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 | 6. (5 points) Find all pairs of integers $(x, y)$ for which the following equality holds:
$$
x(x+1)(x+7)(x+8)=y^{2}
$$
# | (1;12),(1;-12),(-9;12),(-9;-12),(0;0),(-8;0),(-4;-12),(-4;12),(-1;0),(-7;0) | 45 | 54 |
math | Find all values of the positive integer $ m$ such that there exists polynomials $ P(x),Q(x),R(x,y)$ with real coefficient satisfying the condition: For every real numbers $ a,b$ which satisfying $ a^m-b^2=0$, we always have that $ P(R(a,b))=a$ and $ Q(R(a,b))=b$. | m = 1 | 77 | 5 |
math | B1. Find all integer solutions of the equation $m^{4}+2 n^{2}=9 m n$. | (-3,-9),(-2,-8),(-2,-1),(0,0),(2,1),(2,8),(3,9) | 25 | 31 |
math | 3. How many noncongruent triangles are there with one side of length 20 , one side of length 17 , and one $60^{\circ}$ angle? | 2 | 39 | 1 |
math | 15.5. (GDR, 74). Determine into how many parts a regular tetrahedron is divided by six planes, each of which passes through one edge and the midpoint of the opposite edge of the tetrahedron. Find the volume of each of the parts if the volume of the tetrahedron is 1. | \frac{1}{24} | 72 | 8 |
math | # Problem 7.4 (7 points)
Find all three-digit numbers that decrease by 6 times after the first digit is erased. # | 120,240,360,480 | 29 | 15 |
math | In a non-zero common difference arithmetic sequence $a_{1} \cdot a_{2}, \cdots, a_{1990}$, $a_{i} a,>0,(i, j=1,2, \cdots, 1990)$. Let $b_{1}=a_{1} \cdot a_{1990}, b_{2}=a_{2} \cdot a_{1989}, \cdots, b_{k}=a_{k} \cdot a_{1990-k+1}(k=1,2, \cdots, 1990)$, then the largest term in $\left\{b_{k}\right\}$ is $\qquad$ . | b_{995} | 159 | 6 |
math | $7 \cdot 67$ Given 11 sets $M_{1}, M_{2}, \cdots, M_{11}$, where each set has exactly 5 elements and for all $i, j, 1 \leqslant i<j \leqslant 11$, there is $M_{i} \cap M_{j} \neq \varnothing$, find the minimum possible value of the maximum number of sets among these sets whose intersection is non-empty. | 4 | 106 | 1 |
math | $ABCD$ is circumscribed in a circle $k$, such that $[ACB]=s$, $[ACD]=t$, $s<t$. Determine the smallest value of $\frac{4s^2+t^2}{5st}$ and when this minimum is achieved. | \frac{4}{5} | 60 | 7 |
math | ## Task Condition
Calculate the volume of the tetrahedron with vertices at points $A_{1}, A_{2}, A_{3}, A_{4_{\text {and }}}$ its height dropped from vertex $A_{4 \text { to the face }} A_{1} A_{2} A_{3}$.
$A_{1}(-3 ; 4 ;-7)$
$A_{2}(1 ; 5 ;-4)$
$A_{3}(-5 ;-2 ; 0)$
$A_{4}(2 ; 5 ; 4)$ | 25\frac{1}{6} | 123 | 9 |
math | 760. What are the sines, cosines, and tangents of the angles $30^{\circ}, 45^{\circ}, 60^{\circ}, 90^{\circ}$? | \begin{aligned}&\sin30=\frac{1}{2},\quad\cos30=\frac{\sqrt{3}}{2},\quad\tan30=\frac{1}{\sqrt{3}}\\&\sin45=\frac{1}{\sqrt{2}},\quad\cos45^ | 48 | 71 |
math | 4. 156 Solve the system of equations
$$\left\{\begin{array}{l}
\frac{4 x^{2}}{1+4 x^{2}}=y \\
\frac{4 y^{2}}{1+4 y^{2}}=z \\
\frac{4 z^{2}}{1+4 z^{2}}=x
\end{array}\right.$$
for all real solutions, and prove that your solution is correct. | (x, y, z)=(0,0,0) \text { and }(x, y, z)=\left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right) | 101 | 48 |
math | 3. Find the value of $\frac{1}{1+1^{2}+1^{4}}+\frac{2}{1+2^{2}+2^{4}}+\frac{3}{1+3^{2}+3^{4}}+\cdots+\frac{100}{1+100^{2}+100^{4}}$.
