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 | Task 11.7. For what least natural $x$ is the expression
$$
\sqrt{29+\sqrt{x}}+\sqrt{29-\sqrt{x}}
$$
an integer? | 400 | 43 | 3 |
math | Let $\triangle A B C$ have internal angles $\angle A, \angle B, \angle C$ with opposite sides $a, b, c$ respectively,
vector $\boldsymbol{m}=(\sin A, b+c)$,
$$
\boldsymbol{n}=(\sin C-\sin B, a-b),
$$
and there exists a real number $\lambda$ such that $\boldsymbol{m}=\lambda \boldsymbol{n}$.
(1) Find the size of $\angl... | (1,2] | 127 | 5 |
math | 8. [6] $A B C D$ is a convex quadrilateral such that $A B<\underline{A D}$. The diagonal $\overline{A C}$ bisects $\angle B A D$, and $m \angle A B D=130^{\circ}$. Let $E$ be a point on the interior of $\overline{A D}$, and $m \angle B A D=40^{\circ}$. Given that $B C=$ $C D=D E$, determine $m \angle A C E$ in degrees. | 55 | 123 | 2 |
math | In triangle $ABC$, let $M$ be the midpoint of $BC$, $H$ be the orthocenter, and $O$ be the circumcenter. Let $N$ be the reflection of $M$ over $H$. Suppose that $OA = ON = 11$ and $OH = 7.$ Compute $BC^2$. | 288 | 73 | 3 |
math | 6.227. $\left\{\begin{array}{l}\sqrt[3]{u+v}+\sqrt[3]{v+w}=3 \\ \sqrt[3]{v+w}+\sqrt[3]{w+u}=1 \\ \sqrt[3]{w+u}+\sqrt[3]{u+v}=0\end{array}\right.$ | (-4;5;3) | 77 | 7 |
math | 2. The natural numbers from 1 to n are divided into two sets. One set contains two of the given numbers, and the second set contains the remaining $n-2$ numbers. The product of the two numbers in the first set is equal to the sum of all the numbers in the second set. Can this division be made for:
a) $n=10$,
b) $n=15 ?... | 6\cdot7=42 | 88 | 7 |
math | 1. Let $i$ be the imaginary unit, $a$ and $b$ be positive integers, and $|(a+i)(2+i)|=\left|\frac{b-i}{2-i}\right|$, then $a+b=$ | 8 | 50 | 1 |
math | 18. C6 (FRA 2) Let \( O \) be a point of three-dimensional space and let \( l_{1}, l_{2}, l_{3} \) be mutually perpendicular straight lines passing through \( O \). Let \( S \) denote the sphere with center \( O \) and radius \( R \), and for every point \( M \) of \( S \), let \( S_{M} \) denote the sphere with center... | \frac{2R}{3} | 258 | 8 |
math | Let $ABC$ be a triangle with $|AB|=|AC|=26$, $|BC|=20$. The altitudes of $\triangle ABC$ from $A$ and $B$ cut the opposite sides at $D$ and $E$, respectively. Calculate the radius of the circle passing through $D$ and tangent to $AC$ at $E$. | \frac{65}{12} | 76 | 9 |
math | Task 1 - 300821 In a garden, there are two barrels of water. Jörg pours as many liters of water from the first barrel into the second barrel as the second barrel already contains. Then, he pours as many liters of water from the second barrel into the first barrel as there are in the first barrel after the previous pour... | 30 | 130 | 2 |
math | Alice thinks of four positive integers $a\leq b\leq c\leq d$ satisfying $\{ab+cd,ac+bd,ad+bc\}=\{40,70,100\}$. What are all the possible tuples $(a,b,c,d)$ that Alice could be thinking of? | (1, 4, 6, 16) | 70 | 13 |
math | Fix an integer $n \geq 2$. An $n\times n$ sieve is an $n\times n$ array with $n$ cells removed so that exactly one cell is removed from every row and every column. A stick is a $1\times k$ or $k\times 1$ array for any positive integer $k$. For any sieve $A$, let $m(A)$ be the minimal number of sticks required to partit... | 2n - 2 | 134 | 7 |
