task_type stringclasses 4
values | problem stringlengths 21 5.23k | answer stringlengths 1 8.29k | problem_tokens int64 11 1.16k | answer_tokens int64 1 2.04k |
|---|---|---|---|---|
math | Find all integer solutions of the equation:
$$
x^{2}+y^{2}=3\left(u^{2}+v^{2}\right)
$$ | (0,0,0,0) | 34 | 9 |
math | Subject 3. Determine the cardinality of the set $A=\left\{\overline{a b} \left\lvert\, \frac{16}{a^{2}+b} \in \mathbb{N}\right.\right\}$. | 8 | 57 | 1 |
math | 31. $\frac{4^{x}-2^{x+1}+8}{2^{1-x}}<8^{x}$. | x>1 | 30 | 3 |
math | 2. Variant 1. Find the number of four-digit numbers where the digit in the units place is exactly 1 more than the digit in the tens place (the number cannot start with zero). | 810 | 40 | 3 |
math | 3.1.15 ** Let $\left\{a_{n}\right\}$ be a geometric sequence with all terms being positive, and
$$
S=a_{1}+a_{2}+\cdots+a_{n}, T=\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}},
$$
find the product of the first $n$ terms of $\left\{a_{n}\right\}$. | a_{1}a_{2}\cdotsa_{n}=(\frac{S}{T})^{\frac{n}{2}} | 106 | 29 |
math | 11. Using 120 unit cubes, form a rectangular prism, and paint its entire surface. The maximum number of unit cubes that have only one face painted is $\qquad$ pieces
Constructing the above text in English, please retain the original text's line breaks and format, and output the translation result directly. | 64 | 67 | 2 |
math | 4. In the box, there are 3 white cups, 3 red cups, and 2 black cups. Sonya took out 5 cups at random. What is the probability that she took out 2 white, 2 red, and 1 black cup? (Round the answer to the nearest hundredth). | 0.32 | 66 | 4 |
math | Find all pairs of integers $a, b$ such that the following system of equations has a unique integral solution $(x , y , z )$ :
$\begin{cases}x + y = a - 1 \\
x(y + 1) - z^2 = b \end{cases}$ | b = \frac{a^2}{4} | 63 | 12 |
math | For integers $a, b, c$ and $d$, it holds that $a>b>c>d$ and
$$
(1-a)(1-b)(1-c)(1-d)=10
$$
What values can the expression $a+b-c-d$ take? | 9 | 57 | 1 |
math | # 7. Variant 1.
100 natural numbers are written in a circle. It is known that among any three consecutive numbers, there is an even number. What is the smallest number of even numbers that can be among the written numbers? | 34 | 51 | 2 |
math | Example 4 Given that the area of quadrilateral $A B C D$ is $9, P$ is a point inside the quadrilateral. Let the centroids of $\triangle A B P, \triangle B C P, \triangle C D P$, and $\triangle D A P$ be $G_{1}, G_{2}, G_{3}, G_{4}$ respectively. Find the area of quadrilateral $G_{1} G_{2} G_{3} G_{4}$. | 2 | 103 | 1 |
math | 3. At present, the exchange rates of the US dollar and the euro are as follows: $D=6$ yuan and $E=7$ yuan. The People's Bank of China determines the yuan exchange rate independently of market conditions and adheres to a policy of approximate equality of currencies. One bank employee proposed the following scheme for ch... | 2 | 247 | 1 |
math | In the triangle $ABC$, point $D$ is given on the extension of side $CA$ beyond $A$, and point $E$ is given on the extension of side $CB$ beyond $B$, such that $AB = AD = BE$. The angle bisectors from $A$ and $B$ of triangle $ABC$ intersect the opposite sides at points $A_1$ and $B_1$, respectively. What is the area of ... | 6 | 129 | 1 |
math | ## Task A-2.1.
