problem stringlengths 15 7.47k | answer stringlengths 1 256 | correctness_count int64 0 8 | category stringclasses 4
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## Task B-1.3.
A ship traveling along a river has covered $24 \mathrm{~km}$ upstream and $28 \mathrm{~km}$ downstream. For this journey, it took half an hour less than for traveling $30 \mathrm{~km}$ upstream and $21 \mathrm{~km}$ downstream, or half an hour more than for traveling $15 \mathrm{~km}$ upstream and $42 \... | v_{R}=4\mathrm{~}/\mathrm{},v_{B}=10\mathrm{~}/\mathrm{} | 0 | all_wrong |
3. (6 points) A construction company was building a tunnel. When $\frac{1}{3}$ of the tunnel was completed at the original speed, they started using new equipment, which increased the construction speed by $20 \%$ and reduced the working hours to $80 \%$ of the original. As a result, it took a total of 185 days to comp... | 180 | 0 | all_wrong |
Prove that number $1$ can be represented as a sum of a finite number $n$ of real numbers, less than $1,$ not necessarily distinct, which contain in their decimal representation only the digits $0$ and/or $7.$ Which is the least possible number $n$? | 8 | 0 | all_wrong |
1. Solve the equation: $\frac{8 x+13}{3}=\frac{19-12 x}{2}-\left(16 x-\frac{7-4 x}{6}\right)$. | \frac{1}{4} | 3 | suitable |
A right-angled triangle has side lengths that are integers. What could be the last digit of the area's measure, if the length of the hypotenuse is not divisible by 5? | 0 | 0 | all_wrong |
Let $a_n\ (n\geq 1)$ be the value for which $\int_x^{2x} e^{-t^n}dt\ (x\geq 0)$ is maximal. Find $\lim_{n\to\infty} \ln a_n.$ | -\ln 2 | 0 | all_wrong |
11. Given that the internal angles $A, B, C$ of $\triangle ABC$ have opposite sides $a, b, c$ respectively, and $\sqrt{3} b \cos \frac{A+B}{2}=c \sin B$.
(1) Find the size of $\angle C$;
(2) If $a+b=\sqrt{3} c$, find $\sin A$. | \sinA=\frac{1}{2}or\sinA=1 | 0 | all_wrong |
Task B-4.2. Let $n$ be the number obtained by writing 2013 zeros between every two digits of the number 14641. Determine all solutions of the equation $x^{4}=n$ in the set $\mathbb{C}$. | x_{1,2}=\(10^{2014}+1),\quadx_{3,4}=\(10^{2014}+1)\cdoti | 0 | all_wrong |
8.59 For each pair of real numbers $x, y$, the function $f$ satisfies the functional equation
$$
f(x)+f(y)=f(x+y)-x y-1 .
$$
If $f(1)=1$, then the number of integers $n$ (where $n \neq 1$) that satisfy $f(n)=n$ is
(A) 0.
(B) 1.
(C) 2.
(D) 3.
(E) infinitely many.
(30th American High School Mathematics Examination, 1979... | 1 | 1 | suitable |
9.6. Find the minimum value of the expression $(\sqrt{2(1+\cos 2 x)}-\sqrt{36-4 \sqrt{5}} \sin x+2) \cdot(3+2 \sqrt{10-\sqrt{5}} \cos y-\cos 2 y) \cdot$ If the answer is not an integer, round it to the nearest integer. | -27 | 0 | all_wrong |
Given $0 \leqslant x, y, z \leqslant 1$, solve the equation:
$$\frac{x}{1+y+z x}+\frac{y}{1+z+x y}+\frac{z}{1+x+y z}=\frac{3}{x+y+z} .$$ | x=y=z=1 | 0 | all_wrong |
## Problem Statement
Calculate the definite integral:
$$
\int_{0}^{3 / 2} \frac{x^{2} \cdot d x}{\sqrt{9-x^{2}}}
$$ | \frac{3\pi}{4}-\frac{9\sqrt{3}}{8} | 0 | all_wrong |
Example 6 The rules of a "level-up game" stipulate: On the $n$-th level, a die must be rolled $n$ times. If the sum of the points obtained from these $n$ rolls is greater than $2^{n}$, the level is considered passed. Questions:
(1) What is the maximum number of levels a person can pass in this game?
