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 | [ Formulas for abbreviated multiplication (other).] Case analysis
Solve the equation $\left(x^{2}-y^{2}\right)^{2}=16 y+1$ in integers.
# | (\1,0),(\4,3),(\4,5) | 42 | 15 |
math | A week ago, Sandy’s seasonal Little League batting average was $360$. After five more at bats this week, Sandy’s batting average is up to $400$. What is the smallest number of hits that Sandy could have had this season? | 12 | 52 | 2 |
math | ## Task 1 - 070731
The sides of a hexagon, in which no side is parallel to another, are extended beyond the vertices.
What is the maximum number of new intersection points that can thus be created? | 9 | 50 | 1 |
math | 11. (5 points) A fast car and a slow car start from locations A and B respectively at the same time, heading towards each other. The fast car travels at 33 kilometers per hour and covers $\frac{4}{7}$ of the total distance before they meet. It is known that the slow car takes 8 hours to travel the entire distance. Ther... | 198 | 91 | 3 |
math | Solve the system of equation
$$x+y+z=2;$$$$(x+y)(y+z)+(y+z)(z+x)+(z+x)(x+y)=1;$$$$x^2(y+z)+y^2(z+x)+z^2(x+y)=-6.$$ | \{(0, 3, -1), (0, -1, 3), (3, 0, -1), (3, -1, 0), (-1, 0, 3), (-1, 3, 0)\} | 58 | 56 |
math | 411. A parallelepiped is composed of identical cubes. Three faces of the parallelepiped, having a common vertex, were painted. It turned out that at least one face of half of all the cubes is painted. How many cubes have painted faces? | 60,72,84,90,120 | 54 | 15 |
math | XXXII - I - Problem 10
Determine all functions $ f $ mapping the set of all rational numbers $ \mathbb{Q} $ to itself that satisfy the following conditions:
a) $ f(1)=2 $,
b) $ f(xy) = f(x)f(y)-f(x+y)+1 $ for $ x, y \in \mathbb{Q} $. | f(x)=x+1 | 82 | 6 |
math | If we divide a three-digit, decimal number by its reverse, the quotient is 3, and the remainder is the sum of the digits of the number. What could this number be? | 441882 | 38 | 6 |
math | 2. (15 points) Given $p, q(q \neq 0)$ are real numbers, the equation $x^{2}-p x+q=0$ has two real roots $\alpha, \beta$, and the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=p, a_{2}=p^{2}-q, a_{n}=p a_{n-1}-q a_{n-2}(n=$ $3,4, \cdots)$.
(1) Find the general term formula of the sequence $\left\{a_{n}\right\}$ (expr... | S_{n}=3-\frac{n+3}{2^{n}} | 185 | 15 |
math | 1. Let the set $P_{n}=\{1,2, \cdots, n\}\left(n \in \mathbf{Z}_{+}\right)$. Denote $f(n)$ as the number of sets $A$ that satisfy the following conditions:
(1) $A \subseteq P_{n}, \bar{A}=P_{n} \backslash A$;
(2) If $x \in A$, then $2 x \notin A$;
(3) If $x \in \bar{A}$, then $2 x \notin \bar{A}$.
Then $f(2018)=$ $\qqua... | 2^{1009} | 143 | 7 |
math | 4. Igor Gorshkov has all seven books about Harry Potter. In how many ways can Igor arrange these seven volumes on three different bookshelves, so that each shelf has at least one book? (Arrangements that differ by the order of books on the shelf are considered different). | 75600 | 59 | 5 |
math | Three. (Full marks 12 points) Solve the equation:
$$
\frac{13 x-x^{2}}{x+1}\left(x+\frac{13-x}{x+1}\right)=42 \text {. }
$$ | x_{1}=1, x_{2}=6, x_{3}=3+\sqrt{2}, x_{4}=3-\sqrt{2} | 53 | 32 |
math | 【Question 11】Satisfy the condition of leaving a remainder of 3 when divided by 7, leaving a remainder of 4 when divided by 9, and being less than 100. The natural numbers are
保留源文本的换行和格式，这里应该是：
【Question 11】Satisfy the condition of leaving a remainder of 3 when divided by 7, leaving a remainder of 4 when divided by 9... | 31,94 | 131 | 5 |
math | 12. Given the ellipse $\frac{x^{2}}{4}+y^{2}=1$ and the hyperbola $x^{2}-\frac{y^{2}}{2}=1$, where $F_{1} 、 F_{2}$ are the foci of the ellipse, and $P$ is an intersection point of the ellipse and the hyperbola. Then $\cos \angle F_{1} P F_{2}$ $=$ . $\qquad$ | -\frac{1}{3} | 102 | 7 |
math | # 8. Variant 1.
