task_type stringclasses 1
value | problem stringlengths 23 3.94k | answer stringlengths 1 231 | problem_tokens int64 8 1.39k | answer_tokens int64 1 200 |
|---|---|---|---|---|
math | ## Task B-2.6.
In the process of simplifying the fraction $\frac{\overline{200 \ldots 0 x}}{\overline{300 \ldots 0 y}}(x, y \neq 0)$, where there are 2020 zeros between 2 and $x$ and between 3 and $y$, Matko neglected the zeros and wrote $\frac{\overline{200 \ldots 0 x}}{\overline{300 \ldots 0 y}}=\frac{\overline{2 x}... | (x,y)\in{(2,3),(4,6),(6,9)} | 170 | 17 |
math | Example 11 Express $\frac{x}{(x-2)\left(x^{2}+2 x+2\right)}$ as partial fractions. | N=\frac{1}{5(x-2)}+\frac{-x+1}{5\left(x^{2}+2 x+2\right)} | 32 | 33 |
math | For real numbers $a$ and $b$, we define $a \nabla b=a b-b a^{2}$. For example, $5 \nabla 4=5(4)-4\left(5^{2}\right)=-80$. Determine the sum of the values of $x$ for which $(2 \nabla x)-8=x \nabla 6$. | \frac{4}{3} | 86 | 7 |
math | 5. Six equal circles $\mathrm{O}, \mathrm{O}_{1}, \mathrm{O}_{2}, \mathrm{O}_{3}, \mathrm{O}_{4}$, $0_{5}$, on the surface of a sphere with radius 1, $\odot O$ is tangent to the other five circles and $\odot \mathrm{O}_{1}$ is tangent to $\odot \mathrm{O}_{2}$, $\odot \mathrm{O}_{2}$ is tangent to $\odot \mathrm{O}_{3}... | r=\frac{1}{2} \sqrt{3-\operatorname{ctg}^{2} \frac{\pi}{5}} | 178 | 29 |
math | An urn contains 15 red, 9 white, and 4 green balls. If we draw 3 balls one after another without replacement, what is the probability that
a) the first is red, the second is white, and the third is green?
b) the 3 drawn balls are red, white, and green, regardless of the order? | \frac{15}{91} | 74 | 9 |
math | 1. Find the smallest four-digit number $n$ for which the system
$$
\begin{aligned}
x^{3}+y^{3}+y^{2} x+x^{2} y & =n, \\
x^{2}+y^{2}+x+y & =n+1
\end{aligned}
$$
has only integer solutions. | 1013 | 78 | 4 |
math | Determine all finite nonempty sets $S$ of positive integers satisfying
$$
\frac{i+j}{(i, j)} \text { is an element of } S \text { for all } i, j \text { in } S
$$
where $(i, j)$ is the greatest common divisor of $i$ and $j$.
Answer: $S=\{2\}$.
# | {2} | 85 | 3 |
math | ## Subject III. (30 points)
a) A three-digit number is reduced by 6 times if the middle digit is removed. Find the number.
b) Find all pairs of non-zero natural numbers whose sum is 111 and for which 111 is divisible by the difference between the two numbers.
Prof. Vasile Şerdean, Gherla Middle School No. 1 | (56,55),(57,54),(74,37) | 83 | 19 |
math | Example 2. Solve the system
$$
\left\{\begin{aligned}
3 x_{1}-2 x_{2}+x_{3} & =-10 \\
2 x_{1}+3 x_{2}-4 x_{3} & =16 \\
x_{1}-4 x_{2}+3 x_{3} & =-18
\end{aligned}\right.