Determine the value of the real parameter $p$ so that the solutions of the equation
$$
(p-3) x^{2}+\left(p^{2}+1\right) x-11 p+18=0
$$
are the lengths of the legs of a right triangle with a hypotenuse of length $\sqrt{17}$. | 2 | 85 | 1 |
math | Twenty-seven balls labelled from $1$ to $27$ are distributed in three bowls: red, blue, and yellow. What are the possible values of the number of balls in the red bowl if the average labels in the red, blue and yellow bowl are $15$, $3$, and $18$, respectively? | \{11, 16, 21\} | 67 | 15 |
math | The sequence $a_{n}$ is defined as follows: $a_{1}=1, a_{2 n}=a_{n}, a_{2 n+1}+a_{n}=1$. What is $a_{2006}$? | 0 | 53 | 1 |
math | 15. Given $A(-2,4) 、 B(3,-1) 、 C(-3,-4)$, and $\overrightarrow{C M}=3 \overrightarrow{C A}, \overrightarrow{C N}=$ $2 \overrightarrow{C B}$, find the coordinates of $M 、 N$ and $\overrightarrow{M N}$. | M(0,20),N(9,2),\overrightarrow{MN}={9,-18} | 81 | 25 |
math | 10. (5 points) There is a group of students forming a hollow square, with the outermost layer consisting of 52 people and the innermost layer consisting of 28 people. The group of students has people. | 160 | 48 | 3 |
math | 2. (10 points) Cars A and B start from points $A$ and $B$ respectively at the same time, heading towards each other. At the start, the speed ratio of cars A and B is 5:4. Shortly after departure, car A gets a flat tire, stops to change the tire, and then continues, increasing its speed by $20 \%$. As a result, 3 hours ... | 52 | 150 | 2 |
math | 5.1. Petya runs twice as fast as Kolya and three times as fast as Masha. On the running track of the stadium, Petya, Kolya, and Masha started at the same time. Petya reached the finish line 12 seconds earlier than Kolya. By how many seconds did Petya finish earlier than Masha? | 24 | 80 | 2 |
math | 2. Compute $\lim _{x \rightarrow 0} \frac{e^{x \cos x}-1-x}{\sin \left(x^{2}\right)}$. | \frac{1}{2} | 37 | 7 |
math | Example 6 Given a quadratic function $f(x)$ satisfies: (1) $f(-1)=0$; (2) $x \leqslant f(x)$ $\leqslant \frac{x^{2}+1}{2}$, and $x \in R$. Find the analytical expression of this quadratic function. | f(x)=\frac{1}{4}x^{2}+\frac{1}{2}x+\frac{1}{4} | 70 | 29 |
math | Problem 2. A segment with a length of 89.6 cm is divided into four parts such that $\frac{1}{2}$ of the first part is equal to $\frac{1}{3}$ of the second part, $\frac{1}{3}$ of the second part is equal to $\frac{1}{4}$ of the third part, and $\frac{1}{4}$ of the third part is equal to $\frac{1}{5}$ of the fourth part.... | 12.8\mathrm{~},19.2\mathrm{~},25.6\mathrm{~},32\mathrm{~} | 115 | 34 |
math | ## Task 24/89
Let $n \in N$ be a truly four-digit number in the decimal system; its representation in the positional system with base $b \neq 10 \quad(b \in N, b \geq 2)$ is 1549.