math | 36. Find the maximum value of the function $y=\frac{x}{a x^{2}+b}(a>0, b>0)$. | \frac{1}{2\sqrt{}} | 33 | 10 |
math | What is the multiplicity of the root 1 of $X^{2 n}-n X^{n+1}+n X^{n}-X^{2}$ for $n \in \mathbb{N}_{\geq 2}$? | 1 | 52 | 1 |
math | We call the [i]tribi [/i] of a positive integer $k$ (denoted $T(k)$) the number of all pairs $11$ in the binary representation of $k$. e.g $$T(1)=T(2)=0,\, T(3)=1, \,T(4)=T(5)=0,\,T(6)=1,\,T(7)=2.$$
Calculate $S_n=\sum_{k=1}^{2^n}T(K)$. | S_n = 2^{n-2}(n-1) | 110 | 15 |
math | Task B-4.6. Determine the equations of all tangents to the ellipse $x^{2}+4 y^{2}=20$ such that the points of tangency bisect the segments that the coordinate axes cut off on these tangents. Calculate the area of the quadrilateral determined by these tangents. | 40 | 66 | 2 |
math | Example 19. Solve the equation
$$
\sqrt{x+11}=x-1
$$ | 5 | 23 | 1 |
math | $=$ 2. Simplify the radical $\sqrt{2(6-2 \sqrt{3}-2 \sqrt{5}+\sqrt{15})}$. | \sqrt{3}+\sqrt{5}-2 | 36 | 11 |
math | \section*{Problem 1 - 311041}
Determine all real numbers \(k\) for which the following statement (1) is true: (1) For every pair \((a ; b)\) of real numbers \(a, b\), it holds that \(a^{2}+b^{2} \geq k \cdot a b\) | -2\leqk\leq2 | 79 | 10 |
math | 10.280. Determine the angles of an isosceles triangle if its area is related to the area of a square constructed on the base as $\sqrt{3}: 12$. | 30,30,120 | 42 | 9 |
math | 【Question 14】
The kindergarten teacher distributed 270 apples, 180 pears, and 235 oranges evenly among the children in the senior class, with the remaining apples, pears, and oranges in the ratio of $3: 2: 1$. There are $\qquad$ children in the senior class. | 29 | 74 | 2 |
math | In trapezoid $ABCD$ with $AD\parallel BC$, $AB=6$, $AD=9$, and $BD=12$. If $\angle ABD=\angle DCB$, find the perimeter of the trapezoid. | 39 | 53 | 2 |
math | 3. Two motorcyclists are moving on a circular path that is $1650 \mathrm{~m}$ long at constant speeds. If the motorcyclists move in opposite directions, they meet every minute, and if they move in the same direction, the motorcyclist with the higher speed catches up to the other motorcyclist every eleven minutes. Deter... | 900\mathrm{~}/\mathrm{},700\mathrm{~}/\mathrm{} | 83 | 22 |
math | Radovan is reading an interesting book. Yesterday he read 15 pages, today another 12 pages. To his surprise, he realized that the sum of the page numbers he read yesterday is the same as the sum of the page numbers he read today. Which page will he start reading tomorrow? (Radovan does not skip any pages or read any pa... | 74 | 95 | 2 |
math | 1. Write the smallest four-digit number in which all digits are different. | 1023 | 15 | 4 |
math | 4. (15 points) A one-kilogram model of a sports car body was made from carbon fiber for aerodynamic studies at a scale of 1:11. What is the mass of the actual body if it is also entirely made of carbon fiber? | 1331 | 55 | 4 |
math | ## Problem Statement
Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.