$$ | (-1,2,-3) | 89 | 7 |
math | 14. Let the larger root of the equation $2002^{2} x^{2}-2003 \times 2001 x-1=0$ be $r$, and the smaller root of the equation $2001 x^{2}-2002 x+1=0$ be $s$. Find the value of $r-s$. | \frac{2000}{2001} | 81 | 13 |
math | Let $\omega = \cos\frac{2\pi}{7} + i \cdot \sin\frac{2\pi}{7},$ where $i = \sqrt{-1}.$ Find the value of the product\[\prod_{k=0}^6 \left(\omega^{3k} + \omega^k + 1\right).\] | 24 | 77 | 2 |
math | 4. Let the side lengths of $\triangle A B C$ be $6, x, 2x$. Then the maximum value of its area $S$ is $\qquad$ . | 12 | 39 | 2 |
math | An ideal gas in equilibrium consists of $N$ molecules in a container of volume $V$. What is the probability that a volume $V^{*}$ of the container is empty? What is this probability if $V^{*}=\frac{V}{N}$, and $N \gg 1$? | e^{-1}\approx0.368 | 64 | 10 |
math | 3. Write down all three-digit numbers whose sum of digits is equal to 9. How many are there? | 45 | 23 | 2 |
math | Q. For integers $m,n\geq 1$, Let $A_{m,n}$ , $B_{m,n}$ and $C_{m,n}$ denote the following sets:
$A_{m,n}=\{(\alpha _1,\alpha _2,\ldots,\alpha _m) \colon 1\leq \alpha _1\leq \alpha_2 \leq \ldots \leq \alpha_m\leq n\}$ given that $\alpha _i \in \mathbb{Z}$ for all $i$
$B_{m,n}=\{(\alpha _1,\alpha _2,\ldots ,\alpha _m) ... | |A_{m,n}| = \binom{m+n-1}{n-1} | 346 | 21 |
math | Given a positive integer $n$, find the smallest value of $\left\lfloor\frac{a_{1}}{1}\right\rfloor+\left\lfloor\frac{a_{2}}{2}\right\rfloor+\cdots+\left\lfloor\frac{a_{n}}{n}\right\rfloor$ over all permutations $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ of $(1,2, \ldots, n)$. Answer: The minimum of such sums is $\left... | k+1 | 168 | 3 |
math | $2 \cdot 29$ Let $T=\left\{9^{k} \mid k\right.$ be an integer, $\left.0 \leqslant k \leqslant 4000\right\}$, it is known that $9^{4000}$ has 3817 digits, and its leftmost digit is 9, how many elements in $T$ have 9 as their leftmost digit? | 184 | 99 | 3 |
math | $\left[\begin{array}{l}\text { Algebraic inequalities (miscellaneous) } \\ {[\quad \text { Case analysis }}}\end{array}\right]$
$x, y>0$. Let $S$ denote the smallest of the numbers $x, 1 / y, y+1 / x$. What is the maximum value that $S$ can take? | \sqrt{2} | 79 | 5 |
math | 7.274. $\left\{\begin{array}{l}\left(\log _{a} x+\log _{a} y-2\right) \log _{18} a=1, \\ 2 x+y-20 a=0 .\end{array}\right.$ | (18),(92) | 66 | 7 |
math | 1. Does there exist a quadratic trinomial $P(x)$ with integer coefficients, such that for any natural number $n$ in decimal notation consisting entirely of the digit 1, the value of the quadratic trinomial $P(n)$ is also a natural number consisting entirely of the digit 1. | P(x)=90x^{2}+20x+1 | 63 | 15 |
math | 5. a square board consists of $2 n \times 2 n$ squares. You want to mark $n$ of these squares so that no two marked squares are in the same or adjacent rows, and so that no two marked squares are in the same or adjacent columns. In how many ways is this possible?