This representation differs from the one in the decimal system exactly at the two middle positions. Calculate $b$ and $n$. | n=1989,b=11 | 89 | 10 |
math | 24. Find the number of functions $f$ from the set $S=\{0,1,2, \ldots, 2020\}$ to itself such that, for all $a, b, c \in S$, all three of the following conditions are satisfied:
(i) If $f(a)=a$, then $a=0$;
(ii) If $f(a)=f(b)$, then $a=b$; and
(iii) If $c \equiv a+b(\bmod 2021)$, then $f(c) \equiv f(a)+f(b)(\bmod 2021)$... | 1845 | 139 | 4 |
math | 5. [5 points] Given the numbers $\log _{\sqrt{x+34}}(2 x+23), \log _{(x+4)^{2}}(x+34), \log _{\sqrt{2 x+23}}(-x-4)$. For which $x$ are two of these numbers equal, and the third one greater than them by 1? | -9 | 84 | 2 |
math | (2) The sequence $\left\{a_{n}\right\}$ is defined as follows: $a_{1}=0, a_{2}=1, a_{n}=\frac{1}{2} n a_{n-1}+$ $\frac{1}{2} n(n-1) a_{n-2}+(-1)^{n}\left(1-\frac{n}{2}\right), n \geqslant 3$. Try to find the simplest expression for $f_{n}=a_{n}+$ $2 \mathrm{C}_{n}^{1} a_{n-1}+3 \mathrm{C}_{n}^{2} a_{n-2}+\cdots+(n-1) \... | 2\cdotn!-(n+1) | 194 | 10 |
math | 1.049. $\frac{2^{-2}+5^{0}}{(0.5)^{-2}-5(-2)^{-2}+\left(\frac{2}{3}\right)^{-2}}+4.75$. | 5 | 53 | 1 |
math | Let $ABC$ be a triangle with $\angle C = 90^o$ and $AC = 1$. The median $AM$ intersects the incircle at the points $P$ and $Q$, with $P$ between $A$ and $Q$, such that $AP = QM$. Find the length of $PQ$. | \sqrt{2\sqrt{5} - 4} | 72 | 13 |
math | 3.364. $\sin \frac{\alpha+\beta}{2}$ and $\cos \frac{\alpha+\beta}{2}$, if $\sin \alpha+\sin \beta=-\frac{21}{65}$;
$$
\cos \alpha+\cos \beta=-\frac{27}{65} ; \frac{5}{2} \pi<\alpha<3 \pi \text { and }-\frac{\pi}{2}<\beta<0
$$ | \sin\frac{\alpha+\beta}{2}=-\frac{7}{\sqrt{130}} | 105 | 24 |
math | Weighted coins numbered $2,3,4, \ldots, 2015$ are tossed. The coin numbered $i$ comes up heads with probability $1 /\left(2 i^{2}\right)$. What is the probability that an odd number of coins come up heads? | \frac{1007}{4030} | 62 | 13 |
math | $3.416 \cos \frac{\pi}{33} \cdot \cos \frac{2 \pi}{33} \cdot \cos \frac{4 \pi}{33} \cdot \cos \frac{8 \pi}{33} \cdot \cos \frac{16 \pi}{33}=\frac{1}{32}$. | \frac{1}{32} | 82 | 8 |
math | Task 1. In a confectionery store, three types of candies are sold: caramels for 3 rubles, toffees for 5 rubles, and chocolates for 10 rubles. Varya wanted to buy exactly 8 candies of each type and took 200 rubles with her. In the morning, she saw announcements in the store: “When paying for three chocolates, get a free... | 72 | 124 | 2 |
math | Find the value of $\cos ^{2} 10^{\circ}+\cos ^{2} 50^{\circ}-\sin 40^{\circ} \sin 80^{\circ}$. | \frac{3}{4} | 49 | 7 |
math | Three, (50 points) Let the number of all positive integers satisfying the following conditions be $N$:
(1) less than or equal to 2,012;
(2) the number of 1s in their binary representation is at least 2 more than the number of 0s.
Find the sum of the digits of $N$.
| 13 | 76 | 2 |
math | The real-valued function $f$ is defined for all positive integers. For any integers $a>1, b>1$ with $d=\operatorname{gcd}(a, b)$, we have
$$
f(a b)=f(d) \cdot\left(f\left(\frac{a}{d}\right)+f\left(\frac{b}{d}\right)\right),
$$
Determine all possible values of $f(2001)$. | \frac{1}{2} | 100 | 7 |
math | 3. (12 points) There are four weights of different masses. Katya weighs the weights in pairs. As a result, she got 1700, 1870, 2110, 2330, and 2500 grams. How many grams does the sixth weighing variant weigh? | 2090 | 71 | 4 |
math | Example 1 Find the equation of the line $l$ passing through the intersection point of the two lines $x-2 y+4=0$ and $x+y-2=0$, and satisfying the following conditions.
