Found problems: 916
Determine all pairs of integers $(x, y)$ such that $2xy^2 + x + y + 1 = x^2 + 2y^2 + xy$.
Find the sum of all integers $n$ that fulfill the equation \[2^n(6-n)=8n.\]
a) Two positive integers are chosen. The sum is revealed to logician $A$, and the sum of squares is revealed to logician $B$. Both $A$ and $B$ are given this information and the information contained in this sentence. The conversation between $A$ and $B$ goes as follows: $B$ starts
B: ` I can't tell what they are.'
A: ` I can't tell what they are.'
B: ` I can't tell what they are.'
A: ` I can't tell what they are.'
B: ` I can't tell what they are.'
A: ` I can't tell what they are.'
B: ` Now I can tell what they are.'
What are the two numbers?
b) When $B$ first says that he cannot tell what the two numbers are, $A$ receives a large amount of information. But when $A$ first says that he cannot tell what the two numbers are, $B$ already knows that $A$ cannot tell what the two numbers are. What good does it do $B$ to listen to $A$?
Find all positive integers $k$ and $\ell$ such that $k^2 -\ell^2 = 1005$.
Solve the equation $2^a-5^b=3$ in positive integers $a,b$.
Solve the equation $x^2+y^4+1=6^z$ in the set of integers.
Solve in nonnegative integers the following equation :
$$21^x+4^y=z^2$$
Show that the following equation has finitely many solutions $(t,A,x,y,z)$ in positive integers
$$\sqrt{t(1-A^{-2})(1-x^{-2})(1-y^{-2})(1-z^{-2})}=(1+x^{-1})(1+y^{-1})(1+z^{-1})$$
Prove that the equation $x^2 + y^2 - z^2 = 1997$ has infinitely many solutions in integers $x$, $y$ and $z$.
(N Vassiliev)
Show that there exist no integer solutions $(x, y, z)$ to the equation
$$x^3+2y^3+4z^3=9$$
Let $ a, b \in \mathbb{Z}$ which are not perfect squares. Prove that if \[ x^2 \minus{} ay^2 \minus{} bz^2 \plus{} abw^2 \equal{} 0\] has a nontrivial solution in integers, then so does \[ x^2 \minus{} ay^2 \minus{} bz^2 \equal{} 0.\]
Determine all prime numbers $p, q$ and $r$ with $p + q^2 = r^4$.
[i](Karl Czakler)[/i]
Find all nonnegative integer solutions $(x,y,z)$ of the equation
$\frac{1}{x+2}+\frac{1}{y+2}=\frac{1}{2} +\frac{1}{z+2}$
Find all pairs of positive integers $(x,y) $ for which $x^3 + y^3 = 4(x^2y + xy^2 - 5) .$
Let $n$ be a natural number. Solve in whole numbers the equation \[x^{n}+y^{n}=(x-y)^{n+1}.\]
Find integer solutions of $x^3+y^3-2xy+x+y+2=0$
Find all solutions for positive integers $(x,y,k,m)$ such that
\[ 20x^k+24y^m = 2024\]
with $k, m > 1$
Let $r$ be a positive integer and let $a_r$ be the number of solutions to the equation $3^x-2^y=r$ ,such that $0\leq x,y\leq 5781$ are integers. What is the maximal value of $a_r$?
Determine all prime numbers $p, q$ and $r$ with $p + q^2 = r^4$.
[i](Karl Czakler)[/i]
Find all positive prime numbers $p,q,r,s$ so that $p^2+2019=26(q^2+r^2+s^2)$.
Find all triples $(a, b, c)$ of positive integers such that $a \leq b$ and \[a!+b!=c^4+2024\]
[i]Proposed by Otgonbayar Uuye.[/i]
Find all pairs of prime numbers $(p, q)$ for which $7pq^2 + p = q^3 + 43p^3 + 1$
Are there integers $k$ and $m$ for which
$$\frac{(k-3)(k-2)(k-1)k+1}{(k+1)(k+2)(k+3)(k+4)+1}=m(m+1)+(m+1)(m+2)+(m+2)m \,\,
?$$
Find all the prime numbers $p$ and $q$ such that $ p^2+q=37q^2+p $.
Clarification: $1$ is not a prime number.
Find all couples of positive integers $m$ and $n$ such that
$$n!+5=m^3$$