Found problems: 744
Given an integer $n \ge 2$, solve in real numbers the system of equations \begin{align*}
\max\{1, x_1\} &= x_2 \\
\max\{2, x_2\} &= 2x_3 \\
&\cdots \\
\max\{n, x_n\} &= nx_1. \\
\end{align*}
For some $k > 0$ the lines $50x + ky = 1240$ and $ky = 8x + 544$ intersect at right angles at the point $(m,n)$. Find $m + n$.
Find all integers $n \geq 3$ for which there exist real numbers $a_1, a_2, \dots a_{n + 2}$ satisfying $a_{n + 1} = a_1$, $a_{n + 2} = a_2$ and
$$a_ia_{i + 1} + 1 = a_{i + 2},$$
for $i = 1, 2, \dots, n$.
[i]Proposed by Patrik Bak, Slovakia[/i]
Determine all real solutions of the system
\begin{align*}
x+y+z &=3, \\
\frac1x+\frac1y+\frac1z &= \frac{5}{12}, \\
x^3+y^3+z^3 &=45.
\end{align*}
Prove that the system of equations
\[2yz+x-y-z=a,\\ 2xz-x+y-z=a,\\ 2xy-x-y+z=a, \]
$a$ being a parameter, cannot have five distinct solutions. For what values of $a$ does this system have four distinct integer solutions?
Find all pairs of real numbers $(x,y)$ satisfying the system of equations:
\begin{align*}\frac{1}{x} + \frac{1}{2y} &= (x^2+3y^2)(3x^2+y^2)\\
\frac{1}{x} - \frac{1}{2y} &= 2(y^4-x^4)\end{align*}
Find all the pairs of real numbers $(x,y)$ that are solutions of the system:
$(x^{2}+y^{2})^{2}-xy(x+y)^{2}=19 $
$| x - y | = 1$
For certain real constants $ p, q, r$, we are given a system of equations
$$\begin{cases} a^2 + b + c = p \\
a + b^2 + c = q \\
a + b + c^2 = r \end{cases}$$
What is the maximum number of solutions of real triplets $(a, b, c)$ across all possible $p, q, r$? Give an example of the $p$, $q$, $r$ that achieves this maximum.
Solve the system of equations in integers:
$$\begin{cases} a^3=abc+2a+2c \\ b^3=abc-c \\ c^3=abc-a+b \end{cases}$$
Find all the solutions of the system
$$\begin{cases} y^2 = x^3 - 3x^2 + 2x \\ x^2 = y^3 - 3y^2 + 2y \end{cases}$$
The cost of $1000$ grams of chocolate is $x$ dollars and the cost of $1000$ grams of potatoes is $y$ dollars, the numbers $x$ and $y$ are positive integers and have not more than $2$ digits. Mother said to Maria to buy $200$ grams of chocolate and $1000$ grams of potatoes that cost exactly $N$ dollars. Maria got confused and bought $1000$ grams of chocolate and $200$ grams of potatoes that cost exactly $M$ dollars ($M >N$). It turned out that the numbers $M$ and $N$ have no more than two digits and are formed of the same digits but in a different order. Find $x$ and $y$.
Let $n$ be a positive integer greater than $2$. Solve in nonnegative real numbers the following system of equations \[x_{k}+x_{k+1}=x_{k+2}^{2}\quad , \quad k=1,2,\cdots,n\] where $x_{n+1}=x_{1}$ and $x_{n+2}=x_{2}$.
Solve the system
$$\begin{cases} x+y+z=1\\ x^5+y^5+z^5=1\end{cases}$$
in real numbers.
In a mathematical contest, three problems, $A,B,C$ were posed. Among the participants ther were 25 students who solved at least one problem each. Of all the contestants who did not solve problem $A$, the number who solved $B$ was twice the number who solved $C$. The number of students who solved only problem $A$ was one more than the number of students who solved $A$ and at least one other problem. Of all students who solved just one problem, half did not solve problem $A$. How many students solved only problem $B$?