(1) Passing through the point $(3,-2)$.
(2) Perpendicular to the line $3 x-4 y+7=0$. | 4x+3y-6=0 | 78 | 9 |
math | Let $k>1$ be an integer. A function $f:\mathbb{N^*}\to\mathbb{N^*}$ is called $k$-[i]tastrophic[/i] when for every integer $n>0$, we have $f_k(n)=n^k$ where $f_k$ is the $k$-th iteration of $f$:
\[f_k(n)=\underbrace{f\circ f\circ\cdots \circ f}_{k\text{ times}}(n)\]
For which $k$ does there exist a $k$-tastrophic funct... | k > 1 | 133 | 5 |
math | 8. In a division equation, if the dividend is 2016 more than the divisor, the quotient is 15, and the remainder is 0, then the dividend is | 2160 | 39 | 4 |
math | Example 6. Factorize in the set of real numbers $R$:
$$
(x+1)(x+2)(x+3)(x+6)-3 x^{2}
$$ | \left(x^{2}+4 x+6\right)(x+4-\sqrt{10})(x+4+\sqrt{10}) | 40 | 32 |
math | 13 A line with a slope of 1 intersects the $x$-axis at point $C$, and intersects a parabola with the $x$-axis as its axis of symmetry, opening to the right, and passing through the point $(-1,1)$ at points $A$ and $B$. $|A B|=5 \sqrt{2}$, and the distance between point $C$ and the vertex of the parabola is $\frac{4}{3}... | y^{2}=3x+4or3y^{2}=25x+28 | 112 | 20 |
math | 19. Let $\triangle A B C$ be an equilateral triangle with side length 16 . Points $D, E, F$ are on $C A, A B$, and $B C$, respectively, such that $D E \perp A E, D F \perp C F$, and $B D=14$. The perimeter of $\triangle B E F$ can be written in the form $a+b \sqrt{2}+c \sqrt{3}+d \sqrt{6}$, where $a, b, c$, and $d$ are... | 31 | 133 | 2 |
math | Example 9 Solve the equation
\[
\begin{array}{l}
\frac{36}{\sqrt{x-2}}+\frac{4}{\sqrt{y-1}}+4 \sqrt{x-2}+\sqrt{y-1}-28 \\
=0 .
\end{array}
\]
(1988, Xuzhou City Junior High School Mathematics Competition) | x=11, y=5 | 84 | 8 |
math | Example 2. Find the integral $\int \frac{d x}{x^{2}-6 x-16}$. | \frac{1}{10}\ln|\frac{x-8}{x+2}|+C | 26 | 21 |
math | 13. Given the sequence $\{\sqrt[n]{n}\}\left(n \in \mathbf{Z}_{+}\right)$. Then the value of the largest term in the sequence is $\qquad$ . | \sqrt[3]{3} | 45 | 7 |
math | The number $734{,}851{,}474{,}594{,}578{,}436{,}096$ is equal to $n^6$ for some positive integer $n$. What is the value of $n$? | 3004 | 68 | 4 |
math | 2. Solve the equation $\left(\cos \frac{2 x}{5}-\cos \frac{2 \pi}{15}\right)^{2}+\left(\sin \frac{2 x}{3}-\sin \frac{4 \pi}{9}\right)^{2}=0$. | \frac{29\pi}{3}+15\pi,\inZ | 64 | 18 |
math | 4B. The function $f$ satisfies the conditions
a) $f(0)=1$
b) For every $n \in \mathbb{N}$, it holds that
$$
1+f(0)+f(1)+\ldots+f(n-1)=f(n)
$$
Calculate the sum
$$
S=f(0)^{2}+f(1)^{2}+\ldots+f(n-1)^{2}+f(n)^{2}
$$ | \frac{1}{3}(4^{n+1}-1) | 105 | 15 |
