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 | Find the sum of all positive integers $n$ such that when $1^3+2^3+3^3+\cdots +n^3$ is divided by $n+5$, the remainder is $17$. | 239 | 48 | 3 |
math | Three, (Full marks 12 points) At the foot of the mountain is a pond, the scene: a steady flow (i.e., the same amount of water flows into the pond from the river per unit time) continuously flows into the pond. The pond contains a certain depth of water. If one Type A water pump is used, it will take exactly 1 hour to p... | 12 | 140 | 2 |
math | Find the minimum number of colors necessary to color the integers from $1$ to $2007$ such that if distinct integers $a$, $b$, and $c$ are the same color, then $a \nmid b$ or $b \nmid c$. | 6 | 58 | 3 |
math | Steve needed to address a letter to $2743$ Becker Road. He remembered the digits of the address, but he forgot the correct order of the digits, so he wrote them down in random order. The probability that Steve got exactly two of the four digits in their correct positions is $\tfrac m n$ where $m$ and $n$ are relatively... | 5 | 85 | 1 |
math | 6.202. $\left\{\begin{array}{l}\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=3, \\ \frac{1}{x y}+\frac{1}{y z}+\frac{1}{z x}=3, \\ \frac{1}{x y z}=1 .\end{array}\right.$ | (1;1;1) | 84 | 7 |
math | Example 4 Find the equation of the line passing through the point $P(6,7)$ and tangent to the hyperbola $\frac{x^{2}}{9}-\frac{y^{2}}{16}=1$. | 5x-3y-9=013x-9y-15=0 | 49 | 20 |
math | Question 53, Let the area of $\triangle A B C$ be $1, \angle A$ be opposite to the side length $a$, try to find the minimum value of $\mathrm{a}^{2}+\frac{1}{\sin \mathrm{A}}$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | 3 | 86 | 1 |
math |
Opgave 2. Vind alle functies $f: \mathbb{R} \rightarrow \mathbb{R}$ waarvoor geldt dat
$$
f(x)=\max _{y \in \mathbb{R}}(2 x y-f(y))
$$
voor alle $x \in \mathbb{R}$.
(In het algemeen betekent de uitdrukking $a=\max _{s \in S} g(s)$ : er geldt $a \geq g(s)$ voor alle $s \in S$ en bovendien is er een $s \in S$ waarvoo... | f(x)=x^{2} | 145 | 7 |
math | 6. We use $S_{k}$ to denote an arithmetic sequence with the first term $k$ and a common difference of $k^{2}$, for example, $S_{3}$ is $3, 12, 21, \cdots$. If 306 is a term in $S_{k}$, the sum of all $k$ that satisfy this condition is $\qquad$. | 326 | 87 | 3 |
math | Task 1. Write the number 100000 as a product of two numbers whose representations contain no digit 0. | 32\cdot3125 | 28 | 8 |
math | Borisov L
The sage S. was told the sum of three natural numbers, while the sage P. was told their product.
- If I had known, - said S., - that your number is greater than mine, I would have immediately named the three numbers.
- My number is less than yours, - replied P., - and the numbers are ..., ... and ... .
Wha... | 1,1,4 | 85 | 5 |
math | Shapovalov A.V.