(2) What is the pro... | \frac{100}{243} | 0 | all_wrong |
2. (9th Canadian Mathematical Competition) $N$ is an integer, its representation in base $b$ is 777. Find the smallest positive integer $b$ such that $N$ is a fourth power of an integer in decimal notation. | 18 | 5 | suitable |
Problem 6. (8 points) In the plane, there is a non-closed, non-self-intersecting broken line consisting of 31 segments (adjacent segments do not lie on the same straight line). For each segment, the line defined by it is constructed. It is possible for some of the 31 constructed lines to coincide. What is the minimum n... | 9 | 8 | all_correct |
7.1. Solve the equation $\frac{n!}{2}=k!+l!$ in natural numbers, where $n!=1 \cdot 2 \cdot \ldots n$. If there are no solutions, write 0; if there is one solution, write $n$; if there are multiple solutions, write the sum of the values of $n$ for all solutions. Recall that a solution is a triplet $(n, k, l)$; if soluti... | 10 | 0 | all_wrong |
Example 1 (Question from the 13th "Hope Cup" Invitational Competition) The real roots of the equations $x^{5}+x+1=0$ and $x+\sqrt[5]{x}+1=0$ are $\alpha, \beta$ respectively, then $\alpha+\beta$ equals ( ).
A. -1
B. $-\frac{1}{2}$
C. $\frac{1}{2}$
D. 1 | -1 | 4 | suitable |
Example 5 Given that $x_{1}, x_{2}, \cdots, x_{10}$ are all positive integers, and $x_{1}+x_{2}+\cdots+x_{10}=2005$, find the maximum and minimum values of $x_{1}^{2}+x_{2}^{2}+\cdots+x_{10}^{2}$. | 402005 | 0 | all_wrong |
Four, (50 points) In an $n \times n$ grid, fill each cell with one of the numbers 1 to $n^{2}$. If no matter how you fill it, there must be two adjacent cells where the difference between the two numbers is at least 1011, find the minimum value of $n$.
---
The translation preserves the original text's formatting and ... | 2020 | 0 | all_wrong |
Augusto has a wire that is $10 \mathrm{~m}$ long. He makes a cut at a point on the wire, obtaining two pieces. One piece has a length of $x$ and the other has a length of $10-x$ as shown in the figure below:
. MATHEMATICE YoUTH CLUE A. 25
B. 31
C. 37
D. 43 | 31 | 1 | suitable |
Problem 3. In the school, there are 50 teachers, of whom 29 drink coffee, 28 drink tea, and 16 do not drink either coffee or tea. How many teachers drink only coffee, and how many drink only tea? | 6 | 0 | all_wrong |
12.180. A side of the triangle is equal to 15, the sum of the other two sides is 27. Find the cosine of the angle opposite the given side, if the radius of the inscribed circle in the triangle is 4. | \frac{5}{13} | 0 | all_wrong |
83. Fill in the following squares with $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ respectively, so that the sum of the two five-digit numbers is 99999. Then the number of different addition equations is $\qquad$. $(a+b$ and $b+a$ are considered the same equation) | 1536 | 0 | all_wrong |
On a board, the numbers from 1 to 2009 are written. A couple of them are erased and instead of them, on the board is written the remainder of the sum of the erased numbers divided by 13. After a couple of repetition of this erasing, only 3 numbers are left, of which two are 9 and 999. Find the third number. | 8 | 0 | all_wrong |
## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty} \frac{(n+1)^{4}-(n-1)^{4}}{(n+1)^{3}+(n-1)^{3}}$ | 4 | 6 | easy |
Condition of the problem
Find the derivative.
$$
y=\frac{1}{24}\left(x^{2}+8\right) \sqrt{x^{2}-4}+\frac{x^{2}}{16} \arcsin \frac{2}{x}, x>0
$$ | \frac{x^{3}-x}{8\sqrt{x^{2}-4}}+\frac{x}{8}\cdot\arcsin\frac{2}{x} | 0 | all_wrong |
\section*{Problem 5 - 071225}
All ordered pairs of real numbers \((x, y)\) are to be determined for which the system of equations
\[
\begin{aligned}
x \cdot\left(a x^{2}+b y^{2}-a\right) & =0 \\
y \cdot\left(a x^{2}+b y^{2}-b\right) & =0
\end{aligned}
\]
is satisfied. Here, \(a\) and \(b\) are real numbers with \(a ... | (0,-1),(0,0),(0,1),(-1,0),(1,0) | 0 | all_wrong |
40. The sum of three consecutive odd numbers is equal to the fourth power of a single-digit number. Find all such triples of numbers. | 25,27,292185,2187,2189 | 0 | all_wrong |
Two is $10 \%$ of $x$ and $20 \%$ of $y$. What is $x - y$?
