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 1. In the first barn, there are 1999 more apples than in the second barn. In which barn will there be more apples and by how many, if we transfer 1000 apples from the first barn to the second barn. | 1 | 55 | 1 |
math | 3-3. A two-digit number, when added to the number written with the same digits but in reverse order, gives a perfect square. Find all such numbers. | 29,38,47,56,65,74,83,92 | 34 | 23 |
math | Task B-4.1. Determine the term in the expansion of the binomial
$$
\left(\sqrt[3]{\frac{2 x}{\sqrt{y}}}-\sqrt{\frac{y}{2 \sqrt[3]{x}}}\right)^{21}, \quad x, y>0
$$
which contains $x$ and $y$ with the same exponent. | -\binom{21}{9}\cdot2^{-\frac{1}{2}}x^{\frac{5}{2}}y^{\frac{5}{2}} | 85 | 37 |
math | Let $a \in \mathbb{N}^{*}, n, m \in \mathbb{N}$. Calculate the GCD of $a^{n}-1$ and $a^{m}-1$. | ^{n\wedge}-1 | 46 | 7 |
math | For a positive integer $n$, let $d\left(n\right)$ be the number of positive divisors of $n$ (for example $d\left(39\right)=4$). Estimate the average value that $d\left(n\right)$ takes on as $n$ ranges from $1$ to $2019$. An estimate of $E$ earns $2^{1-\left|A-E\right|}$ points, where $A$ is the actual answer.
[i]2019 ... | \frac{15682}{2019} | 125 | 14 |
math | ## 4. Submerged Bodies
Marta is playing with models of geometric bodies by submerging them in water in a container shaped like a cube. If she places a cube and a cylinder in the container, the water level rises $5 \mathrm{~cm}$ more than if she only places a pyramid. If she places a cube and a pyramid in the container... | 150 | 159 | 3 |
math | 9. Given two points $M(-5,0)$ and $N(5,0)$. If there exists a point $P$ on a line such that $|P M|-|P N|=6$, then the line is called a "harmonious line". Given the following lines:
(1) $y=x-1$;
(2) $y=2$;
(3) $y=\frac{5}{3} x$;
(4) $y=2 x+1$,
which of these lines are harmonious lines? $\qquad$ | (1),(2) | 121 | 5 |
math | 10.276. A right triangle $ABC$ is divided by the altitude $CD$, drawn to the hypotenuse, into two triangles $BCD$ and $ACD$. The radii of the circles inscribed in triangles $BCD$ and $ACD$ are 4 and $3 \text{ cm}$, respectively. Find the distance between their centers. | 5\sqrt{2} | 80 | 6 |
math | 11. Stocks of "Maloney & Co". On Monday, April 7, the stock of the "Maloney & Co" Trading House was worth 5 Anchurian dollars. Over the next six days, the stock did not fall in price or even rose, and by Sunday, April 13, the stock price reached 5 dollars and 14 cents. For the entire following week, the stocks did not ... | B | 244 | 1 |
math | 8. A batch of goods is transported from city $A$ to city $B$ by 17 freight trains at a uniform speed of $v$ kilometers per hour. The railway line between $A$ and $B$ is 400 kilometers long. For safety, the distance between two freight trains must not be less than $\left(\frac{v}{20}\right)^{2}$ kilometers. If the lengt... | 8 | 126 | 1 |
math | Example 4 Given $\cdots 1 \leqslant a \leqslant 1$, the inequality $\left(\frac{1}{2}\right)^{x^{2}+a x}<\left(\frac{1}{2}\right)^{2 x+a-1}$ always holds. Find the range of values for $x$. | x>2 \text{ or } x<0 | 75 | 11 |
math | 4. In how many ways can the number 210 be factored into the product of four natural numbers? The order of the factors does not matter.
$(12$ points) | 15 | 39 | 2 |
math | Jay notices that there are $n$ primes that form an arithmetic sequence with common difference $12$. What is the maximum possible value for $n$?