55 boxers participated in a tournament with a "loser leaves" system. The fights proceeded sequentially. It is known that in each match, the number of previous victories of the participants differed by no more than 1. What is the maximum number of fights the tournament winner could have conducted? | 8 | 66 | 1 |
math | Square $ABCD$ has center $O$, $AB=900$, $E$ and $F$ are on $AB$ with $AE<BF$ and $E$ between $A$ and $F$, $m\angle EOF =45^\circ$, and $EF=400$. Given that $BF=p+q\sqrt{r}$, wherer $p,q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p+q+r$. | 307 | 113 | 3 |
math | 1. Solve the equation $\frac{1}{x+y+z}=\overline{0, x y z}$ (here $x, y$ and $z-$ are some digits). | (1;2;5) | 39 | 7 |
math | 6. Find a two-digit number if it is known that the sum of its digits is 13, and the difference between the sought number and the number written with the same digits but in reverse order is a two-digit number with 7 units. | 85 | 51 | 2 |
math | In a triangle $ABC$ such that $\angle{BAC}=90^\circ, \ |\overrightarrow{AB}|=1,\ |\overrightarrow{AC}|=\sqrt{3}$, a point $P$ inside the $\triangle{ABC}$ satisfies $\frac{\overrightarrow{PA}}{|\overrightarrow{PA}|}+\frac{\overrightarrow{PB}}{|\overrightarrow{PB}|}+\frac{\overrightarrow{PC}}{|\overrightarrow{PC}|}=\ove... | |\overrightarrow{PA}| = \frac{\sqrt{7}}{7}, \ |\overrightarrow{PB}| = \frac{2\sqrt{7}}{7}, \ |\overrightarrow{PC}| = \frac{4\sqrt{7}}{7} | 156 | 58 |
math | 4A. Silvia chose three natural numbers $a, b$, and $c$. Petar found that the value of $a+\frac{b}{c}$ is 101, Lina found that the value of $\frac{a}{c}+b$ is 68, and Maria found that the value of $\frac{a+b}{c}$ is $k$. Determine $a, b, c$, and $k$. | =96,b=60,=12,k=13 | 93 | 15 |
math | 13. Find the range of the function $f(x)=|\sin x|+|\cos x|$.
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly.
However, since the request is to translate the given text, here is the translation:
13. Find the range of the functi... | [1,\sqrt{2}] | 89 | 7 |
math | 11. Find the last three digits of $2016^{3}+2017^{3}+2018^{3}+\ldots+3014^{3}$. | 625 | 45 | 3 |
math | Example 6. A person wrote 6 letters to 6 different people and prepared 6 envelopes with the recipients' addresses written on them. How many ways are there to place the letters into the envelopes so that no letter matches the recipient on the envelope? (Polish Competition Question) | 265 | 58 | 3 |
math | 1. If positive numbers $m, n$ satisfy $m+4 \sqrt{m n}-$ $2 \sqrt{m}-4 \sqrt{n}+4 n=3$, then $\frac{\sqrt{m}+2 \sqrt{n}}{\sqrt{m}+2 \sqrt{n}+3}=$ | \frac{1}{2} | 69 | 7 |
math | ## 3. How many are there?
How many three-digit numbers are there for which the sum of two digits is twice the third?
## Result: $\quad 121$ | 121 | 38 | 3 |
math | For each integer $n\geq 3$, find the least natural number $f(n)$ having the property
$\star$ For every $A \subset \{1, 2, \ldots, n\}$ with $f(n)$ elements, there exist elements $x, y, z \in A$ that are pairwise coprime. | \left\lfloor \frac{n}{2} \right\rfloor + \left\lfloor \frac{n}{3} \right\rfloor - \left\lfloor \frac{n}{6} \right\rfloor + 1 | 74 | 53 |
math | $\mathrm{Na}$ two families with a total of six children, aged $2, 3, 4, 5, 6$ and 8 years, met. The sum of the ages of the children from one family was the same as the sum of the ages of the children from the other family.
How old could the children from each family have been? Determine all possibilities.