math | 32nd Putnam 1971 Problem B2 Let X be the set of all reals except 0 and 1. Find all real valued functions f(x) on X which satisfy f(x) + f(1 - 1/x) = 1 + x for all x in X. | f(x)=\frac{x^3-x^2-1}{2x^2-2x} | 64 | 22 |
math | 【Question 18】
There are two cars leaving the station one after another heading to the dock. The speed of the first car is 90 kilometers per hour. At 8:32, the distance of the first car from the station is 1.2 times the distance of the second car from the station; by 9:20, the distance of the first car from the station ... | 30 | 121 | 2 |
math | Positive integers $a_0<a_1<\dots<a_n$, are to be chosen so that $a_j-a_i$ is not a prime for any $i,j$ with $0 \le i <j \le n$. For each $n \ge 1$, determine the smallest possible value of $a_n$. | 4n + 1 | 68 | 7 |
math | Task 3. Sasha chose five numbers from the numbers 1, 2, 3, 4, 5, 6, and 7 and told Anna their product. Based on this information, Anna realized that she could not uniquely determine the parity of the sum of the numbers chosen by Sasha. What number did Sasha tell Anna? (20 points) | 420 | 75 | 3 |
math | Find out the maximum possible area of the triangle $ABC$ whose medians have lengths satisfying inequalities $m_a \le 2, m_b \le 3, m_c \le 4$. | 4 | 42 | 1 |
math | ## C3
For a given positive integer n, two players $A$ and B play the following game: Given is pile of $\boldsymbol{\Omega}$ stones. The players take turn alternatively with A going first. On each turn the player is allowed to take one stone, a prime number of stones, or a multiple of $n$ stones. The winner is the one ... | n-1 | 107 | 3 |
math | Let $p>2$ be a prime. How many residues $\pmod p$ are both squares and squares plus one? | \frac{p + (-1)^{\frac{p-1}{2}} + 2}{4} | 26 | 23 |
math | $$
\begin{array}{l}
\frac{a}{a^{3}+a^{2} b+6 b^{2}+b^{3}}+\frac{b}{a^{3}-a^{2} b+a b^{2}-b^{3}} \\
+\frac{1}{a^{2}-b^{2}}-\frac{1}{b^{2}+a^{2}}-\frac{a^{2}+3 b^{2}}{a^{4}-b^{4}} .
\end{array}
$$ | 0 | 115 | 1 |
math | Problem 8.4. In a class, there are 20 students. Thinking about which girls to send a Valentine's card to on February 14, each boy made a list of all the girls in the class he finds attractive (possibly an empty list). It is known that there do not exist three boys whose lists have the same number of girls. What is the ... | 6 | 88 | 1 |
math | # Task 5. (20 points)
Find all values of the parameter $a$ for which the roots $x_{1}$ and $x_{2}$ of the equation
$$
2 x^{2}-2016(x-2016+a)-1=a^{2}
$$
satisfy the double inequality $x_{1}<a<x_{2}$.
# | \in(2015,2017) | 82 | 13 |
math | 8. [6] Suppose a real number $x>1$ satisfies
$$
\log _{2}\left(\log _{4} x\right)+\log _{4}\left(\log _{16} x\right)+\log _{16}\left(\log _{2} x\right)=0 \text {. }
$$
Compute
$$
\log _{2}\left(\log _{16} x\right)+\log _{16}\left(\log _{4} x\right)+\log _{4}\left(\log _{2} x\right) .
