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 | Begunni A.
What can be the product of several different prime numbers if it is divisible by each of them, decreased by 1?
Find all possible values of this product.
# | 6,42,1806 | 39 | 9 |
math | 11.13. Solve the equation
$$
3-7 \cos ^{2} x \sin x-3 \sin ^{3} x=0
$$ | \frac{\pi}{2}+2k\pi(-1)^{k}\frac{\pi}{6}+k\pi | 38 | 28 |
math | 5. Let the function $f(n)$ be defined on the set of positive integers, for any positive integer $n$, we have $f(f(n))=4 n+3$, and for any non-negative integer $k$, we have
$$
f\left(2^{k}\right)=2^{k+1}+1 \text {. }
$$
Then $f(2303)=$ | 4607 | 86 | 4 |
math | 1. Given three non-zero real numbers $a, b, c$, the set $A=$ $\left\{\frac{a+b}{c}, \frac{b+c}{a}, \frac{c+a}{b}\right\}$. Let $x$ be the sum of all elements in set $A$, and $y$ be the product of all elements in set $A$. If $x=2 y$, then the value of $x+y$ is $\qquad$ | -6 | 101 | 2 |
math | If $a$ and $b$ satisfy the equations $a +\frac1b=4$ and $\frac1a+b=\frac{16}{15}$, determine the product of all possible values of $ab$.
| 1 | 49 | 1 |
math | 4. (2005 Croatian Mathematical Olympiad) Find all positive integers $n$ such that $2^{4}+2^{7}+2^{n}$ is a perfect square. | 8 | 41 | 1 |
math | 9. Two circles are concentric. The area of the ring between them is $A$. In terms of $A$, find the length of the longest chord contained entirely within the ring. | 2\sqrt{\frac{A}{\pi}} | 38 | 11 |
math | Let $f(x)=\tfrac{x+a}{x+b}$ for real numbers $x$ such that $x\neq -b$. Compute all pairs of real numbers $(a,b)$ such that $f(f(x))=-\tfrac{1}{x}$ for $x\neq0$. | (-1, 1) | 64 | 6 |
math | Example 3. Find the derivative of the scalar field $u=x z^{2}+2 y z$ at the point $M_{0}(1,0,2)$ along the circle
$$
\left\{\begin{array}{l}
x=1+\cos t \\
y=\sin t-1 \\
z=2
\end{array}\right.
$$ | -4 | 79 | 2 |
math | ## Task 3 - 340533
Annette, Bernd, Christiane, Dieter, and Ruth are playing the following game: the four children except Ruth agree that one of them will hide a letter and that each of these children will make three statements, at least two of which are true.
Ruth, who only knows these rules and the statements of the... | Dieter | 310 | 2 |
math | 13. $(x, y)=\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\left(x_{1}<x_{2}\right)$ are two integer solutions of the equation $x^{2}-y^{2}-2 x+6 y-8=0$. In the Cartesian coordinate system, point $A\left(x_{1}, y_{1}\right)$ and point $B\left(x_{2}, y_{2}\right)$ are symmetric about $P(1,3)$, and point $C(5,-1)$. I... | (-1,1) | 166 | 5 |
math | 1. Find all polynomials satisfying $(x-1) \cdot p(x+1)-(x+2) \cdot P(x) \equiv 0, x \in \mathbf{R}$.
(1976 New York Competition Problem) | p(x)=(x^3-x) | 53 | 8 |
math | XLVIII OM - I - Problem 7
Calculate the upper limit of the volume of tetrahedra contained within a sphere of a given radius $ R $, one of whose edges is the diameter of the sphere. | \frac{1}{3}R^3 | 46 | 10 |
math | Find all positive integers $w$, $x$, $y$ and $z$ which satisfy $w! = x! + y! + z!$. | (w, x, y, z) = (3, 2, 2, 2) | 32 | 23 |
math | 2. Determine all values of $m, n, p$ such that $p^{n}+144=m^{2}$, where $m$ and $n$ are positive integers and $p$ is a prime number. | (13,2,5),(20,8,2),(15,4,3) | 49 | 22 |
math | }
Find all natural $n>2$, for which the polynomial $x^{n}+x^{2}+1$ is divisible by the polynomial $x^{2}+x+1$.