(E. Novotná... | )8 | 87 | 2 |
math | 3 Let $n$ be a given positive integer, $n \geqslant 3$, and for $n$ given real numbers $a_{1}, a_{2}, \cdots, a_{n}$, let $m$ be the minimum value of $\left|a_{i}-a_{j}\right|(1 \leqslant i<j \leqslant n)$. Find the maximum value of $m$ under the condition $a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}=1$. | \sqrt{\frac{12}{(n-1) n(n+1)}} | 124 | 18 |
math | Example $8 n^{2}(n \geqslant 4)$ positive numbers are arranged in several rows and columns:
\begin{tabular}{llllll}
$a_{11}$ & $a_{12}$ & $a_{13}$ & $a_{14}$ & $\ldots$ & $a_{1 n}$, \\
$a_{21}$ & $a_{22}$ & $a_{23}$ & $a_{24}$ & $\ldots$ & $a_{2 n}$, \\
$a_{31}$ & $a_{32}$ & $a_{33}$ & $a_{34}$ & $\ldots$ & $a_{3 n}$, ... | 2-\frac{1}{2^{n-1}}-\frac{n}{2^{n}} | 328 | 20 |
math | 6. Let the sequence of positive integers $a_{1} 、 a_{2} 、 a_{3} 、 a_{4}$ be a geometric sequence, with the common ratio $r$ not being an integer and $r>1$. The smallest value that $a_{4}$ can take in such a sequence is $\qquad$ . | 27 | 74 | 2 |
math | 12. Ivan the Tsarevich's Arrows (from 8th grade. 3 points). Ivan the Tsarevich is learning to shoot a bow. He put 14 arrows in his quiver and went to shoot at pine cones in the forest. He knocks down a pine cone with a probability of 0.1, and for each pine cone he knocks down, the Frog-Princess gives him 3 more arrows.... | 20 | 122 | 2 |
math | 1. Let $n \geqslant 4, x_{1}, x_{2}, \cdots, x_{n}$ be $n$ non-negative real numbers, satisfying $x_{1}+x_{2}+\cdots+x_{n}=1$. Find the maximum value of
$$
x_{1} x_{2} x_{3}+x_{2} x_{3} x_{4}+\cdots+x_{n} x_{1} x_{2}
$$
(Han Jingjun, problem contributor) | (\sum_{i=1}^{n}x_{i}x_{i+1}x_{i+2})_{\max}={\begin{pmatrix}\frac{1}{16},&n=4;\\\frac{1}{25},&n=5;\\\frac{1}{27},&n\geqslant60\end{pmatrix}.} | 116 | 86 |
math | For which $k$ the number $N = 101 ... 0101$ with $k$ ones is a prime? | k = 2 | 30 | 5 |
math | Find all triples $(a, b, p)$ of positive integers, where $p$ is a prime number, such that $a^p - b^p = 2013$. | (337, 334, 2) | 40 | 14 |
math | Example 6 Let $S$ be the set of all real numbers greater than -1, determine all functions $f: S \rightarrow S$, such that the following two conditions are satisfied:
(i) For all $x$ and $y$ in $S$, we have
$$
f(x+f(y)+x f(y))=y+f(x)+y f(x) \text {; }
$$
(ii) In each of the intervals $(-1,0)$ and $(0,+\infty)$, $\frac{f... | f(x)=-\frac{x}{1+x} | 125 | 11 |
math | 6. In the Cartesian coordinate system, the "rectilinear distance" between points $P\left(x_{1}, y_{1}\right)$ and $Q\left(x_{2}, y_{2}\right)$ is defined as
$$
d(P, Q)=\left|x_{1}-x_{2}\right|+\left|y_{1}-y_{2}\right| \text {. }
$$
If point $C(x, y)$ has equal rectilinear distances to $A(1,3)$ and $B(6,9)$, where the ... | 5(\sqrt{2}+1) | 173 | 9 |
math | 8.323. $\frac{5 \sin x-5 \tan x}{\sin x+\tan x}+4(1-\cos x)=0$. | \\arccos\frac{1}{4}+2\pin,n\inZ | 36 | 19 |
math | 3. What is the greatest value that the sum $S_{n}$ of the first $n$ terms of an arithmetic progression can take, given that the sum $S_{3}=327$ and the sum $S_{57}=57$? | 1653 | 55 | 4 |
math | G7.3-G7.4 Chicken eggs cost $\$ 0.50$ each, duck eggs cost $\$ 0.60$ each and goose eggs cost $\$ 0.90$ each. A man sold $x$ chicken eggs, $y$ duck eggs, $z$ goose eggs and received $\$ 60$. If $x, y, z$ are all positive numbers with $x+y+z=100$ and two of the values $x, y, z$ are equal,
G7.3 find $x$.