(1 mark)
Find the value of $\frac{1}{1+1^{2}+1^{4}}+\frac{2}{1+2^{2}+2^{4}}+\frac{3}{1+3^{2}+3^{4}}+\cdots+\frac{100}{1+100^{2}+100^{4}}$. | \frac{5050}{10101} | 163 | 14 |
math | Find the maximum possible value of the inradius of a triangle whose vertices lie in the interior, or on the boundary, of a unit square. | \frac{\sqrt{5}-1}{4} | 29 | 11 |
math | 71.
Let's start with a simple case. A major theft was committed at a warehouse. The criminal (or criminals) transported the stolen goods by car. Suspicion fell on three repeat offenders A, B, and C, who were brought to Scotland Yard for interrogation. The following was established:
1) No one other than A, B, and C w... | A | 191 | 1 |
math | Lajcsi and Pali are discussing how often in lottery draws three numbers contain the same digit. After a brief calculation, Lajcsi says, "Out of 100 draws, on average, nearly 7 will have at least three numbers containing the digit 8." To this, Pali replies, "I once made a mistake in a similar calculation and I think you... | 5.3 | 168 | 3 |
math | 6-0. Quadrilateral $A B C D$ is inscribed in a circle, and the areas of triangles $A B C$ and $A C D$ are equal. Three sides of the quadrilateral are 5, 8, and 10. Find all possible values of the length of the fourth side. | 4;6.25;16 | 68 | 9 |
math | T a s k 2. Find the natural roots of the equation
$17(x y z t+x y+x t+z t+1)-54(y z t+y+t)=0$. | 3,1,=2,5 | 41 | 8 |
math | \section*{Problem 3B - 321043B}
Determine all integers \(a, b\) for which the division of the polynomial
\[
f(x)=x^{4}+2 a x^{3}+2 x^{2}+x-2
\]
by the polynomial
\[
g(x)=x^{2}+a x+b
\]
results in a polynomial \(h(x)\) with no remainder (such that for every number \(x\) for which \(g(x) \neq 0\), the equation \(f(x... | (1,-1)(1,2) | 133 | 9 |
math | ## Problem Statement
Are the vectors $c_{1 \text { and }} c_{2}$, constructed from vectors $a \text{ and } b$, collinear?
$a=\{3 ; 4 ;-1\}$
$b=\{2 ;-1 ; 1\}$
$c_{1}=6 a-3 b$
$c_{2}=b-2 a$ | c_{1}=-3\cdotc_{2} | 79 | 12 |
math | Example 3.1 Let $f(n)=3^{n}$, find $\Delta^{k} f(n) \quad(k \geqslant 1)$. | \Delta^{k}f(n)=2^{k}\cdot3^{n} | 36 | 17 |
math | Example 1 Given the sequence $\left\{x_{n}\right\}$, and
$$
x_{n+1}=\frac{x_{n}+(2-\sqrt{3})}{1-(2-\sqrt{3}) x_{n}} \text {. }
$$
Find the value of $x_{1001}-x_{401}$. | 0 | 79 | 1 |
math | ## Task 2 - 330522
Rolf is looking for four-digit numbers in which no two digits are the same. The difference between the tens and the hundreds digit should be 3, and the difference between the hundreds and the thousands digit should be 4.