# | 3k+1,k\in{N} | 43 | 10 |
math | 30. Find the units digit of $2013^{1}+2013^{2}+2013^{3}+\cdots+2013^{2013}$. | 3 | 47 | 1 |
math | 469. Solve the equation:
$$
x^{4}+5 x^{3}-3 x^{2}-35 x-28=0
$$
(solved by elementary means). | x_1=-1,x_2=-4,x_{3,4}=\\sqrt{7} | 42 | 21 |
math | 6. Given three distinct positive integers $a, b, c$ form a geometric sequence, and their sum is 111. Then $\{a, b, c\}=$ $\qquad$ | \{1,10,100\},\{27,36,48\} | 43 | 24 |
math | 16. The last digit of a six-digit number was moved to the beginning (for example, $456789 \rightarrow$ 945678), and the resulting six-digit number was added to the original number. Which numbers from the interval param 1 could have resulted from the addition? In the answer, write the sum of the obtained numbers.
| par... | 1279267 | 177 | 7 |
math | LIX OM - I - Task 9
Determine the smallest real number a with the following property:
For any real numbers $ x, y, z \geqslant a $ satisfying the condition $ x + y + z = 3 $
the inequality holds | -5 | 57 | 2 |
math | 1. The price of a mathematics book in bookstore A at the beginning of January 2012 was higher than the price of the same book in bookstore B. On February 15, 2012, the price of the book in bookstore A decreased by $30\%$, while the price of the same book in bookstore B increased by $30\%$. Then, on March 15, 2012, the ... | 300 | 193 | 3 |
math | 10. (14 points) Given a real number $a>1$. Find the minimum value of the function
$$
f(x)=\frac{(a+\sin x)(4+\sin x)}{1+\sin x}
$$ | \frac{5(a+1)}{2} | 50 | 11 |
math | Let $a,b$ be positive integers. Find, with proof, the maximum possible value of $a\lceil b\lambda \rceil - b \lfloor a \lambda \rfloor$ for irrational $\lambda$. | a + b - \gcd(a, b) | 47 | 11 |
math | [ Combinatorics of orbits ]
[ Product rule ]
$p$ - a prime number. How many ways are there to color the vertices of a regular $p$-gon using $a$ colors?
(Colorings that can be matched by rotation are considered the same.)
# | \frac{^{p}-}{p}+ | 56 | 10 |
math | 10. $a, b, m, n$ satisfy: $a m m^{2001}+b n^{2001}=3 ; a m^{20002}+b n^{2002}=7 ; a m^{2003}+$ $b n^{2003}=24 ; a m^{21004}+b m^{2004}=102$. Then the value of $m^{2}(n-1)$ is $\qquad$ . | 6 | 117 | 1 |
math | 1. Susie thinks of a positive integer $n$. She notices that, when she divides 2023 by $n$, she is left with a remainder of 43 . Find how many possible values of $n$ there are. | 19 | 51 | 2 |
math | 3.1. Along the groove, there are 100 multi-colored balls arranged in a row with a periodic repetition of colors in the following order: red, yellow, green, blue, purple. What color is the ball at the $78-$th position?