$M_{1}(2 ;-4 ;-3)$
$M_{2}(5 ;-6 ; 0)$
$M_{3}(-1 ; 3 ;-3)$
$M_{0}(2 ;-10 ; 8)$ | \frac{73}{\sqrt{83}} | 87 | 12 |
math | 1. A compact disc has the shape of a circle of diameter 5 inches with a 1-inch-diameter circular hole in the center. Assuming the capacity of the CD is proportional to its area, how many inches would need to be added to the outer diameter to double the capacity? | 2 | 58 | 1 |
math | 13.161. The cost of transporting a ton of cargo from point $M$ to point $N$ by rail is $b$ rubles more expensive than by water. How many tons of cargo can be transported from $M$ to $N$ by rail for $a$ rubles, if by water the same amount can transport $k$ tons more than by rail? | \frac{-+\sqrt{b^{2}k^{2}+4k}}{2b} | 81 | 22 |
math | ## Aufgabe $3 / 72$
Gegeben seien die vier Scheitelpunkte einer Ellipse. Man konstruiere unter ausschließlicher Verwendung von Zirkel und Lineal das der Ellipse umbeschriebene Quadrat.
| \sqrt{^{2}+b^{2}} | 55 | 11 |
math | 20. N6 (POL) ${ }^{\mathrm{IMO} 06}$ Let $p$ be an odd prime. Find the number of $p$-element subsets $A$ of $\{1,2, \ldots, 2 p\}$ such that the sum of all elements of $A$ is divisible by $p$. | \frac{1}{p}\left(\binom{2 p}{p}-2\right)+2 | 75 | 22 |
math | 11. In the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, the measure of the dihedral angle $A-B D_{1}-A_{1}$ is | 60 | 48 | 2 |
math | Given a positive integer $A$ written in decimal expansion: $(a_{n},a_{n-1}, \ldots, a_{0})$ and let $f(A)$ denote $\sum^{n}_{k=0} 2^{n-k}\cdot a_k$. Define $A_1=f(A), A_2=f(A_1)$. Prove that:
[b]I.[/b] There exists positive integer $k$ for which $A_{k+1}=A_k$.
[b]II.[/b] Find such $A_k$ for $19^{86}.$ | 19 | 128 | 4 |
math | 2. Let the function $f(x)=x^{3}+3 x^{2}+6 x+14$, and $f(a)=1, f(b)=19$, then $a+b=$ | -2 | 44 | 2 |
math | [ Average values ]
Three pirates divided the diamonds they had obtained during the day in the evening: twelve each for Bill and Sam, and the rest for John, who could not count. At night, Bill stole one diamond from Sam, Sam stole one from John, and John stole one from Bill. As a result, the average weight of Bill's di... | 9 | 101 | 1 |
math | 16. If the function $y=f(x)$ for every value $x_{1}$ in its domain, there exists a unique $x_{2}$ in its domain such that $f\left(x_{1}\right) f\left(x_{2}\right)=1$, then the function is called a "dependent function". Given the following propositions:
(1) $y=\frac{1}{x^{2}}$ is a dependent function;
(2) $y=\sqrt{2}+\s... | (2)(3) | 215 | 5 |
math | Let $ABC$ be a triangle (right in $B$) inscribed in a semi-circumference of diameter $AC=10$. Determine the distance of the vertice $B$ to the side $AC$ if the median corresponding to the hypotenuse is the geometric mean of the sides of the triangle. | \frac{5}{2} | 67 | 7 |
math | ## Task Condition
Based on the definition of the derivative, find $f^{\prime}(0)$:
$$
f(x)=\left\{\begin{array}{c}
\frac{2^{\operatorname{tg} x}-2^{\sin x}}{x^{2}}, x \neq 0 \\
0, x=0
\end{array}\right.