When calculating these differences, the order of the two digits involved shoul... | 112 | 103 | 3 |
math | 13.313. The denominator of the fraction is less than the square of its numerator by 1. If 2 is added to both the numerator and the denominator, the value of the fraction will be greater than $1 / 4$; if 3 is subtracted from both the numerator and the denominator of the original fraction, the value of the fraction will ... | \frac{4}{15} | 91 | 8 |
math | $3.420 \sin 10^{\circ} \cdot \sin 20^{\circ} \cdot \sin 30^{\circ} \cdot \sin 40^{\circ} \cdot \sin 50^{\circ} \cdot \sin 60^{\circ} \cdot \sin 70^{\circ} \cdot \sin 80^{\circ}=\frac{3}{256} \cdot$ | \frac{3}{256} | 104 | 9 |
math | (7) Let $f(x)=a x+b$, where $a, b$ are real numbers, $f_{1}(x)=f(x), f_{n+1}(x)=$ $f\left(f_{n}(x)\right), n=1,2,3, \cdots$, if $f_{7}(x)=128 x+381$, then $a+b=$ $\qquad$ . | 5 | 92 | 1 |
math | 1. The range of real numbers $x$ that satisfy $\sqrt{1+x}+\frac{2}{5}<\sqrt{3-x}$ is $\qquad$ | -1\leqslantx<\frac{11}{25} | 36 | 18 |
math | 7. Variant 1.
In a convex $n$-gon, a diagonal is highlighted. The highlighted diagonal is intersected by exactly 14 other diagonals of this $n$-gon. Find the sum of all possible values of $n$. A vertex of the $n$-gon is not considered an intersection. | 28 | 68 | 2 |
math | 5. Given the number $800 \ldots 008$ (80 zeros). It is required to replace some two zeros with non-zero digits so that after the replacement, the resulting number is divisible by 198. In how many ways can this be done? | 14080 | 60 | 5 |
math | 6.137. $\frac{x}{x+1}+\frac{x+1}{x+2}+\frac{x+2}{x}=\frac{25}{6}$. | x_{1}=1,x_{2,3}=-2\\frac{2\sqrt{7}}{7} | 41 | 25 |
math | 1. If $\frac{a+b}{b}=1.4$, what is $\frac{b-a}{a}$? | \frac{3}{2} | 26 | 7 |
math | For each positive integer $k$, let $d(k)$ be the number of positive divisors of $k$ and $\sigma(k)$ be the sum of positive divisors of $k$. Let $\mathbb N$ be the set of all positive integers. Find all functions $f: \mathbb{N} \to \mathbb N$ such that \begin{align*}
f(d(n+1)) &= d(f(n)+1)\quad \text{and} \\
f(\si... | f(n) = n | 132 | 6 |
math | A tailor met a tortoise sitting under a tree. When the tortoise was the tailor’s age, the tailor was only a quarter of his current age. When the tree was the tortoise’s age, the tortoise was only a seventh of its current age. If the sum of their ages is now $264$, how old is the tortoise? | 77 | 74 | 2 |
math | Let $A B C$ be a triangle with circumradius $R$, perimeter $P$ and area $K$. Determine the maximum value of $K P / R^{3}$. | \frac{27}{4} | 38 | 8 |
math | Determine all pairs of positive integers $(m,n)$ such that m is but divisible by every integer from $1$ to $n$ (inclusive), but not divisible by $n + 1, n + 2$, and $n + 3$. | (m, 1) | 53 | 7 |
math | ## Task B-1.4.