$$
\text { (4-5 grades) }
$$ | green | 69 | 1 |
math | Problem 1. Student Vasya, who lives in the countryside, arrives at the station by train every evening after classes at 6 PM. By this time, his father picks him up by car and takes him home. One day, Vasya's last class at the institute was canceled, and he arrived at the station an hour earlier. Unfortunately, he forgot... | 17:50 | 117 | 5 |
math | II. (Total 50 points) Let $x_{1}, x_{2}, \cdots, x_{n}$ and $a_{1}, a_{2}, \cdots, a_{n}$ be two sets of real numbers satisfying the following conditions:
(1) $\sum_{i=1}^{n} x_{i}=0$;
(2) $\sum_{i=1}^{n}\left|x_{i}\right|=1$;
(3) $a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{n}\left(a_{1}>a_{n}, n \geqslant 2\r... | \frac{1}{2} | 212 | 7 |
math | 12. (3 points) Two natural numbers $a$ and $b$, their least common multiple is 60. Then, the difference between these two natural numbers has $\qquad$ possible numerical values. | 23 | 44 | 2 |
math | II. (40 points) Let $p$ be a prime number, and the sequence $\left\{a_{n}\right\}$ satisfies $a_{0}=0, a_{1}=1$, and for any non-negative integer $n$,
$$
a_{n+2}=2 a_{n+1}-p a_{n} \text {. }
$$
If -1 is a term in the sequence $\left\{a_{n}\right\}$, find all possible values of $p$. | p=5 | 109 | 3 |
math | $\begin{array}{c}8 \cdot 9 \text { - Let } \quad a_{n}=\frac{1}{(n+1) \sqrt{n}+n \sqrt{n+1}}, n=1,2,3, \cdots \text { Find } \\ a_{1}+a_{2}+\cdots+a_{99} .\end{array}$ | \frac{9}{10} | 86 | 8 |
math | 7.4. Given nine cards with the numbers $5,5,6,6,6,7,8,8,9$ written on them. From these cards, three three-digit numbers $A, B, C$ were formed, each with all three digits being different. What is the smallest value that the expression $A+B-C$ can have? | 149 | 75 | 3 |
math | Circles $\mathcal{P}$ and $\mathcal{Q}$ have radii $1$ and $4$, respectively, and are externally tangent at point $A$. Point $B$ is on $\mathcal{P}$ and point $C$ is on $\mathcal{Q}$ so that line $BC$ is a common external tangent of the two circles. A line $\ell$ through $A$ intersects $\mathcal{P}$ again at $D$ and in... | 129 | 176 | 3 |
math | 6. If the function $y=3 \sin x-4 \cos x$ attains its maximum value at $x_{0}$, then the value of $\tan x_{0}$ is $\qquad$. | -\frac{3}{4} | 45 | 7 |
math | How many positive integers less than $1998$ are relatively prime to $1547$? (Two integers are relatively prime if they have no common factors besides 1.) | 1487 | 39 | 4 |
math | 1. (16 points) Does there exist a real number $a$, such that the line $y=a x+1$ intersects the hyperbola $3 x^{2}-y^{2}=1$ at two points $A$ and $B$, and the circle with diameter $AB$ passes exactly through the origin of the coordinate system? | a= \pm 1 | 72 | 6 |
math | Example 3. Let $x, y$ satisfy $3 x^{2}+2 y^{2}=6 x$, find the maximum value of $x^{2}+y^{2}$:
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | 4 | 67 | 1 |
math | 5. (6 points) If $a<\frac{1}{\frac{1}{2011}+\frac{1}{2012}+\frac{1}{2013}+\frac{1}{2014}+\frac{1}{2015}}<a+1$, then the natural number $a=$ $\qquad$ | 402 | 81 | 3 |
math | Example 1 (1979 Yunnan Province Mathematics Competition Question) A four-digit number, its unit digit and hundred digit are the same. If the digits of this four-digit number are reversed (i.e., the thousand digit and unit digit are swapped, the hundred digit and ten digit are swapped), the new number minus the original... | 1979 | 84 | 4 |
math | Let triangle $ABC$ with incenter $I$ and circumcircle $\Gamma$ satisfy $AB = 6\sqrt{3}, BC = 14,$ and $CA = 22$. Construct points $P$ and $Q$ on rays $BA$ and $CA$ such that $BP = CQ = 14$. Lines $PI$ and $QI$ meet the tangents from $B$ and $C$ to $\Gamma$, respectively, at points $X$ and $Y$. If $XY$ can be expressed ... | 31 | 160 | 2 |
math | In the garden of Anya and Vitya, there were 2006 rose bushes. Vitya watered half of all the bushes, and Anya watered half of all the bushes. It turned out that exactly three bushes, the most beautiful ones, were watered by both Anya and Vitya. How many rose bushes remained unwatered? | 3 | 77 | 1 |
math | II. (50 points) Find the largest real number $m$ such that the inequality
$$
\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+m \leqslant \frac{1+x}{1+y}+\frac{1+y}{1+z}+\frac{1+z}{1+x}
$$
holds for any positive real numbers $x$, $y$, $z$ satisfying $x y z=x+y+z+2$. | \frac{3}{2} | 105 | 7 |
math | Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that for every prime number $p$ and natural number $x$,
$$\{ x,f(x),\cdots f^{p-1}(x) \} $$
is a complete residue system modulo $p$. With $f^{k+1}(x)=f(f^k(x))$ for every natural number $k$ and $f^1(x)=f(x)$.
