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 | 33. Let $f: \mathbb{N} \rightarrow \mathbb{Q}$ be a function, where $\mathbb{N}$ denotes the set of natural numbers, and $\mathbb{Q}$ denotes the set of rational numbers. Suppose that $f(1)=\frac{3}{2}$, and
$$
f(x+y)=\left(1+\frac{y}{x+1}\right) f(x)+\left(1+\frac{x}{y+1}\right) f(y)+x^{2} y+x y+x y^{2}
$$
for all nat... | 4305 | 145 | 4 |
math | 15. Find the real numbers $a, b, c$ that satisfy the following inequalities: $4(ab + bc + ca) - 1 \geqslant a^2 + b^2 + c^2 \geqslant 3(a^3 + b^3 + c^3) \cdot$ (2005 Australian National Training Team Problem) | a=b=c=\frac{1}{3} | 80 | 10 |
math | In the interior of a 7-sided convex polygon, we select 10 points such that in the 17-element set $H$ consisting of the 7 vertices of the polygon and the selected points, no 3 points lie on the same line. We then partition the heptagon into triangles such that the set of vertices of the triangles is exactly $H$. What ca... | 25 | 87 | 2 |
math | 4. The numbers $2 x, 1, y-1$ form an arithmetic sequence in the given order, and $y+3,|x+1|+|x-1|$, $\cos (\arccos x)$ form a geometric sequence in the given order. Then $x+y+x y=$ | 3 | 66 | 1 |
math | 8. Find the smallest real number $m$, such that for any positive integers $a, b, c$ satisfying $a+b+c=1$, the inequality $m\left(a^{3}+b^{3}+c^{3}\right) \geqslant 6\left(a^{2}+b^{2}+c^{2}\right)+1$ holds.
(2006 China Southeast Mathematical Olympiad) | 27 | 93 | 2 |
math | 1. The product of three natural numbers is 240. The product of the first and second number is 60, and the product of the first and third number is 24. What are these numbers? | =6,b=10,=4 | 46 | 9 |
math | 4. A kilo of sausages was placed on a straight line between a dog in a kennel and a cat. The animals simultaneously rushed to the sausages. The cat runs twice as fast as the dog, but eats twice as slowly. Upon reaching the sausages, both ate without fighting and ate an equal amount. It is known that the cat could eat a... | 5x=7y | 118 | 5 |
math | 146. From a group of 15 people, four participants for the relay race $800+400+200+100$ are selected. In how many ways can the athletes be arranged for the relay stages? | 32760 | 53 | 5 |
math | Example 2. For $\triangle A B C=$ with sides $a, b, c$, construct squares outward on each side, with areas sequentially $S_{a}, S_{b}, S_{c}$. If $a+b+c=18$, find the minimum value of $S_{\mathrm{a}}+S_{\mathrm{b}}+S_{\mathrm{c}}$. | 108 | 83 | 3 |
math | 1. determine all positive real solutions of the following system of equations:
$$
\begin{aligned}
& a=\max \left\{\frac{1}{b}, \frac{1}{c}\right\} \\
& b=\max \left\{\frac{1}{c}, \frac{1}{d}\right\} \\
& c=\max \left\{\frac{1}{d}, \frac{1}{e}\right\} \\
& d=\max \left\{\frac{1}{e}, \frac{1}{f}\right\} \\
& e=\max \lef... | (,b,,,e,f)=(x,x,\frac{1}{x},x,x,\frac{1}{x}),\quadx\geq1 | 176 | 32 |
math | Example 3 Given that $a$ is a positive integer. If the roots of the equation
$$
x^{3}+(a+17) x^{2}+(38-a) x-56=0
$$
are all integers, find the value of $a$ and the integer roots of the equation. | a=12, \text{ roots } 1, -2, -28 | 69 | 19 |
math | 1. Real numbers $x, y, z$ (non-zero) satisfy the equation: $x+y=z$. Find all possible values that the expression $\frac{z}{x}+\frac{y}{z}-\frac{x}{y}+\frac{x}{z}+\frac{z}{y}-\frac{y}{x}$ can take. | 3 | 74 | 1 |
math | ## Task Condition
Find the derivative.
$$
y=\frac{3^{x}(4 \sin 4 x+\ln 3 \cdot \cos 4 x)}{16+\ln ^{2} 3}
$$ | 3^{x}\cdot\cos4x | 49 | 9 |
math | Determine all positive integers $n$ satisfying the following condition: for every monic polynomial $P$ of degree at most $n$ with integer coefficients, there exists a positive integer $k\le n$ and $k+1$ distinct integers $x_1,x_2,\cdots ,x_{k+1}$ such that \[P(x_1)+P(x_2)+\cdots +P(x_k)=P(x_{k+1})\].
