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 | 4. (8 points) A furnace of molten iron solidifies into an iron block, its volume decreases by $\frac{1}{34}$; then, if this iron block melts back into molten iron (without any loss), its volume increases by $\frac{(}{()}$. | \frac{1}{33} | 59 | 8 |
math | For real numbers $x, y$ and $z$ it holds that $x y z=1$. Calculate the value of the expression
$$
\frac{x+1}{x y+x+1}+\frac{y+1}{y z+y+1}+\frac{z+1}{z x+z+1}
$$ | 2 | 70 | 1 |
math | 9. Let the positive integer $n$ satisfy $31 \mid\left(5^{n}+n\right)$. Then the minimum value of $n$ is $\qquad$ . | 30 | 42 | 2 |
math | Three, (50 points) Find all functions $f: \mathbf{R}_{+} \rightarrow \mathbf{R}_{+}$ such that for all positive numbers $x, y$, the following always holds:
$$
x y(f(x)+f(y)) \doteq(x+y) f(f(x) y) .
$$ | f(x)=x | 72 | 4 |
math | ## Task A-2.3.
Each of the four walls of the room needs to be painted with one color so that adjacent walls are not the same color. If we have three different colors available, in how many ways can the room be painted? It is not necessary to use all the colors. | 18 | 61 | 2 |
math | A sequence of real numbers $\{a_n\}_{n = 1}^\infty (n=1,2,...)$ has the following property:
\begin{align*}
6a_n+5a_{n-2}=20+11a_{n-1}\ (\text{for }n\geq3).
\end{align*}
The first two elements are $a_1=0, a_2=1$. Find the integer closest to $a_{2011}$. | 40086 | 110 | 5 |
math | # Task 2. (20 points)
Find the maximum possible value of the ratio of a three-digit number to the sum of its digits.
# | 100 | 31 | 3 |
math | Let $a$ and $b$ be such that $a+b=13$ and $a b=41$. Calculate $a^{3}+b^{3}$. | 598 | 38 | 3 |
math | 4.033. In an infinite geometric progression with positive terms and a common ratio $|q|<1$, the sum of the first three terms is 10.5, and the sum of the progression is 12. Find the progression. | 6,3,\frac{3}{2},\ldots | 54 | 13 |
math | How many ordered triples $(a, b, c)$ of odd positive integers satisfy $a + b + c = 25?$ | 78 | 27 | 2 |
math | Example 2 Find three real numbers $x, y, z$ such that they simultaneously satisfy the following equations:
$$
\begin{array}{l}
2 x+3 y+z=13, \\
4 x^{2}+9 y^{2}+z^{2}-2 x+15 y+3 z=82 .
\end{array}
$$
(1992, Friendship Cup International Mathematics Competition) | x=3, y=1, z=4 | 91 | 11 |
math | Example 6 Let $f(x)$ be a function defined on $\mathbf{R}$. If $f(0)=2008$, and for any $x \in \mathbf{R}$, it satisfies
$f(x+2)-f(x) \leqslant 3 \times 2^{x}$,
$f(x+6)-f(x) \geqslant 63 \times 2^{x}$,
then $f(2008)=$ $\qquad$
(2008, National High School Mathematics Joint Competition) | 2^{2008}+2007 | 122 | 12 |
math | 23. Find the largest real number $p$ such that all three roots of the equation below are positive integers:
$$
5 x^{3}-5(p+1) x^{2}+(71 p-1) x+1=66 p .
$$ | 76 | 56 | 2 |
math | 3.1. Philatelist Andrey decided to distribute all his stamps equally into 2 envelopes, but it turned out that one stamp was left over. When he distributed them equally into 3 envelopes, one stamp was again left over; when he distributed them equally into 5 envelopes, 3 stamps were left over; finally, when he tried to d... | 223 | 128 | 3 |
math | All three vertices of an equilateral triangle are on the parabola $ y \equal{} x^2$, and one of its sides has a slope of 2. The x-coordinates of the three vertices have a sum of $ m/n$, where $ m$ and $ n$ are relatively prime positive integers. What is the value of $ m \plus{} n$?