$(\mathrm {A}) \ 1 \qquad (\mathrm {B}) \ 2 \qquad (\mathrm {C})\ 5 \qquad (\mathrm {D}) \ 10 \qquad (\mathrm {E})\ 20$ | 10 | 8 | all_correct |
1. (6 points) Today is January 31, 2015, welcome to the 2015 "Spring Cup" Final. The calculation result of the expression $\frac{\frac{2015}{1}+\frac{2015}{0.31}}{1+0.31}$ is | 6500 | 2 | suitable |
Consider a regular hexagon with an incircle. What is the ratio of the area inside the incircle to the area of the hexagon?
| \frac{\pi \sqrt{3}}{6} | 3 | suitable |
Let $ABC$ be a triangle with centroid $G$. Determine, with proof, the position of the point $P$ in the plane of $ABC$ such that $AP{\cdot}AG + BP{\cdot}BG + CP{\cdot}CG$ is a minimum, and express this minimum value in terms of the side lengths of $ABC$. | \frac{a^2 + b^2 + c^2}{3} | 0 | all_wrong |
The function $f$ maps the set of positive integers into itself, and satisfies the equation
$$
f(f(n))+f(n)=2 n+6
$$
What could this function be? | f(n)=n+2 | 1 | suitable |
30. Find the remainder when the 2018-digit number $\underbrace{\overline{55 \cdots}}_{2018 \text { 555 }}$ is divided by 13. | 3 | 2 | suitable |
1. A line $l$ intersects a hyperbola $c$, then the maximum number of intersection points is ( ).
A. 1
B. 2
C. 3
D. 4 | B | 0 | all_wrong |
1. Let the universal set be the set of real numbers. If $A=\{x \mid \sqrt{x-2} \leqslant 0\}, B=\left\{x \mid 10^{x^{2}-2}=10^{x}\right\}$, then $A \cap \bar{B}$ is
A. $\{2\}$
B. $\{-1\}$
C. $\{x \mid x \leqslant 2\}$
D. $\varnothing$ | D | 0 | all_wrong |
7.242. $\left(16 \cdot 5^{2 x-1}-2 \cdot 5^{x-1}-0.048\right) \lg \left(x^{3}+2 x+1\right)=0$. | 0 | 6 | easy |
Example 1 In $\triangle ABC$, it is known that $x \sin A + y \sin B + z \sin C = 0$. Find the value of $(y + z \cos A)(z + x \cos B)(x + y \cos C) + (y \cos A + z)(z \cos B + x)(x \cos C + y)$. | 0 | 8 | all_correct |
In order for Mateen to walk a kilometer (1000m) in his rectangular backyard, he must walk the length 25 times or walk its perimeter 10 times. What is the area of Mateen's backyard in square meters?
$\text{(A)}\ 40 \qquad \text{(B)}\ 200 \qquad \text{(C)}\ 400 \qquad \text{(D)}\ 500 \qquad \text{(E)}\ 1000$ | 400 | 6 | easy |
11.005. The plane angle at the vertex of a regular triangular pyramid is $90^{\circ}$. Find the ratio of the lateral surface area of the pyramid to the area of its base. | \sqrt{3} | 0 | all_wrong |
The knights in a certain kingdom come in two colors. $\frac{2}{7}$ of them are red, and the rest are blue. Furthermore, $\frac{1}{6}$ of the knights are magical, and the fraction of red knights who are magical is $2$ times the fraction of blue knights who are magical. What fraction of red knights are magical?