$$ | \ln\sqrt{2} | 82 | 7 |
math | 65. Let $a_{1}, a_{2}, \cdots, a_{n}$ be positive real numbers, and $n>2$ be a given positive integer. Find the largest positive number $K$ and the smallest positive number $G$ such that the following inequality holds: $K<\frac{a_{1}}{a_{1}+a_{2}}+\frac{a_{2}}{a_{2}+a_{3}}+\cdots+\frac{a_{n}}{a_{n}+a_{1}}<G$. (1991 Jap... | 1 < S_n < n-1 | 131 | 8 |
math | 10. (15 points) For a positive integer $n$, denote
$$
n!=1 \times 2 \times \cdots \times n \text {. }
$$
Find all positive integer tuples $(a, b, c, d, e, f)$ such that
$$
a!=b!+c!+d!+e!+f!,
$$
and $\quad a>b \geqslant c \geqslant d \geqslant e \geqslant f$. | (3,2,1,1,1,1),(5,4,4,4,4,4) | 111 | 25 |
math | 8、Car A and Car B start from A and B respectively at the same time, heading towards each other, and meet after 4 hours. Car A then takes another 3 hours to reach B. If Car A travels 20 kilometers more per hour than Car B, the distance between A and B is ( ) kilometers. | 560 | 68 | 3 |
math | A TGV departs from Paris at $150 \mathrm{~km} / \mathrm{h}$ heading towards Marseille, which is $800 \mathrm{~km}$ away. An intercity train departs from Marseille at $50 \mathrm{~km} / \mathrm{h}$ towards Paris at the same time. A swallow perched on the TGV takes off at this moment, at $200 \mathrm{~km} / \mathrm{h}$, ... | 800\mathrm{~} | 151 | 8 |
math | 1. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x, y \in \mathbb{R}$
$$
f(x f(y)-y f(x))=f(x y)-x y
$$
(Dušan Djukić) | f(x)=xf(x)=|x| | 68 | 9 |
math | 2A. Jane read five books. From these five books, 5 sets of four books can be formed. The four books in each of these sets together had 913, 973, 873, 1011, and 1002 pages. How many pages did each of the five books have? | =182,b=191,=320,=220,e=280 | 73 | 24 |
math | 4. Two cyclists are training on a circular stadium. In the first two hours, Ivanov lapped Petrov by 3 laps. Then Ivanov increased his speed by 10 km/h, and as a result, after 3 hours from the start, he lapped Petrov by 7 laps. Find the length of the lap. | 4 | 71 | 1 |
math | Let's determine the distinct digits $A, B, C, D, E, F$ such that the following equalities hold:
$$
\begin{aligned}
& A B C^{2}=D A E C F B \quad \text { and } \\
& C B A^{2}=E D C A B F
\end{aligned}
$$ | A=3,B=6,C=4,D=1,E=2,F=9 | 75 | 18 |
math | 7-5. In a row, there are 1000 toy bears. The bears can be of three colors: white, brown, and black. Among any three consecutive bears, there is a toy of each color. Iskander is trying to guess the colors of the bears. He made five guesses:
- The 2nd bear from the left is white;
- The 20th bear from the left is brown;
... | 20 | 167 | 2 |
math | 12. There is a four-digit number $A$, rearranging the digits of the four-digit number (none of which are 0) to form the largest possible number is 7668 greater than $A$, and the smallest possible number is 594 less than $A$, then $A=$ $\qquad$ . | 1963 | 70 | 4 |
math | 11.7. Given a polynomial
$$
P(x)=a_{2 n} x^{2 n}+a_{2 n-1} x^{2 n-1}+\ldots+a_{1} x+a_{0}
$$
where each coefficient $a_{i}$ belongs to the interval $[100,101]$. For what minimal $n$ can such a polynomial have a real root? (I. Bogdanov, K. Sukhov) | 100 | 104 | 3 |
math | ## Task 17/78
For the equation $x^{n}+p x-q=0, n \geq 2$, it is known that it has a positive rational solution and that $p$ and $q$ are prime numbers. Determine the solution as well as $p$ and $q$! | x_{0}=1,p=2,q=3 | 68 | 11 |
math | Problem 4. Consider the function $f: \mathbb{N}_{0} \rightarrow \mathbb{N}_{0}$, where $\mathbb{N}_{0}$ is the set of all non-negative integers, defined by the following conditions:
(i) $f(0)=0$,
(ii) $f(2 n)=2 f(n)$ and
(iii) $f(2 n+1)=n+2 f(n)$ for all $n \geq 0$.