The length of $\overline{A B}$ is sequentially divided from vertex $A$ by points $T_{1}, T_{2}, T_{3}, \ldots, T_{2016}$ into 2017 equal parts. If the coordinates of points $T_{3}(5,-1)$ and $T_{4}(8,-3)$ are known, determine the coordinates of points $A$ and $B$. Can the difference in coordinates for a... | T_{405}(1211,-805) | 150 | 15 |
math | Example 10. (IMO6-1(1)) Find all positive integers $n$ such that $2^{n}-1$ is divisible by 7. | n = 3k | 35 | 5 |
math | Calculer
$$
\sum_{i=1}^{n} \sum_{j=1}^{n} i^{2} j^{2}
$$ | (\frac{n(n+1)(2n+1)}{6})^{2} | 34 | 18 |
math | Example 15 Given a positive integer $n$ and a positive number $M$, for all arithmetic sequences $a_{1}, a_{2}, a_{3}, \cdots$ satisfying the condition $a_{1}^{2}+$ $a_{n+1}^{2} \leqslant M$, find the maximum value of $S=a_{n+1}+a_{n+2}+\cdots+a_{2 n+1}$. | \frac{(n+1)}{2} \sqrt{10M} | 99 | 17 |
math | ## PROBLEM 4
Given the sets: $A=\left\{x \in N^{*} \mid 2^{*} x \leq 8\right\}$,
$$
\begin{aligned}
& B=\left\{x \in N^{*} \mid x=2^{n-1}, n \in A\right\}, \\
& C=\{x \in N \mid x=m-n, m \in B, n \in A, m>n\} .
\end{aligned}
$$
a) Determine the elements of the sets $A, B$ and $C$;
b) Determine the elements of the s... | {3,5,6,7,8} | 190 | 11 |
math | Problem 10.4. Find all values of the real parameter $a$ such that the number of the solutions of the equation
$$
3\left(5 x^{2}-a^{4}\right)-2 x=2 a^{2}(6 x-1)
$$
does not exceed the number of the solutions of the equation
$$
2 x^{3}+6 x=\left(3^{6 a}-9\right) \sqrt{2^{8 a}-\frac{1}{6}}-(3 a-1)^{2} 12^{x}
$$
Ivan L... | \frac{1}{3} | 131 | 7 |
math | Question 75, Given real numbers $a \geq b \geq c \geq d, a+b+c+d=9, a^{2}+b^{2}+c^{2}+d^{2}=21$, find the minimum possible value of $\mathrm{ab}-\mathrm{cd}$. | 2 | 69 | 1 |
math | 6. Let point $A(2,0)$, and $B$ be a point on the elliptical arc
$$
\frac{x^{2}}{4}+\frac{y^{2}}{3}=1(x>0, y>0)
$$
Draw a perpendicular from point $B$ to the $y$-axis, and let the foot of the perpendicular be $C$. Then the maximum value of the area of quadrilateral $O A B C$ is $\qquad$ | \frac{9}{4} | 105 | 7 |
math | Exercise 5. A player has four black cards and three red cards, all distinct. In how many ways can he order them so that two successive cards are not both red? | 1440 | 36 | 4 |
math | Problem 4. Determine the family of primitives
$$
I=\int \frac{x \ln \left(1+\sqrt{1+x^{2}}\right)}{\sqrt{1+x^{2}}} d x
$$
## NOTE:
Working time 3 hours.
All subjects are mandatory.
Each problem will be graded from 1 to 7 points (1 point is given automatically).
## National Mathematics Olympiad Local stage, Febru... | (1+\sqrt{1+x^{2}})\ln(1+\sqrt{1+x^{2}})-\sqrt{1+x^{2}}+\mathcal{C} | 108 | 37 |
math | \section*{Problem 1 - 071021}
In
\begin{tabular}{cccccc}
& \(\mathrm{F}\) & \(\mathrm{U}\) & \(\mathrm{E}\) & \(\mathrm{N}\) & \(\mathrm{F}\) \\
+ & & \(\mathrm{Z}\) & \(\mathrm{W}\) & \(\mathrm{E}\) & \(\mathrm{I}\) \\
\hline \(\mathrm{S}\) & \(\mathrm{I}\) & \(\mathrm{E}\) & \(\mathrm{B}\) & \(\mathrm{E}\) & \(\ma... | 0 | 203 | 1 |
math | 9. For all real numbers $x$, let
$$
f(x)=\frac{1}{\sqrt[2011]{1-x^{2011}}} .