$ \textbf{(A)}\ 14\qquad
\textbf{... | 14 | 147 | 2 |
math | 10. (2006 National High School Mathematics League Zhejiang Preliminary Contest Question) $\max _{a, b, \cdots \in \mathbf{R}^{+}} \min \left\{\frac{1}{a}, \frac{1}{b^{2}}, \frac{1}{c^{3}}, a+b^{2}+c^{3}\right\}=$ | \sqrt{3} | 87 | 5 |
math | 18.3.19 $\star \star$ Find all positive integer triples $(a, b, c)$ that satisfy $a^{2}+b^{2}+c^{2}=2005$ and $a \leqslant b \leqslant c$. | (23,24,30),(12,30,31),(9,30,32),(4,30,33),(15,22,36),(9,18,40),(4,15,42) | 62 | 60 |
math | In Yang's number theory class, Michael K, Michael M, and Michael R take a series of tests. Afterwards, Yang makes the following observations about the test scores:
(a) Michael K had an average test score of $90$, Michael M had an average test score of $91$, and Michael R had an average test score of $92$.
(b) Michael... | 413 | 170 | 3 |
math | ## Task 6A - 241246A
Investigate whether there are 40 consecutive natural numbers that are all less than $10^{9}$ and are not prime numbers. | 2\cdot3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot19\cdot43+k | 43 | 31 |
math | No math tournament exam is complete without a self referencing question. What is the product of
the smallest prime factor of the number of words in this problem times the largest prime factor of the
number of words in this problem | 1681 | 44 | 4 |
math | 9. Let sets $A$ and $B$ satisfy:
$$
A \cup B=\{1,2, \cdots, 10\}, A \cap B=\varnothing \text {. }
$$
If the number of elements in set $A$ is not an element of $A$, and the number of elements in set $B$ is not an element of $B$, then the number of all different sets $A$ that satisfy the conditions is $\qquad$ | 186 | 102 | 3 |
math | 10. (12 points) In the Sheep Sheep Sports Meet, Happy Sheep, Boiling Sheep, Lazy Sheep, Warm Sheep, and Big Bad Wolf participated in a 400-meter race. After the race, the five of them discussed the results.
First place said: “Happy Sheep ran faster than Lazy Sheep.”
Second place said: “I ran faster than Warm Sheep.”
Th... | 2 | 143 | 1 |
math | Five, $f$ is a mapping from the set of natural numbers $N$ to set $A$. If for $x$, $y \in \mathbf{N}$, $x-y$ is a prime number, then $f(x) \neq f(y)$. How many elements does $A$ have at least? | 4 | 70 | 1 |
math | 1. Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$, such that for all real numbers $x, y$, we have
$$
f\left(x^{2}+f(x) f(y)\right)=x f(x+y) .
$$ | f(x)=0,f(x)=x,f(x)=-x | 62 | 13 |
math | 8. Positive integers $a_{1}, a_{2}, \ldots, a_{7}, b_{1}, b_{2}, \ldots, b_{7}$ satisfy $2 \leq a_{i} \leq 166$ and $a_{i}^{b_{i}} \equiv a_{i+1}^{2}(\bmod 167)$ for each $1 \leq i \leq 7$ (where $a_{8}=a_{1}$ ). Compute the minimum possible value of $b_{1} b_{2} \cdots b_{7}\left(b_{1}+b_{2}+\cdots+b_{7}\right.$ ). | 675 | 153 | 3 |
math | [ Mathematical logic (other).]
Lilac. In a vase, there is a bouquet of 7 white and blue lilac branches. It is known that 1) at least one branch is white, 2) out of any two branches, at least one is blue. How many white branches and how many blue branches are in the bouquet?