(a) Determine the three sets $L:=\{n \mid f(n)f(n+1)\... | a_{k}=k2^{k-1}-2^{k}+1 | 182 | 17 |
math | 3. Let the function $f(x)=\frac{1}{2}+\log _{2} \frac{x}{1-x}$, define $S_{n}$ $=\sum_{i=1}^{n-1} f\left(\frac{i}{n}\right)$, where $n \in \mathbf{N}^{*}$, and $n \geqslant 2$, then $S_{n}=$ | \frac{n-1}{2} | 94 | 8 |
math | 1. 169 Determine all positive integer triples $(a, b, c)$ such that $a^{2}=2^{b}+c^{4}$ and only one of $a$, $b$, $c$ is odd. | (,b,)=(3\cdot2^{2n},4n+3,2^{n}) | 50 | 22 |
math | 4. A farmer has a flock of $n$ sheep, where $2000 \leq n \leq 2100$. The farmer puts some number of the sheep into one barn and the rest of the sheep into a second barn. The farmer realizes that if she were to select two different sheep at random from her flock, the probability that they are in different barns is exact... | 2025 | 98 | 4 |
math | Let $a$ be a positive integer that is a multiple of 5 and such that $a+1$ is a multiple of 7, $a+2$ is a multiple of 9, and $a+3$ is a multiple of 11. Determine the smallest possible value of $a$. | 1735 | 65 | 4 |
math | Consider those functions \( f: \mathbb{N} \rightarrow \mathbb{N} \) which satisfy the condition
\[ f(m+n) \geq f(m) + f(f(n)) - 1 \]
for all \( m, n \in \mathbb{N} \). Find all possible values of \( f(2007) \). \((\mathbb{N}\) denotes the set of all positive integers.) (Bulgaria) Answer. 1, 2, .., 2008. | 1, 2, \ldots, 2008 | 120 | 14 |
math | 24. Let $a$ and $b$ be two integers. Suppose that $\sqrt{7-4 \sqrt{3}}$ is a root of the equation $x^{2}+a x+b=0$. Find the value of $b-a$. | 5 | 55 | 1 |
math | 【Example 8】 How many terms are there in the expansion of an $n$-order determinant?
| n! | 22 | 2 |
math | 119. For what values of $n$ is the sum $n^{2}+(n+1)^{2}+(n+2)^{2}+$ $(n+3)^{2}$ divisible by 10 ( $n$ is a natural number)? | 5t+1 | 58 | 4 |
math | Solve the following system of equations:
$$
\begin{aligned}
& x_{1}+2 x_{2}+3 x_{3}+\ldots+9 x_{9}+10 x_{10}=55, \\
& x_{2}+2 x_{3}+3 x_{4}+\ldots+9 x_{10}+10 x_{1}=55, \\
& \vdots \\
& x_{10}+2 x_{1}+3 x_{2}+\ldots+9 x_{8}+10 x_{9}=55 .