G7.4 find $y$. | 60 | 129 | 2 |
math | 1. Let $p$ and $q$ be natural numbers and $\left(a_{n}\right)_{n=1}^{\infty}$ be an arithmetic progression for which $a_{p}$ is equal to $q$, and $a_{q}$ is equal to $p$.
Calculate the value of $a_{n}$. | a_{n}=p+q-(n-1) | 72 | 12 |
math | Five football teams held a tournament - each team played against each other once. 3 points were awarded for a win, 1 point for a draw, and no points for a loss. Four teams scored 1, 2, 5, and 7 points respectively. How many points did the fifth team score?
# | 12 | 66 | 2 |
math | Problem 4. Given a quadratic trinomial $g(x)=x^{2}+a x+b$, which has exactly one root. Find the coefficients $a$ and $b$, if it is known that the polynomial $g\left(x^{5}+2 x-1\right)+g\left(x^{5}+3 x+1\right)$ also has exactly one root. | =74,b=1369 | 83 | 9 |
math | 23. How many ordered pairs of integers $(m, n)$ where $0<m<n<2008$ satisfy the equation $2008^{2}+m^{2}=2007^{2}+n^{2} ?$ | 3 | 55 | 1 |
math | 7. Students have three types of number cards: $1$, $2$, and $3$, with many cards of each type. The teacher asks each student to take out two or three number cards to form a two-digit or three-digit number. If at least three students form the same number, then these students have at least people. | 73 | 69 | 2 |
math | In a group of $n > 20$ people, there are some (at least one, and possibly all) pairs of people that know each other. Knowing is symmetric; if Alice knows Blaine, then Blaine also knows Alice. For some values of $n$ and $k,$ this group has a peculiar property: If any $20$ people are removed from the group, the number of... | n \ge 381 | 157 | 7 |
math | ## Task B-2.5.
In how many ways can three numbers be selected from the set $\{1,2,3, \ldots, 12\}$ such that their sum is divisible by 3? | 76 | 47 | 2 |
math | \section*{Problem 3 - V11133}
Given are two fixed points \(A\) and \(B\) with the distance \(e\).
a) Where do all points \(F\) lie for which the squares of their distances from \(A\) and \(B\) have the fixed sum \(s\)?
b) Are there such points for any choice of \(e\) and \(s\)? | x^{2}+y^{2}=\frac{}{2}-\frac{e^{2}}{4} | 85 | 25 |
math | ## Task 4.
Determine all pairs $(k, n)$ of natural numbers that satisfy the equation
$$
1!+2!+\ldots+k!=1+2+\ldots+n
$$
(We denote by $b!$ the product of the first $b$ natural numbers. For example, $4!=1 \cdot 2 \cdot 3 \cdot 4=24$.) | (1,1),(2,2),(5,17) | 87 | 14 |
math | A2. Jakob is reading a book with 630 pages. On the first day, he read a third of the book. The sum of the numbers that marked the pages he read on the second day was 4410. How many pages does Jakob have left to finish the book? (The book starts with page number 1.)
(A) 210
(B) 211
(C) 230
(D) 390
(E) 400 | 400 | 108 | 3 |
math | ## Task Condition
Compose the equation of the normal to the given curve at the point with abscissa $x_{0}$.
$$
y=\frac{x^{3}+2}{x^{3}-2}, x_{0}=2
$$ | \frac{3}{4}\cdotx+\frac{1}{6} | 52 | 16 |
math | 2.228. $\sqrt{\left(y^{2}+\frac{4}{y^{2}}\right)^{2}-8 \cdot\left(y+\frac{2}{y}\right)^{2}+48}$. | (y-\frac{2}{y})^{2} | 52 | 11 |
math | Example 10 Find the integer solution of $1215 x \equiv 560(\bmod 2755)$. | x \equiv 200,751,1302,1853,2404 \quad(\bmod 2755) | 31 | 37 |
math | 13.109. Two cyclists set off simultaneously towards each other from two places, the distance between which is 270 km. The second cyclist travels 1.5 km less per hour than the first, and meets him after as many hours as the first cyclist travels in kilometers per hour. Determine the speed of each cyclist. | 12 | 70 | 2 |
math | Example 1. Find the volume of the body bounded by the surfaces
$$
x=17 \sqrt{2 y}, \quad x=2 \sqrt{2 y}, \quad z=1 / 2-y, \quad z=0
$$ | 1 | 55 | 1 |
math | 7. Let $p$ be a prime number, and $q=4^{p}+p^{4}+4$ is also a prime number. Find the value of $p+q$.