[i]Proposed by James Lin[/i] | 5 | 42 | 1 |
math | 90. In the cube $A B C D A_{1} B_{1} C_{1} D_{1}$, point $M$ is taken on $A C$ and point $N$ is taken on the diagonal $B D_{1}$ of the cube such that $\widehat{N M C}=$ $=60^{\circ}, \widehat{M N B}=45^{\circ}$. In what ratio do points $M$ and $N$ divide the segments $A C$ and $B D_{1}$? | |AM|:|MC|=2-\sqrt{3},|BN|:|ND_{1}|=2 | 119 | 24 |
math | Three, write 10 different
natural numbers, such that each of them is a divisor of the sum of these 10 numbers (explain the reason why the 10 natural numbers written meet the conditions of the problem). | 1,2,3,6,12,24,48,96,192,384 | 47 | 27 |
math | Given acute triangle $\triangle ABC$ in plane $P$, a point $Q$ in space is defined such that $\angle AQB = \angle BQC = \angle CQA = 90^\circ.$ Point $X$ is the point in plane $P$ such that $QX$ is perpendicular to plane $P$. Given $\angle ABC = 40^\circ$ and $\angle ACB = 75^\circ,$ find $\angle AXC.$ | 140^\circ | 99 | 5 |
math | 8. (10 points) A deck of playing cards, excluding the joker, 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 4 cards Feifei draws is exactly 34, then among them, there are $\qquad$ cards ... | 4 | 106 | 1 |
math | 7. Given the sequence $\left\{a_{n}\right\}$ with the general term
$$
a_{n}=n^{4}+6 n^{3}+11 n^{2}+6 n \text {. }
$$
Then the sum of the first 12 terms $S_{12}=$ $\qquad$ | 104832 | 74 | 6 |
math | Portraits of famous scientists hang on a wall. The scientists lived between 1600 and 2008, and none of them lived longer than 80 years. Vasya multiplied the years of birth of these scientists, and Petya multiplied the years of their death. Petya's result is exactly $ 5\over 4$ times greater than Vasya's result. What m... | 5 | 108 | 1 |
math | $\begin{aligned} & \text { [ Examples and counterexamples. Constructions ] } \\ & \text { [Numerical inequalities. Comparing numbers.] }\end{aligned}$
The sum of several numbers is 1. Can the sum of their squares be less than 0.1? | 1/11<0.1 | 62 | 8 |
math | Task B-1.5. In a container, which is not full to the brim, there is a solution containing $85 \%$ alcohol. We fill the container to the brim with a solution containing $21 \%$ alcohol and mix it thoroughly. If we then pour out as much liquid as we added and repeat the process (again adding a $21 \%$ alcohol solution), ... | 77 | 124 | 2 |
math | 6. All natural numbers, the sum of the digits of each of which is equal to 5, were arranged in ascending order. What number is in the 125th place
# | 41000 | 40 | 5 |
math | A triangle has sides of integer lengths, the radius of its inscribed circle is one unit. What is the radius of the circumscribed circle? | \frac{5}{2} | 30 | 7 |
math | ## Task B-3.1.