Find all triples $ (x,y,z)$ of real numbers that satisfy the system of equations
\[ \begin{cases}x^3 \equal{} 3x\minus{}12y\plus{}50, \\ y^3 \equal{} 12y\plus{}3z\minus{}2, \\ z^3 \equal{} 27z \plus{} 27x. \end{cases}\]
[i]Razvan Gelca.[/i]
Find all real solutions to
\[x^{3}+3x-3+\ln{(x^{2}-x+1)}=y,\]
\[y^{3}+3y-3+\ln{(y^{2}-y+1)}=z,\]
\[z^{3}+3z-3+\ln{(z^{2}-z+1)}=x.\]
P6. Determine all integer pairs $(x, y)$ satisfying the following system of equations.
\[ \begin{cases}
x + y - 6 &= \sqrt{2x + y + 1} \\
x^2 - x &= 3y + 5
\end{cases} \]
P7. Determine the sum of all (positive) integers $n \leq 2019$ such that $1^2 + 2^2 + 3^2 + \cdots + n^2$ is an odd number and $1^1 + 2^2 + 3^3 + \cdots + n^n$ is also an odd number.
P8. Two quadrilateral-based pyramids where the length of all its edges are the same, have their bases coincide, forming a new 3D figure called "8-plane" (octahedron). If the volume of such "8-plane" (octahedron) is $a^3\sqrt{2}$ cm$^3$, determine the volume of the largest sphere that can be fit inside such "8-plane" (octahedron).
P9. Six-digit numbers $\overline{ABCDEF}$ with distinct digits are arranged from the digits 1, 2, 3, 4, 5, 6, 7, 8 with the rule that the sum of the first three numbers and the sum of the last three numbers are the same. Determine the probability that such arranged number has the property that either the first or last three digits (might be both) form an arithmetic sequence or a geometric sequence.
[hide=Remarks (Answer spoiled)]It's a bit ambiguous whether the first or last three digits mentioned should be in that order, or not. If it should be in that order, the answer to this problem would be $\frac{1}{9}$, whereas if not, it would be $\frac{1}{3}$. Some of us agree that the correct interpretation should be the latter (which means that it's not in order) and the answer should be $\frac{1}{3}$. However since this is an essay problem, your interpretation can be written in your solution as well and it's left to the judges' discretion to accept your interpretation, or not. This problem is very bashy.[/hide]
P10. $X_n$ denotes the number which is arranged by the digit $X$ written (concatenated) $n$ times. As an example, $2_{(3)} = 222$ and $5_{(2)} = 55$. For $A, B, C \in \{1, 2, \ldots, 9\}$ and $1 \leq n \leq 2019$, determine the number of ordered quadruples $(A, B, C, n)$ satisfying:
\[ A_{(2n)} = 2 \left ( B_{(n)} \right ) + \left ( C_{(n)} \right )^2. \]
Let $x$ and $y$ be real numbers satisfying the conditions $x + y = 4$ and $xy = 3$. Compute the value of $(x - y)^2$.
A. $0$ B. $1$ C. $4$ D. $9$ E.$ -1$
What is the smallest $n$ for which there is a solution to $$\begin{cases} \sin x_1 + \sin x_2 + ... + \sin x_n = 0 \\ \sin x_1 + 2 \sin x_2 + ... + n \sin x_n = 100 \end{cases}$$ ?
For real numbers $x, y$ and $z$ it is known that $$\begin{cases} x + y = 2 \\ xy = z^2 + 1\end {cases}$$
Find the value of the expression $x^2 + y^2+ z^2$.
Find all $(a, b, c, d)\in \mathbb{R}$ satisfying
\[\begin{aligned}
\begin{cases}
a+b+c+d=0,\\
a^2+b^2+c^2+d^2=12,\\
abcd=-3.\\
\end{cases}
\end{aligned}\]
Find how many different solutions depending on $a$ has the system of equations :
$$\begin{cases} x+z=2a
\\ y+u+xz=a-3
\\ yz+xu=2a
\\ yu=1
\end{cases}$$
Let $a_1,\ldots,a_n\ (n>2)$ be real numbers with at most one zero. Solve the system
\begin{align*}
x_1x_2 &= a_1, \\
x_2x_3 &= a_2, \\
&\ \vdots \\
x_{n-1}x_n &= a_{n-1}, \\
x_nx_1 &\ge a_n.
\end{align*}
[b]a)[/b] Solve the system
$$ \left\{\begin{matrix} 3^y-4^x=11\\ \log_4{x} +\log_3 y =3/2\end{matrix}\right. $$
[b]b)[/b] Solve the equation $ \quad 9^{\log_5 (x-2)} -5^{\log_9 (x+2)} = 4. $
Find all triples $(x,y, z)$ of real (but not necessarily positive) numbers satisfying
$3(x^2 + y^2 + z^2) = 1$ , $x^2y^2 + y^2z^2 + z^2x^2 = xyz(x + y + z)^3$.