$$
Evaluate $(f(f(\ldots(f(2011)) \ldots)))^{2011}$, where $f$ is applied 2010 times. | 2011^{2011} | 78 | 10 |
math | Find the area of a triangle with angles $\frac{1}{7} \pi$, $\frac{2}{7} \pi$, and $\frac{4}{7} \pi $, and radius of its circumscribed circle $R=1$. | \frac{\sqrt{7}}{4} | 53 | 10 |
math | Using only once each of the digits $1, 2, 3, 4, 5, 6, 7$ and $ 8$, write the square and the cube of a positive integer. Determine what that number can be. | 24 | 51 | 2 |
math | 8. Given that $a, b, c, d$ are all prime numbers (allowing $a, b, c, d$ to be the same), and $a b c d$ is the sum of 35 consecutive positive integers. Then the minimum value of $a+b+c+d$ is $\qquad$ . | 22 | 69 | 2 |
math | 5. In $\triangle A B C$, let $D$ and $E$ be the trisection points of $B C$, with $D$ between $B$ and $E$, $F$ be the midpoint of $A C$, and $G$ be the midpoint of $A B$. Let $H$ be the intersection of line segments $E G$ and $D F$. Find the ratio $E H$ : $H G$. | EH:HG=2:3 | 94 | 7 |
math | 7・2 Betya uses a computer in a store, and must pay according to the following standards: for each number input to be multiplied by 3, he has to pay 5 kopecks, and for adding 4 to any number, he has to pay 2 kopecks, but inputting 1 into the computer is free. Betya hopes to calculate from 1 to 1981 with the least amount... | 42 | 121 | 2 |
math | 6. (8 points) Let for positive numbers $x, y, z$ the following system of equations holds:
$$
\left\{\begin{array}{l}
x^{2}+x y+y^{2}=75 \\
y^{2}+y z+z^{2}=16 \\
z^{2}+x z+x^{2}=91
\end{array}\right.
$$
Find the value of the expression $x y+y z+x z$. | 40 | 101 | 2 |
math | When $a, b$ take what values, the equation
$$
x^{2}+2(1+a) x+\left(3 a^{2}+4 a b+4 b^{2}+2\right)=0
$$
has real roots? | =1,b=-\frac{1}{2} | 57 | 11 |
math | 13.001. Of the four given numbers, the first three are in the ratio $1 / 5: 1 / 3: 1 / 20$, and the fourth number is $15 \%$ of the second number. Find these numbers, given that the second number is 8 more than the sum of the others. | 48;80;12;12 | 74 | 11 |
math | A librarian receives 130 Mathematics books and 195 Portuguese books. She wants to arrange them on shelves, placing an equal number of books on each shelf, without mixing Mathematics and Portuguese books on the same shelf. How many books should she place on each shelf to minimize the number of shelves used? | 65 | 63 | 2 |
math | 4. (20 points) A ball was thrown from the surface of the Earth at an angle of $45^{\circ}$ with a speed of $v_{0}=20 \mathrm{M} / \mathrm{s}$. How long will it take for the velocity vector of the ball to turn by an angle of $90^{\circ}$? Neglect air resistance. The acceleration due to gravity is $g=10 \mathrm{M} / \mat... | 2.83 | 106 | 4 |
math | 10.5 On an $8 \times 8$ chessboard, there are 16 rooks placed in 16 squares. How many pairs of rooks can attack each other (rooks can attack each other if they are in the same row or the same column with no other rooks between them)? | 16 | 66 | 2 |
math | Example 7 Given $a+b+c=1$,
$$
\frac{1}{a+1}+\frac{1}{b+3}+\frac{1}{c+5}=0 \text {. }
$$
Find the value of $(a+1)^{2}+(b+3)^{2}+(c+5)^{2}$. (2017, National Junior High School Mathematics League (Grade 8)) | 100 | 94 | 3 |
math | We call a positive integer [i]alternating[/i] if every two consecutive digits in its decimal representation are of different parity.