Each of the 10 students came up with 5 natural numbers. It turned out that each number was thought of by at least three students. What is the maximum number of different numbers that could have been thought of? | 16 | 53 | 2 |
math | 6. (7 points) If the number $A$ is written on the board, you can add any of its divisors, except 1 and $A$ itself. Can you get 1234321 from $A=4$? Answer: Yes. | 1234321 | 58 | 7 |
math | Example 3 Find the Möbius transform $F(n)$ of $\Omega(n)$.
| \frac{1}{2} \Omega(n) \tau(n) | 19 | 15 |
math | 29. (8th grade) When asked what his ticket number was, someone answered like this: "If all six two-digit numbers that can be formed from the digits of the number are added together, then half of the resulting sum will be exactly my ticket number." Determine the ticket number. | 198 | 60 | 3 |
math | 12. Let $f(x)=\frac{1+10 x}{10-100 x}$. Set $f^{n}=f \circ f \circ \cdots \circ f$. Find the value of
$$
f\left(\frac{1}{2}\right)+f^{2}\left(\frac{1}{2}\right)+f^{3}\left(\frac{1}{2}\right)+\cdots+f^{6000}\left(\frac{1}{2}\right)
$$ | 595 | 113 | 3 |
math | 3. A book has a total of 61 pages, sequentially numbered as 1, 2, ..., 61. Someone, while adding these numbers, mistakenly reversed the digits of two two-digit page numbers (a two-digit number of the form $\overline{a b}$ was treated as $\overline{b a}$), resulting in a total sum of 2008. Therefore, the maximum sum of ... | 68 | 227 | 2 |
math | ## Task A-3.2.
Determine all natural numbers $a$ and $b$ such that
$$
a^{2}=4 b+3 \cdot \text{LCM}(a, b)
$$
where $\text{LCM}(m, n)$ denotes the least common multiple of the numbers $m$ and $n$. | (7q,7q^{2})foranynaturalq,(4q,q^{2})foranyoddnaturalq | 73 | 26 |
math | 21. $y=\frac{3}{x^{2}-4}$. | (-\infty,-2)\cup(-2,2)\cup(2,\infty) | 17 | 20 |
math | Example 5 Find the last three digits of $1 \times 3 \times 5 \times \cdots \times 1997$. | 375 | 32 | 3 |
math | ## Task Condition
Calculate the area of the parallelogram constructed on vectors $a$ and $b$.
$a=4 p+q$
$b=p-q$
$|p|=7$
$|q|=2$
$(\widehat{p, q})=\frac{\pi}{4}$ | 35\sqrt{2} | 62 | 7 |
math | 1. There are 10 balls each of red, white, and black. If 12 balls are drawn from them, but it is required that all three colors of balls are included, how many different ways are there to draw the balls? | 55 | 51 | 2 |
math | Find all polynomials $P(x)$ with real coefficents, such that for all $x,y,z$ satisfying $x+y+z=0$, the equation below is true: \[P(x+y)^3+P(y+z)^3+P(z+x)^3=3P((x+y)(y+z)(z+x))\] | P(x) = 0, \quad P(x) = x, \quad P(x) = -x | 71 | 23 |
math | Find all real numbers $x, y, z$ satisfying:
$$
\left\{\begin{aligned}
(x+1) y z & =12 \\
(y+1) z x & =4 \\
(z+1) x y & =4
\end{aligned}\right.
$$
Proposition 1.27 (Vieta's Formulas). Let $P(x)=a x^{2}+b x+c$ be a real polynomial of degree 2 (with $a \neq 0$) having $z_{1}$ and $z_{2}$ as real roots. Then $z_{1} z_{2}... | (2,-2,-2)(\frac{1}{3},3,3) | 252 | 18 |
math | Let $x$, $y$, $z$ be arbitrary positive numbers such that $xy+yz+zx=x+y+z$.