[i]Proposed by IndoMathXdZ[/i] | f(x) = x + 1 | 113 | 9 |
math | 10. (10 points) $n$ pirates divide gold coins. The 1st pirate takes 1 coin first, then takes $1 \%$ of the remaining coins; then, the 2nd pirate takes 2 coins, then takes $1 \%$ of the remaining coins; the 3rd pirate takes 3 coins, then takes $1 \%$ of the remaining coins; ... the $n$-th pirate takes $n$ coins, then ta... | 9801 | 142 | 4 |
math | $11$ theatrical groups participated in a festival. Each day, some of the groups were scheduled to perform while the remaining groups joined the general audience. At the conclusion of the festival, each group had seen, during its days off, at least $1$ performance of every other group. At least how many days did the fes... | 6 | 69 | 1 |
math | 37. Find the smallest $a \in \mathbf{N}^{*}$, such that the following equation has real roots:
$$
\cos ^{2} \pi(a-x)-2 \cos \pi(a-x)+\cos \frac{3 \pi x}{2 a} \cdot \cos \left(\frac{\pi x}{2 a}+\frac{\pi}{3}\right)+2=0 .
$$ | 6 | 92 | 1 |
math | Four. (15 points) The number of elements in set $S$ is denoted as $|S|$, and the number of subsets of set $S$ is denoted as $n(S)$. Given three non-empty finite sets $A$, $B$, and $C$ that satisfy the condition:
$$
\begin{array}{l}
|A|=|B|=2019, \\
n(A)+n(B)+n(C)=n(A \cup B \cup C) .
\end{array}
$$
Determine the maxim... | 2018 | 136 | 4 |
math | 10.060. The radii of the inscribed and circumscribed circles of a right triangle are 2 and 5 cm, respectively (Fig. 10.59). Find the legs of the triangle. | 6;8 | 49 | 3 |
math | 29. In the right-angled $\triangle A B C$, find the largest positive real number $k$ such that the inequality $a^{3}+b^{3}+c^{3} \geqslant k(a+$ $b+c)^{3}$ holds. (2006 Iran Mathematical Olympiad) | \frac{1}{\sqrt{2}(1+\sqrt{2})^{2}} | 69 | 19 |
math | Find all positive integers $x , y$ which are roots of the equation
$2 x^y-y= 2005$
[u] Babis[/u] | (x, y) = (1003, 1) | 38 | 14 |
math | Example 11 A convex polyhedron's surface is composed of 12 squares, 8 regular hexagons, and 6 regular octagons. At each vertex, one square, one octagon, and one hexagon intersect. Among the line segments connecting the vertices of the polyhedron, how many are inside the polyhedron, not on the faces or edges of the poly... | 840 | 85 | 3 |
math | Problem 7. On an island, there live knights who always tell the truth, and liars who always lie. One day, 15 natives, among whom were both knights and liars, stood in a circle, and each said: "Of the two people standing opposite me, one is a knight, and the other is a liar." How many of them are knights? | 10 | 78 | 2 |
math | Problem 4. Solve the equation:
$$
(x+1)^{2}+(x+3)^{2}+(x+5)^{2}+\ldots+(x+2021)^{2}=x^{2}+(x-2)^{2}+(x-4)^{2}+\ldots+(x-2020)^{2}
$$ | -0.5 | 82 | 4 |
math | 8. $n$ chess players participate in a chess tournament, with each pair of players competing in one match. The rules are: the winner gets 1 point, the loser gets 0 points, and in the case of a draw, both players get 0.5 points. If it is found after the tournament that among any $m$ players, there is one player who has w... | 2m-3 | 180 | 4 |
math | Find the $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $f(x-f(y))=1-x-y$. | f(x)=\frac{1}{2}-x | 31 | 11 |
math | G2.1 If $p=2-2^{2}-2^{3}-2^{4}-\ldots-2^{9}-2^{10}+2^{11}$, find the value of $p$. | 6 | 49 | 1 |
math | Example 1 Color each vertex of a square pyramid with one color, and make the endpoints of the same edge have different colors. If only 5 colors are available, then the total number of different coloring methods is $\qquad$
(1995, National High School Mathematics Competition)
The modified solution better illustrates the... | 420 | 69 | 3 |
math | 9. Team A and Team B each send out 7 players to participate in a Go chess tournament according to a pre-arranged order. Both sides start with the No. 1 player competing, the loser is eliminated, and the winner then competes with the No. 2 player of the losing side, ..., until all players of one side are eliminated, and... | 3432 | 100 | 4 |
math | 3. (3 points) If $a$ represents a three-digit number, and $b$ represents a two-digit number, then, the minimum value of $a+b$ is $\qquad$ the maximum value of $a+b$ is $\qquad$ , the minimum value of $a-b$ is $\qquad$ , and the maximum value of $a-b$ is $\qquad$ . | 110,1098,1,989 | 84 | 14 |
math | ## Task 1 - 020611
Inge asks her brother Klaus, who helped with the potato harvest at a LPG with his class during the autumn holidays, about the result of the harvest assistance.