# | 1 | 73 | 1 |
math | Example 9. It is known that the variance of each of the given independent random variables does not exceed 4. Determine the number of such variables for which the probability of the deviation of the arithmetic mean of the random variable from the arithmetic mean of their mathematical expectations by no more than 0.25 e... | n\geq6400 | 69 | 8 |
math | 6 Let $\alpha$ be a given positive real number, find all functions $f: \mathbf{N}^{*} \rightarrow \mathbf{R}$, such that for any positive integers $k, m$ satisfying $\alpha m \leqslant k<(\alpha+1) m$, we have
$$
f(k+m)=f(k)+f(m) .
$$
(Chen Yonggao) | f(n)= | 90 | 3 |
math | A Vinczlér. A poor Vinczlér was just milking, when the landowner approached and asked for 4 messzely of must to drink. The poor Vinczlér only had two measuring vessels, one 3 messzely, the other 5 messzely, the question is, how did he still manage to give 4 messzely of drink to the lord? 1[^0]
[^0]: 1* We are happ... | 4 | 116 | 1 |
math | Example 9. Find the residue of the function
$$
f(z)=e^{1 / z^{2}} \cos z
$$
at the point $z=0$. | 0 | 38 | 1 |
math | 1. Determine all integers $x$ such that $x^{2}+3 x+24$ is a perfect square. | x\in{-23,-8,5,20} | 27 | 14 |
math | Example 35 (1999 Shanghai High School Competition Question) Let $a, b, c, d$ be four distinct real numbers such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=4$, and $a c=b d$. Find the maximum value of $\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}$. | -12 | 103 | 3 |
math | The real numbers $x_{1}, x_{2}, x_{3}, \ldots, x_{n}$ are the consecutive terms of an arithmetic sequence. If
$$
\frac{x_{2}}{x_{1}+x_{3}}+\frac{x_{3}}{x_{2}+x_{4}}+\frac{x_{4}}{x_{3}+x_{5}}+\cdots+\frac{x_{n-2}}{x_{n-3}+x_{n-1}}+\frac{x_{n-1}}{x_{n-2}+x_{n}}=1957
$$
what is the value of $n$ ? | 3916 | 147 | 4 |
math | Task 1. (5 points) Solve the equation $x^{6}-20 x^{2}-\sqrt{21}=0$. | {\\sqrt[4]{21}} | 30 | 9 |
math | 9. A. If $y=\sqrt{1-x}+\sqrt{x-\frac{1}{2}}$ has a maximum value of $a$ and a minimum value of $b$, then the value of $a^{2}+b^{2}$ is $\qquad$. | \frac{3}{2} | 59 | 7 |
math | ## Problem Statement
Find the coordinates of point $A$, which is equidistant from points $B$ and $C$.
$A(0 ; 0 ; z)$
$B(-6 ; 7 ; 5)$
$C(8 ;-4 ; 3)$ | A(0;0;5.25) | 60 | 11 |
math | 9.2 Find all real numbers $x$ that satisfy the inequality
$$\sqrt{3-x}-\sqrt{x+1}>\frac{1}{2}$$ | -1 \leqslant x<1-\frac{\sqrt{31}}{8} | 36 | 21 |
math | 5. Out of two hundred ninth-grade students, 80% received excellent grades on the first exam, 70% on the second exam, and 59% on the third exam. What is the smallest number of participants who could have received excellent grades on all three exams?
Answer: 18. | 18 | 65 | 2 |
math | 7. Bruno set a four-digit even number as the unlock code for his mobile phone. To avoid forgetting the number, he wrote down the following:
- All digits are different, and the sum of all digits is 15.
- The unit digit is three times smaller than the thousands digit.
- The tens digit is smaller than the unit digit.
De... | 6702 | 84 | 4 |
math | 13. (10 points) In the equation below, $A, B, C, D, E, F, G, H, I$ each represent different digits from $1 \sim 9$.
$$
\overline{\mathrm{ABCD}}+\overline{\mathrm{EF}} \times \overline{\mathrm{GH}}-I=X
$$
Then the minimum value of $X$ is . $\qquad$ | 2369 | 94 | 4 |
math | Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ with the property
$$
f(f(x)+y)=2 x+f(f(y)-x)
$$
for all $x, y \in \mathbb{R}$. | f(x)=x+c | 58 | 5 |
math | 11、Divide a rope of length 31 cm into three segments, with each segment's length being an integer. Any two segments can be taken to form the length and width of a rectangle, resulting in three possible rectangles. The maximum value of the sum of the areas of these three rectangles is $\qquad$ square centimeters. | 320 | 70 | 3 |
math | Example 4 A group of tourists is traveling by car, with the requirement that the number of passengers in each car is equal. At first, each car carries 22 people, leaving 1 person unable to board. If one car leaves empty, then all the tourists can be evenly distributed among the remaining cars. It is known that each car... | 24, 529 | 93 | 7 |
math | # 1. Option 1.