[i]Note.[/i] A po... | n = 2 | 126 | 5 |
math | Example 6. Find the static moments with respect to the coordinate planes and the coordinates of the center of gravity of a homogeneous triangular plate $x+y+z=a, x \geqslant 0, y \geqslant 0, z \geqslant 0$. | M_{xy}=M_{yz}=M_{xz}=\frac{\sqrt{3}}{6}^{3},x_{}=y_{}=z_{}=\frac{}{3} | 60 | 42 |
math | Find all the triples of positive integers $(a, b, c)$ for which the number
$$
\frac{(a+b)^{4}}{c}+\frac{(b+c)^{4}}{a}+\frac{(c+a)^{4}}{b}
$$
is an integer and $a+b+c$ is a prime.
# | (1,1,1),(1,2,2),(2,3,6) | 73 | 19 |
math | 16. Let $O$ be the centre of the circumcircle of $\triangle A B C, P$ and $Q$ the midpoints of $A O$ and $B C$, respectively. Suppose $\angle C B A=4 \angle O P Q$ and $\angle A C B=6 \angle O P Q$. Find the size of $\angle O P Q$ in degrees. | 12 | 83 | 2 |
math | Example 2 Let $n=1999$. Find the value of the following expression:
$$
\frac{1}{2^{n}}\left(1-3 \mathrm{C}_{n}^{2}+3^{2} \mathrm{C}_{n}^{4}-3^{3} \mathrm{C}_{n}^{6}+\cdots-3^{999} \mathrm{C}_{n}^{1998}\right) \text {. }
$$ | \frac{1}{2} | 106 | 7 |
math | Problem 2. There are 22 batteries, 15 of which are charged and 7 are discharged. The camera works with three charged batteries. You can insert any three batteries into it and check if it works. How can you guarantee to turn on the camera in 10 such attempts? | 10 | 62 | 2 |
math | 6. In the Thrice-Tenth Kingdom, there are 17 islands, each inhabited by 119 people. The inhabitants of the kingdom are divided into two castes: knights, who always tell the truth, and liars, who always lie. During the census, each person was first asked: "Excluding you, are there an equal number of knights and liars on... | 1013 | 180 | 4 |
math | 11. Let the line $y=k x+b$ intersect the curve $y=x^{3}-x$ at three distinct points $A, B, C$, and $|A B|=|B C|=2$. Then the value of $k$ is $\qquad$. | 1 | 58 | 1 |
math |
Problem 3. Let $n>1$ be an integer. Find the number of the permutations $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ of the numbers $1,2, \ldots, n$ with the following property: there exists only one index $i \in\{1,2, \ldots, n-$ 1\} such that $a_{i}>a_{i+1}$.
| 2^{n}-n-1 | 99 | 7 |
math | 1. The sum of a set of numbers is the sum of all its elements. Let $S$ be a set of positive integers not exceeding 15, such that the sums of any two disjoint subsets of $S$ are not equal. Moreover, among all sets with this property, the sum of $S$ is the largest. Find the sum of the set $S$.