## Solution: | (n+1)^{2}n! | 68 | 9 |
math | Circle $o$ contains the circles $m$ , $p$ and $r$, such that they are tangent to $o$ internally and any two of them are tangent between themselves. The radii of the circles $m$ and $p$ are equal to $x$ . The circle $r$ has radius $1$ and passes through the center of the circle $o$. Find the value of $x$ . | \frac{8}{9} | 87 | 7 |
math | In triangle $ABC$, find the smallest possible value of $$|(\cot A + \cot B)(\cot B +\cot C)(\cot C + \cot A)|$$ | \frac{8\sqrt{3}}{9} | 38 | 12 |
math | Example 2. Find the sum $S_{n}=1+(1+2)+(1+2+3)$ $+\cdots+(1+2+3+\cdots+n)$. | \frac{n(n+1)(n+2)}{6} | 40 | 14 |
math | 1.200 people stand in a circle, some of whom are honest people, and some are liars. Liars always tell lies, while honest people tell the truth depending on the situation. If both of his neighbors are honest people, he will definitely tell the truth; if at least one of his neighbors is a liar, he may sometimes tell the ... | 150 | 129 | 3 |
math | 28. Suppose $a \neq 0, b \neq 0, c \neq 0$ and $\frac{0}{b}=\frac{b}{c}=\frac{1}{a}$. Find the value of $\frac{a+b-c}{a-b+c}$. | 1 | 65 | 1 |
math | Determine all pairs $(n, k)$ of non-negative integers such that
$$
2023+2^{n}=k^{2}
$$ | (1,45) | 32 | 6 |
math | There are $n$ students standing in line positions $1$ to $n$. While the teacher looks away, some students change their positions. When the teacher looks back, they are standing in line again. If a student who was initially in position $i$ is now in position $j$, we say the student moved for $|i-j|$ steps. Determine the... | \left\lfloor \frac{n^2}{2} \right\rfloor | 87 | 19 |
math | *2. Let $M, N$ be two points on the line segment $AB$, $\frac{AM}{MB}=\frac{1}{4}, \frac{AN}{NB}=\frac{3}{2}$. Construct any right triangle $\triangle ABC$ with $AB$ as the hypotenuse. Then construct $MD \perp BC$ at $D$, $ME \perp AC$ at $E$, $NF \perp BC$ at $F$, and $NG \perp AC$ at $G$. The maximum possible value o... | \frac{10-4 \sqrt{3}}{5} | 140 | 15 |
math | You are trying to maximize a function of the form $f(x, y, z) = ax + by + cz$, where $a$, $b$, and $c$ are constants. You know that $f(3, 1, 1) > f(2, 1, 1)$, $f(2, 2, 3) > f(2, 3, 4)$, and $f(3, 3, 4) > f(3, 3, 3)$. For $-5 \le x,y,z \le 5$, what value of $(x,y,z)$ maximizes the value of $f(x, y, z)$? Give your answer... | (5, -5, 5) | 156 | 9 |
math | ## Task 16/81
Determine all ordered pairs of natural numbers $(n ; m)$ that satisfy the equation $2^{n}+65=m^{2}$! | (4,9),(10,33) | 39 | 11 |
math | Example 1. Find the integral $\int \frac{\sqrt{x+9}}{x} d x$. | 2\sqrt{x+9}+3\ln|\frac{\sqrt{x+9}-3}{\sqrt{x+9}+3}|+C | 23 | 32 |
math | 471. A motorcyclist left A for B and at the same time a pedestrian set off from B to A. Upon meeting the pedestrian, the motorcyclist gave him a ride, brought him to A, and immediately set off again for B. As a result, the pedestrian reached A 4 times faster than he would have if he had walked the entire way. How many ... | 2.75 | 98 | 4 |
math | Find all sequences of integer $x_1,x_2,..,x_n,...$ such that $ij$ divides $x_i+x_j$ for any distinct positive integer $i$, $j$. | x_i = 0 | 42 | 6 |
math | Exercise 3. Find all integers $n \geqslant 3$ such that, if $a_{1}, \ldots, a_{n}$ are strictly positive real numbers such that $\max \left(a_{1}, \ldots, a_{n}\right) \leqslant n \cdot \min \left(a_{1}, \ldots, a_{n}\right)$, then there necessarily exist three of these real numbers that are the lengths of the sides of... | n\geqslant13 | 119 | 8 |