Klaus replies: "In total, 15000 dt of potatoes were harvested. $\frac{1}{5}$ of this amount was collected by us students, $\frac{1}{3}$ of ... | 5000\mathrm{dt} | 146 | 9 |
math | Let $p$ be an odd prime number. Let $S$ be the sum of all primitive roots modulo $p$. Show that if $p-1$ isn't squarefree (i. e., if there exist integers $k$ and $m$ with $k>1$ and $p-1=k^2m$), then $S \equiv 0 \mod p$.
If not, then what is $S$ congruent to $\mod p$ ? | S \equiv 0 \mod p | 99 | 9 |
math | 6. Given the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ with the left vertex $A$ and the right focus $F$, let $P$ be any point on the hyperbola in the first quadrant. If $\angle P F A=2 \angle F A P$ always holds, then the eccentricity of the hyperbola is $\qquad$ | 2 | 95 | 1 |
math | 9. $[\boldsymbol{7}] g$ is a twice differentiable function over the positive reals such that
$$
\begin{aligned}
g(x)+2 x^{3} g^{\prime}(x)+x^{4} g^{\prime \prime}(x) & =0 \quad \text { for all positive reals } x . \\
\lim _{x \rightarrow \infty} x g(x) & =1
\end{aligned}
$$
Find the real number $\alpha>1$ such that $g... | \frac{6}{\pi} | 124 | 8 |
math | 7. What is $3 x+2 y$, if $x^{2}+y^{2}+x y=28$ and $8 x-12 y+x y=80$?
The use of a pocket calculator or any manuals is not allowed. | 0 | 57 | 1 |
math | 10. The general solution of the equation $\cos \frac{x}{4}=\cos x$ is $\qquad$ , and in $(0,24 \pi)$ there are $\qquad$ distinct solutions. | 20 | 46 | 2 |
math | 5. The number of non-empty subsets of the set $\{1,2, \cdots, 2016\}$ whose elements sum to an odd number is $\qquad$ | 2^{2015} | 40 | 7 |
math | 6. The range of the function $f(x)=\sqrt{3 x-6}+\sqrt{3-x}$ is
$\qquad$ | [1,2] | 31 | 5 |
math | 211. How many digits does the number $2^{100}$ have? | 31 | 19 | 2 |
math | Example 3 Given $n$ positive integers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying $x_{1}+x_{2}+\cdots+x_{n}=2020$. Find the maximum value of the product $x_{1} x_{2} \cdots x_{n}$ of these $n$ positive integers.