(4th American Invitational ... | 61 | 88 | 2 |
math | Let's calculate as simply as possible the value of
$$
a^{3}+b^{3}+3\left(a^{3} b+a b^{3}\right)+6\left(a^{3} b^{2}+a^{2} b^{3}\right)
$$
if $a+b=1$. | 1 | 69 | 1 |
math | Let $P$ be a point chosen uniformly at random in the interior of the unit square with vertices at $(0,0), (1,0), (1,1)$, and $(0,1)$. The probability that the slope of the line determined by $P$ and the point $\left(\frac58, \frac38 \right)$ is greater than or equal to $\frac12$ can be written as $\frac{m}{n}$, where $... | 171 | 116 | 3 |
math | 4. The quiz participants were asked four questions: 90 participants answered the first question correctly, 50 answered the second, 40 answered the third, and 20 answered the fourth, and no one was able to answer more than two questions correctly. What is the minimum number of participants in the quiz under these condit... | 100 | 69 | 3 |
math | 109. In the equation $x^{2}-2 x+a=0$, the square of the difference of the roots is 20. Find $a$. | -4 | 35 | 2 |
math | Find the maximum number of different integers that can be selected from the set $ \{1,2,...,2013\}$ so that no two exist that their difference equals to $17$. | 1010 | 42 | 4 |
math | Four. (50 points) On a plane, there are $n$ points, no three of which are collinear. Each pair of points is connected by a line segment, and each line segment is colored either red or blue. A triangle with all three sides of the same color is called a "monochromatic triangle." Let the number of monochromatic triangles ... | \frac{k(k-1)(k-2)}{3} | 141 | 14 |
math | ## Task 1 - V10811
The new double-decker articulated train of our State Railway weighs a total of $129 \mathrm{Mp}$ (unladen weight) and has seating for 640 passengers. An old-style D-train car weighs $40 \mathrm{Mp}$ and offers seating for 64 passengers.
By what percentage is the "seat weight" (unladen weight per se... | 67.75 | 111 | 5 |
math | 10. (3 points)
Find the smallest natural number that starts with the digit 3 and which, after erasing this digit, decreases by 25 times. | 3125 | 35 | 4 |
math | Daniel has a (mostly) standard deck of 54 cards, consisting of 4 suits each containing the ranks 1 to 13 as well as 2 jokers. Daniel plays the following game: He shuffles the deck uniformly randomly and then takes all of the cards that end up strictly between the two jokers. He then sums up the ranks of all the cards h... | \frac{77}{689} | 167 | 10 |
math | # Problem 5. (3 points)
In trapezoid $ABCD$, a point $X$ is taken on the base $AD$ such that segments $XB$ and $XC$ divide the trapezoid into three triangles that are similar to each other but pairwise unequal and non-isosceles. The side $AB$ has a length of 6. Find $AX \cdot DX$. | 36 | 85 | 2 |
math | 2. Let $\triangle A B C$ be a right triangle with legs $A B=6$ and $B C=8$. Let $R$ and $r$ be the circumradius and the inradius of $\triangle A B C$, respectively. Find the sum of $R$ and $r$. | 7 | 64 | 1 |
math | Let $S$ be the set of all 3-tuples $(a, b, c)$ of positive integers such that $a + b + c = 2013$. Find $$\sum_{(a,b,c)\in S} abc.$$ | \binom{2015}{5} | 53 | 11 |
math | 13.330. Two trucks were supposed to transport a certain cargo in 6 hours. The second truck was delayed in the garage, and when it arrived at the loading site, the first truck had already transported $3 / 5$ of the total cargo; the remaining part of the cargo was transported by the second truck, and the entire cargo was... | 10 | 97 | 2 |
math | Example 1.2 Let $n$ be a positive integer greater than 1, find the number of ordered pairs of positive integers $(x, y)$ that satisfy the condition $x+y \leqslant n$. | \frac{n(n-1)}{2} | 46 | 10 |
math | In a convex $n$-sided polygon, all the diagonals are drawn and no three of them pass through a point. Find a formula for the number of regions formed inside the polygon. | 1+(\begin{pmatrix}n\\2\end{pmatrix})-n+(\begin{pmatrix}n\\4\end{pmatrix}) | 40 | 33 |
math | 2. Given numbers $x, y, z \in\left[0, \frac{\pi}{2}\right]$. Find the minimum value of the expression
$$
A=\cos (x-y)+\cos (y-z)+\cos (z-x)
$$ | 1 | 57 | 1 |
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 | 10. (20 points) Let the function
$$
f(x)=\log _{a}(1+a x)-\log _{a}(1-a x),
$$
where $a>0$, and $a \neq 1$.
(1) When $a>1$, if the inequality
$$
f(x) \geqslant \log _{a} 8 x
$$
holds for all $x$, find the range of values for $a$;
(2) If $f\left(x_{0}\right)=x_{0}-1$, prove: $\left|x_{0}\right|<1$. | 24-16\sqrt{2} | 139 | 10 |
math | 9. Solve the congruence equation $x^{2} \equiv 59(\bmod 125)$ | x \equiv \pm 53\left(\bmod 5^{3}\right) | 26 | 20 |
math | Konagin S.