Prove that $$\frac{1}{x^2+y+1} + \frac{1}{y^2+z+1} + \frac{1}{z^2+x+1} \leq 1$$.
When does equality occur?
[i]Proposed by Marko Radovanovic[/i] | 1 | 95 | 1 |
math | 4. In rectangle $A B C D$, $A B=2, A D=1$, point $P$ on segment $D C$ and point $Q$ on the extension of $C B$ satisfy the condition $|\overrightarrow{D P}|=|\overrightarrow{B Q}|$, then the minimum value of $\overrightarrow{P A} \cdot \overrightarrow{P Q}$ is $\qquad$ . | \frac{3}{4} | 91 | 7 |
math | ## Task B-4.1.
In the set of natural numbers, solve the equation $5^{x}+5^{y}+5^{z}=18775$, where $x<y<z$. How many triangles have side lengths that are numbers, not necessarily distinct, from the set $\{x, y, z\}$? | 8 | 73 | 1 |
math | 5. Solve the inequality $\frac{x+3-3 \sqrt{x+1}}{x^{2}-4 x}>0$ (10 points) | x\in[-1;0)\cup(0;3)\cup(4;+\infty) | 33 | 22 |
math | 8. Given $a, b \in \mathbf{Z}$, and $a+b$ is a root of the equation
$$
x^{2}+a x+b=0
$$
Then the maximum possible value of $b$ is $\qquad$ | 9 | 57 | 1 |
math | 2. Matvey decided to start eating properly and every day he ate one bun less and one pear more than the previous day. In total, during the time of proper nutrition, he ate 264 buns and 187 pears. How many days was Matvey on a proper diet? | 11 | 63 | 2 |
math | Let [i]Revolution[/i]$(x) = x^3 +Ux^2 +Sx + A$, where $U$, $S$, and $A$ are all integers and $U +S + A +1 = 1773$. Given that [i]Revolution[/i] has exactly two distinct nonzero integer roots $G$ and $B$, find the minimum value of $|GB|$.
[i]Proposed by Jacob Xu[/i]
[hide=Solution]
[i]Solution.[/i] $\boxed{392}$
Notic... | 392 | 308 | 3 |
math | Example 1. To determine the parameters $a_{1}, a_{2}$, and $a_{3}$ in the formula $y=a_{1} x^{2}+a_{2} x+a_{3}$, values of $y$ were measured at different values of $x$. The obtained sample is
$$
\begin{array}{rrrrrrrrrr}
x_{k} & -1 & -0.75 & -0.5 & -0.25 & 0 & 0.25 & 0.5 & 0.75 & 1 \\
y_{k} & 6.01 & 5.07 & 4.30 & 3.56... | 0.96x^{2}-1.97x+3.07 | 249 | 18 |
math | 13. $\cos 20^{\circ} \cdot \cos 40^{\circ} \cdot \cos 60^{\circ} \cdot \cos 80^{\circ}=$ | \frac{1}{16} | 46 | 8 |
math | 41*. Factor the polynomial with real coefficients:
$$
x\left(y^{2}-z^{2}\right)+y\left(z^{2}-x^{2}\right)+z\left(x^{2}-y^{2}\right)
$$ | (x-y)(y-z)(z-x) | 52 | 9 |
math | 1. Given $\min _{x \in R} \frac{a x^{2}+b}{\sqrt{x^{2}+1}}=3$.
