the award ceremony of a mathematics competition, each prize winner congratulated every other with a handshake. In total, 91 handshakes were performed, and exactly one handshake took place with each congratulation.
Determine the number of prize winners of the competition from this... | 14 | 73 | 2 |
math | By what should the expression
$$
\sqrt[3]{5 \sqrt{3}-3 \sqrt{7}}
$$
be multiplied so that the value of the product is 2? | \sqrt[3]{\frac{2}{3}(5\sqrt{3}+3\sqrt{7})} | 40 | 26 |
math | Let $[x]$ be the integer part of a number $x$, and $\{x\}=x-[x]$. Solve the equation
$$
[x] \cdot\{x\}=1991 x .
$$ | -\frac{1}{1992} | 48 | 10 |
math | The positive integers $a$ and $b$ are such that the numbers $15a+16b$ and $16a-15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares? | 231361 | 60 | 6 |
math | Alice and Bob are each secretly given a real number between 0 and 1 uniformly at random. Alice states, “My number is probably greater than yours.” Bob repudiates, saying, “No, my number is probably greater than yours!” Alice concedes, muttering, “Fine, your number is probably greater than mine.” If Bob and Alice are pe... | \frac{11}{12} | 93 | 9 |
math |
Problem 11.1. Consider the function $f(x)=\sqrt{x}+\sqrt{x-4}-$ $\sqrt{x-1}-\sqrt{x-3}, x \geq 4$.
a.) Find $\lim _{x \rightarrow \infty} f(x)$.
b.) Prove that $f(x)$ is an increasing function.
c.) Find the number of real roots of the equation $f(x)=a \sqrt{\frac{x-3}{x}}$, where $a$ is a real parameter.
| 2(1-\sqrt{3})\leq0 | 114 | 12 |
math | 10. Given $A_{1}, A_{2}, \cdots, A_{n}$ are $n$ non-empty subsets of the set $A=\{1,2,3, \cdots, 10\}$, if for any $i, j \in$ $\{1,2,3, \cdots, n\}$, we have $A_{i} \cup A_{j} \neq A$, then the maximum value of $n$ is $\qquad$. | 511 | 108 | 3 |
math | 26. How many five-digit numbers are there that contain at least one digit 3 and are multiples of 3?
Please retain the original text's line breaks and format, and output the translation result directly. | 12504 | 43 | 5 |
math | B2. Maja found out that in the store, she gets one additional chocolate if she buys five chocolates. She rushed to the store and bought 32 chocolates.
a) How many chocolates did she actually pay for?
b) One chocolate bar cost 119 SIT, and Maja paid with a 5000 SIT banknote. How many tolars did the saleswoman return t... | 27 | 87 | 2 |
math | ## Task 5 - 231245
Determine all functions $f$ that are defined for all non-zero real numbers $x$ and satisfy the following conditions:
(1) For all $x_{1}, x_{2}$ with $x_{1} \neq 0, x_{2} \neq 0, x_{1}+x_{2} \neq 0$, $f\left(\frac{1}{x_{1}+x_{2}}\right)=f\left(\frac{1}{x_{1}}\right)+f\left(\frac{1}{x_{2}}\right)$.
... | f(r)=\frac{1}{r} | 262 | 10 |
math | 8.2. A duckling and a gosling were competing in a triathlon. The distance consisted of equally long running, swimming, and flying segments. The duckling ran, swam, and flew at the same speed. The gosling ran twice as slow as the duckling, but swam twice as fast. Who and by how many times faster did they fly, if they st... | 2 | 89 | 1 |
math | 3. How many solutions in natural numbers does the equation
$$
(2 x+y)(2 y+x)=2017^{2017} ?