math | In a board game, we have 11 red, 7 blue, and 20 green tokens. The bank exchanges one red and one blue token for two green tokens, one red and one green token for two blue tokens, and one blue and one green token for two red tokens. During the exchanges, we aim to have all tokens of the same color. Which color will this... | blue | 81 | 1 |
math | 5. [6] What is the sum of all integers $x$ such that $|x+2| \leq 10$ ? | -42 | 31 | 3 |
math | 5. The integer part $[x]$ of a number $x$ is defined as the greatest integer $n$ such that $n \leqslant x$, for example, $[10]=10,[9.93]=9,\left[\frac{1}{9}\right]=0,[-1.7]=-2$. Find all solutions to the equation $\left[\frac{x+3}{2}\right]^{2}-x=1$. | 0 | 97 | 1 |
math | 2. (2nd Quiz, 2nd Question, provided by Huang Yumin) Given a sequence of positive numbers $a_{1}, a_{2}, a_{3}, \cdots$, satisfying $a_{n+1}$ $=\frac{1}{a_{1}+a_{2}+\cdots+a_{n}}(n \in N)$. Find $\lim _{n \rightarrow \infty} \sqrt{n} a_{n}$. | \frac{\sqrt{2}}{2} | 99 | 10 |
math | 3. There is a pile of 100 matches. Petya and Vasya take turns, starting with Petya. Petya can take one, three, or four matches on his turn. Vasya can take one, two, or three matches on his turn. The player who cannot make a move loses. Which of the players, Petya or Vasya, can win regardless of the opponent's play? | Vasya | 92 | 3 |
math | 10.068. The length of the base of the triangle is 36 cm. A line parallel to the base divides the area of the triangle in half. Find the length of the segment of this line enclosed between the sides of the triangle. | 18\sqrt{2} | 53 | 7 |
math | 4. Given that $\triangle A B C$ is an isosceles right triangle, $\angle A=90^{\circ}$, and $\overrightarrow{A B}=\boldsymbol{a}+\boldsymbol{b}$, $\overrightarrow{A C}=a-b$. If $a=(\cos \theta, \sin \theta)(\theta \in \mathbf{R})$, then the area of $\triangle A B C$ is $\qquad$ . | 1 | 102 | 1 |
math | 【Question 1】
On a 200-meter circular track, two people, A and B, start from the same position at the same time, running in a clockwise direction. It is known that A runs 6 meters per second, and B runs 4 meters per second. How many times does A overtake B in 16 minutes? | 9 | 74 | 1 |
math | 5. Let $x, y$ be real numbers. Then the maximum value of $\frac{2 x+\sqrt{2} y}{2 x^{4}+4 y^{4}+9}$ is $\qquad$ . | \frac{1}{4} | 49 | 7 |
math | From the conversation of the sailors, we noted the following detail this year:
Cook: When I was as old as the sailor is now, I was twice as old as he is.
Engineer: I am only 4 years older than the sailor.
Sailor: Only the cook's age is an odd number, and the least common multiple of our three ages is the captain's b... | 1938 | 87 | 4 |
math | 6. If $\frac{z-1}{z+1}(z \in \mathbf{C})$ is a pure imaginary number, then the minimum value of $\left|z^{2}-z+2\right|$ is | \frac{\sqrt{14}}{4} | 49 | 11 |
math | 9. (2000 Shanghai Competition Problem) Let $a_{1} a_{2} a_{3} a_{4} a_{5}$ be a permutation of $1,2,3,4,5$, satisfying that for any $1 \leqslant i \leqslant 4$, $a_{1} a_{2} a_{3} \cdots a_{i}$ is not any permutation of $1,2, \cdots, i$. Find the number of such permutations. | 71 | 111 | 2 |
math | ## Task Condition
Find the differential $d y$.