$$ | -\frac{1}{4} | 137 | 7 |
math | \section*{Problem 4 - 141214}
For all real triples \((a, b, c)\), it is to be investigated whether the system of equations
\[
x y^{2} z^{3}=a, \quad ; \quad x^{2} y^{3} z=b, \quad ; \quad x^{3} y z^{2}=c
\]
\[
(*, * *, * * *)
\]
1) has no,
2) exactly one,
3) exactly two,
4) more than two but finitely many,
5) i... | Inthecase0,nosolution.In | 150 | 9 |
math | If $f(x)$ is a linear function with $f(k)=4, f(f(k))=7$, and $f(f(f(k)))=19$, what is the value of $k$ ? | \frac{13}{4} | 43 | 8 |
math | 4. (3 points) Given a triangle $A B C$ with angle $C=120^{\circ}$. Point $D$ is the foot of the perpendicular dropped from point $C$ to side $A B$; points $E$ and $F$ are the feet of the perpendiculars dropped from point $D$ to sides $A C$ and $B C$ respectively. Find the perimeter of triangle $A B C$, if it is known t... | 16+8\sqrt{3} | 120 | 9 |
math | # Problem 5. (3 points)
In trapezoid $A B C D$, the lateral side $A B$ is equal to the diagonal $A C$. On the smaller arc $A D$ of the circumscribed circle of triangle $A B D$, a point $E$ is chosen such that $A B=A E$. Find the angle $\angle C E D$. | 90 | 81 | 2 |
math | ## Problem Statement
Find the coordinates of point $A$, which is equidistant from points $B$ and $C$.
$A(0 ; 0 ; z)$
$B(7 ; 0 ;-15)$
$C(2 ; 10 ;-12)$ | A(0;0;-4\frac{1}{3}) | 62 | 14 |
math | Grogg takes an a × b × c rectangular block (where a, b, c are positive integers), paints
the outside of it purple, and cuts it into abc small 1 × 1 × 1 cubes. He then places all the
small cubes into a bag, and Winnie reaches in and randomly picks one of the small cubes.
If the probability that Winnie picks a totally un... | 120, 160, 240, 360 | 104 | 18 |
math | 3. Given that $p$ is a prime number, and the equation
$$
x^{2}+p x-444 p=0
$$
has two integer roots. Then $p=$ $\qquad$ | 37 | 48 | 2 |
math | ## Problem Statement
Find the derivative.
$$
y=\sqrt{(4+x)(1+x)}+3 \ln (\sqrt{4+x}+\sqrt{1+x})
$$ | \sqrt{\frac{4+x}{1+x}} | 37 | 11 |
math | 1. Calculate: $1 \frac{3}{8} \cdot \frac{2}{5}-\frac{2}{5}: 2 \frac{2}{7}$. | \frac{3}{8} | 39 | 7 |
math | 14. In 1993, American mathematician F. Smarandache proposed many number theory problems, attracting the attention of scholars both at home and abroad. One of these is the famous Smarandache function. The Smarandache function of a positive integer \( n \) is defined as:
\[
S(n)=\min \left\{m \left| m \in \mathbb{N}^{*},... | 8 | 217 | 1 |
math | 5. Let $a_{1}, a_{2}, \cdots, a_{n}$ be given real numbers, not all zero. If real numbers $r_{1}, r_{2}, \cdots, r_{n}$ satisfy the inequality
$$
\sum_{k=1}^{n} r_{k}\left(x_{k}-a_{k}\right) \leqslant\left(\sum_{k=1}^{n} x_{k}^{2}\right)^{\frac{1}{2}}-\left(\sum_{k=1}^{n} a_{k}^{2}\right)^{\frac{1}{2}}
$$
for any real... | r_{i}=\frac{a_{i}}{\sqrt{\sum_{k=1}^{n}a_{k}^{2}}},i=1,2,\cdots,n | 194 | 40 |
math | 1. Let $n$ be a natural number, $a, b$ be positive real numbers, and satisfy $a+b=2$, then the minimum value of $\frac{1}{1+a^{n}}+\frac{1}{1+b^{n}}$ is | 1 | 56 | 1 |
math | Task A-3.5. (4 points)
If the lengths of the sides of a triangle satisfy the equation $\frac{a+b}{b+c}=\frac{c}{a-b}$, determine the measure of the largest angle of this triangle. | 120 | 52 | 3 |
math | Suppose $a_1,a_2,\ldots,a_n>0 $, where $n>1$ and $\sum_{i=1}^{n}a_i=1$.