Find all positive integers $n$ such that $n$ has a multiple which is alternating. | 20 \nmid n | 46 | 6 |
math | 6.146. $\frac{(x-1)(x-2)(x-3)(x-4)}{(x+1)(x+2)(x+3)(x+4)}=1$. | 0 | 45 | 1 |
math | 18. Among all tetrahedra with edge lengths 2, 3, 3, 4, 5, 5, what is the maximum volume? Prove your conclusion.
(1983 National Competition Problem) | \frac{8\sqrt{2}}{3} | 50 | 12 |
math | Let $n\ge 4$ be a positive integer and let $M$ be a set of $n$ points in the plane, where no three points are collinear and not all of the $n$ points being concyclic. Find all real functions $f:M\to\mathbb{R}$ such that for any circle $\mathcal{C}$ containing at least three points from $M$, the following equality holds... | f(P) = 0 | 122 | 7 |
math | 1. $\arctan x+\arctan \frac{1-x}{1+x}=(x>-1)$.
Translate the above text into English, keeping the original text's line breaks and format, and output the translation result directly.
1. $\arctan x+\arctan \frac{1-x}{1+x}=(x>-1)$. | \frac{\pi}{4} | 76 | 7 |
math | (Hungary 2017)(D) Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that, for all $x, y \in \mathbb{R}$,
$$
f(x y-1)+f(x) f(y)=2 x y-1
$$ | f(x)=xorf(x)=-x^2 | 70 | 11 |
math | Initial 74. Given isosceles $\triangle A B C$ with vertex angle $A$ being $108^{\circ}, D$ is a point on the extension of $A C$, and $A D=B C, M$ is the midpoint of $B D$. Find the degree measure of $\angle C M A$. | 90^\circ | 72 | 4 |
math | G3.3 Let $x$ and $y$ be positive real numbers with $x<y$. If $\sqrt{x}+\sqrt{y}=1$ and $\sqrt{\frac{x}{y}}+\sqrt{\frac{y}{x}}=\frac{10}{3}$ and $x<y$, find the value of $y-x$. | \frac{1}{2} | 72 | 7 |
math | ## Task B-2.2.
Determine the complex number $z$ such that
$$
|z+2|=|1-\bar{z}| \quad \text{and} \quad \operatorname{Re}\left(\frac{z}{2+3 i}\right)=\frac{1}{13}
$$ | -\frac{1}{2}+\frac{2}{3}i | 70 | 15 |
math | 73. Find three numbers such that the sums of all three and each pair are squares. | I=80,II=320,III=41 | 19 | 15 |
math | In a triangle $ABC$ with $ \angle A = 36^o$ and $AB = AC$, the bisector of the angle at $C$ meets the oposite side at $D$. Compute the angles of $\triangle BCD$. Express the length of side $BC$ in terms of the length $b$ of side $AC$ without using trigonometric functions. | BC = \frac{b}{2} (\sqrt{5} - 1) | 80 | 19 |
math | 8. Let $a$ be a given positive real number, $n$ be a given integer greater than 1, and real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfy
$$
\begin{array}{l}
x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=a . \\
\text { Then }\left(x_{1}-x_{2}\right)^{2}+\left(x_{1}-x_{3}\right)^{2}+\cdots+\left(x_{1}-x_{n}\right)^{2}+ \\
\left(... | na | 206 | 1 |
math | A group of people is lined up in [i]almost-order[/i] if, whenever person $A$ is to the left of person $B$ in the line, $A$ is not more than $8$ centimeters taller than $B$. For example, five people with heights $160, 165, 170, 175$, and $180$ centimeters could line up in almost-order with heights (from left-to-right) o... | 89 | 214 | 2 |
math | 5. The solution to the equation $\frac{a x^{2}}{x-1}-2 a=a^{2}+1$ with respect to $x$ is $\qquad$ . | x_{1}=a+1, x_{2}=1+\frac{1}{a} | 41 | 20 |
math | ## Task Condition
Find the derivative.