Find the value of the expression $101+102-103-104+105+106-107-108+\ldots-619$? | 100 | 55 | 3 |
math | 4. Find the smallest four-digit number, the product of all digits of which is equal to 512.
Answer. 1888 | 1888 | 31 | 4 |
math | 9.1. Two given quadratic trinomials $f(x)$ and $g(x)$ each have two roots, and the equalities $f(1)=g(2)$ and $g(1)=f(2)$ hold. Find the sum of all four roots of these trinomials. | 6 | 64 | 1 |
math | 15. (15 points) Two cars, A and B, are traveling towards each other on a road parallel to a railway. A 180-meter-long train is moving in the same direction as car A at a speed of 60 kilometers/hour. The train takes 5 minutes from the moment it catches up with car A to the moment it meets car B. If the train takes 30 se... | 1.25 | 117 | 4 |
math | ## Task $2 / 88$
The smallest two natural numbers $n$ are sought, which are the sum of both two and three consecutive square numbers. | 365 | 33 | 3 |
math | Example 7 Given $a=\sqrt{5}-1$. Then the value of $2 a^{3}+7 a^{2}-2 a$ -12 is $\qquad$ [7]
(2010) "Mathematics Weekly" Cup National Junior High School Mathematics Competition) | 0 | 62 | 1 |
math | 23. Let $x$, $y$, and $a$ be real numbers. And satisfy $x+y=x^{3}+y^{3}=$ $x^{5}+y^{5}=a$. Find all possible values of $a$.
(Greece for the 43rd IMO selection exam) | -2, -1, 0, 1, 2 | 66 | 14 |
math | 8.1. There are 15 rectangular sheets of paper. In each move, one of the sheets is chosen and divided by a straight cut, not passing through the vertices, into two sheets. After 60 moves, it turned out that all the sheets are either triangles or hexagons. How many hexagons? | 25 | 68 | 2 |
math | Find a function $f$ such that $f(t^2 +t +1) = t$ for all real $t \ge 0$ | f(m) = \frac{-1 + \sqrt{4m - 3}}{2} | 31 | 22 |
math | 6. A Herd of Elephants. Springs are bubbling at the bottom of the lake. A herd of 183 elephants could drink it dry in one day, while a herd of 37 elephants would take 5 days. How many days would it take for 1 elephant to drink the lake? | 365 | 65 | 3 |
math | 9. The line $l$ intersects the circle $x^{2}+y^{2}=a^{2}$ at points $P, Q$, and intersects the hyperbola $x^{2}-y^{2}=a^{2}$ at points $M, N$, such that $P, Q$ trisect the segment $M N$. Find the equation of the line $l$. | \\frac{2\sqrt{5}}{5}xor\\frac{2\sqrt{5}}{5} | 82 | 25 |
math | 9. In the arithmetic sequences $3,10,17, \cdots, 2005$ and $3,8, 13, \cdots, 2003$, the number of terms that have the same value is $\qquad$. | 58 | 59 | 2 |
math | ## Task 1.
Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $x$ and $y$,
$$
f\left(x^{2}+f(y)\right)=\left(f(x)+y^{2}\right)^{2}
$$ | f(x)=x^{2} | 70 | 7 |
math | Let $\mathbb{N}$ be the set of all positive integers. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that the number $(f(m)+n)(m+f(n))$ is a square for all $m, n \in \mathbb{N}$. Answer. All functions of the form $f(n)=n+c$, where $c \in \mathbb{N} \cup\{0\}$. | f(n) = n + c | 102 | 7 |
math | A drawer contains a mixture of red socks and blue socks, at most $1991$ in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly $\frac{1}{2}$ that both are red or both are blue. What is the largest possible number of red socks in the drawer that is consist... | 990 | 78 | 3 |
math | 3. (8 points) A bag of rice, Liu Bei can finish it alone in 5 days, Guan Yu can finish it alone in 3 days; a bag of wheat, Guan Yu can finish it alone in 5 days, Zhang Fei can finish it alone in 4 days. Liu Bei's daily food intake is less than Zhang Fei's daily food intake by $\qquad$ $\%$. | 52 | 87 | 2 |
math | Example 2 Given that $\alpha$ is a root of the equation $x^{2}-x-1=0$. Try to find the value of $\alpha^{18}+323 \alpha^{-6}$.