(1) Find the range of $b$;
(2) For a given $b$, find $a$. | a=\frac{b-\sqrt{b^{2}-9}}{2} | 60 | 17 |
math | Problem 5. Timofey placed 10 grid rectangles on a grid field, with areas of $1, 2, 3, \ldots, 10$ respectively. Some of the rectangles overlapped each other (possibly completely, or only partially). After this, he noticed that there is exactly one cell covered exactly once; there are exactly two cells covered exactly t... | 5 | 146 | 1 |
math | 1.7. Calculate the determinant
$$
\Delta=\left|\begin{array}{cccc}
3 & 1 & -1 & 2 \\
-3 & 1 & 4 & -5 \\
2 & 0 & 1 & -1 \\
3 & -5 & 4 & -4
\end{array}\right|
$$ | 40 | 74 | 2 |
math | Example 2. If $a, b$ are both positive real numbers, and $\frac{1}{a}-\frac{1}{b}-$ $\frac{1}{a+b}:=0$, then $\left(\frac{b}{a}\right)^{3}+\left(\frac{a}{b}\right)^{3}=$ $\qquad$ | 2 \sqrt{5} | 77 | 6 |
math | 5.5. Four dolls and five robots cost 4100 rubles, while five dolls and four robots cost 4000. How much does one doll cost? | 400 | 38 | 3 |
math | 5. Find all triples of real numbers $a, b, c$, for which
$$
27^{a^{2}+b+c+1}+27^{b^{2}+c+a+1}+27^{c^{2}+a+b+1}=3
$$ | =b==-1 | 64 | 3 |
math | 3.262. $\left(\operatorname{tg} 255^{\circ}-\operatorname{tg} 555^{\circ}\right)\left(\operatorname{tg} 795^{\circ}+\operatorname{tg} 195^{\circ}\right)$.
3.262. $\left(\tan 255^{\circ}-\tan 555^{\circ}\right)\left(\tan 795^{\circ}+\tan 195^{\circ}\right)$. | 8\sqrt{3} | 124 | 6 |
math | \section*{Problem 6}
Find all integers \(x, y\) satisfying \(x^{2}+x=y^{4}+y^{3}+y^{2}+y\).
| x,y=-1,1;0,-1;-1,0;0,0;-6,2;5,2 | 42 | 26 |
math | 53rd Putnam 1992 Problem A6 Four points are chosen independently and at random on the surface of a sphere (using the uniform distribution). What is the probability that the center of the sphere lies inside the resulting tetrahedron? Solution | \frac{1}{8} | 53 | 7 |
math | (6) For $n \in \mathbf{N}^{*}$, if $n \cdot 2^{n}+1$ is a multiple of 3, then the set of remainders when $n$ is divided by 6 is $\qquad$. | {1,2} | 58 | 5 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty}\left(\frac{n+5}{n-7}\right)^{\frac{n}{6}+1}$ | e^2 | 45 | 3 |
math | ## Task Condition
Find the derivative.
$$
y=\frac{1}{\sqrt{2}} \ln \left(\sqrt{2} \tan x+\sqrt{1+2 \tan^{2} x}\right)
$$ | \frac{1}{\cos^{2}x\sqrt{1+2\tan^{2}x}} | 49 | 24 |
math | Find all triplets of positive rational numbers $(m,n,p)$ such that the numbers $m+\frac 1{np}$, $n+\frac 1{pm}$, $p+\frac 1{mn}$ are integers.
[i]Valentin Vornicu, Romania[/i] | \left(\frac{1}{2}, \frac{1}{2}, 4\right), \left(\frac{1}{2}, 1, 2\right), (1, 1, 1) | 62 | 47 |
math | Mom cooked homemade currant syrup and poured it into bottles. The bottles were of two types: small with a volume of $500 \mathrm{ml}$ and large with a volume of $750 \mathrm{ml}$. In the end, she had 12 empty small bottles left, and all other bottles were completely filled. Then Mom realized that she could have poured ... | 8 | 130 | 1 |
math | # 8. Variant 1.