The expression $\cos ^{2} x+\cos ^{2}\left(\frac{\pi}{6}+x\right)-2 \cos \frac{\pi}{6} \cdot \cos x \cdot \cos \left(\frac{\pi}{6}+x\right)$ has the same value for all real numbers $x$. Calculate that value. | \frac{1}{4} | 84 | 7 |
math | In a tennis tournament, each player advances to the next round only in case of victory. If it is not possible for an even number of players to always advance to the next round, the tournament organizers decide which rounds certain players should play. For example, a seeded player can, at the discretion of the organizer... | 2010 | 125 | 4 |
math | (7) Let the function $f(x)=\left\{\begin{array}{l}\frac{1}{p}\left(x=\frac{q}{p}\right), \\ 0\left(x \neq \frac{q}{p}\right),\end{array}\right.$ where $p, q$ are coprime (prime), and $p \geqslant 2$. Then the number of $x$ values that satisfy $x \in[0,1]$, and $f(x)>\frac{1}{5}$ is $\qquad$ | 5 | 123 | 1 |
math | The 17th question: Given an integer $n \geq 2$, it is known that real numbers $x_{1}, x_{2}, \ldots, x_{n}$ satisfy $\sum_{i=1}^{n} i x_{i}=1$,
(1) Find the minimum value of $\sum_{k=1}^{n} x_{k}^{2}+\sum_{1 \leq i<j \leq n} x_{i} x_{j}$;
(2) Find the minimum value of $\sum_{k=1}^{n} x_{k}^{2}+\sum_{1 \leq i<j \leq n}\... | \frac{12}{n(n+2)(3n+1)} | 162 | 16 |
math | A frog jumps from vertex to vertex of triangle $A B C$, moving each time to one of the adjacent vertices.
In how many ways can it get from $A$ to $A$ in $n$ jumps? | a_{n}=\frac{2^{n}+2\cdot(-1)^{n}}{3} | 45 | 24 |
math | 3. In a triangle, $\beta=74^{\circ} 18^{\prime}$ and $\gamma=38^{\circ} 46^{\prime}$, and $|A C|-|A B|=2.5 \mathrm{~cm}$. Calculate the lengths of the sides $|A B|$ and $|A C|$ and round the result to two decimal places. Draw a sketch. | b=7.1\mathrm{~},=4.6\mathrm{~} | 90 | 19 |
math | Given a continuous function $ f(x)$ such that $ \int_0^{2\pi} f(x)\ dx \equal{} 0$.
Let $ S(x) \equal{} A_0 \plus{} A_1\cos x \plus{} B_1\sin x$, find constant numbers $ A_0,\ A_1$ and $ B_1$ for which $ \int_0^{2\pi} \{f(x) \minus{} S(x)\}^2\ dx$ is minimized. | A_0 = 0, A_1 = \frac{1}{\pi} \int_0^{2\pi} f(x) \cos x \, dx, B_1 = \frac{1}{\pi} \int_0^{2\pi} f(x) \sin x \, dx | 111 | 69 |
math | 5.18. In parallelogram $A B C D$, point $K$ is the midpoint of side $B C$, and point $M$ is the midpoint of side $C D$. Find $A D$, if $A K=6$ cm, $A M=3$ cm, and $\angle K A M=60^{\circ}$. | 4 | 78 | 1 |
math | 1074*. Find all such rational numbers $x$ and $y$ for which the numbers $x+\frac{1}{y}$ and $y+\frac{1}{x}$ are natural. | 1,1;2,\frac{1}{2};\frac{1}{2},2 | 43 | 20 |
math | $(MON 1)$ Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other di... | 1944 | 67 | 4 |
math | 6. (4 points) Given three circles with radii 2, 3, and 5, each pair of circles touching each other externally at points $A, B$, and $C$. Find the radius of the inscribed circle of triangle $A B C$. | \frac{9-\sqrt{21}}{2\sqrt{7}}=\frac{9\sqrt{7}-7\sqrt{3}}{14} | 56 | 36 |
math | Given $a$, $b$, and $c$ are complex numbers satisfying
\[ a^2+ab+b^2=1+i \]
\[ b^2+bc+c^2=-2 \]
\[ c^2+ca+a^2=1, \]
compute $(ab+bc+ca)^2$. (Here, $i=\sqrt{-1}$) | \frac{-11 - 4i}{3} | 78 | 12 |
math | Establish necessary and sufficient conditions on the constant $k$ for the existence of a continuous real valued function $f(x)$ satisfying $$f(f(x))=kx^9$$ for all real $x$. | k \geq 0 | 42 | 7 |
math | 2.1. The segment $A B$ is divided into three equal parts by points $C(3,4)$ and $D(5,6)$. Find the coordinates of points $A$ and $B$. | A(1,2),B(7,8) | 46 | 12 |
math | 6-12 $f(n)$ is a function defined on the set of positive integers, taking non-negative integer values, and for all $m, n$ we have:
$$
\begin{array}{l}
f(m+n)-f(m)-f(n)=0 \text { or } 1 ; \\
f(2)=0, f(3)>0, f(9999)=3333 .