Translating the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 5796 | 75 | 4 |
math | On a semicircle of diameter $AB$ and center $C$, consider variable points $M$ and $N$ such that $MC \perp NC$. The circumcircle of triangle $MNC$ intersects $AB$ for the second time at $P$. Prove that $\frac{|PM-PN|}{PC}$ constant and find its value. | \sqrt{2} | 78 | 6 |
math | A train traveling between $A$ and $B$ arrives 20 minutes earlier in $B$ if its speed exceeds the scheduled speed by $5 \mathrm{~km}$ per hour; however, it arrives 25 minutes late if its speed is $5 \mathrm{~km}$ per hour less than the scheduled speed. What is the scheduled speed? | 45 | 75 | 2 |
math | Determine the number of all positive ten-digit integers with the following properties:
- The number contains each of the digits 0, 1, 2, ..., 8, and 9 exactly once.
- Each digit, except for the 9, has a neighboring digit that is greater than it.
(Note. For example, in the number 1230, the digits 1 and 3 are the neigh... | 256 | 139 | 3 |
math | 20th VMO 1982 Problem B1 Find all positive integer solutions to 2 a + 2 b + 2 c = 2336. | {,b,}={5,8,11} | 37 | 13 |
math | # Problem 5. (3 points)
In trapezoid $ABCD$, a point $X$ is taken on the base $BC$ such that segments $XA$ and $XD$ 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 5. Find $XC \cdot BX$.
Answer: 25 | 25 | 91 | 2 |
math | There is a safe that can be opened by entering a secret code consisting of $n$ digits, each of them is $0$ or $1$. Initially, $n$ zeros were entered, and the safe is closed (so, all zeros is not the secret code).
In one attempt, you can enter an arbitrary sequence of $n$ digits, each of them is $0$ or $1$. If the ente... | n | 153 | 2 |
math | Consider the sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1,...$ Find $n$ such that the fi rst $n$ terms sum up
to $2010$. | 1027 | 74 | 4 |
math | Example 41 (12th CMO Question) Let real numbers $x_{1}, x_{2}, \cdots, x_{1997}$ satisfy the following conditions:
(1) $-\frac{1}{\sqrt{3}} \leqslant x_{i} \leqslant \sqrt{3}(i=1,2, \cdots, 1997)$;
(2) $x_{1}+x_{2}+\cdots+x_{1997}=-318 \sqrt{3}$, find the maximum value of $x_{1}^{12}+x_{2}^{12}+\cdots+x_{1997}^{12}$. | 189548 | 161 | 6 |
math | 4. Find the probability that a number randomly taken from the interval $[0 ; 5]$ is a solution to the equation $\sin (x+|x-\pi|)+2 \sin ^{2}(x-|x|)=0$.
# | \frac{\pi}{5} | 54 | 7 |
math | B4. An ant walks from the bottom left corner of a $10 \times 10$ square grid to the diagonally-opposite corner, always walking along grid lines and taking as short a route as possible. Let $N(k)$ be the number of different paths that ant could follow if it makes exactly $k$ turns. Find $N(6)-N(5)$. | 3456 | 81 | 4 |
math | 3. Define the function $f$ on the positive integers such that
$$
f(1)=1, \quad f(3)=3
$$
and
$$
\begin{aligned}
f(2 n) & =f(n) \\
f(4 n+1) & =2 f(2 n+1)-f(n) \\
f(4 n+3) & =3 f(2 n+1)-2 f(n)
\end{aligned}
$$
for all positive integers $n$.
Determine the number of integers $n$ in the range $1 \leq n \leq 1988$ for w... | 92 | 147 | 2 |
math | Exercise 1. Calculate the number
$(1+11+21+31+41+51+61+71+81+91)+(9+19+29+39+49+59+69+79+89+99)$.