The text above is translated into English, keeping the original text's line breaks and format. | 152 | 61 | 3 |
math | 8 Let $p$ be a given positive even number, and the set $A_{p}=\left\{x \mid 2^{p}<x<2^{p+1}, x=\right.$ $3 m, m \in \mathbf{N}\}$ . The sum of all elements in $A_{p}$ is $\qquad$ . | 2^{2p-1}-2^{p-1} | 76 | 13 |
math | ## Task 2 - 260832
Determine all pairs $(p ; q)$ of prime numbers that satisfy the following conditions (1), (2), (3)!
(1) The difference $q-p$ is greater than 0 and less than 10.
(2) The sum $p+q$ is the square of a natural number $n$.
(3) If you add the number $n$ to the sum of $p$ and $q$, you get 42. | (17,19) | 110 | 7 |
math | ## Task B-2.4.
Andro and Leo are guessing a four-digit number that the other has thought of. In doing so, they must uncover some information about their number. Andro's number has all different digits, and the first and last digits are odd numbers. Leo's number does not have to have all different digits, but all of it... | 881 | 107 | 3 |
math | A natural number was sequentially multiplied by each of its digits. The result was 1995. Find the original number.
# | 57 | 27 | 2 |
math | 1. [6] Let $m>1$ be a fixed positive integer. For a nonempty string of base-ten digits $S$, let $c(S)$ be the number of ways to split $S$ into contiguous nonempty strings of digits such that the base-ten number represented by each string is divisible by $m$. These strings are allowed to have leading zeroes.
In terms of... | 02^{n}forallnonnegativeintegern | 146 | 11 |
math | The distance from a fixed point $P$ on the plane to two vertices $A, B$ of an equilateral triangle $A B C$ are $A P=2 ; B P=3$. Determine the maximum value that the segment $P C$ can have.
# | 5 | 57 | 1 |
math | 3. Given the polynomial
$$
\begin{aligned}
a_{0}+ & \left(a_{1}+4\right) x+ \\
& \left(a_{2}-10\right) x^{2}+\left(a_{3}+6\right) x^{3}+\left(a_{4}-1\right) x^{4}+ \\
& \left(a_{5}-1\right) x^{5}+a_{6} x^{6}+\cdots+a_{2 \alpha 5} x^{2 \omega 5}
\end{aligned}
$$
can be divided by $x^{2}+3 x-2$, and $\alpha^{2}+3 \alpha... | 0 | 214 | 1 |
math | \section*{Task 3 - 320923}
When refueling an oldtimer with a two-stroke engine that requires a fuel-oil mixture of \(1: 50\), 7 liters of fuel without oil were mistakenly added first.
How many liters of the still available mixture with a ratio of \(1: 33\) need to be added now to achieve the correct mixture ratio of ... | 14 | 108 | 2 |
math | Find the functions $f: \mathbb{R}_{+}^{*} \rightarrow \mathbb{R}_{+}^{*}$ satisfying, for all $x, y>0$,
$$
f(2 x f(3 y))+f\left(27 y^{3} f(2 x)\right)=6 x y+54 x y^{3}
$$
Solved by Gä̈tan Dautzenberg and Georges Tézé | f(x)=x | 100 | 4 |
math | What is the largest possible value of the expression $$gcd \,\,\, (n^2 + 3, (n + 1)^2 + 3 )$$ for naturals $n$?