\end{array}
$$
Try to find: $f(1982)$. | 660 | 109 | 3 |
math | 356. Find: a) variance; b) standard deviation of the exponential distribution given by the probability density function: $f(x)=\lambda \mathrm{e}^{-\lambda x}$ for $x \geqslant 0 ; f(x)=0$ for $x<0$. | D(X)=\frac{1}{\lambda^2},\sigma(X)=\frac{1}{\lambda} | 63 | 25 |
math | 2. (10 points) Calculate:
$$
10 \times 9 \times 8 + 7 \times 6 \times 5 + 6 \times 5 \times 4 + 3 \times 2 \times 1 - 9 \times 8 \times 7 - 8 \times 7 \times 6 - 5 \times 4 \times 3 - 4 \times 3 \times 2
$$ | 132 | 100 | 3 |
math | 10. Write the number 7 using five twos and arithmetic operation signs. Find several solutions. | 7 | 21 | 1 |
math | By how many zeros does the number 2012! end? | 501 | 15 | 3 |
math | Problem 4. Place digits in the positions of $a$ and $b$, in the four-digit numbers, so that the sum $\overline{323 a}+\overline{b 410}$ is divisible by 9. Determine all possible solutions. | (0,5),(1,4),(4,1),(2,3),(3,2),(5,9),(9,5),(6,8),(8,6),(7,7) | 57 | 41 |
math | 202. Simplify the sum
$$
\left[\frac{n+1}{2}\right]+\left[\frac{n+2}{4}\right]+\left[\frac{n+4}{8}\right]+\ldots+\left[\frac{n+2^{k}}{2^{k+1}}\right]+\ldots
$$
where $n$ is a positive integer. | n | 81 | 1 |
math | 8. Given that $f(x)$ is a function defined on $\mathbf{R}$, and satisfies $f(x+2)[1-f(x)]=1+f(x), f(1)=$ 9997, then the value of $f(2009)$ is $\qquad$ . | 9997 | 66 | 4 |
math | 5. Find all solutions of the equation $|\sin 2 x-\cos x|=|| \sin 2 x|-| \cos x||$ on the interval $(-2 \pi ; 2 \pi]$. Answer: $(-2 \pi ;-\pi] \cup[0 ; \pi] \cup\left\{-\frac{\pi}{2} ; \frac{3 \pi}{2} ; 2 \pi\right\}$. | (-2\pi;-\pi]\cup[0;\pi]\cup{-\frac{\pi}{2};\frac{3\pi}{2};2\pi} | 98 | 36 |
math | 1. Calculate the sum $S=S_{1}+S_{2}$, where $S_{1}$ and $S_{2}$ are given by
$$
S_{1}=\frac{1}{\log _{\operatorname{tg} 1^{\circ}} 2}+\frac{2}{\log _{\operatorname{tg} 2^{\circ}} 2^{2}}+\ldots+\frac{44}{\log _{\operatorname{tg} 44^{\circ}} 2^{44}} \text { and } S_{2}=\frac{46}{\log _{\operatorname{tg} 46^{\circ}} 2^{4... | 0 | 215 | 1 |
math | ### 3.34. Compute the integral
$$
\int_{L} \frac{\sin z}{z\left(z-\frac{\pi}{2}\right)} d z
$$
where $L-$ is a rectangle bounded by the following lines: $x=2, x=-1, y=2, y=-1$. | 4i | 72 | 2 |
math | ## Task A-2.2.