\end{aligned}
$$ | x_{1}=x_{2}=\cdots=x_{10}=1 | 139 | 17 |
math | Determine all integers $n$ for which there exist an integer $k\geq 2$ and positive integers $x_1,x_2,\hdots,x_k$ so that
$$x_1x_2+x_2x_3+\hdots+x_{k-1}x_k=n\text{ and } x_1+x_2+\hdots+x_k=2019.$$ | [2018, 1009 \cdot 1010] | 87 | 20 |
math | 3. The range of the function $f(x)=\sqrt{2 x-x^{2}}+x$ is $\qquad$ | [0,\sqrt{2}+1] | 28 | 10 |
math | 12. If $16^{\sin ^{2} x}+16^{\cos ^{2} x}=10$, then $\cos 4 x=$
$\qquad$ . | -\frac{1}{2} | 44 | 7 |
math | 1. Determine all four-digit numbers $n=\overline{a b c d}$ such that inserting a digit 0 in any position results in a multiple of 7. | 7000,7007,7070,7077,7700,7707,7770,7777 | 36 | 39 |
math | B4. The equation $\sin x=\frac{x}{2021 \pi}$ has exactly $n$ solutions. Find $n$. | 4043 | 30 | 4 |
math | Let’s call a positive integer [i]interesting[/i] if it is a product of two (distinct or equal) prime numbers. What is the greatest number of consecutive positive integers all of which are interesting? | 3 | 43 | 1 |
math | # Task No. 7.1
## Condition:
Artist Ivan Konstantinovich decided to sell several of his paintings at the Broken Auction. The rules of the Broken Auction are as follows: first, Ivan Konstantinovich names a certain starting price for his painting, after which the participants who want to buy this painting begin to bid ... | 6 | 216 | 1 |
math | [Example 2.4.7] Find the number of all natural numbers $n$, $4 \leqslant n \leqslant 1023$, such that $n$ in binary representation does not have three consecutive identical digits. | 228 | 54 | 3 |
math | Find the dihedral angles of a trihedral angle, the plane angles of which are $90^{\circ}, 90^{\circ}$ and $\alpha$.
# | \alpha,90,90 | 39 | 8 |
math | ## Task A-4.1.
Determine all complex numbers $a$ for which all coefficients of the polynomial
$$
P(x)=(x-a)\left(x-a^{2}\right)\left(x-a^{3}\right)\left(x-a^{4}\right)
$$
are real. | \in\mathbb{R}\cup{i,-i}\cup{\cos(\frac{2k\pi}{5})+i\sin(\frac{2k\pi}{5}):k=1,2,3,4} | 61 | 50 |
math | 1. Find all real numbers $a$ and $b$
$$
\left\{\begin{array}{l}
\left|\frac{x^{y}-1}{x^{y}+1}\right|=a \\
x^{2}+y^{2}=b
\end{array}\right.
$$
given that $x>0$, there is a unique solution. | 00<b\leq1 | 79 | 7 |
math | Determine the largest positive integer $n$ for which there exists a set $S$ with exactly $n$ numbers such that [list][*] each member in $S$ is a positive integer not exceeding $2002$, [*] if $a,b\in S$ (not necessarily different), then $ab\not\in S$. [/list] | 1958 | 77 | 6 |
math | 4. For any integer $n(n \geqslant 4)$, consider $m$ subsets $A_{1}, A_{2}, \cdots, A_{m}$ of the set $\{1,2, \cdots, n\}$, where $A_{1}$ has one element, $A_{2}$ has two elements, $\cdots \cdots A_{m}$ has $m$ elements, and no subset is contained in another subset. Find the maximum value of $m$.
| n-2 | 109 | 3 |
math | 5-35 (1) Find the possible minimum value of the polynomial $P(x, y)=4+x^{2} y^{4}+x^{4} y^{2}-3 x^{2} y^{2}$.