Only a numerical answer is expected here. | 1000 | 76 | 4 |
math | 9. (2003 Taiwan Training Problem) Find all functions $f: \mathbf{N} \rightarrow \mathbf{N}$, for all $m, n \in \mathbf{N}$ satisfying $f\left(m^{2}+n^{2}\right)=$ $f^{2}(m)+f^{2}(n)$ and $f(1)>0$. | f(n)=n | 84 | 4 |
math | 7. The minimum value of the function $f(x)=\sum_{k=1}^{2013}|x-k|$ is | 1013042 | 29 | 7 |
math | 2. The digits $1,2,3,4,5, A$ and $B$ are all different and nonzero. Each of the two six-digit integers 'A12345' and ' $12345 A$ ' is divisible by $B$.
Find all possible pairs of values of $A$ and $B$. | A=9,B=7 | 74 | 6 |
math | In a hospital, there are 7 patients (Doc, Grumpy, Happy, Sleepy, Bashful, Sneezy, and Dopey) who need to be assigned to 3 doctors (Huey, Dewey, and Louie). In how many ways can the patients be assigned to the doctors so that each patient is assigned to exactly one doctor and each doctor is assigned at least one patient... | 1806 | 85 | 4 |
math | Express $\log _{x y} z$ as a function of $\log _{x} z$ and $\log _{y} z$. | \frac{(\log_{x}z)\cdot(\log_{y}z)}{\log_{x}z+\log_{y}z} | 31 | 31 |
math | We know the following about three natural numbers:
a) all three are different;
b) their sum is 406;
c) their greatest common divisor is a prime number greater than 2 (denoted by $p$);
d) if the individual numbers are divided by $p$, we again get three prime numbers (let these be $p_{1}, p_{2}, p_{3}$). What are the... | 14,21,371;14,91,301;14,133,259;58,145,203 | 100 | 41 |
math | 4. [3] Suppose that $a, b, c, d$ are real numbers satisfying $a \geq b \geq c \geq d \geq 0, a^{2}+d^{2}=1, b^{2}+c^{2}=1$, and $a c+b d=1 / 3$. Find the value of $a b-c d$. | \frac{2\sqrt{2}}{3} | 84 | 12 |
math | Example 10. The random variable $X$ is distributed according to the normal law, with $M(X)=10, \quad D(X)=4$. Find:
$$
\text { 1) } P(12<X<14) ; 2) P(8<X<12)
$$ | 0.1359050.682690 | 67 | 16 |
math | 2. (10 points) In each cell of a $50 \times 50$ square, a number is written that is equal to the number of $1 \times 16$ rectangles (both vertical and horizontal) in which this cell is an end cell. In how many cells are numbers greater than or equal to 3 written? | 1600 | 73 | 4 |
math | 6.054. $\sqrt{5+\sqrt[3]{x}}+\sqrt{5-\sqrt[3]{x}}=\sqrt[3]{x}$. | 64 | 36 | 2 |
math | 7. Given $x, y, z \in \mathbf{R}$, then $\sum \frac{x^{2}}{(3 x-2 y-z)^{2}}$ has the minimum value of $\qquad$ ("sum" indicates cyclic sum).