On the Island of Misfortune, there live knights who always tell the truth, and liars who always lie. One day, 2023 natives, among whom $N$ are liars, stood in a circle and each said: "Both of my neighbors are liars." How many different values can $N$ take? | 337 | 77 | 3 |
math | 12. Let the set $M=\{1,2, \cdots, 1000\}$, and for any non-empty subset $X$ of $M$, let $\alpha_{X}$ denote the sum of the largest and smallest numbers in $X$. Then, the arithmetic mean of all such $\alpha_{X}$ is $\qquad$ . | 1001 | 77 | 4 |
math | Let set $A=\{1,2,\ldots,n\} ,$ and $X,Y$ be two subsets (not necessarily distinct) of $A.$ Define that $\textup{max} X$ and $\textup{min} Y$ represent the greatest element of $X$ and the least element of $Y,$ respectively. Determine the number of two-tuples $(X,Y)$ which satisfies $\textup{max} X>\textup{min} Y.$ | 2^{2n} - (n+1)2^n | 98 | 13 |
math | 2. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ so that
$$
(x+y)(f(x)-f(y))=(x-y) f(x+y)
$$
for all $x, y \in \mathbb{R}$. | f(x)=ax+bx^2 | 61 | 8 |
math | 29*. a) How many different positive integer solutions does the equation
$$
x_{1}+x_{2}+x_{3}+\ldots+x_{m}=n
$$
have?
b) How many different non-negative integer solutions does the equation
$$
x_{1}+x_{2}+x_{3}+\ldots+x_{m}=n
$$
have?
Note. A particular case of problem 29a) (corresponding to $m=3$) is problem 25.
... | C_{n+-1}^{-} | 380 | 8 |
math | ## Task Condition
Calculate approximately using the differential.
$y=\frac{x+\sqrt{5-x^{2}}}{2}, x=0.98$ | 1.495 | 33 | 5 |
math | 2. From $1,2,3, \cdots, 14$ select $a_{1}, a_{2}, a_{3}$ in ascending order, such that $a_{2}-a_{1} \geqslant 3$, $a_{3}-a_{2} \geqslant 3$ are simultaneously satisfied. The number of all different selection methods that meet the above requirements is $\qquad$. | 120 | 93 | 3 |
math | 4. Determine all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ that satisfy the condition:
$$
x + \sqrt{x} - f(x) = \sqrt{f(f(x))}, \quad \forall x \in \mathbb{N}
$$
Gabriel Daniilescu, Brăila | f(n)=n,\foralln\in\mathbb{N} | 73 | 15 |
math | C5 (8-1, USSR) In a middle school mathematics competition, three problems $A, B, C$ were given, with 25 students participating. Each student solved at least one of the problems. Among the students who did not solve problem $A$, the number of students who solved problem $B$ is twice the number of students who solved pro... | 6 | 146 | 1 |
math | ## Task 14/81
In a plane, let equilateral triangles $D_{i}$ with side lengths $a_{i}=2 i-1, i=1 ; 2 ; 3 ; \ldots$ be arranged along a line $g$ such that the "right" vertex of triangle $D_{k}$ coincides with the "left" vertex of triangle $D_{k+1}$ and that the third vertices all lie in the same half-plane generated by ... | y^{2}=3(x-\frac{1}{4}) | 115 | 13 |
math | 4. Through a point $A$ on the line $x=-4$, draw a tangent line $l$ to the parabola $C$: $y^{2}=2 p x(p>0)$, with the point of tangency being $B(1,2)$. Line $l$ intersects the $x$-axis at point $D$. $P$ is a moving point on the parabola $C$ different from point $B$. $\overrightarrow{P E}=\lambda_{1} \overrightarrow{P A}... | y^{2}=\frac{8}{3}(x+\frac{1}{3})(y\neq\frac{4}{3}) | 210 | 30 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty} \frac{n \sqrt[3]{5 n^{2}}+\sqrt[4]{9 n^{8}+1}}{(n+\sqrt{n}) \sqrt{7-n+n^{2}}}$ | \sqrt{3} | 65 | 5 |
math | ## Task B-4.4.