Determine all real solutions of the equation
$$
4 x^{2}-20\lfloor x\rfloor+9=0
$$
where $\lfloor x\rfloor$ denotes the greatest integer not greater than $x$. | \frac{1}{2}\sqrt{11},\quad\frac{1}{2}\sqrt{31},\quad\frac{1}{2}\sqrt{51},\quad\frac{1}{2}\sqrt{71} | 59 | 54 |
math | 4. The distance between the foci of the conic section
$$
(3 x+4 y-13)(7 x-24 y+3)=200
$$
is $\qquad$ | 2\sqrt{10} | 46 | 7 |
math | 8. Arrange all positive integers whose sum of digits is 10 in ascending order to form a sequence $\left\{a_{n}\right\}$, if $a_{n}=2017$, then $n=$ $\qquad$ | 120 | 52 | 3 |
math | 4. On the side $B C$ of triangle $A B C$, a point $M$ is taken such that $B M: M C=3: 7$. The bisector $B L$ of the given triangle and the segment $A M$ intersect at point $P$ at an angle of $90^{\circ}$.
a) Find the ratio of the area of triangle $A B P$ to the area of quadrilateral $L P M C$.
b) On the segment $M C$... | 39:161,\arccos\sqrt{\frac{13}{15}} | 159 | 21 |
math | Problem 11.2. Find all values of the real parameter $a$ such that the equation
$$
\lg (a x+1)=\lg (x-1)+\lg (2-x)
$$
has exactly one solution.
Aleksander Ivanov | \in(-1,-\frac{1}{2}]\cup{3-2\sqrt{3}} | 57 | 23 |
math | 2. Suppose Bag A contains 4 white balls, 5 red balls, and 6 black balls; Bag B contains 7 white balls, 6 red balls, and 2 black balls. If one ball is drawn from each bag, then the probability that the two balls are of different colors is $\qquad$ . | \frac{31}{45} | 67 | 9 |
math | 1. The function $y=\log _{a}\left(x^{2}-a x+2\right)$ is always positive on $[2,+\infty)$. Then the range of the real number $a$ is $\qquad$ | 1<a<\frac{5}{2} | 52 | 10 |
math | 5. Let $S=\{-3,-2,1,2,3,4\}$, and take any two numbers $a, b(a \neq b)$ from $S$. Let $g(a, b)$ be the minimum value of the function $f(x)=x^{2}-(a+b) x+a b$ with respect to the variable $x$, then the maximum value of $g(a, b)$ is $\qquad$. | -\frac{1}{4} | 94 | 7 |
math | $\underline{\text { Tolkpy A.K. }}$
A circle is divided into seven arcs such that the sum of any two adjacent arcs does not exceed $103^{\circ}$.
Name the largest number $A$ such that in any such division, each of the seven arcs contains at least $A^{\circ}$. | 51 | 70 | 2 |
math | ## 186. Math Puzzle $11 / 80$
Jens is supposed to choose the pot with the largest capacity from two cylindrical cooking pots.
The brown pot is twice as high as the blue one, but the blue pot is $11 / 2 \mathrm{times}$ as wide as the brown one.
Which cooking pot should Jens choose? | V_{1}>V_{2} | 76 | 8 |
math | 7. Find the maximum value of the expression
$$
\left(x_{1}-x_{2}\right)^{2}+\left(x_{2}-x_{3}\right)^{2}+\ldots+\left(x_{2010}-x_{2011}\right)^{2}+\left(x_{2011}-x_{1}\right)^{2}
$$
for $x_{1}, \ldots, x_{2011} \in[0 ; 1]$.