For each $i=1,2,\ldots,n $, let $b_i=\frac{a^2_i}{\sum\limits_{j=1}^{n}a^2_j}$. Prove that \[\sum_{i=1}^{n}\frac{a_i}{1-a_i}\le \sum_{i=1}^{n}\frac{b_i}{1-b_i} .\]
When does equality occur ? | \sum_{i=1}^{n} \frac{a_i}{1 - a_i} \leq \sum_{i=1}^{n} \frac{b_i}{1 - b_i} | 136 | 46 |
math | The sum of the two perfect cubes that are closest to $500$ is $343+512=855$. Find the sum of the two perfect cubes that are closest to $2008$. | 3925 | 49 | 4 |
math | 6. Find all prime pairs $(p, q)$ such that $p q \mid\left(5^{p}-2^{p}\right)\left(5^{q}-2^{q}\right)$. | (3,3),(3,13),(13,3) | 44 | 15 |
math | 4. In an equilateral triangle $A B C$ the side is equal to $a$. On side $B C$ lies point $D$, and on $A B$ point $E$ such that $|B D|=\frac{1}{3} a,|A E|=|D E|$. Find the length of $C E$. | \frac{13}{15} | 74 | 9 |
math | 11. (20 points) Find all positive integers $n \geqslant 3$, such that the following proposition holds: If $a_{1}, a_{2}, \cdots, a_{n}$ form an arithmetic sequence, and if $a_{1}+2 a_{2}+\cdots+n a_{n}$ is a rational number, then at least one of $a_{1}, a_{2}, \cdots, a_{n}$ is a rational number. | n\equiv1(\bmod3) | 105 | 9 |
math | 60. Solve the matrix equation
$$
\left(\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right) x=\binom{7}{17}
$$ | \binom{3}{2} | 47 | 8 |
math | \section*{Problem 6 - 031216}
It is
\[
\frac{26}{65}=\frac{2 \emptyset}{\phi 5}=\frac{2}{5}
\]
Thus, for this fraction, it is allowed to "cancel" the digit 6. For which fractions with two-digit numerators and denominators is such "cancellation" of any digit of the numerator against a digit of the denominator permitt... | \frac{16}{64},\frac{26}{65},\frac{19}{95},\frac{49}{98},\frac{64}{16},\frac{65}{26},\frac{95}{19},\frac{98}{49} | 107 | 72 |
math | 886. Find the mass of the arc $A B$ of the curve $y=\ln x$, if at each point of it the linear density is proportional to the square of the abscissa of the point; $x_{A}=1, x_{B}=3$. | \frac{k}{3}(10\sqrt{10}-2\sqrt{2}) | 59 | 20 |
math | On average, how many times does one need to roll a fair die so that the sum of the numbers rolled is at least 3? | 1.36 | 28 | 4 |
math | 2. Given two non-zero numbers. If you add one to each of them, as well as subtract one from each of them, then the sum of the reciprocals of the four resulting numbers will be equal to 0. What number can result if you subtract the sum of their reciprocals from the sum of the original numbers? Find all possibilities. (S... | 0 | 79 | 1 |
math | 35th CanMO 2003 Problem 1 The angle between the hour and minute hands of a standard 12-hour clock is exactly 1 o . The time is an integral number n of minutes after noon (where 0 < n < 720). Find the possible values of n. | 262or458 | 65 | 7 |
math | \left.\begin{array}{l}{[\text { Angles subtending equal arcs and equal chords }]} \\ {[\text { Inscribed angle subtending the diameter }]\end{array}\right]
A circle passes through vertices $A$ and $C$ of triangle $ABC$, intersecting side $AB$ at point $E$ and side $BC$ at point $F$. Angle $AEC$ is five times the angle... | 3 | 121 | 1 |
math | Find the smallest multiple of 9 that does not have any odd digits. | 288 | 15 | 3 |
math | Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function which satisfies the following:
[list][*] $f(m)=m$, for all $m\in\mathbb{Z}$;[*] $f(\frac{a+b}{c+d})=\frac{f(\frac{a}{c})+f(\frac{b}{d})}{2}$, for all $a, b, c, d\in\mathbb{Z}$ such that $|ad-bc|=1$, $c>0$ and $d>0$;[*] $f$ is monotonically increasing.[/list]
(a) ... | f(x) = a_0 - 2 \sum_{n \geq 1} \frac{(-1)^n}{2^{A_n}} | 170 | 34 |
math | Let
$$
R=3 \times 9+4 \times 10+5 \times 11+\cdots+2003 \times 2009
$$
$$
S=1 \times 11+2 \times 12+3 \times 13+\cdots+2001 \times 2011
$$
(a) Which is the larger number: $\mathrm{R}$ or $S$?