$$
y=\ln |\cos \sqrt{x}|+\sqrt{x} \tan \sqrt{x}
$$ | \frac{}{2\cos^{2}\sqrt{x}} | 34 | 13 |
math | 6. Given $a=\frac{11 \times 66+12 \times 67+13 \times 68+14 \times 69+15 \times 70}{11 \times 65+12 \times 66+13 \times 67+14 \times 68+15 \times 69} \times 100$, find: the integer part of $a$ is
what? | 101 | 108 | 3 |
math | 3. (3 points) Solve the equation of the form $f(f(x))=x$, given that $f(x)=x^{2}+2 x-4$
# | \frac{1}{2}(-1\\sqrt{17}),\frac{1}{2}(-3\\sqrt{13}) | 37 | 30 |
math | Exercise 8. Let $a$ and $b$ be two real numbers. We define the sequences $\left(a_{n}\right)$ and $\left(b_{n}\right)$ by $a_{0}=a, b_{0}=b$ and for all $n$ a natural number, $a_{n+1}=a_{n}+b_{n}$ and $b_{n+1}=a_{n} b_{n}$. Determine all pairs $(a, b)$ such that $\mathrm{a}_{2022}=\mathrm{a}_{0}$ and $\mathrm{b}_{2022}... | (,0) | 141 | 4 |
math | There are $12$ dentists in a clinic near a school. The students of the $5$th year, who are $29$, attend the clinic. Each dentist serves at least $2$ students. Determine the greater number of students that can attend to a single dentist . | 7 | 59 | 1 |
math | $2 \cdot 73$ Let positive integers $a, b$ make $15a + 16b$ and $16a - 15b$ both squares of positive integers. Find the smallest value that the smaller of these two squares can take. | 481^2 | 58 | 5 |
math | C4 For any positive integer $n$, an $n$-tuple of positive integers $\left(x_{1}, x_{2}, \cdots, x_{n}\right)$ is said to be supersquared if it satisfies both of the following properties:
(1) $x_{1}>x_{2}>x_{3}>\cdots>x_{n}$.
(2) The sum $x_{1}^{2}+x_{2}^{2}+\cdots+x_{k}^{2}$ is a perfect square for each $1 \leq k \leq ... | 24 | 290 | 2 |
math | Kirienko d:
Sasha and Masha each thought of a natural number and told them to Vasya. Vasya wrote the sum of the numbers on one piece of paper and their product on another, then hid one of the papers and showed the other (which had the number 2002 written on it) to Sasha and Masha. Seeing this number, Sasha said that h... | 1001 | 119 | 4 |
math | 8. (6 points) A roll of wire, the first time used half of the total length plus 3 meters, the second time used half of the remaining minus 10 meters, the third time used 15 meters, and finally 7 meters were left. How long was the roll of wire originally? | 54 | 65 | 2 |
math | 1. On the island of knights and liars (liars always lie, knights always tell the truth), each resident supports exactly one football team. In a survey, all residents of the island participated. To the question "Do you support 'Rostov'?", 40% of the residents answered "Yes". To a similar question about 'Zенit', 30% answ... | 30 | 116 | 2 |
math | 4. Given that $[x]$ represents the greatest integer not exceeding $x$, the number of integer solutions to the equation $3^{2 x}-\left[10 \cdot 3^{x+1}\right]+ \sqrt{3^{2 x}-\left[10 \cdot 3^{x+1}\right]+82}=-80$ is $\qquad$ | 2 | 83 | 1 |
math | # 2. Solve the inequality:
$$
\{x\}([x]-1)<x-2,
$$
where $[x]$ and $\{x\}$ are the integer and fractional parts of the number $\boldsymbol{x}$, respectively (9 points).
# | x\geq3 | 58 | 5 |
math | Example 2. There are $n$ points on a plane, where any three points can be covered by a circle of radius 1, but there are always three points that cannot be covered by any circle of radius less than 1. Find the minimum radius of a circle that can cover all $n$ points. | 1 | 65 | 1 |
math | 4. 59 The equation $z^{6}+z^{3}+1=0$ has a complex root, and on the complex plane, the argument $\theta$ of this root lies between $90^{\circ}$ and $180^{\circ}$. Find the degree measure of $\theta$.