(Adapted from the 2008 National Junior High School Mathematics Competition Tianjin Preliminary Round) | 2^{2}\times3^{672} | 103 | 11 |
math | II. (50 points) Let $x_{i} \in[1,3], i=1,2, \cdots$, 2007, and $\sum_{i=1}^{2007} x_{i}=5988$. Try to find the maximum value of $\sum_{i=1}^{2007} x_{i}^{2008}$ and the values of $x_{i}(i=1,2, \cdots, 2007)$ at this time. | 16+1990 \times 3^{2008}+2^{2008} | 118 | 25 |
math | 1. A wooden cuboid with edges $72 \mathrm{dm}, 96 \mathrm{dm}$ and $120 \mathrm{dm}$ is divided into equal cubes, with the largest possible edge, such that the edge length is a natural number. The resulting cubes are arranged one on top of another, forming a column. How many meters high is the resulting column? | 144 | 80 | 3 |
math | 36. The front tires of a car wear out after 25,000 km, while the rear tires wear out after 15,000 km. When is it advisable to swap the tires to ensure they wear out evenly? (Assume that the tires are swapped once, although drivers do this more frequently in practice.) | 9375 | 71 | 4 |
math | 2. (HUN) For which real numbers $x$ does the following inequality hold: $$ \frac{4 x^{2}}{(1-\sqrt{1+2 x})^{2}}<2 x+9 ? $$ | -\frac{1}{2} \leq x < \frac{45}{8} \text{ and } x \neq 0 | 49 | 31 |
math | 7. (10 points) A deck of playing cards, excluding the jokers, has 4 suits totaling 52 cards, with each suit having 13 cards, numbered from 1 to 13. Feifei draws 2 hearts, 3 spades, 4 diamonds, and 5 clubs. If the sum of the face values of the spades is 11 times the sum of the face values of the hearts, and the sum of t... | 101 | 136 | 3 |
math | 9. Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=1, a_{n+1} a_{n-1}=a_{n} a_{n-1}+a_{n}^{2}\left(n \in \mathbf{N}^{*}, n \geqslant 2\right), \frac{a_{n+1}}{a_{n}}=k n+1$. Let $f(n)=\left\{\begin{array}{l}a_{n}+5 a_{n-1}\left(n=2 l-1, l \in \mathbf{N}^{*}\right) \\ 3 a_{n}+2 a_{n-1}\left(n=2 l, l \in \mathb... | 11 | 223 | 2 |
math | 2. (6 points) $2012 \times 2012 - 2013 \times 2011=$ | 1 | 32 | 1 |
math | Let $ABC$ be a triangle in which $\measuredangle{A}=135^{\circ}$. The perpendicular to the line $AB$ erected at $A$ intersects the side $BC$ at $D$, and the angle bisector of $\angle B$ intersects the side $AC$ at $E$.
Find the measure of $\measuredangle{BED}$. | 45^\circ | 80 | 4 |
math | 40. Find the best constant $k$ such that the following inequality holds for all positive real numbers $a, b, c$.
$$\frac{(a+b)(b+c)(c+a)}{a b c}+\frac{k(a b+b c+c a)}{a^{2}+b^{2}+c^{2}} \geq 8+k \quad(\text{Pham Kim Hung})$$ | 4 \sqrt{2} | 89 | 6 |
math | 7.2. Vasya left the village of Vasilki and walked to the village of Romashki. At the same time, Roma left the village of Romashki for the village of Vasilki. The distance between the villages is 36 km, Vasya's speed is 5 km/h, and Roma's speed is 4 km/h. At the same time as Vasya, Dima left Vasilki on a bicycle at a sp... | 36 | 163 | 2 |
math | 6. (8 points) Let for positive numbers $x, y, z$ the system of equations holds:
$$
\left\{\begin{array}{l}
x^{2}+x y+y^{2}=108 \\
y^{2}+y z+z^{2}=16 \\
z^{2}+x z+x^{2}=124
\end{array}\right.
$$
Find the value of the expression $x y+y z+x z$. | 48 | 102 | 2 |
math | 4. The positive integers $x, y, z$ satisfy
$$
x+2 y=z, \quad x^{2}-4 y^{2}+z^{2}=310 .
$$
Find all possible values of the product $xyz$. | 11935,2015 | 54 | 10 |
math | Bakayev E.v.
Petya places 500 kings on the cells of a $100 \times 50$ board so that they do not attack each other. And Vasya places 500 kings on the white cells (in a chessboard coloring) of a $100 \times 100$ board so that they do not attack each other. Who has more ways to do this? | Vasya | 93 | 3 |
math | \section*{Problem 2 - 281032}
Determine all pairs \((f, g)\) of functions \(f\) and \(g\), which are defined for all real numbers and satisfy the following conditions (1) to (4)!
(1) \(f\) is a quadratic function, in whose representation \(y=f(x)\) the coefficient of \(x^{2}\) is 1.
(2) For all real \(x\), \(f(x+1)=... | \begin{aligned}&1)\quadf(x)=x^{2}-16x+64\quad;\quad(x)=x^{2}-14x+49\\&2)\quadf(x)=x^{2}-8x+16\quad;\quad(x)=x^{2}-6x+9\end{aligned} | 136 | 73 |
math | ## Task B-1.1.