$\textbf{... | \frac{7}{27} | 2 | suitable |
A father wants to divide his property among his children: first, he gives 1000 yuan and one-tenth of the remaining property to the eldest child, then 2000 yuan and one-tenth of the remaining property to the second child, then 3000 yuan and one-tenth of the remaining property to the third child, and so on. It turns out ... | 9 | 1 | suitable |
9. Given is a regular tetrahedron of volume 1 . We obtain a second regular tetrahedron by reflecting the given one through its center. What is the volume of their intersection? | \frac{1}{2} | 1 | suitable |
Let's determine all the triples of numbers $(x, y, m)$ for which
$$
-2 x + 3 y = 2 m, \quad x - 5 y = -11
$$
and $x$ is a negative integer, $y$ is a positive integer, and $m$ is a real number. | (x,y,)=(-1,2,4)or(-6,1,7.5) | 0 | all_wrong |
## Zadatak B-1.2.
Na slici su prikazani pravilni peterokut. $A B C D E$ i kvadrat $A B F G$. Odredite mjeru kuta $F A D$.
| 27 | 0 | all_wrong |
The equation of line $\ell_1$ is $24x-7y = 319$ and the equation of line $\ell_2$ is $12x-5y = 125$. Let $a$ be the number of positive integer values $n$ less than $2023$ such that for both $\ell_1$ and $\ell_2$ there exists a lattice point on that line that is a distance of $n$ from the point $(20,23)$. Determine $a$.... | 6 | 8 | all_correct |
Let's determine $m$ such that the expression
$$
m x^{2}+(m-1) x+m-1
$$
is negative for all values of $x$.
---
Determine $m$ so that the expression
$$
m x^{2}+(m-1) x+m-1
$$
is negative for all values of $x$. | <-\frac{1}{3} | 0 | all_wrong |
2. How many integers $b$ exist such that the equation $x^{2}+b x-9600=0$ has an integer solution that is a multiple of both 10 and 12? Specify the largest possible $b$. | 9599 | 0 | all_wrong |
Example 1 The range of the function $y=-x^{2}-2 x+3(-5 \leqslant x \leqslant 0)$ is $(\quad)$.
(A) $(-\infty, 4]$
(B) $[3,12]$
(C) $[-12,4]$
(D) $[4,12]$ | C | 0 | all_wrong |
4.206 There are two forces $f_{1}$ and $f_{2}$ acting on the origin $O$ of the coordinate axis,
$$\begin{array}{l}
\vec{f}_{1}=\overrightarrow{O A}=\sqrt{2}\left(\cos 45^{\circ}+i \sin 45^{\circ}\right) \\
\vec{f}_{2}=\overrightarrow{O B}=2\left[\cos \left(-30^{\circ}\right)+i \sin \left(-30^{\circ}\right)\right]
\end{... | 2.1 | 0 | all_wrong |
6. Given that $\overrightarrow{O A} \perp \overrightarrow{O B}$, and $|\overrightarrow{O A}|=|\overrightarrow{O B}|=24$. If $t \in[0,1]$, then
$$
|t \overrightarrow{A B}-\overrightarrow{A O}|+\left|\frac{5}{12} \overrightarrow{B O}-(1-t) \overrightarrow{B A}\right|
$$
the minimum value is ( ).
(A) $2 \sqrt{193}$
(B) 2... | 26 | 0 | all_wrong |
All three vertices of $\bigtriangleup ABC$ lie on the parabola defined by $y=x^2$, with $A$ at the origin and $\overline{BC}$ parallel to the $x$-axis. The area of the triangle is $64$. What is the length of $BC$?
$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 16$ | 8 | 7 | easy |
1B. If for the non-zero real numbers $a, b$ and $c$ the equalities $a^{2}+a=b^{2}, b^{2}+b=c^{2}$ and $c^{2}+c=a^{2}$ hold, determine the value of the expression $(a-b)(b-c)(c-a)$. | 1 | 0 | all_wrong |
V-2 If one side of the rectangle is reduced by $3 \mathrm{~cm}$, and the other side is reduced by $2 \mathrm{~cm}$, we get a square whose area is $21 \mathrm{~cm}^{2}$ less than the area of the rectangle. Calculate the dimensions of the rectangle.
 Li Shuang rides a bike at a speed of 320 meters per minute from location $A$ to location $B$. On the way, due to a bicycle malfunction, he pushes the bike and walks for 5 minutes to a place 1800 meters from $B$ to repair the bike. After 15 minutes, he continues towards $B$ at 1.5 times his original ridin... | 72 | 0 | all_wrong |
53. How many four-digit numbers contain at least one even digit? | 8375 | 7 | easy |
Let $a_1,a_2,\ldots,a_n$ be a permutation of the numbers $1,2,\ldots,n$, with $n\geq 2$. Determine the largest possible value of the sum \[ S(n)=|a_2-a_1|+ |a_3-a_2| + \cdots + |a_n-a_{n-1}| . \]
[i]Romania[/i] | \left\lfloor \frac{n^2}{2} \right\rfloor - 1 | 0 | all_wrong |
There is a regular $17$-gon $\mathcal{P}$ and its circumcircle $\mathcal{Y}$ on the plane.