Find $x_{1000}$, if $x_{1}=4, x_{2}=6$, and for any natural $n \geq 3$, $x_{n}$ is the smallest composite number greater than $2 x_{n-1}-x_{n-2}$. | 501500 | 68 | 6 |
math | 5. We will call a number greater than 25 semi-prime if it is the sum of some two different prime numbers. What is the maximum number of consecutive semi-prime numbers that can be semi-prime? | 5 | 45 | 1 |
math | 2. 2.9 * The range of negative values of $a$ that make the inequality $\sin ^{2} x+a \cos x+a^{2} \geqslant 1+\cos x$ hold for all $x \in \mathbf{R}$ is $\qquad$ . | \leqslant-2 | 65 | 7 |
math | [ Divisibility of numbers. General properties ]
Find the smallest natural $n$ for which $(n+1)(n+2)(n+3)(n+4)$ is divisible by 1000. | 121 | 45 | 3 |
math | 11. Given $\tan \alpha+\log _{2}(2 \tan \alpha-6)=8$, $\tan \beta+2^{\tan \beta-1}=5$. Then $\tan \alpha+\tan \beta$ equals $\qquad$ . | 8 | 56 | 1 |
math | 5. Let $a, b, c$ be non-negative real numbers, satisfying $a+b+c=8, ab+bc+ca=16$. Let $m=\min \{ab, bc, ca\}$, then the maximum value of $m$ is $\qquad$ . | \frac{16}{9} | 62 | 8 |
math | Example 11. Among 25 students in a group, where 10 are girls, 5 tickets are being raffled. Find the probability that 2 of the ticket winners will be girls. | \frac{195}{506}\approx0.385 | 44 | 17 |
math | Let $w_{1}$ and $w_{2}$ denote the circles $x^{2}+y^{2}+10x-24y-87=0$ and $x^{2}+y^{2}-10x-24y+153=0$, respectively. Let $m$ be the smallest positive value of $a$ for which the line $y=ax$ contains the center of a circle that is externally tangent to $w_{2}$ and internally tangent to $w_{1}$. Given that $m^{2}=p/q$, wh... | 169 | 142 | 3 |
math | 6. Convex quadrilateral $A B C D$ satisfies $\angle C A B=\angle A D B=30^{\circ}, \angle A B D=77^{\circ}, B C=C D$, and $\angle B C D=n^{\circ}$ for some positive integer $n$. Compute $n$. | 68 | 69 | 2 |
math | 6. (USS) ${ }^{1 \mathrm{MO4}}$ Let $A$ be the sum of the digits of the number $16^{16}$ and $B$ the sum of the digits of the number $A$. Find the sum of the digits of the number $B$ without calculating $16^{16}$. | 7 | 74 | 1 |
math | Four, (Full marks 20 points, (1) 10 points, (2) 10 points) A total of 240 students participating in a large group performance stand in a row facing the coach, and report numbers in sequence from left to right as $1,2,3,4,5, \cdots$. The coach asks all students to remember their reported numbers and perform the followin... | 118 | 181 | 3 |
math | ## 188. Math Puzzle $1 / 81$
A service car (passenger car) was driven $1992 \mathrm{~km}$ in one month; of this, $379 \mathrm{~km}$ was city driving. The fuel consumption limit is eleven liters per $100 \mathrm{~km}$ for city driving and ten liters per $100 \mathrm{~km}$ for long-distance driving.
At the beginning of... | 22.59 | 166 | 5 |
math | 3. Let $x, y, z \in [0,1]$, then the maximum value of $M=\sqrt{|x-y|}+\sqrt{|y-z|}+\sqrt{|z-x|}$ is $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | \sqrt{2}+1 | 75 | 7 |
math | Determine the largest positive integer $n$ for which $7^{n}$ is a divisor of the integer $\frac{200 !}{90 ! 30 !}$.