Determine the largest natural number $n$ such that $\frac{500!}{7^{n}}$ is a natural number. | 82 | 37 | 2 |
math | 9. (16 points) Let $n$ be an odd number $(n>1)$. The $n$ roots of the equation $z^{n}=1$ are $1, x_{1}, x_{2}, \cdots, x_{n-1}$. Find the value of the expression $\sum_{i=1}^{n-1} \frac{1}{1+x_{i}}$. | \frac{n-1}{2} | 88 | 8 |
math | Task B-3.2. (20 points) Solve the inequality
$$
\log _{x} 2 \cdot \log _{2 x} 2 \cdot \log _{2}(16 x)<1
$$ | x\in\langle0,\frac{1}{4}\rangle\cup\langle\frac{1}{2},1\rangle\cup\langle4,\infty\rangle | 52 | 38 |
math | 5. Draw the tangent line to the curve $y=3x-x^{3}$ through the point $A(2,-2)$. Then the equation of the tangent line is $\qquad$ . | y=-2 \text{ or } 9x+y-16=0 | 42 | 17 |
math | 4. If the sum of 12 distinct positive integers is 2016, then the maximum value of the greatest common divisor of these positive integers is $\qquad$
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | 24 | 62 | 2 |
math | ## Zadatak B-3.2.
Riješite sustav jednadžbi
$$
\begin{aligned}
& y^{2020 x}=10 \\
& 6060 x+2 \log y=7
\end{aligned}
$$
| (x,y)=(\frac{1}{1010},\sqrt{10})(x,y)=(\frac{1}{6060},1000) | 62 | 37 |
math | 7.3. How many five-digit natural numbers exist, each of which has adjacent digits with different parity | 5625 | 21 | 4 |
math | 8. $11 n^{2}(n \geqslant 4)$ positive numbers are arranged in $n$ rows and $n$ columns:
$$
\begin{array}{l}
a_{11} a_{12} a_{13} a_{14} \cdots a_{1 n} \\
a_{21} a_{22} a_{23} a_{24} \cdots a_{2 n} \\
a_{31} a_{32} a_{33} a_{34} \cdots a_{3 n} \\
a_{41} a_{42} a_{43} a_{44} \cdots a_{4 n} \\
\cdots \cdots \\
a_{n 1} a_{... | 2-\frac{n+2}{2^{n}} | 293 | 11 |
math | ## Task 3 - 331213
In a beauty contest for poodles, Asta, Benno, Cäsar, and Dolly face a jury of four members. Each jury member votes for one of the dogs by raising a card with the initial letter of the dog's name. As a rule for evaluating this voting result, it was stipulated: If two dogs clearly receive more votes t... | 172 | 205 | 3 |
math | Can we find positive reals $a_1, a_2, \dots, a_{2002}$ such that for any positive integer $k$, with $1 \leq k \leq 2002$, every complex root $z$ of the following polynomial $f(x)$ satisfies the condition $|\text{Im } z| \leq |\text{Re } z|$,
\[f(x)=a_{k+2001}x^{2001}+a_{k+2000}x^{2000}+ \cdots + a_{k+1}x+a_k,\] w... | a_1, a_2, \dots, a_{2002} | 173 | 19 |
math | The fourth-degree equation $x^4-x-504=0$ has $4$ roots $r_1$, $r_2$, $r_3$, $r_4$. If $S_x$ denotes the value of ${r_1}^4+{r_2}^4+{r_3}^4+{r_4}^4$, compute $S_4$.