The vertices of $\mathcal{P}$ are coloured in such a way that $A,B \in \mathcal{P}$ are of diff erent colour, if the shorter arc connecting $A$ and $B$ on $\mathcal{Y}$ has $2^k+1$ vertices, for some $k \in \mathbb{N},$ includin... | 4 | 0 | all_wrong |
10.319. The diagonals of an isosceles trapezoid are perpendicular to each other, and its area is $a^{2}$. Determine the height of the trapezoid. | a | 6 | easy |
8. Find the last four digits of $7^{7^{-7}}$ (100 sevens).
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | 2343 | 0 | all_wrong |
1. Let $S=\{1,2, \cdots, n\}, A$ be an arithmetic sequence with at least two terms, a positive common difference, all of whose terms are in $S$, and such that adding any other element of $S$ does not form an arithmetic sequence with the same common difference as $A$. Find the number of such $A$. (Here, a sequence with ... | [\frac{n^{2}}{4}] | 0 | all_wrong |
Let $a_1, a_2, a_3, a_4$ be integers with distinct absolute values. In the coordinate plane, let $A_1=(a_1,a_1^2)$, $A_2=(a_2,a_2^2)$, $A_3=(a_3,a_3^2)$ and $A_4=(a_4,a_4^2)$. Assume that lines $A_1A_2$ and $A_3A_4$ intersect on the $y$-axis at an acute angle of $\theta$. The maximum possible value for $\tan \theta$ ca... | 503 | 0 | all_wrong |
Solve the triangle whose area $t=357.18 \mathrm{~cm}^{2}$, where the ratio of the sides is $a: b: c=4: 5: 6$.
| =24,b=30,=36,\alpha=4124^{\}34^{\\},\beta=5546^{\}16^{\\},\gamma=8249^{\}10^ | 0 | all_wrong |
Find all triples $(m,p,q)$ where $ m $ is a positive integer and $ p , q $ are primes.
\[ 2^m p^2 + 1 = q^5 \] | (1, 11, 3) | 1 | suitable |
15. As shown in the figure, the area of square $\mathrm{ABCD}$ is 196 square centimeters, and it contains two partially overlapping smaller squares. The larger of the two smaller squares has an area that is 4 times the area of the smaller one, and the overlapping area of the two squares is 1 square centimeter. Therefor... | 72 | 0 | all_wrong |
25. Anna, Bridgit and Carol run in a $100 \mathrm{~m}$ race. When Anna finishes, Bridgit is $16 \mathrm{~m}$ behind her and when Bridgit finishes, Carol is $25 \mathrm{~m}$ behind her. The girls run at constant speeds throughout the race. How far behind was Carol when Anna finished?
A $37 \mathrm{~m}$
B $41 \mathrm{~m}... | 37\mathrm{~} | 0 | all_wrong |
## Problem 1
Perform the calculations:
a) $7 \cdot 147 - 7 \cdot 47$ (1p)
b) $(2+4+6+8+\cdots+50)-(1+3+5+7+\cdots+49)$ (2p)
c) $10 \cdot 9^{2} : 3^{2} - 3^{4} \quad(2 \text{p})$
d) $(\overline{a b} + \overline{b c} + \overline{c a}) : (a + b + c) \quad$ (2p) | 700 | 2 | suitable |
1. The curve $(x+2 y+a)\left(x^{2}-y^{2}\right)=0$ represents three straight lines intersecting at one point on the plane if and only if
A. $a=0$
B. $a=1$
C. $a=-1$
D. $a \in \mathbf{R}$ | 0 | 3 | suitable |
9.27 In the metro train at the initial stop, 100 passengers entered. How many ways are there to distribute the exit of all these passengers at the next 16 stops of the train? | 16^{100} | 5 | suitable |
Of the 36 students in Richelle's class, 12 prefer chocolate pie, 8 prefer apple, and 6 prefer blueberry. Half of the remaining students prefer cherry pie and half prefer lemon. For Richelle's pie graph showing this data, how many degrees should she use for cherry pie?