(2) Prove: this polynomial cannot be expressed as a sum of squares of polynomials in variables $x, y$. | 3 | 75 | 1 |
math | What are the integers $a$ between 1 and 105 such that $35 \mid a^{3}-1$? | 1,11,16,36,46,51,71,81,86 | 29 | 25 |
math | 2. If real numbers $x, y$ satisfy
$$
\begin{array}{l}
\frac{x}{2^{10}+5^{3}}+\frac{y}{2^{10}+6^{3}}=1, \\
\frac{x}{3^{10}+5^{3}}+\frac{y}{3^{10}+6^{3}}=1,
\end{array}
$$
then $x+y=$ $\qquad$ (the result should be written in the form of a power). | 2^{10}+3^{10}+5^{3}+6^{3} | 115 | 21 |
math | $\mathbf{R}^{*}$, such that for any non-zero real numbers $x, y$ satisfying $x^{2}+y \neq 0$, we have $f\left(x^{2}+y\right)=f^{2}(x)+\frac{f(x y)}{f(x)}$. | f(x)=x | 70 | 4 |
math | 4. If the complex coefficient equation with respect to $x$
$$
(1+2 \mathrm{i}) x^{2}+m x+1-2 \mathrm{i}=0
$$
has real roots, then the minimum value of the modulus of the complex number $m$ is $\qquad$. | 2 | 65 | 1 |
math | Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ that satisfy
- $f(p)>0$ for all prime numbers $p$,
- $p \mid (f(x) + f(p))^{f(p)} - x$ for all $x \in \mathbb{Z}$ and all prime numbers $p$. | f(x)=x | 77 | 4 |
math | Let $ABCD$ be a quadrilateral with side lengths $AB = 2$, $BC = 3$, $CD = 5$, and $DA = 4$. What is the maximum possible radius of a circle inscribed in quadrilateral $ABCD$? | \frac{2\sqrt{30}}{7} | 56 | 13 |
math | ## Task Condition
Find the derivative.
$$
y=\ln \left(5 x+\sqrt{25 x^{2}+1}\right)-\sqrt{25 x^{2}+1} \cdot \operatorname{arctg} 5 x
$$ | -\frac{25x\cdot\operatorname{arctg}5x}{\sqrt{25x^{2}+1}} | 59 | 31 |
math | Let's determine the distinct digits $A, B, C$ if the product of the number " $\overline{A B}$ whole, $C$ tenth" and $C$, rounded to the nearest integer, results in the number $\overline{B C}$: $\overline{A B, C} \cdot C \approx \overline{B C}$. | A=1,B=4,C=3 | 78 | 9 |
math | Let G, O, D, I, and T be digits that satisfy the following equation:
\begin{tabular}{ccccc}
&G&O&G&O\\
+&D&I&D&I\\
\hline
G&O&D&O&T
\end{tabular}
(Note that G and D cannot be $0$, and that the five variables are not necessarily different.)
Compute the value of GODOT. | 10908 | 93 | 5 |
math | 3. For each $n \in \mathbf{N}^{\cdot}$, solve the equation
$$
\sin x \sin 2 x \cdots \sin n x+\cos x \cos 2 x \cdots \cos n x=1
$$ | 2\pior2k\pi+\frac{\pi}{2},,k\in{Z}whenn=1;\,2\pi,\in{Z}whenn=4-2orn=4+1,\in{N}^{*};\ | 59 | 56 |
math | $ABC$ is a triangle
$E, F$ are points in $AB$, such that $AE = EF = FB$
$D$ is a point at the line $BC$ such that $ED$ is perpendiculat to $BC$
$AD$ is perpendicular to $CF$.
The angle CFA is the triple of angle BDF. ($3\angle BDF = \angle CFA$)
Determine the ratio $\frac{DB}{DC}$.