[i]2015 CCA Math Bonanza Individual Round #10[/i] | 2016 | 108 | 4 |
math | Points $E$ and $F$ are chosen on sides $BC$ and $CD$ respectively of rhombus $ABCD$ such that $AB=AE=AF=EF$, and $FC,DF,BE,EC>0$. Compute the measure of $\angle ABC$. | 80^\circ | 59 | 4 |
math | Let $a$, $b$, and $c$ be pairwise distinct positive integers, which are side lengths of a triangle. There is a line which cuts both the area and the perimeter of the triangle into two equal parts. This line cuts the longest side of the triangle into two parts with ratio $2:1$. Determine $a$, $b$, and $c$ for which the ... | 504 | 83 | 3 |
math | 9.12 How many different four-digit numbers can be formed from the digits $0,1,2,3$, if each digit appears only once in the representation of the number? | 18 | 38 | 2 |
math | 2. (17 points) The medians drawn from vertices $A$ and $B$ of triangle $ABC$ are perpendicular to each other. Find the area of the square with side $AB$, if $BC=28, AC=44$.
# | 544 | 56 | 3 |
math | 6. Find the greatest possible value of $\gcd(x+2015 y, y+2015 x)$, given that $x$ and $y$ are coprime numbers. | 4060224 | 42 | 7 |
math | ## Task $4 / 90$
Let $a$ be a real number. Solve the following system of equations in real numbers $x, y, z$. Which values of $a$ are excluded?
$$
\begin{array}{r}
a x+y+z=1 \\
x+a y+z=a \\
x+y+a z=a^{2}
\end{array}
$$ | -\frac{+1}{+2};\quad\frac{1}{+2};\quad\frac{(+1)^{2}}{+2} | 79 | 34 |
math | ## Problem Statement
Write the decomposition of vector $x$ in terms of vectors $p, q, r$:
$x=\{13 ; 2 ; 7\}$
$p=\{5 ; 1 ; 0\}$
$q=\{2 ;-1 ; 3\}$
$r=\{1 ; 0 ;-1\}$ | 3p+q-4r | 74 | 7 |
math | 10. (12 points) There are two docks, $A$ and $B$, on a river, with $A$ upstream and $B$ downstream. There are two boats, Boat A and Boat B, where Boat A's speed in still water is twice that of Boat B. Boat A and Boat B start from docks $A$ and $B$ respectively at the same time, heading towards each other. When Boat A s... | 40 | 202 | 2 |
math | 2. Given a positive integer $k(1 \leqslant k \leqslant 9)$. Let $\underset{n \uparrow \cdots k}{k k \cdots}$ denote the $n$-digit positive integer in decimal notation where each digit is $k$. If for any positive integer $n$, the function $f(x)$ satisfies
$$
f(\underbrace{\overline{k k \cdots k}}_{n \uparrow})=\underbra... | f(x)=\frac{9}{k} x^{2}+2 x | 134 | 17 |
math | Example 10 Find the largest integer $n$ such that all non-zero solutions of the equation $(z+1)^{n}=z^{n}+1$ lie on the unit circle. | 7 | 41 | 1 |
math | SUBIECTUL IV: On the right of $d$, points $A_{0}, A_{1}, \ldots, A_{50}$ are considered in this order, such that $A_{0} A_{1}=1 \mathrm{~cm}, A_{1} A_{2}=3 \mathrm{~cm}, A_{2} A_{3}=5 \mathrm{~cm}, \ldots, A_{49} A_{50}=99 \mathrm{~cm}$. Let $O$ be the midpoint of the segment $\left[A_{0} A_{50}\right]$.