# | 2010 | 113 | 4 |
math | 4. How many numbers divisible by 4 and less than 1000 do not contain any of the digits $6,7,8,9$ or 0. | 31 | 37 | 2 |
math | 11. (10 points) Calculate: $\frac{\left(1+\frac{1}{2}\right)^{2} \times\left(1+\frac{1}{3}\right)^{2} \times\left(1+\frac{1}{4}\right)^{2} \times\left(1+\frac{1}{5}\right)^{2} \times \cdots \times\left(1+\frac{1}{10}\right)^{2}}{\left(1-\frac{1}{2^{2}}\right) \times\left(1-\frac{1}{3^{2}}\right) \times\left(1-\frac{1}{... | 55 | 201 | 2 |
math | 4. [4 points] Solve the system of equations
$$
\left\{\begin{array}{l}
3 x^{2}+3 y^{2}-x^{2} y^{2}=3 \\
x^{4}+y^{4}-x^{2} y^{2}=31
\end{array}\right.
$$ | (\sqrt{5};\\sqrt{6}),(-\sqrt{5};\\sqrt{6}),(\sqrt{6};\\sqrt{5}),(-\sqrt{6};\\sqrt{5}) | 73 | 42 |
math | A number $ x$ is uniformly chosen on the interval $ [0,1]$, and $ y$ is uniformly randomly chosen on $ [\minus{}1,1]$. Find the probability that $ x>y$. | \frac{3}{4} | 44 | 7 |
math | 9. Let $x, y \in \mathbf{R}$ satisfy $x-6 \sqrt{y}-4 \sqrt{x-y}+12=0$, then the range of values for $x$ is | 14-2\sqrt{13}\leqx\leq14+2\sqrt{13} | 47 | 25 |
math | Let $n$ be a positive integer, and let $a>0$ be a real number. Consider the equation:
\[ \sum_{i=1}^{n}(x_i^2+(a-x_i)^2)= na^2 \]
How many solutions ($x_1, x_2 \cdots , x_n$) does this equation have, such that:
\[ 0 \leq x_i \leq a, i \in N^+ \] | 2^n | 101 | 2 |
math | Let $E$ be the set of all bijective mappings from $\mathbb R$ to $\mathbb R$ satisfying
\[f(t) + f^{-1}(t) = 2t, \qquad \forall t \in \mathbb R,\]
where $f^{-1}$ is the mapping inverse to $f$. Find all elements of $E$ that are monotonic mappings. | f(x) = x + c | 83 | 7 |
math | In the final of a play competition for March 8, two plays made it to the final. In the first play, $n$ students from class 5A participated, and in the second play, $n$ students from class 5B participated. At the play, $2n$ mothers of all $2n$ students were present. The best play is chosen by the mothers' votes. It is k... | 1-2^{n-n} | 159 | 7 |
math | ## Task 2 - 270622
(a) In a competition, exactly four teams $A, B, C$, and $D$ participated, and each of these teams played exactly one game against each of the other teams. List these games!
(b) In another competition, each of the participating teams played exactly one game against each of the other participating te... | 7 | 115 | 1 |
math | 8. Let the rational number $r=\frac{p}{q} \in(0,1)$, where $p, q$ are coprime positive integers, and $pq$ divides 3600. The number of such rational numbers $r$ is $\qquad$ . | 112 | 62 | 3 |
math | Task 5. Find all triples $(a, b, c)$ of positive integers with $a+b+c=10$ such that there exist $a$ red, $b$ blue, and $c$ green points (all distinct) in the plane with the following properties:
- for each red point and each blue point, we consider the distance between these two points; the sum of all these distances ... | (8,1,1) | 147 | 7 |
math | Find the sum of the smallest and largest possible values for $x$ which satisfy the following equation.
$$9^{x+1} + 2187 = 3^{6x-x^2}.$$ | 5 | 44 | 1 |
math | Example 5. One root of the equation $3 x^{2}-5 x+a=0$ is greater than -2 and less than $0_{3}$, the other is much greater than 1 and less than 3. Find the range of real values for a. | -12<\mathrm{a}<0 | 58 | 10 |
math | 5. The numbers $1,2,3, \ldots, 99$ are written on the board. Petya and Vasya are playing a game, with Petya starting. Each move involves erasing three numbers that sum to 150. The player who cannot make a move loses. Which player can win, regardless of how the opponent plays? | Petya | 79 | 3 |
math | Example 14 Given the equations in $x$: $4 x^{2}-8 n x- 3 n=2$ and $x^{2}-(n+3) x-2 n^{2}+2=0$. Does there exist a value of $n$ such that the square of the difference of the two real roots of the first equation equals an integer root of the second equation? If it exists, find such $n$ values; if not, explain the reason.... | n=0 | 117 | 3 |
math | Example 13 Given $\alpha, \beta \in\left(0, \frac{\pi}{2}\right)$, and $\sin \beta=2 \cos (\alpha+\beta) \cdot \sin \alpha\left(\alpha+\beta \neq \frac{\pi}{2}\right)$, find the maximum value of $\tan \beta$.