(Note: If $n$ is a positive integer, the symbol $n$ ! (read " $n$ factorial") is used to represent the product of the integers from 1 to $n$. That is, $n !=n(n-1)(n-2) \cdots(3)(2)(1)$. For example, $5 !... | 15 | 131 | 2 |
math | 1. It is known that $\operatorname{tg}(\alpha+2 \gamma)+\operatorname{tg} \alpha+\frac{5}{2} \operatorname{tg}(2 \gamma)=0, \operatorname{tg} \gamma=-\frac{1}{2}$. Find $\operatorname{ctg} \alpha$. | \frac{1}{3} | 74 | 7 |
math | For which $k \in \mathbb{N}$ do there exist non-negative integers $a$ and $b$ such that
$$
2^{a} 3^{b}=k(k+1) ?
$$ | 1,2,3,8 | 46 | 7 |
math | 7.072. $10^{\frac{2}{x}}+25^{\frac{1}{x}}=4.25 \cdot 50^{\frac{1}{x}}$. | x_{1}=-\frac{1}{2};x_{2}=\frac{1}{2} | 47 | 23 |
math | In the class, there are fewer than 30 people. The probability that a randomly chosen girl is an excellent student is $3 / 13$, and the probability that a randomly chosen boy is an excellent student is $4 / 11$. How many excellent students are there in the class? | 7 | 62 | 1 |
math | 31. Given that $\alpha$ is an acute angle satisfying
$$
\sqrt{369-360 \cos \alpha}+\sqrt{544-480 \sin \alpha}-25=0
$$
find the value of $40 \tan \alpha$. | 30 | 65 | 2 |
math | When Meena turned 16 years old, her parents gave her a cake with $n$ candles, where $n$ has exactly 16 different positive integer divisors. What is the smallest possible value of $n$? | 120 | 49 | 3 |
math | G7.3 and G7.4 $\quad A, B$ are positive integers less than 10 such that $21 A 104 \times 11=2 B 8016 \times 9$.
G7.3 Find $A$.
G7.4 Find $B$. | A=1,B=5 | 69 | 6 |
math | 7. The vertices of a triangle are $P(-8$, 5), $Q(-15,-19), R(1,-7), \angle P$ bisector can be expressed as $a x+2 y+c=0$, find $a+c$,
| 89 | 57 | 2 |
math | 4. Given the function $f(x)=a x+b, g(x)=\left\{\begin{array}{l}f(x), 0 \leq x \leq a \\ f(f(x)), x>a\end{array}\right.$, if for any positive number $a$, the function $g(x)$ is strictly monotonically increasing on $(0,+\infty)$, then the minimum value of the constant $b$ is | \frac{1}{4} | 94 | 7 |
math | 7. Mental arithmetic. Square 85 in your mind. | 7225 | 13 | 4 |
math | 10. Let $[x]$ denote the greatest integer not exceeding the real number $x$. Then the value of $\sum_{k=1}^{2008}\left[\frac{2008 k}{2009}\right]$ is $\qquad$ . | 2015028 | 60 | 7 |
math | 19. [10] Let $A B C D$ be a regular tetrahedron, and let $O$ be the centroid of triangle $B C D$. Consider the point $P$ on $A O$ such that $P$ minimizes $P A+2(P B+P C+P D)$. Find $\sin \angle P B O$. | \frac{1}{6} | 79 | 7 |
math | 4. Place the natural numbers $1,2, \cdots, 2 n$ randomly on a circle. It is found that among all sets of three consecutive numbers, there are $a$ sets where all three numbers are odd, $b$ sets where exactly two numbers are odd, $c$ sets where only one number is odd, and $d$ sets where all three numbers are even. If $a ... | -3 | 110 | 2 |
math | Example 1.13. Find $\int \sin 7 x \cos 3 x d x$. | -\frac{1}{20}\cos10x-\frac{1}{8}\cos4x+C | 23 | 23 |
math | The triangle with side lengths $3, 5$, and $k$ has area $6$ for two distinct values of $k$: $x$ and $y$. Compute $ |x^2 -y^2|$. | 36 | 47 | 2 |
math | Problem 4.7. At the festival of namesakes, 45 Alexanders, 122 Borises, 27 Vasily, and several Gennady arrived. At the beginning of the festival, all of them stood in a row so that no two people with the same name stood next to each other. What is the minimum number of Gennady that could have arrived at the festival? | 49 | 85 | 2 |
math | 11.016. The base of a quadrilateral pyramid is a rectangle with a diagonal equal to $b$, and the angle between the diagonals is $60^{\circ}$. Each of the lateral edges forms an angle of $45^{\circ}$ with the base plane. Find the volume of the pyramid. | \frac{b^{3}\sqrt{3}}{24} | 69 | 15 |
math | 5. (10 points) The distance from home to work is $s=3 \kappa$ km. At the moment Ivan left work, his favorite dog ran out of the house towards him. They met at a distance of a quarter of the entire path from work. The dog instantly turned around and ran back home. Upon reaching home, he instantly turned around again and... | 9\kappa | 111 | 4 |
math | 5.103 Suppose there are 100 mutually hostile countries on Mars. To maintain peace, it is decided to form several alliances, with each alliance including at most 50 countries, and any two countries must belong to at least one alliance. Try to answer the following questions:
(1) What is the minimum number of alliances ne... | 6 | 106 | 1 |
math | ## Task B-3.4.