a. Determine ... | 35,O\in[A_{30}A_{40}] | 302 | 15 |
math | 10.2. Find all sets of natural numbers $x_{1}, x_{2}, \ldots, x_{20}$ such that
$$
x_{i+2}^{2}=\operatorname{LCM}\left(x_{i+1}, x_{i}\right)+\operatorname{LCM}\left(x_{i}, x_{i-1}\right)
$$
for $i=1,2, \ldots, 20$, where $x_{0}=x_{20}, x_{21}=x_{1}, x_{22}=x_{2} . \quad$ (P. Kozlov) | x_{1}=x_{2}=\ldots=x_{20}=2 | 141 | 17 |
math | 7.247. $\sqrt{\log _{5} x}+\sqrt[3]{\log _{5} x}=2$. | 5 | 31 | 1 |
math | For the numbers $a, b, c$, it holds that
$$
\frac{-a+b+c}{a}=\frac{a-b+c}{b}=\frac{a+b-c}{c}
$$
What values can the expression
$$
p=\frac{(a+b)(b+c)(c+a)}{a b c}
$$
take? | -1or8 | 75 | 4 |
math | Find the last three digits of the number $2003^{2002^{2001}}$. | 241 | 25 | 3 |
math | 2. Determine the remainder of the division of the number $3^{100}$ by 13. | 3 | 23 | 1 |
math | 12. (10 points) Cut a pentagon along a straight line into two polygons, then cut one of the polygons along a straight line into two parts, resulting in three polygons, and then cut one of the polygons along a straight line into two parts, $\cdots$, and so on. To have 20 pentagons among the resulting polygons, what is t... | 38 | 84 | 2 |
math | The real numbers $a_{1},a_{2},\ldots ,a_{n}$ where $n\ge 3$ are such that $\sum_{i=1}^{n}a_{i}=0$ and $2a_{k}\le\ a_{k-1}+a_{k+1}$ for all $k=2,3,\ldots ,n-1$. Find the least $f(n)$ such that, for all $k\in\left\{1,2,\ldots ,n\right\}$, we have $|a_{k}|\le f(n)\max\left\{|a_{1}|,|a_{n}|\right\}$. | \frac{n+1}{n-1} | 151 | 10 |
math | For each positive integer $k$, let $t(k)$ be the largest odd divisor of $k$. Determine all positive integers $a$ for which there exists a positive integer $n$ such that all the differences
$$ t(n+a)-t(n), \quad t(n+a+1)-t(n+1), \quad \ldots, \quad t(n+2 a-1)-t(n+a-1) $$
are divisible by 4. Answer. $a=1,3$, or 5. | a=1,3,5 | 108 | 7 |
math | Simplify the following expression:
$$
\frac{\frac{a+b}{1-a b}+\frac{c-a}{1+c a}}{1-\frac{a+b}{1-a b} \cdot \frac{c-a}{1+c a}}
$$
Explain the result using trigonometric identities. | \frac{b+}{1-} | 66 | 9 |
math | Example 9 Given $a_{n}-7 a_{n-1}+10 a_{n-2}=3^{n}, a_{0}$ $=0, a_{1}=1$. Find $a_{n}$. | a_{n}=\frac{8}{3} \cdot 2^{n}+\frac{11}{6} \cdot 5^{n}-\frac{9}{2} \cdot 3^{n} | 50 | 47 |
math | ## Problem Statement
Find the coordinates of point $A$, which is equidistant from points $B$ and $C$.
$A(x ; 0 ; 0)$
$B(4 ; 6 ; 8)$
$C(2 ; 4 ; 6)$ | A(15;0;0) | 61 | 9 |
math | Example 14. Find the number of integer solutions $(x, y, z)$ that satisfy $0<x<y<z$, and $\sqrt{1984}=\sqrt{ } x+\sqrt{ } y+\sqrt{ } z$. | 2 | 52 | 1 |
math | The elliptic curve $y^2=x^3+1$ is tangent to a circle centered at $(4,0)$ at the point $(x_0,y_0)$. Determine the sum of all possible values of $x_0$. | \frac{1}{3} | 52 | 7 |
math | Example 2 Let $x \in \mathbf{R}$, try to find the minimum value of the function $f(x)=\left(x^{2}+4 x+5\right)\left(x^{2}+4 x+2\right)+2 x^{2}+8 x+1$. | -9 | 66 | 2 |
math | 26. (ITA) For every integer $n \geq 2$ determine the minimum value that the sum $a_{0}+a_{1}+\cdots+a_{n}$ can take for nonnegative numbers $a_{0}, a_{1}, \ldots, a_{n}$ satisfying the condition $$ a_{0}=1, \quad a_{i} \leq a_{i+1}+a_{i+2} \quad \text { for } i=0, \ldots, n-2 $$ ### 3.39 The Thirty-Ninth IMO | \frac{F_{n+2} - 1}{F_{n}} | 128 | 17 |
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