Determine all natural numbers $x, y, z$ for which
$$
4 x^{2}+45 y^{2}+9 z^{2}-12 x y-36 y z=25
$$
where $x<y<z$. | (1,2,3),(1,2,5),(2,3,6) | 65 | 19 |
math | 1. Given real numbers $x, y$ satisfy $3 x^{2}+4 y^{2}=1$. If $|3 x+4 y-\lambda|+|\lambda+7-3 x-4 y|$ is independent of $x, y$, then the range of values for $\lambda$ is | [\sqrt{7}-7,-\sqrt{7}] | 66 | 12 |
math | 2. Solve the inequality $\frac{\sqrt{x}-6}{2-\sqrt{x+4}} \geq 2+\sqrt{x+4}$. | x\in(0;4] | 32 | 8 |
math | 11. The Ace of Diamonds (From 8th grade, 2 points). Thirty-six players are playing a game: from a deck of 36 cards, they take turns drawing a random card. If a player draws the ace of diamonds, the player wins; if another card is drawn, the player returns it to the deck, and the next player draws a card. They draw card... | Thebetofeachsubsequentplayershouldbeintheof35:36tothebetofthepreviousone. | 198 | 28 |
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 | 3. Mowgli asked five monkeys to bring him some nuts. The monkeys collected an equal number of nuts and carried them to Mowgli. On the way, they quarreled, and each monkey threw one nut at each of the others. As a result, they brought Mowgli half as many nuts as they had collected. How many nuts did Mowgli receive? Be s... | 20 | 85 | 2 |
math | 【Question 3】
Cut a cube into 27 identical smaller cubes. The sum of the surface areas of these smaller cubes is 432 square centimeters larger than the surface area of the large cube. Then, the volume of the large cube is $\qquad$ cubic centimeters. | 216 | 62 | 3 |
math | $\square$ Example 1 Let real numbers $a_{1}, a_{2}, \cdots, a_{100}$ satisfy $a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{100} \geqslant 0, a_{1}+$ $a_{2} \leqslant 100, a_{3}+a_{4}+\cdots+a_{100} \leqslant 100$, determine the maximum value of $a_{1}^{2}+a_{2}^{2}+\cdots+a_{100}^{2}$, and find the sequence $a_{... | 10000 | 192 | 5 |
math | 1B. Determine all pairs of natural numbers $(a, b)$ such that $a \leq b$ and
$$
\left(a+\frac{6}{b}\right)\left(b+\frac{6}{a}\right)=25
$$ | (1,4),(2,2),(1,9),(3,3) | 54 | 17 |
math | 4. Triangle $A B C$ is similar to the triangle formed by its altitudes. Two sides of triangle $A B C$ are 4 cm and 9 cm. Find the third side. | 6 | 42 | 1 |
math | 3. (8 points) The teacher has a calculator, and each student comes up in turn to input a natural number. The first student inputs 1, the second student inputs 2, the third student inputs 3. Since the buttons 4 and 8 on the calculator are broken, the next student can only input 5, and the following students input 6, 7, ... | 155 | 135 | 3 |
math | 409. Several merchants contributed to a common fund 100 times as many rubles as there were merchants. They sent a trusted person to Venice, who received a number of rubles from each hundred rubles, twice the number of merchants. The question is: how many merchants were there, if the trusted person received 2662 rubles? | 11 | 75 | 2 |
math | 6-159 Functions $f, g: R \rightarrow R$ are not constant and satisfy
$$
\begin{array}{l}
f(x+y) \equiv f(x) g(y)+g(x) f(y), \\
g(x+y) \equiv g(x) g(y)-f(x) f(y),
\end{array} \quad x, y \in R .
$$
Find all possible values of $f(0)$ and $g(0)$. | f(0)=0,(0)=1 | 101 | 9 |
math | ## 19. Red and Green
On a large one-way street, two traffic lights are located one after another; each traffic light is designed so that the period when the green light is on constitutes two-thirds of the total operating time of the traffic light. A motorist noticed that when he, moving at a normal speed, passes the f... | \frac{1}{2} | 123 | 7 |
math | 6. Let $a$ and $b$ be real numbers such that the equation $x^{4} + a x^{3} + b x^{2} + a x + 1 = 0$ has at least one real root. For all such pairs of real numbers $(a, b)$, find the minimum value of $a$. (15th IMO Problem) | -2 | 79 | 2 |
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