What is the area of a triangle whose sum of the squares of the side lengths is 2016, and the sum of the cotangents of the angles is 18? | 28 | 47 | 2 |
math | 15.15 A paper punch can be placed at any point in the plane, and when it operates, it can punch out points at an irrational distance from it. What is the minimum number of paper punches needed to punch out all points in the plane?
(51st Putnam Mathematical Competition, 1990) | 3 | 68 | 1 |
math | Question 30, Let the function $\mathrm{f}(\mathrm{x})=\mathrm{x}-\ln (\mathrm{ax}+2 \mathrm{a}+1)+2$, if for any $\mathrm{x} \geq-2$, it always holds that $\mathrm{f}(\mathrm{x}) \geq 0$, try to find the range of the real number $\mathrm{a}$. | \in[0,1] | 87 | 7 |
math | 5. Nine pairwise noncongruent circles are drawn in the plane such that any two circles intersect twice. For each pair of circles, we draw the line through these two points, for a total of $\binom{9}{2}=36$ lines. Assume that all 36 lines drawn are distinct. What is the maximum possible number of points which lie on at ... | 462 | 84 | 3 |
math | ## Zadatak A-1.3.
Dan je trokut $A B C$ u kojem je $\varangle B A C=45^{\circ},|A B|=4,|A C|=3 \sqrt{2}$. Neka su $\overline{A D}$ i $\overline{B E}$ visine tog trokuta. Okomica na $\overline{A B}$ kroz točku $E$ siječe dužinu $\overline{A D}$ u točki $P$.
Odredi $|E P|$.
| \frac{4}{3} | 128 | 7 |
math | Find all real numbers a such that all solutions to the quadratic equation $ x^2 \minus{} ax \plus{} a \equal{} 0$ are integers. | a = 0, 4 | 34 | 8 |
math | 5. How many natural numbers less than 100000 are divisible by 4 and in whose decimal representation only the digits $0,1,2,3$ and 5 participate? (Digits can repeat and not all of them need to appear in the representation of such a number.)
## Second Grade - B category | 624 | 68 | 3 |
math | An arithmetic sequence with common difference $r$ is a sequence of numbers $s_{0}, s_{1}, s_{2}, \ldots$ that satisfies for all indices $k, s_{k+1}=s_{k}+r$. Determine an expression for $s_{n}$ in terms of $s_{0}, r$, and $n$. | s_{n}=s_{0}+nr | 75 | 10 |
math | 8. In $\triangle A B C$, $a, b, c$ are the sides opposite to angles $A, B, C$ respectively. Given $a+c=2 b, A-C=\frac{\pi}{3}, \sin B=$ $\qquad$ . | \frac{\sqrt{39}}{8} | 57 | 11 |
math | 2.236. $\frac{\sqrt{z^{2}-1}}{\sqrt{z^{2}-1}-z} ; \quad z=\frac{1}{2}\left(\sqrt{m}+\frac{1}{\sqrt{m}}\right)$ | \frac{-1}{2},if\in(0;1);\frac{1-}{2},if\in[1;\infty) | 58 | 31 |
math | Example 11 Try to find all positive integers $a$ such that the equation $a x^{2}+2(2 a-1) x+4(a-3)=0$ has at least one integer solution. | a=1,3,6,10 | 47 | 10 |
math | 4. Given the function $f(x)=\frac{2 x^{3}-6 x^{2}+13 x+10}{2 x^{2}-9 x}$
Determine all positive integers $x$ for which $f(x)$ is an integer. | 5,10 | 57 | 4 |
math | 7. (10 points) On the board, 31 ones are written. Every minute, Karlson erases two arbitrary numbers and writes their sum on the board, and then eats a number of candies equal to the product of the two erased numbers. What is the maximum number of candies he could have eaten in 31 minutes? | 465 | 70 | 3 |
math | $7.9 \quad 3 \cdot 5^{2x-1}-2 \cdot 5^{x-1}=0.2$.