[hide]original wording]Kāda ir izteiksmes LKD (n2 + 3, (n + 1)2 + 3) lielākā iespējamā vērtība naturāliem n? [/hide] | 13 | 102 | 2 |
math | 9. (16 points) Let $x \in(0,1]$. Find the range of the function $y=$ $\frac{3 x^{6}+15 x^{2}+2}{2 x^{6}+15 x^{4}+3}$. | \left(\frac{2}{3}, \frac{3}{2}\right] | 61 | 18 |
math | 5. (15 points) A massive vertical plate is fixed on a car moving at a speed of $4 \mathrm{M} / \mathrm{c}$. A ball is flying towards it at a speed of $5 \mathrm{M} / \mathrm{c}$ relative to the Earth. Determine the speed of the ball relative to the Earth after a perfectly elastic normal collision. | 13\mathrm{M}/\mathrm{} | 80 | 10 |
math | 2. Solve the equation
$$
\log _{3 x+4}(2 x+1)^{2}+\log _{2 x+1}\left(6 x^{2}+11 x+4\right)=4
$$ | \frac{3}{4} | 53 | 7 |
math |
6. Find all ordered triples $(x, y, z)$ of mutually distinct real numbers which satisfy the set equation
$$
\{x, y, z\}=\left\{\frac{x-y}{y-z}, \frac{y-z}{z-x}, \frac{z-x}{x-y}\right\} .
$$
| (,-\frac{1}{1+},-\frac{1+}{}),(1,-2,-\frac{1}{2}),(-\frac{1}{2},1,-2),(-2,-\frac{1}{2},1) | 71 | 53 |
math | A sequence of numbers $1, 4, 7, 10, \cdots, 697, 700$ follows the rule: the first number is 1, and each subsequent number is 3 more than the previous one, up to 700. If all these numbers are multiplied together, find the number of trailing zeros in the resulting product (for example, the number of trailing zeros in 120... | 60 | 103 | 2 |
math | Condition of the problem
Find the derivative.
$y=\ln ^{2}(x+\cos x)$ | \frac{1-\sinx}{x+\cosx}\cdot2\ln(x+\cosx) | 22 | 22 |
math | 3. Solve the system $\left\{\begin{array}{l}2 x+y+8 \leq 0, \\ x^{4}+2 x^{2} y^{2}+y^{4}+9-10 x^{2}-10 y^{2}=8 x y .\end{array}\right.$ | (-3,-2) | 72 | 5 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty} \sqrt{n+2}(\sqrt{n+3}-\sqrt{n-4})
$$ | \frac{7}{2} | 44 | 7 |
math | 7、A river flows at a uniform speed, with docks A and B located upstream and downstream, respectively, 200 kilometers apart. Boats A and B depart from docks A and B simultaneously and head towards each other. After meeting, they continue to their respective destinations, immediately turn around, and meet again on their ... | 14 | 148 | 2 |
math | Example 7 Given $f: \mathbf{R}_{+} \rightarrow \mathbf{R}_{+}$, and satisfies the conditions:
(1) For any $x, y \in \mathbf{R}_{+}$, $f(x f(y))=y f(x)$;
(2) As $x \rightarrow+\infty$, $f(x) \rightarrow 0$.
Try to find the function $f(x)$. | f(x)=\frac{1}{x} | 95 | 10 |
math | Example 4 The increasing sequence $1,3,4,9,10,12,13, \cdots$ consists of some positive integers, which are either powers of 3 or sums of several different powers of 3. Find the 100th term of this sequence. | 981 | 63 | 3 |
math | 9.112. For what values of $a$ is the inequality $\frac{a x}{x^{2}+4}<1$ true? If $x>0$, then we have $a x < x^{2} + 4$. This inequality holds for any values of $x \in R$ if $D=a^{2}-36<0$, $a^{2}<36$, $-6<a<6$.
Answer: $a \in(-6 ; 6)$.