The equation $z^{6}+z^{3}+1=0$ has a complex root, and on the complex plane, the argument $\theta$ of this root lies be... | 160^{\circ} | 132 | 7 |
math | 5. For any $x \in\left[-\frac{\pi}{6}, \frac{\pi}{2}\right]$, the inequality
$$
\sin ^{2} x+a \sin x+a+3 \geqslant 0
$$
always holds. Then the range of the real number $a$ is $\qquad$ | a \geqslant -2 | 75 | 8 |
math | Find the smallest positive integer $ n$ such that $ 107n$ has the same last two digits as $ n$. | 50 | 27 | 2 |
math | ## 32. What time is it?
To answer this question, it is enough to add two fifths of the time that has passed since midnight to the time remaining until noon. | 7:30 | 38 | 4 |
math | 18. Master Li made 8 identical rabbit lanterns in three days, making at least 1 per day. Master Li has ( ) different ways to do this. | 21 | 35 | 2 |
math | Mark's teacher is randomly pairing his class of $16$ students into groups of $2$ for a project. What is the probability that Mark is paired up with his best friend, Mike? (There is only one Mike in the class)
[i]2015 CCA Math Bonanza Individual Round #3[/i] | \frac{1}{15} | 68 | 8 |
math | 9. (10 points) Given that 7 red balls and 5 white balls weigh 43 grams, and 5 red balls and 7 white balls weigh 47 grams, then 4 red balls and 8 white balls weigh $\qquad$ grams. | 49 | 57 | 2 |
math | Let $x_1$, $x_2$, and $x_3$ be the roots of the polynomial $x^3+3x+1$. There are relatively prime positive integers $m$ and $n$ such that $\tfrac{m}{n}=\tfrac{x_1^2}{(5x_2+1)(5x_3+1)}+\tfrac{x_2^2}{(5x_1+1)(5x_3+1)}+\tfrac{x_3^2}{(5x_1+1)(5x_2+1)}$. Find $m+n$. | 10 | 136 | 2 |
math | Beroov S.L.
The numbers $a$ and $b$ are such that each of the two quadratic trinomials $x^{2} + a x + b$ and $x^{2} + b x + a$ has two distinct roots, and the product of these trinomials has exactly three distinct roots. Find all possible values of the sum of these three roots. | 0 | 82 | 1 |
math | 10.29 Try to find the smallest natural number that satisfies the following property: its first digit is 4, but when the first digit is moved to the end, its value becomes $\frac{1}{4}$ of the original.
(46th Moscow Mathematical Olympiad, 1983) | 410256 | 65 | 6 |
math | 7. Given a function $f(x)$ defined on $\mathbf{R}$ with period $T$ that satisfies: $f(1+x)=f(1-x)$ and $f(8+x)=f(8-x)$, the maximum value of $T$ is $\qquad$ . | 14 | 62 | 2 |
math | Four siblings are sitting down to eat some mashed potatoes for lunch: Ethan has 1 ounce of mashed potatoes, Macey has 2 ounces, Liana has 4 ounces, and Samuel has 8 ounces. This is not fair. A blend consists of choosing any two children at random, combining their plates of mashed potatoes, and then giving each of those... | \frac{1}{54} | 109 | 8 |
math | 3. (5 points) If the product of 6 consecutive odd numbers is 135135, then the sum of these 6 numbers is $\qquad$ | 48 | 37 | 2 |
math | 2. In the interval $0 \leq x \leq \pi$ find the solutions to the equation
$$
\frac{1}{\sin x}-\frac{1}{\cos x}=2 \sqrt{2}
$$ | \frac{3\pi}{4},\frac{\pi}{12} | 51 | 17 |
math | Let $a_1,a_2,a_3,\dots,a_6$ be an arithmetic sequence with common difference $3$. Suppose that $a_1$, $a_3$, and $a_6$ also form a geometric sequence. Compute $a_1$. | 12 | 57 | 2 |
math | 14. Decode the record ${ }_{* *}+{ }_{* * *}={ }_{* * * *}$, if it is known that both addends and the sum will not change if read from right to left. | 22+979=1001 | 48 | 11 |
math | 3. Find the number of natural numbers $k$, not exceeding 267000, such that $k^{2}-1$ is divisible by 267. | 4000 | 38 | 4 |
math | 1. In the field of real numbers, solve the equation
$$
4 x^{4}-12 x^{3}-7 x^{2}+22 x+14=0,
$$
knowing that it has four distinct real roots, and the sum of two of them is equal to the number 1. | \frac{1}{2}+\sqrt{2},\frac{1}{2}-\sqrt{2},1+\sqrt{3},1-\sqrt{3} | 68 | 36 |
math | Example 3. Find $\lim _{x \rightarrow 0} \frac{\ln \left(\sin ^{2} x+e^{x}\right)-x}{\ln \left(x^{2}+e^{2 x}\right)-2 x}$. | 1 | 57 | 1 |
math | 9. (10 points) From the ten digits $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$, select nine digits to form a two-digit number, a three-digit number, and a four-digit number, such that the sum of these three numbers equals 2010. The digit that was not selected is . $\qquad$ | 6 | 89 | 1 |
math |
N4. Find all triples of integers $(a, b, c)$ such that the number
$$
N=\frac{(a-b)(b-c)(c-a)}{2}+2
$$
is a power of 2016 .