(b) Calculate the difference between the larger and the smaller. | 32016 | 115 | 5 |
math | Find the number of positive integers less than $1000$ that can be expressed as the difference of two integral powers of $2.$ | 50 | 29 | 2 |
math | Task 1. A number was multiplied by the sum of its digits and the result was 2008. Find this number.
$[4$ points] (I. V. Yashchenko) | 251 | 44 | 3 |
math | 1388. Write the first five terms of the series given the general term $a_{n}=\frac{1}{n(n+1)}$. | \frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\frac{1}{4\cdot5}+\frac{1}{5\cdot6} | 33 | 50 |
math | # 3. CONDITION
Perpendiculars $B E$ and $D F$, dropped from vertices $B$ and $D$ of parallelogram $A B C D$ to sides $A D$ and $B C$ respectively, divide the parallelogram into three parts of equal area. On the extension of diagonal $B D$ beyond vertex $D$, segment $D G$ is laid off, equal to segment $B D$. Line $B E$... | 1:1 | 119 | 3 |
math | 8,9 |
Four spheres of radius 1 touch each other pairwise. Find the height of the cone that contains these spheres such that all of them touch the lateral surface and three of them touch the base of the cone. | 1+2\sqrt{\frac{2}{3}}+\sqrt{3} | 45 | 17 |
math | 11. (6 points) A car rally has two equal-distance courses. The first course starts on a flat road, 26 kilometers from the midpoint it begins to climb; 4 kilometers after passing the midpoint, it is all downhill; The second course also starts on a flat road, 4 kilometers from the midpoint it begins to descend, 26 kilome... | 92 | 176 | 2 |
math | 2. In a company consisting of 2010 people, each person is either a knight, who always tells the truth, or a liar, who always lies. The people in the company are numbered from 1 to 2010. For each $k=1,2, \ldots, 2010$, the person with number $k$ said the following: "The number of liars among us is a divisor of $k$." How... | 2009 | 110 | 4 |
math | 4. (7 points) On the board, 40 ones are written. Every minute, Karlson erases two arbitrary numbers and writes their sum on the board, and then eats a number of candies equal to the product of the two erased numbers. What is the maximum number of candies he could have eaten in 40 minutes? | 780 | 69 | 3 |
math | 9.184. $\log _{0.25}\left|\frac{2 x+1}{x+3}+\frac{1}{2}\right|>\frac{1}{2}$. | x\in(-\frac{4}{3};-1)\cup(-1;-\frac{1}{2}) | 45 | 25 |
math | 3. Find the last three digits of the non-zero natural number $n$, knowing that when $29 n$ is divided by 250, the remainder is 67, and when $23 n$ is divided by 200, the remainder is 29.
(Gazeta Matematica) | 123 | 68 | 3 |
math | 3. In $\triangle A B C$, $G$ is the centroid, and $I$ is the intersection of the angle bisectors of $\angle B$ and $\angle C$. If $I G / / B C$, and $B C=5$, then $A B+A C$ $=$ . $\qquad$ | 10 | 68 | 2 |
math | Find the largest natural number, all digits in the decimal representation of which are different and which
is reduced by 5 times if the first digit is erased. | 3750 | 32 | 4 |
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