$$
y=\frac{x \cdot \sqrt{x+1}}{x^{2}+x+1}
$$ | \frac{-x^{3}-x^{2}+3x+2}{2\sqrt{x+1}(x^{2}+x+1)^{2}} | 33 | 36 |
math | 6. The solution set of the system of equations in $x, y, z, w$
$$
\left\{\begin{array}{l}
x^{2}+2 y^{2}+2 z^{2}+w^{2}=43, \\
y^{2}+z^{2}+w^{2}=29, \\
5 z^{2}-3 w^{2}+4 x y+12 y z+6 z x=95
\end{array}\right.
$$
is $\qquad$ . | (1,2,3,4),(1,2,3,-4),(-1,-2,-3,4),(-1,-2,-3,-4) | 117 | 35 |
math | 4. Determine the maximum value of the difference $x-y$ if
$$
2\left(x^{2}+y^{2}\right)=x+y
$$ | \frac{1}{2} | 35 | 7 |
math | 10. Given an arithmetic sequence $\left\{a_{n}\right\}$ with a common difference $d$ not equal to 0, and a geometric sequence $\left\{b_{n}\right\}$ with a common ratio $q$ that is a positive rational number less than 1. If $a_{1}=d, b_{1}=d^{2}$, and $\frac{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}{b_{1}+b_{2}+b_{3}}$ is a posit... | \frac{1}{2} | 138 | 7 |
math | A perpendicular bisector plane intersects the faces of a corner at areas that are in the ratio $16: 21: 28$. What are the lengths of the edges if the diagonal of the bisector plane is $29 \mathrm{~cm}$? | 16 | 56 | 2 |
math | 9. (16 points) Let $a \geqslant 0$, for any $m, x (0 \leqslant m \leqslant a, 0 \leqslant x \leqslant \pi)$, we have
$$
|\sin x - \sin (x+m)| \leqslant 1 \text{.}
$$
Find the maximum value of $a$. | \frac{\pi}{3} | 93 | 7 |
math | A triangle $ABC$ with $AC=20$ is inscribed in a circle $\omega$. A tangent $t$ to $\omega$ is drawn through $B$. The distance $t$ from $A$ is $25$ and that from $C$ is $16$.If $S$ denotes the area of the triangle $ABC$, find the largest integer not exceeding $\frac{S}{20}$ | 10 | 88 | 2 |
math | A cylindrical log has diameter $ 12$ inches. A wedge is cut from the log by making two planar cuts that go entirely through the log. The first is perpendicular to the axis of the cylinder, and the plane of the second cut forms a $ 45^\circ$ angle with the plane of the first cut. The intersection of these two planes has... | 216 | 114 | 3 |
math | I bought a lottery ticket, the sum of the digits of its five-digit number turned out to be equal to the age of my neighbor. Determine the number of this ticket, given that my neighbor solved this problem without difficulty.
# | 99999 | 46 | 5 |
math | 2. (5 points) Two different natural numbers end with 7 zeros and have exactly 72 divisors. Find their sum. | 70000000 | 28 | 8 |
math | Let $\Omega$ be a circle with radius $18$ and let $\mathcal{S}$ be the region inside $\Omega$ that the centroid of $\triangle XYZ$ sweeps through as $X$ varies along all possible points lying outside of $\Omega$, $Y$ varies along all possible points lying on $\Omega$ and $XZ$ is tangent to the circle. Compute the great... | 904 | 116 | 3 |
math | 10.3. The sum $1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{45}$ is represented as a fraction with the denominator $45!=1 \cdot 2 \cdots 45$. How many zeros (in decimal notation) does the numerator of this fraction end with? | 8 | 75 | 1 |
math | (Example 3 Given that the function $y=f(x)$ has an inverse function within its domain $(-\infty, 0]$, and $f(x-1)=x^{2}-2 x$, find the value of $f^{-1}\left(-\frac{1}{2}\right)$. | -\frac{\sqrt{2}}{2} | 65 | 10 |
math | ## Task 19/73
Determine all prime numbers $p$ for which $P=20 p^{2}+1$ is a prime number. | 3 | 36 | 1 |
math | 5. In a group of 2017 people, any two people have exactly one common friend (excluding the two people themselves). Determine the minimum possible value of the difference between the number of friends of the person with the most friends and the person with the least friends in this group. | 2014 | 59 | 4 |
math | Three numbers form a geometric progression; if we subtract 16 from the third number, we get an arithmetic progression. If we then subtract 2 from the second term of this arithmetic progression, we get a geometric progression again. What are these numbers? | x_1=1,y_1=5,z_1=25 | 51 | 16 |
math | Example 5 If $k$ is a positive integer, and the two roots of the linear equation $(k-1) x^{2}-p x+k=0$ are both positive integers, then the value of $k^{p k}\left(p^{p}+k^{k}\right)+(p+k)$ is $\qquad$ . | 1989 | 71 | 4 |
math | Problem 7.2. Vlad and Dima decided to earn some money. Each of them decided to deposit 3000 rubles in the bank and withdraw all the money after a year.