9.113. Find the domain of the function $f$, if $f(x)=\sqrt{\log _{0.5}\left(x^{2}-... | x\in[-5;-3)\cup(3;5] | 150 | 14 |
math | In Mr. Goat's garden, several cherry trees were in bloom. On each cherry tree, three sparrows were sitting, and one more was sitting on the fence. Mr. Goat's dog scared them away and the sparrows flew off. After a while, they all returned and settled on the cherry trees. The cherry tree under which the dog was sleeping... | 5 | 107 | 1 |
math | 178 In $\triangle A B C$, the sides opposite to angles $A, B, C$ are $a, b, c$ respectively. If $c-a$ equals the height $h$ from $A C$, then $\left(\cos \frac{A}{2}-\sin \frac{A}{2}\right) \cdot\left(\sin \frac{C}{2}+\cos \frac{C}{2}\right)=$ $\qquad$ . | 1 | 101 | 1 |
math |
NT 2. Find all positive integers $a, b, c$ and $p$, where $p$ is a prime number, such that
$$
73 p^{2}+6=9 a^{2}+17 b^{2}+17 c^{2}
$$
| (,b,,p)\in{(1,1,4,2),(1,4,1,2)} | 64 | 24 |
math | 1. Given that for every pair of real numbers $x, y$, the function $f$ satisfies $f(x)+f(y)=f(x+y)-x y-1$. If $f(1)=1$, then the number of integers $n$ that satisfy $f(n)=n$ is $\qquad$ . | 2 | 67 | 1 |
math | 4. Let $f(x)$ be a polynomial function of degree 2016 whose 2016 zeroes have a sum of $S$. Find the sum of the 2016 zeroes of $f(2 x-3)$ in terms of $S$. | \frac{1}{2}S+3024 | 58 | 13 |
math | 1. A Pythagorean triangle is a right-angled triangle where all three sides are integers. The most famous example is the triangle with legs 3 and 4 and hypotenuse 5.
Determine all Pythagorean triangles for which the area is equal to twice the perimeter. | (9,40,41),(10,24,26),(12,16,20) | 59 | 27 |
math | # Problem 2. (Folklore)
In a box, there are balls of two colors: blue and red (both colors are present). It is known that there are more blue balls, and two balls of the same color can be drawn with the same probability as two balls of different colors. What can the difference between the number of blue and red balls ... | Anynaturalgreaterthan1 | 85 | 6 |
math | For which $ p$ prime numbers, there is an integer root of the polynominal $ 1 \plus{} p \plus{} Q(x^1)\cdot\ Q(x^2)\ldots\ Q(x^{2p \minus{} 2})$ such that $ Q(x)$ is a polynominal with integer coefficients? | p = 2 | 70 | 5 |
math | 23. Consider the 800 -digit integer
$$
234523452345 \cdots 2345 .
$$
The first $m$ digits and the last $n$ digits of the above integer are crossed out so that the sum of the remaining digits is 2345 . Find the value of $m+n$. | 130 | 81 | 3 |
math | 1. A vessel with a capacity of 10 liters is filled with air containing $24\%$ oxygen. A certain volume of air was pumped out of the vessel and the same volume of argon was added. Then, the same volume of the mixture as the first time was pumped out and again the same volume of argon was added. In the new mixture, $11.7... | 3 | 119 | 1 |
math | Example 1: An $8 \times 8$ chessboard is formed by 9 horizontal lines and 9 vertical lines, creating a total of $r$ rectangles, of which $s$ are squares. The value of $\frac{s}{r}$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are positive integers, and $\frac{m}{n}$ is a reduced fraction. Find the valu... | 125 | 117 | 3 |