| \frac{5}{49} | 55 | 8 |
math | 10. The sum of the maximum and minimum values of the function $y=\sin x+\sqrt{2+\cos ^{2} x}$ is $\qquad$ . | 2 \sqrt{2} | 37 | 6 |
math | 4B. Let $z$ be a complex number such that $z+\frac{1}{z}=1$. Calculate the value of the expression
$$
z^{1993}+z^{1994}+z^{1995}+z^{1996}+z^{1997}
$$ | -1 | 73 | 2 |
math | 11th Brazil 1989 Problem 3 Let Z be the integers. f : Z → Z is defined by f(n) = n - 10 for n > 100 and f(n) = f(f(n+11)) for n ≤ 100. Find the set of possible values of f. | 91 | 71 | 2 |
math | Find all integer solutions $x$ of the equation: $x^{3}+(x+1)^{3}+(x+2)^{3}=(x+3)^{3}$ | 3 | 40 | 1 |
math | 2. Find all real roots of the equation
$(x+1)^{5}+(x+1)^{4}(x-1)+(x+1)^{3}(x-1)^{2}+(x+1)^{2}(x-1)^{3}+(x+1)(x-1)^{4}+(x-1)^{5}=0$ | 0 | 81 | 1 |
math | Starting with a natural number, Márcio replaces this number with the sum of its digits, obtaining a new number, with which he repeats the process, until he finally arrives at a number with only one digit. For example, Márcio replaces 1784102 with 23 and then with 8. He also applies this process to lists of $N$ natural ... | 9,8,4,2232445 | 225 | 13 |
math | 4. On the shores of a circular island (viewed from above), there are cities $A, B, C$, and $D$. A straight asphalt road $A C$ divides the island into two equal halves. A straight asphalt road $B D$ is shorter than road $A C$ and intersects it. The speed of a cyclist on any asphalt road is 15 km/h. The island also has s... | 450 | 157 | 3 |
math | 3. Given $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$ are positive integers. The set of sums obtained by taking any four of these numbers is $\{44, 45, 46, 47\}$. Then these five numbers are $\qquad$ | 10, 11, 11, 12, 13 | 69 | 18 |
math | \section*{Problem 1 - 291021}
Determine all pairs \((x ; y)\) of real numbers \(x\) and \(y\) that satisfy the following system of equations (1), (2):
\[
\begin{array}{r}
\frac{x}{y}=\frac{4}{9} \\
\frac{x+\sqrt{x}}{y+\sqrt{y}}=\frac{1}{2}
\end{array}
\] | (4;9) | 101 | 5 |
math | Given a positive integer $ n\geq 2$, let $ B_{1}$, $ B_{2}$, ..., $ B_{n}$ denote $ n$ subsets of a set $ X$ such that each $ B_{i}$ contains exactly two elements. Find the minimum value of $ \left|X\right|$ such that for any such choice of subsets $ B_{1}$, $ B_{2}$, ..., $ B_{n}$, there exists a subset $ Y$ of $ X$ s... | 2n - 1 | 168 | 7 |
math | 13. Let complex numbers $z_{1}, z_{2}$ satisfy $\left|z_{1}\right|=\left|z_{1}+z_{2}\right|=3,\left|z_{1}-z_{2}\right|=3 \sqrt{3}$, find the value of $\log _{3}\left|\left(z_{1} \bar{z}_{2}\right)^{2000}+\left(\bar{z}_{1} z_{2}\right)^{2000}\right|$. | 4000 | 116 | 4 |
math | $1 \cdot 34$ Let $a_{1}=1, a_{2}=3$, and for all positive integers $n$,
$$
a_{n+2}=(n+3) a_{n+1}-(n+2) a_{n} .
$$
Find all values of $n$ for which $a_{n}$ is divisible by 11.
(7th Balkan Mathematical Olympiad, 1990) | n=4,n=8,n\geqslant10 | 98 | 14 |
math | 4. In rectangle $A B C D$, it is known that $A B=3, B C=1$, and a moving point $P$ is on side $C D$. Let $\angle P A B=\alpha, \angle P B A=\beta$, then the maximum value of $\frac{\overrightarrow{P A} \cdot \overrightarrow{P B}}{\cos (\alpha+\beta)}$ is $\qquad$ . | -3 | 93 | 2 |
math | [ Cross-sectional Area ]
All edges of a regular quadrilateral pyramid are equal to $a$. A plane is drawn through one side of the base and the midpoint of one of the opposite lateral edges. Find the area of the resulting cross-section.
# | \frac{3^{2}\sqrt{11}}{16} | 50 | 16 |
math | 2. Solve the inequality $\sqrt{x^{2}-1} \leqslant \sqrt{5 x^{2}-1-4 x-x^{3}}$. | (-\infty;-1]\cup{2} | 35 | 11 |
math | ## Zadatak B-3.4.
Riješite jednadžbu
$$
\frac{\sin \left(x-\frac{2015 \pi}{2}\right)-\cos ^{3} x}{\sin (2 x+2015 \pi)}=\frac{1}{2}
$$
| nosolution | 72 | 2 |
math | Solve the system of equations for positive real numbers:
$$\frac{1}{xy}=\frac{x}{z}+ 1,\frac{1}{yz} = \frac{y}{x} + 1, \frac{1}{zx} =\frac{z}{y}+ 1$$ | x = y = z = \frac{1}{\sqrt{2}} | 66 | 16 |
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