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Topic
First Poster
Last Poster
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)
Checking a summand property for integers sufficiently large.
DinDean 1
N
by Double07
For any fixed integer 
, prove that there exists a positive integer 
, such that for any integer 
, can be expressed by a sum of positive integers 
's as
![\[n=a_1+a_2+\dots+a_m,\]](//latex.artofproblemsolving.com/6/c/2/6c2a979a5ff6c40bf2d7152c8c32088fc36f848b.png)
where 
, 
, 
, 
.
, prove that there exists a positive integer 
, such that for any integer 
, can be expressed by a sum of positive integers 
's as![\[n=a_1+a_2+\dots+a_m,\]](http://latex.artofproblemsolving.com/6/c/2/6c2a979a5ff6c40bf2d7152c8c32088fc36f848b.png)
where 
, 
, 
, 
.
1 reply
real+ FE
pomodor_ap 4
N
by jasperE3
Source: Own, PDC001-P7
Let 
be a function such that
for all 
. Determine all such functions 
.
be a function such that
for all 
. Determine all such functions 
.
4 replies
FE solution too simple?
Yiyj1 8
N
by lksb
Source: 101 Algebra Problems from the AMSP
Find all functions 
such that the equality 
holds for all pairs of real numbers 
.
I feel like my solution is too simple. Is there something I did wrong or something I missed?
such that the equality 
holds for all pairs of real numbers 
.
is an obvious solution. Now, let 
.
or 
.
.I feel like my solution is too simple. Is there something I did wrong or something I missed?
8 replies
Polynomials in Z[x]
BartSimpsons 16
N
by bin_sherlo
Source: European Mathematical Cup 2017 Problem 4
Find all polynomials with integer coefficients such that 
and 
is a square of an integer for all nonnegative integers 
.
Remark: For a nonnegative integer and an integer , 
is defined as follows: 
if 
and 
if 
.
Proposed by Adrian Beker.
with integer coefficients such that 
and 
is a square of an integer for all nonnegative integers 
.Remark: For a nonnegative integer
and an integer , 
is defined as follows: 
if 
and 
if 
.Proposed by Adrian Beker.
16 replies