---
The translation maintains the original text's format and line breaks as requested. | \frac{\sqrt{3}}{3} | 92 | 10 |
math | G1.2 Let $a_{1}, a_{2}, a_{3}, \ldots$ be an arithmetic sequence with common difference 1 and $a_{1}+a_{2}+a_{3}+\ldots+a_{100}=2012$. If $P=a_{2}+a_{4}+a_{6}+\ldots+a_{100}$, find the value of $P$. | 1031 | 95 | 4 |
math | The area of the right triangle ABC ( $\angle C=90^{\circ}$ ) is 6, and the radius of the circumscribed circle around it is $\frac{5}{2}$. Find the radius of the circle inscribed in this triangle.
# | 1 | 56 | 1 |
math | Adámek was recalculating his collection of rainbow marbles. He found that he could divide them into equal piles in several ways. If he divided them into three piles, there would be $\mathrm{v}$ more marbles in each pile than if he divided them into four piles.
How many rainbow marbles did Adámek have?
(E. Semerádová)... | 96 | 95 | 2 |
math | Given a circle of radius $r$ and a tangent line $\ell$ to the circle through a given point $P$ on the circle. From a variable point $R$ on the circle, a perpendicular $RQ$ is drawn to $\ell$ with $Q$ on $\ell$. Determine the maximum of the area of triangle $PQR$. | \frac{r^2 \sqrt{3}}{8} | 74 | 14 |
math | 3. (3 points) Some integers, when divided by $\frac{5}{7}, \frac{7}{9}, \frac{9}{11}, \frac{11}{13}$ respectively, yield quotients that, when expressed as mixed numbers, have fractional parts of $\frac{2}{5}, \frac{2}{7}, \frac{2}{9}, \frac{2}{11}$ respectively. The smallest integer greater than 1 that satisfies these ... | 3466 | 107 | 4 |
math | 7. If the integer $k$ is added to $36,300,596$, respectively, the results are the squares of three consecutive terms in an arithmetic sequence, find the value of $k$. | 925 | 46 | 3 |
math | 13. In $\triangle A B C$, $\angle B=\frac{\pi}{4}, \angle C=\frac{5 \pi}{12}, A C$ $=2 \sqrt{6}, A C$'s midpoint is $D$. If a line segment $P Q$ of length 3 (point $P$ to the left of point $Q$) slides on line $B C$, then the minimum value of $A P+D Q$ is $\qquad$. | \frac{\sqrt{30}+3\sqrt{10}}{2} | 104 | 19 |
math | 8.065. $\sin 2z + \cos 2z = \sqrt{2} \sin 3z$. | z_{1}=\frac{\pi}{20}(8n+3);z_{2}=\frac{\pi}{4}(8k+1),n,k\inZ | 29 | 38 |
math | Task 3. A landscaping team worked on a large and a small football field, with the area of the large field being twice the area of the small field. In the part of the team that worked on the large field, there were 6 more workers than in the part that worked on the small field. When the landscaping of the large field wa... | 16 | 102 | 2 |
math | 12. The marked price of a product is $p$ \% higher than the cost. When the product is sold at a discount, to avoid a loss, the discount on the selling price (i.e., the percentage reduction) must not exceed $d$ \%. Then $d$ can be expressed in terms of $p$ as $\qquad$ | \frac{100 p}{100+p} | 73 | 13 |
math | [The angle between two chords and two secants] [Sum of the angles of a triangle. Theorem about the external angle.]