$$ | 0 | 35 | 1 |
math | 4. A sequence of numbers is defined by the rule: "Increase the number by 3, then double the result obtained." Determine the first and sixth terms of this sequence if its third term is 2014. | 499,16154 | 46 | 9 |
math | 10,11
Given a parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$. On the rays $C_{1} C, C_{1} B_{1}$, and $C_{1} D_{1}$, segments $C_{1} M, C_{1} N$, and $C_{1} K$ are laid out, equal to
$5 / 2 C C_{1}, 5 / 2 C_{1} B_{1}$,
$5 / 2 C_{1} D_{1}$, respectively. In what ratio does the plane passing through points $M, N, K$ di... | 1:47 | 179 | 4 |
math | 5. Now define an operation * :
When $m$ and $n$ are both positive odd numbers or both positive even numbers,
$$
m * n=m+n \text {; }
$$
When one of $m$ and $n$ is a positive odd number and the other is a positive even number,
$$
m * n=m \cdot n \text {. }
$$
Then, the number of elements in the set $M=\{(a, b) \mid a... | 41 | 132 | 2 |
math | A positive integer is [i]bold[/i] iff it has $8$ positive divisors that sum up to $3240$. For example, $2006$ is bold because its $8$ positive divisors, $1$, $2$, $17$, $34$, $59$, $118$, $1003$ and $2006$, sum up to $3240$. Find the smallest positive bold number. | 1614 | 102 | 4 |
math | 10. (ROM 4) Consider two segments of length $a, b(a>b)$ and a segment of length $c=\sqrt{a b}$. (a) For what values of $a / b$ can these segments be sides of a triangle? (b) For what values of $a / b$ is this triangle right-angled, obtuse-angled, or acute-angled? | 1<k<\frac{3+\sqrt{5}}{2}, \quad k=\frac{1+\sqrt{5}}{2}, \quad 1<k<\frac{1+\sqrt{5}}{2}, \quad \frac{1+\sqrt{5}}{2}<k<\frac{3+\sqrt{5}}{2} | 84 | 76 |
math | Example 13 Given that the domain of the function $f(x)$ is $[0,1]$, and it satisfies the following conditions:
( I ) For any $x \in[0,1]$, we always have $f(x) \geqslant 3$;
( II ) $f(1)=4$;
(III) If $x_{1} \geqslant 0, x_{2} \geqslant 0, x_{1}+x_{2} \leqslant 1$, then $f\left(x_{1}+x_{2}\right) \geqslant f\left(x_{1}\... | 4 | 284 | 1 |
math | A block $Z$ is formed by gluing one face of a solid cube with side length 6 onto one of the circular faces of a right circular cylinder with radius $10$ and height $3$ so that the centers of the square and circle coincide. If $V$ is the smallest convex region that contains Z, calculate $\lfloor\operatorname{vol}V\rfloo... | 2827 | 98 | 4 |
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 | 4. When written in ascending order, the nine internal angles from three particular triangles form a sequence where the difference between any adjacent pair of numbers in the sequence is a constant $d$. One of the angles measures $42^{\circ}$. Find all possible values of the size of the largest of the nine angles. | 78,84,96 | 65 | 8 |
math | 6. Let $n \geqslant 3, b_{n}$ be the number of subsets of the set $\{1, 2, \cdots, n\}$ that have the following property: any two elements in these subsets (which must contain at least two elements) have an absolute difference greater than 1. Then the value of $b_{10}$ is $\qquad$ | 133 | 84 | 3 |
math | Calculate:
1. $\sum_{k=0}^{n}\binom{n}{k}$
2. $\sum_{k=0}^{n} k\binom{n}{k}$ | n2^{n-1} | 41 | 7 |
math | 7. A meeting is attended by 24 representatives, and between any two representatives, they either shake hands once or do not shake hands at all. After the meeting, it is found that there were a total of 216 handshakes, and for any two representatives $P$ and $Q$ who have shaken hands, among the remaining 22 representati... | 864 | 133 | 3 |
math | 8. The number of positive integer solutions $(x, y, z)$ to the equation $x+y+z=2010$ that satisfy $x \leqslant y \leqslant z$ is $\qquad$ . | 336675 | 51 | 6 |
math | 2. Find all triples of natural numbers $a, b$, and $c$, for which the numbers $a^{2}+1$ and $b^{2}+1$ are prime and $\left(a^{2}+1\right)\left(b^{2}+1\right)=c^{2}+1$. | (1,2,3)(2,1,3) | 69 | 13 |
math | * Given a four-digit number $N$ is a perfect square, and each digit of $N$ is less than 7. Increasing each digit by 3 results in another four-digit number that is also a perfect square. Find $N$. | 1156 | 50 | 4 |
math | 10.316. In a right-angled triangle, the distance from the midpoint of the hypotenuse to one of the legs is 5 cm, and the distance from the midpoint of this leg to the hypotenuse is 4 cm. Calculate the area of the triangle. | \frac{200}{3} | 60 | 9 |
math | 1. The range of the function $f(x)=\frac{\left(x-x^{3}\right)\left(1-6 x^{2}+x^{4}\right)}{\left(1+x^{2}\right)^{4}}$ is . $\qquad$ | [-\frac{1}{8},\frac{1}{8}] | 58 | 15 |
math | 8. Find the last four digits of $7^{7^{-7}}$ (100 sevens).
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | 2343 | 48 | 4 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.