Vlad chose the deposit "Confidence": the amount increases by $20\%$ over the year, but the bank charges a $10\%$ fee upon withdrawal.
Dima chose the deposit "Reliabil... | 120 | 188 | 3 |
math | 26 Determine all non-empty subsets $A, B, C$ of $\mathbf{N}^{*}$ such that:
(1) $A \cap B=B \cap C=C \cap A=\varnothing$;
(2) $A \cup B \cup C=\mathbf{N}^{*}$;
(3) For all $a \in A, b \in B, c \in C$, we have $a+c \in A, b+c \in B, a+b \in C$ | ({3k-2\midk\in{Z}^{+}},{3k-1\midk\in{Z}^{+}},{3k\midk\in{Z}^{+}}) | 111 | 47 |
math | Example 7: There are 4 red cards, 3 blue cards, 2 yellow cards, and 1 white card. Cards of the same color are indistinguishable. Questions:
(1) How many ways are there to arrange these 10 cards in a row from left to right?
(2) How many ways are there to arrange the cards so that the first 3 cards from the left are of t... | 525 | 90 | 3 |
math | G6.3 Find $d$, if
$$
d=\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{1994}\right)\left(\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{1995}\right)-\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{1995}\right)\left(\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{1994}\right)
$$ | \frac{1}{1995} | 140 | 10 |
math | 421 ** Find the maximum value of the function $y=\frac{(x-1)^{5}}{(10 x-6)^{9}}$ for $x>1$.
| \frac{1}{2^{5}\cdot9^{9}} | 41 | 14 |
math | 11. Given the complex number $z$ satisfies $|z|=1$, then the minimum value of $\left|z^{2}-2 z+3\right|$ is $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | \frac{2\sqrt{6}}{3} | 66 | 12 |
math | 4. (13 points) In a dance ensemble, there are 8 boys and 20 girls. Some of them form mixed (boy and girl) dance pairs. It is known that in each pair, at least one of the partners does not belong to any other pair. What is the maximum number of dance pairs that can be formed in this ensemble? | 26 | 74 | 2 |
math | ## Task 4 - 200924
a) Determine all natural numbers $n \geq 3$ for which the following statement is true!
"Every prism that has a convex n-gon as its base has exactly 20n diagonals."
b) Determine, for each prism for which the statement in a) is true, the number of face diagonals and the number of space diagonals!
H... | n=12,spacediagonals=108,facediagonals=132 | 118 | 23 |
math | The cashier at the gallery sells tickets to visitors with a number according to the order in which they came that day. The first visitor gets a ticket with the number 1, the second with the number 2, etc. However, during the day, the yellow paper on which the tickets were printed ran out, so the cashier had to continue... | 82 | 149 | 2 |
math | The numbers $x, y$ and $z$ are such that $\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}=1$. What values can the expression $\frac{x^{2}}{y+z}+\frac{y^{2}}{z+x}+\frac{z^{2}}{x+y}$ take? | 0 | 78 | 1 |
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