$ \text{(A)}\ 10\qquad\text{(B)}\ 20\qquad\text{(... | 50 | 8 | all_correct |
## Problem 4
Given the numbers $1,2,3, \ldots, 1000$. Find the largest number $m$ with the property that by removing any $m$ numbers from these 1000 numbers, among the $1000-m$ remaining numbers, there exist two such that one divides the other.
Selected problems by Prof. Cicortas Marius
Note: a) The actual working t... | 499 | 2 | suitable |
The function $f: \mathbb{R}\rightarrow \mathbb{R}$ is such that $f(x+1)=2f(x)$ for $\forall$ $x\in \mathbb{R}$ and $f(x)=x(x-1)$ for $\forall$ $x\in (0,1]$. Find the greatest real number $m$, for which the inequality $f(x)\geq -\frac{8}{9}$ is true for
$\forall$ $x\in (-\infty , m]$.
| \frac{7}{3} | 0 | all_wrong |
18.3.19 $\star \star$ Find all positive integer triples $(a, b, c)$ that satisfy $a^{2}+b^{2}+c^{2}=2005$ and $a \leqslant b \leqslant c$. | (23,24,30),(12,30,31),(9,30,32),(4,30,33),(15,22,36),(9,18,40),(4,15,42) | 0 | all_wrong |
5. Through the vertex $M$ of some angle, a circle is drawn, intersecting the sides of the angle at points $N$ and $K$, and the bisector of this angle at point $L$. Find the sum of the lengths of segments $M N$ and $M K$, if the area of $M N L K$ is 49, and the angle $L M N$ is $30^{\circ}$. | 14\sqrt[4]{3} | 0 | all_wrong |
4. Let $A$ and $B$ be $n$-digit numbers, where $n$ is odd, which give the same remainder $r \neq 0$ when divided by $k$. Find at least one number $k$, which does not depend on $n$, such that the number $C$, obtained by appending the digits of $A$ and $B$, is divisible by $k$. | 11 | 2 | suitable |
7. Given that $z$ is a complex number, and $|z|=1$. When $\mid 1+z+$ $3 z^{2}+z^{3}+z^{4}$ | takes the minimum value, the complex number $z=$ $\qquad$ or . $\qquad$ | -\frac{1}{4}\\frac{\sqrt{15}}{4}i | 0 | all_wrong |
5. Given positive real numbers $a$ and $b$ satisfy $a+b=1$, then $M=$ $\sqrt{1+a^{2}}+\sqrt{1+2 b}$ the integer part is | 2 | 6 | easy |
1. Arrange the consecutive natural numbers from 1 to 99 in sequence to form a large number:
1234567891011…979899,
By extracting four consecutive digits, you can get a four-digit number, such as 5678, 1011, etc. Among the four-digit numbers obtained by this method, the largest is . $\qquad$ | 9909 | 0 | all_wrong |
8. (15 points) A thin ray of light falls on a thin converging lens at a distance of $x=10$ cm from its optical center. The angle between the incident ray and the plane of the lens $\alpha=45^{\circ}$, and the angle between the refracted ray and the plane of the lens $\beta=30^{\circ}$. Determine its focal length. | 13.7 | 0 | all_wrong |
8. If $n$ is a natural number less than 50, find all values of $n$ such that the values of the algebraic expressions $4n+5$ and $7n+6$ have a common divisor greater than 1. | 7,18,29,40 | 1 | suitable |
9. (16 points) Let the real number $t \in [0, \pi]$. If the equation $\cos (x+t)=1-\cos x$ has a solution for $x$, find the range of values for $t$.
| \in[0,\frac{2\pi}{3}] | 0 | all_wrong |
5. 5.1. Find the largest natural number in which all digits are different and any two adjacent digits differ by 6 or 7. | 60718293 | 0 | all_wrong |
15. Lucia and Carla are playing bingo against each other (with no other opponents). Each has a card with 15 numbers; the two cards have exactly 4 numbers in common. What is the probability that, after 89 numbers have been drawn, neither of them has yet completed a bingo?
