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On existence of infinitely many positive integers satisfying
shivangjindal 22
N
by atdaotlohbh
Source: European Girls' Mathematical Olympiad-2014 - DAY 1 - P3
We denote the number of positive divisors of a positive integer 
by 
and the number of distinct prime divisors of by 
. Let 
be a positive integer. Prove that there exist infinitely many positive integers 
such that 
and 
does not divide 
for any positive integers 
satisfying 
.
by 
and the number of distinct prime divisors of by 
. Let 
be a positive integer. Prove that there exist infinitely many positive integers 
such that 
and 
does not divide 
for any positive integers 
satisfying 
.
22 replies
standard Q FE
jasperE3 3
N
by ErTeeEs06
Source: gghx, p19004309
Find all functions 
such that for any 
:
such that for any 
:
3 replies
![\[
\begin{cases}
x - y z = 1,\\[2pt]
y - z x = 2,\\[2pt]
z - x y = 4.
\end{cases}
\]](http://latex.artofproblemsolving.com/b/2/d/b2d94e49491a71f86109103f985fe3f238cf4d95.png)
Find all functions
Pirkuliyev Rovsen 2
N
by ErTeeEs06
Source: Cup in memory of A.N. Kolmogorov-2023
Find all functions 
such that 
for all 
such that 
for all 
2 replies
Circumcircle excircle chaos
CyclicISLscelesTrapezoid 25
N
by bin_sherlo
Source: ISL 2021 G8
Let 
be a triangle with circumcircle 
and let 
be the 
-excircle. Let 
and 
be the intersection points of and . Let 
and 
be the projections of onto the tangent lines to at and respectively. The tangent line at to the circumcircle of the triangle 
intersects the tangent line at to the circumcircle of the triangle 
at a point 
. Prove that 
.
be a triangle with circumcircle 
and let 
be the 
-excircle. Let 
and 
be the intersection points of and . Let 
and 
be the projections of onto the tangent lines to at and respectively. The tangent line at to the circumcircle of the triangle 
intersects the tangent line at to the circumcircle of the triangle 
at a point 
. Prove that 
.
25 replies
and 
: Prove that :
Combo problem
soryn 2
N
by Anulick
The school A has m1 boys and m2 girls, and ,the school B has n1 boys and n2 girls. Each school is represented by one team formed by p students,boys and girls. If f(k) is the number of cases for which,the twice schools has,togheter k girls, fund f(k) and the valute of k, for which f(k) is maximum.
2 replies
Calculate the distance of chess king!!
egxa 4
N
by Primeniyazidayi
Source: All Russian 2025 9.4
A chess king was placed on a square of an 
board and made 
moves so that it visited all squares and returned to the starting square. At every moment, the distance from the center of the square the king was on to the center of the board was calculated. A move is called 
if this distance becomes smaller after the move. Find the maximum possible number of pleasant moves. (The chess king moves to a square adjacent either by side or by corner.)
board and made 
moves so that it visited all squares and returned to the starting square. At every moment, the distance from the center of the square the king was on to the center of the board was calculated. A move is called 
if this distance becomes smaller after the move. Find the maximum possible number of pleasant moves. (The chess king moves to a square adjacent either by side or by corner.)4 replies
As some nations like to say "Heavy theorems mostly do not help"
Assassino9931 9
N
by EVKV
Source: European Mathematical Cup 2022, Senior Division, Problem 2
We say that a positive integer is lovely if there exist a positive integer and (not necessarily distinct) positive integers 
, 
, 
, 
such that 
and 
for 
.
a) Are there infinitely many lovely numbers?
b) Is there a lovely number, greater than
, which is a perfect square of an integer?
is lovely if there exist a positive integer and (not necessarily distinct) positive integers 
, 
, 
, 
such that 
and 
for 
.a) Are there infinitely many lovely numbers?
b) Is there a lovely number, greater than
, which is a perfect square of an integer?
9 replies
be an odd prime. Prove that:![\[\displaystyle\sum_{k=1}^{p-1}k^{2p-1} \equiv \frac{p(p+1)}{2} \pmod{p^2}\]](http://latex.artofproblemsolving.com/a/5/5/a55e75d4d9769ceec3a3ce359aac676c7c7875fd.png)