| (,b,)=(k+2,k+1,k),k\in\mathbb{Z} | 54 | 22 |
math | 19th Swedish 1979 Problem 1 Solve the equations: x 1 + 2 x 2 + 3 x 3 + ... + (n-1) x n-1 + n x n = n 2 x 1 + 3 x 2 + 4 x 3 + ... + n x n-1 + x n = n-1 3 x 1 + 4 x 2 + 5 x 3 + ... + x n-1 + 2 x n = n-2 ... (n-1) x 1 + n x 2 + x 3 + ... + (n-3) x n-1 + (n-2) x n = 2 n x 1 + x 2 + 2 x 3 + ... + (n-2) x n-1 + (n-1) x n = 1... | x_1=\frac{2}{n}-1,\,x_2=x_3=\ldots=x_n=\frac{2}{n} | 193 | 31 |
math | Ana, Beto, Carlos, Diana, Elena and Fabian are in a circle, located in that order. Ana, Beto, Carlos, Diana, Elena and Fabian each have a piece of paper, where are written the real numbers $a,b,c,d,e,f$ respectively.
At the end of each minute, all the people simultaneously replace the number on their paper by the sum ... | 0 | 213 | 1 |
math | In the acute-angled triangle $ABC$, the altitudes $BP$ and $CQ$ were drawn, and the point $T$ is the intersection point of the altitudes of $\Delta PAQ$. It turned out that $\angle CTB = 90 {} ^ \circ$. Find the measure of $\angle BAC$.
(Mikhail Plotnikov) | 45^\circ | 77 | 4 |
math | Let $n \geqslant 2$. For an $n$-tuple of ordered real numbers
$$
\begin{array}{l}
A=\left(a_{1}, a_{2}, \cdots, a_{n}\right), \\
\text { let } \quad b_{k}=\max _{1 \leqslant i \leqslant k}, k=1,2, \cdots, n .
\end{array}
$$
The array $B=\left(b_{1}, b_{2}, \cdots, b_{n}\right)$ is called the "innovation array" of $A$;... | n-\frac{n-1}{1+\frac{1}{2}+\cdots+\frac{1}{n-1}} | 213 | 27 |
math | 1. Find the integer part of $(\sqrt{3}+1)^{6}$.