(A) $2 / 45$
(B) $3 / 89$
(C) $1 / 15$
(D) $1 / ... | \frac{2}{45} | 0 | all_wrong |
## Task 2 - 080522
In a warehouse, there are three times as many kilograms of wheat as in a second one. After removing $85000 \mathrm{~kg}$ from the first and $5000 \mathrm{~kg}$ from the second, the stocks were equal.
How many tons of wheat were in the first and how many in the second warehouse before the removal? | 120 | 3 | suitable |
$17 \cdot 27$ As shown in the figure, in $\triangle A B C$, $\angle A=42^{\circ}$. If the trisectors of $\angle B$ and $\angle C$ intersect at $D, E$. Then the degree measure of $\angle B D C$ is
(A) $67^{\circ}$.
(B) $84^{\circ}$.
(C) $88^{\circ}$.
(D) $110^{\circ}$.
(China Beijing Junior High School Mathematics Compe... | 88 | 4 | suitable |
Example 5. Expand the function
$$
f(z)=\frac{z}{z^{2}-2 z-3}
$$
into a Taylor series in the neighborhood of the point $z_{0}=0$ using expansion (12), and find the radius of convergence of the series. | 1 | 4 | suitable |
7. The set
$\left\{x \left\lvert\,-1 \leqslant \log _{\frac{1}{x}} 10<-\frac{1}{2}\right., x \in \mathbf{N}\right\}$ has $\qquad$ proper subsets. | 2^{90}-1 | 0 | all_wrong |
7. Given the equality
$$
\left(a^{m_{1}}-1\right) \ldots\left(a^{m_{n}}-1\right)=\left(a^{k_{1}}+1\right) \ldots\left(a^{k_{l}}+1\right)
$$
where $a, n, l$ and all exponents are natural numbers, and $a>1$. Find all possible values of the number $a$. | =2=3 | 0 | all_wrong |
Find two such common fractions - one with a denominator of 8, the other with a denominator of 13 - so that they are not equal, but the difference between the larger and the smaller of them is as small as possible.
# | \frac{3}{8},\frac{5}{13} | 0 | all_wrong |
Let $a_1,a_2,a_3,a_4,a_5$ be distinct real numbers. Consider all sums of the form $a_i + a_j$ where $i,j \in \{1,2,3,4,5\}$ and $i \neq j$. Let $m$ be the number of distinct numbers among these sums. What is the smallest possible value of $m$? | 7 | 3 | suitable |
149. Two equally matched opponents are playing chess. Find the most probable number of wins for any chess player if $2 N$ decisive (without draws) games will be played. | N | 6 | easy |
3. Given positive numbers $a, b, c$ satisfying $a+b+c=1$,
then $\sqrt{\frac{(a+b c)(b+c a)}{c+a b}}+\sqrt{\frac{(b+c a)(c+a b)}{a+b c}}+\sqrt{\frac{(c+a b)(a+b c)}{b+c a}}=$ $\qquad$ | 2 | 4 | suitable |
# Problem 3.
A polynomial $P(x)$ with integer coefficients has the properties
$$
P(1)=2019, \quad P(2019)=1, \quad P(k)=k,
$$
where the number $k$ is an integer. Find this number $k$.
# | 1010 | 0 | all_wrong |
15.15 A paper punch can be placed at any point in the plane, and when it operates, it can punch out points at an irrational distance from it. What is the minimum number of paper punches needed to punch out all points in the plane?
(51st Putnam Mathematical Competition, 1990) | 3 | 1 | suitable |
Bogganov I.I.
Given an infinite supply of white, blue, and red cubes. Any \$N\$ of them are arranged in a circle. A robot, starting at any point on the circle, moves clockwise and, until only one cube remains, repeatedly performs the following operation: it destroys the two nearest cubes in front of it and places a ne... | 2^k | 1 | suitable |
$7 \cdot 74$ There are several locks, and now six people each hold a part of the keys. It is known that any two people trying to open the locks together will have exactly one lock that they cannot open, while any three people can open all the locks. How many locks are there at least?
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Note: The translation keeps t... | 15 | 0 | all_wrong |
9. Point $Y$ lies on line segment $X Z$ such that $X Y=5$ and $Y Z=3$. Point $G$ lies on line $X Z$ such that there exists a triangle $A B C$ with centroid $G$ such that $X$ lies on line $B C, Y$ lies on line $A C$, and $Z$ lies on line $A B$. Compute the largest possible value of $X G$. | \frac{20}{3} | 0 | all_wrong |
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