%Edited!% | \frac{7}{2} | 107 | 7 |
math | (9) (15 points) On the ellipse $x^{2}+4 y^{2}=4 x$, find the coordinates of the point $P$ that makes $z=x^{2}-y^{2}$ attain its maximum and minimum values. | P(\frac{2}{5},\\frac{3}{5})forminimumz=-\frac{1}{5},P(4,0)formaximumz=16 | 53 | 40 |
math | Example 1. $\mathrm{AC}$ and $\mathrm{CE}$ are two diagonals of a regular hexagon $\mathrm{ABCDEF}$, and points $\mathrm{M}$ and $\mathrm{N}$ internally divide $\mathrm{AC}$ and $\mathrm{CE}$, respectively, such that $\mathrm{M}: \mathrm{AC} = \mathrm{CN}: \mathrm{CE} = \mathrm{r}$. If points $\mathrm{B}$, $\mathrm{M}$... | \frac{\sqrt{3}}{3} | 132 | 10 |
math | 1. On the board, 2020 quadratic equations are written:
$$
\begin{gathered}
2020 x^{2} + b x + 2021 = 0 \\
2019 x^{2} + b x + 2020 = 0 \\
2018 x^{2} + b x + 2019 = 0 \\
\ldots \\
x^{2} + b x + 2 = 0
\end{gathered}
$$
(each subsequent equation is obtained from the previous one by decreasing the leading coefficient and ... | 2021 | 159 | 4 |
math | 1B. Find all three-digit numbers $A$, for which the arithmetic mean of the numbers obtained by permuting the digits of the number $A$, is equal to the number $A$. | 111,222,333,444,555,666,777,888,999,407,518,629,370,481,592 | 40 | 59 |
math | Find the smallest integer $k > 1$ for which $n^k-n$ is a multiple of $2010$ for every integer positive $n$. | 133 | 35 | 3 |
math | 8. Given positive integers $a, b, c, x, y, z$ satisfying
$$
a \geqslant b \geqslant c \geqslant 1, x \geqslant y \geqslant z \geqslant 1 \text {, }
$$
and $\left\{\begin{array}{l}2 a+b+4 c=4 x y z, \\ 2 x+y+4 z=4 a b c .\end{array}\right.$
then the number of six-tuples $(a, b, c, x, y, z)$ that satisfy the conditions ... | 0 | 144 | 1 |
math | 7. The number of apples produced by a group of farmers is less than 1000. It is known that they shared the apples in the following way. In turn, each farmer took from the collection of apples either exactly one-half or exactly one-third of the apples remaining in the collection. No apples were cut into pieces. After ea... | 12 | 183 | 2 |
math | 34 chameleons live on an island. At the beginning, there are 7 yellow, 10 red, and 17 green ones. When two chameleons of different colors meet, they simultaneously adopt the third color. One day, Darwin arrives on the island and observes that all the chameleons are of the same color. What is this color? | green | 79 | 1 |
math | 14. Little Marco goes up and down from one floor to another on the escalator of a shopping center, one step at a time. If he proceeds in the direction of the escalator's movement at a constant speed relative to it (i.e., the time interval between steps is constant), he steps on 15 steps. If he proceeds in the opposite ... | 21 | 112 | 2 |
math | 1. A three-digit number is 29 times the sum of its digits. Then this three-digit number is $\qquad$ | 261 | 27 | 3 |
math | ## Task 1 - 250711
In a bag, there is $1 \mathrm{~kg}$ of sugar. Using a balance scale with two weighing pans (each large enough for $1 \mathrm{~kg}$ of loose sugar) and exactly one $50 \mathrm{~g}$ weight, $300 \mathrm{~g}$ of sugar should be weighed out.
Show that this is possible with only three weighings! | 300\mathrm{~} | 97 | 8 |
math | 16. Let $F_{1}$ and $F_{2}$ be the left and right foci, respectively, of the hyperbola
$$
\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \quad(a>0, b>0)
$$
$O$ is the origin, point $P$ is on the left branch of the hyperbola, point $M$ is on the right directrix, and it satisfies
$$
F_{1} \boldsymbol{O}=\boldsymbol{P M}, O P... | y= \pm \sqrt{5} x-3 | 308 | 12 |
math | 5. Determine the angle between the tangents to the parabola $y^{2}=4 x$ at the points of its intersection $\mathrm{s}$
(8) with the line $2 x+y-12=0$. | 45 | 49 | 2 |
math | 12. There are five unequal non-zero natural numbers, the smallest of which is 7. If one of them is reduced by 20, and the other four numbers are all increased by 5, the resulting numbers are still these five numbers. The sum of these five numbers is $\qquad$ . | 85 | 64 | 2 |
math | 5. Find all pairs $(x ; y)$ of natural numbers for which both numbers $x^{2}+8 y ; y^{2}-8 x$ are perfect squares. | (n;n+2),(7;15),(33;17),(45;23) | 37 | 22 |
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