Translate the above text into English, keeping the original text's line breaks and format, and output the translation result directly.
$7.9 \quad 3 \cdot 5^{2x-1}-2 \cdot 5^{x-1}=0.2$. | 0 | 90 | 1 |
math | Example 10 Given $\left(x^{2}-x+1\right)^{6}=a_{12} x^{12}+$
$$
\begin{array}{l}
a_{11} x^{11}+\cdots+a_{2} x^{2}+a_{1} x+a_{0} \text {. Then } a_{12}+a_{10} \\
+\cdots+a_{2}+a_{0}=\ldots
\end{array}
$$
(Fifth National Partial Provinces Junior High School Mathematics Correspondence Competition) | 365 | 126 | 3 |
math | 4. Solve the system $\left\{\begin{array}{l}x^{2}+y^{2} \leq 2, \\ 81 x^{4}-18 x^{2} y^{2}+y^{4}-360 x^{2}-40 y^{2}+400=0 .\end{array}\right.$ | (-\frac{3}{\sqrt{5}};\frac{1}{\sqrt{5}}),(-\frac{3}{\sqrt{5}};-\frac{1}{\sqrt{5}}),(\frac{3}{\sqrt{5}};-\frac{1}{\sqrt{5}}),(\frac{3}{\sqrt{5}};\frac{1}{} | 80 | 83 |
math | 4. When $n$ is a positive integer, the function $f$ satisfies
$$
f(n+3)=\frac{f(n)-1}{f(n)+1}, f(1) \neq 0 \text {, and } f(1) \neq \pm 1 \text {. }
$$
Then $f(1) f(2023)=$ $\qquad$ | -1 | 88 | 2 |
math | 3. $[\mathbf{3}]$ How many diagonals does a regular undecagon (11-sided polygon) have? | 44 | 28 | 2 |
math | 9.1.1. (12 points) Find the minimum value of the function $f(x)=\operatorname{tg}^{2} x+3 \operatorname{tg} x+6 \operatorname{ctg} x+4 \operatorname{ctg}^{2} x-1$ on the interval $\left(0 ; \frac{\pi}{2}\right)$. | 3+6\sqrt{2}\approx11.49 | 85 | 14 |
math | Joãozinho wrote the numbers from 1 to 100000 on the board, then started replacing each number with the sum of its digits and repeated this process until he obtained a list of 100000 single-digit numbers. For example, starting with the number 7234 we get $7+2+3+4=16$ and $1+6=7$.
(a) What number is in the place of the ... | 8,11111,1 | 132 | 9 |
math | Problem 8.2. In a volleyball tournament for the Euro- African cup the European teams are 9 more than the African teams. Every two teams met exactly once and the European teams gained 9 times more points than the African teams (the winner takes 1 point and the loser takes 0 point). What are the maximum possible points g... | 11 | 79 | 2 |
math | In the coordinate plane, the line passing through points $(2023,0)$ and $(-2021,2024)$ also passes through $(1,c)$ for a constant $c$. Find $c$.
[i]Proposed by Andy Xu[/i] | 1012 | 59 | 4 |
math | 8. We will call two numbers almost equal to each other if they are equal to each other or differ from each other by no more than one. A rectangular grid with sides of natural numbers a and b is such that it is impossible to cut out a rectangle along the grid lines, the area of which is almost equal to half the area of ... | 4 | 101 | 1 |
math | ## Condition of the problem
Find the derivative.
$$
y=\operatorname{ctg} \sqrt[3]{5}-\frac{1}{8} \cdot \frac{\cos ^{2} 4 x}{\sin 8 x}
$$ | \frac{1}{4\sin^{2}4x} | 55 | 14 |
math | 2. Determine all pairs $(m, p)$ of natural numbers such that $p$ is a prime number and
$$
2^{m} p^{2}+27
$$
is the cube of a natural number. | (1,7) | 48 | 5 |
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