math | The complex numbers $z=x+i y$, where $x$ and $y$ are integers, are called Gaussian integers. Determine the Gaussian integers that lie inside the circle described by the equation
$$
x^{2}+y^{2}-4 x-10 y+20=0
$$
in their geometric representation. | \begin{pmatrix}3i&4i&5i&6i&7i;\\1+3i&1+4i&1+5i&1+6i&1+7i\\2+3i&2+4i&2+5i&2+6i&2+7i\\3+3i&3+4i | 69 | 81 |
math | 4. In $\triangle A B C$, $A D$ is the median on side $B C$, $A B=\sqrt{2}$, $A D=\sqrt{6}$, $A C=\sqrt{26}$. Then $\angle A B C=$ $\qquad$ | 60^{\circ} | 61 | 6 |
math | 7. For any $x \in \mathbf{R}, f(x)=|\sin x|$. When $n \leqslant x<n+1$ ( $n$ is an integer), $g(x)=x-n$, then among $f(x), g(x), f(x)+g(x), f(x) g(x)$, the number of periodic functions is $\qquad$ | 2 | 82 | 1 |
math | Between 100 and 1000, which numbers are divisible by 7 and, when divided by either 4 or 9, leave a remainder of 3? | 147,399,651,903 | 38 | 15 |
math | Suppose we list the decimal representations of the positive even numbers from left to right. Determine the $2015^{th}$ digit in the list. | 8 | 32 | 1 |
math | 10. Given the sequence $\left\{a_{n}\right\}$ satisfies: $a_{1}=1, a_{n+1}=a_{n}+a_{n}^{2}\left(n \in \mathbf{N}^{*}\right)$. Let
$$
S_{n}=\frac{1}{\left(1+a_{1}\right)\left(1+a_{2}\right) \cdots\left(1+a_{n}\right)}, \quad T_{n}=\sum_{k=1}^{n} \frac{1}{1+a_{k}}
$$
Find the value of $S_{n}+T_{n}$. | 1 | 148 | 1 |
math | In an enterprise, no two employees have jobs of the same difficulty and no two of them take the same salary. Every employee gave the following two claims:
(i) Less than $12$ employees have a more difficult work;
(ii) At least $30$ employees take a higher salary.
Assuming that an employee either always lies or always te... | 42 | 84 | 2 |
math | ### 9.305 Find $a$, for which the inequality
$x^{2}-2^{a+2} \cdot x-2^{a+3}+12>0$ is true for any $x$. | \in(-\infty;0) | 49 | 9 |
math | 2. Let $k$ be a real number, and the quadratic equation $x^{2}+k x+k+1=0$ has two real roots $x_{1}$ and $x_{2}$. If $x_{1}+2 x_{2}^{2}=k$, then $k$ equals $\qquad$ . | 5 | 72 | 1 |
math | Let $\mathbb{Z}_{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that
$$ m^{2}+f(n) \mid m f(m)+n $$
for all positive integers $m$ and $n$. (Malaysia) | f(n)=n | 80 | 4 |
math | 183 If $a, b, c \in [0,1]$, then the maximum value of the ternary function
$$
f(a, b, c)=a(1-a+a \cdot b)(1-a b+a b c)(1-c)
$$
is = | \frac{8}{27} | 60 | 8 |
math | 2. Given that for any real number $x$ we have $a \cos x + b \cos 2x \geqslant -1$.
Then the maximum value of $a + b$ is $\qquad$ | 2 | 49 | 1 |
math | 1. Let the sequence $\left\{\frac{1}{(n+1) \sqrt{n}+n \sqrt{n+1}}\right\}$ have the sum of its first $n$ terms as $S_{n}$. Then the number of rational terms in the first 2016 terms of the sequence $\left\{S_{n}\right\}$ is | 43 | 81 | 2 |
math | 950. Solve the inequality in natural numbers
$$
x y < x + 2 y
$$ | (x;1)(x\in\mathbb{N});(1;y)(y\in\mathbb{N},y\neq1);(2;y)(y\in\mathbb{N},y\neq1);(3;2) | 23 | 56 |
math | ## Task B-3.2.
Solve the system of equations in the set of positive real numbers
$$
\begin{gathered}
\sqrt[x-y]{x+y}=2 \sqrt{3} \\
(x+y) \cdot 2^{y-x}=3
\end{gathered}
$$ | 7,5 | 65 | 3 |
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