Vertices $B, C, D$ of quadrilateral $A B C D$ are located on a circle with center $O$, which intersects side
$A B$ at point $F$, and side $A D$ at point $E$. It is known that angle $B A D$ is a right an... | \frac{3\pi}{7} | 133 | 9 |
math | Question 82: Find all real-coefficient polynomials $P$ such that for any $x \in \mathbb{R}$, we have $P(x) P(x+1) = P(x^2 - x + 3)$. | P(x)=(x^{2}-2x+3)^{n},n\in\mathbb{N} | 53 | 24 |
math | How many of the first 1000 positive integers can be expressed as:
$$
\lfloor 2 x\rfloor+\lfloor 4 x\rfloor+\lfloor 6 x\rfloor+\lfloor 8 x\rfloor \text { of }
$$
form?
where $\mathbf{x}$ is some real number, and $\mathrm{Lz}\rfloor$ denotes the greatest integer not exceeding $z$. | 600 | 94 | 3 |
math | 3. In $\triangle A B C$, $A B=2, A C=1, B C=\sqrt{7}$, $O$ is the circumcenter of $\triangle A B C$, and $\overrightarrow{A O}=\lambda \overrightarrow{A B}+\mu \overrightarrow{A C}$. Then $\lambda+\mu=$ $\qquad$ . | \frac{13}{6} | 80 | 8 |
math | The sum of the digits of all counting numbers less than 13 is
$$
1+2+3+4+5+6+7+8+9+1+0+1+1+1+2=51
$$
Find the sum of the digits of all counting numbers less than 1000. | 13500 | 71 | 5 |
math | 3. How many solutions in natural numbers does the equation
$$
(2 x+y)(2 y+x)=2017^{2017} ?
$$ | 0 | 35 | 1 |
math | 3. Find the sum of the first 10 elements that are found both in the arithmetic progression $\{5,8,11,14, \ldots\}$ and in the geometric progression $\{10,20,40,80, \ldots\} \cdot(10$ points) | 6990500 | 69 | 7 |
math | $11 \cdot 28$ in which base, $4 \cdot 13=100$?
(Kyiv Mathematical Olympiad, 1953) | 6 | 39 | 1 |
math | Determine the number of integers $n$ such that $1 \leqslant n \leqslant 10^{10}$, and such that for all $k=1,2, \ldots, 10$, the integer $n$ is divisible by $k$. | 3968253 | 63 | 7 |
math | ## Task 2 - 060912
If we form the cross sum of a natural number and then (if possible) the cross sum of this number again, etc., we finally obtain a single-digit number, which we will call the "final cross sum." By definition, the cross sum of a single-digit number is set to be the number itself.
Calculate how many n... | 111 | 100 | 3 |
math | Example 3. Find the zeros of the function $f(z)=1+\operatorname{ch} z$ and determine their order. | z_{k}=i\pi(2k+1)\quad(k=0,\1,\ldots) | 28 | 23 |
math | 【Question 12】Given three natural numbers $1,2,3$, perform an operation on these three numbers, replacing one of the numbers with the sum of the other two, and perform this operation 9 times. After these operations, the maximum possible value of the largest number among the three natural numbers obtained is $\qquad$ _. | 233 | 70 | 3 |
math | $2 \cdot 56$ Find the smallest natural number, such that when its last digit is moved to the first position, the number is multiplied by 5. | 142857 | 35 | 6 |
math | 12. An ant is at a vertex of a tetrahedron. Every minute, it randomly moves to one of the adjacent vertices. What is the probability that after one hour, it stops at the original starting point? $\qquad$ . | \frac{3^{59}+1}{4\cdot3^{59}} | 51 | 19 |
math | Example 2 Let $x, y, z$ be real numbers, not all zero. Find the maximum value of $\frac{x y+2 y z}{x^{2}+y^{2}+z^{2}}$.
| \frac{\sqrt{5}}{2} | 49 | 10 |
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