untranslated text remains unchanged. | 415 | 25 | 3 |
math | Consider a $2 \times n$ grid where each cell is either black or white, which we attempt to tile with $2 \times 1$ black or white tiles such that tiles have to match the colors of the cells they cover. We first randomly select a random positive integer $N$ where $N$ takes the value $n$ with probability $\frac{1}{2^n}$. ... | \frac{9}{23} | 117 | 8 |
math | 10,11
In the cube $A B C D A 1 B 1 C 1 D 1$, where $A A 1, B B 1, C C 1$ and $D D 1$ are parallel edges, the plane $P$ passes through point $D$ and the midpoints of edges $A 1 D 1$ and $C 1 D 1$. Find the distance from the midpoint of edge $A A 1$ to the plane $P$, if the edge of the cube is 2. | 1 | 119 | 1 |
math | Solve the following equation:
$$
\frac{\log \left(35-x^{3}\right)^{3}}{\log (5-x)}=9
$$ | x_{1}=2,x_{2}=3 | 36 | 10 |
math | Example 1 Find all positive integer solutions of the system of equations
$$
\left\{\begin{array}{l}
a^{3}-b^{3}-c^{3}=3 a b c, \\
a^{2}=2(b+c)
\end{array}\right.
$$ | (a, b, c)=(2,1,1) | 60 | 12 |
math | Condition of the problem
Calculate the limit of the function:
$\lim _{x \rightarrow 0} \frac{1-\cos 2 x}{\cos 7 x-\cos 3 x}$ | -\frac{1}{10} | 43 | 8 |
math | 3. Given $x, y>0$. If
$$
f(x, y)=\left(x^{2}+y^{2}+2\right)\left(\frac{1}{x+y}+\frac{1}{x y+1}\right) \text {, }
$$
then the minimum value of $f(x, y)$ is | 4 | 75 | 1 |
math | 2. find all polynomials $P$ with real coefficients so that the following equation holds for all $x \in \mathbb{R}$:
$$
(x-2) P(x+2)+(x+2) P(x-2)=2 x P(x)
$$ | P(x)=b(x-2)(x)(x+2)+ | 57 | 14 |
math | Starting with a positive integer $M$ written on the board , Alice plays the following game: in each move, if $x$ is the number on the board, she replaces it with $3x+2$.Similarly, starting with a positive integer $N$ written on the board, Bob plays the following game: in each move, if $x$ is the number on the board, he... | 10 | 118 | 2 |
math | 3. Find all natural numbers $n \geq 2$, for which the equality $4 x_{n}+2 y_{n}=20 n^{2}+13 n-33$ holds, where $x_{n}=1 \cdot 2+2 \cdot 3+\cdots+(n-1) \cdot n, y_{n}=1^{2}+2^{2}+3^{2}+\cdots+(n-1)^{2}$.
(20 points) | 11 | 110 | 2 |
math | 14. Let $f(x, y)$ be a bivariate polynomial, and satisfy the following conditions:
(1) $f(1,2)=2$; (2) $y f(x, f(x, y))=x f(f(x, y), y)=(f(x, y))^{2}$, determine all such $f(x, y)$. | f(x,y)=xy | 77 | 5 |
math | 5. The giants were prepared 813 burgers, among which are cheeseburgers, hamburgers, fishburgers, and chickenburgers. If three of them start eating cheeseburgers, then in that time two giants will eat all the hamburgers. If five take on eating hamburgers, then in that time six giants will eat all the fishburgers. If sev... | 252 | 140 | 3 |
math | 3. In a psychiatric hospital, there is a chief doctor and many lunatics. During the week, each lunatic bit someone (possibly even themselves) once a day. At the end of the week, it turned out that each of the patients had two bites, and the chief doctor had a hundred bites. How many lunatics are there in the hospital | 20 | 72 | 2 |
math | 4. Consider the numbers $A=1+2+3+\ldots+2016^{2}$ and $B=2+4+6+\ldots+2016^{2}$.
a) Find the last digit for each of the numbers $A$ and $B$.
b) Show that $A-B$ is a perfect square.
The problems were proposed by: Prof. Mariana Guzu, School "D. Zamfirescu" Prof. Marius Mohonea, C.N. "Unirea"
NOTE: Working time: 2 hou... | (1008\cdot2016)^2 | 170 | 13 |
math | 4. Find the positive integer solutions for
$$\left\{\begin{array}{l}
5 x+7 y+2 z=24 \\
3 x-y-4 z=4
\end{array}\right.$$ | x=3, y=1, z=1 | 48 | 11 |
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