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Find all functions $f$: \(\mathbb{R}\) \(\rightarrow\) \(\mathbb{R}\) such : $f(
guramuta 3
N
by nabodorbuco2
Find all functions 
: 
such :
: 
such :
3 replies
with 
.![\[
\frac{a^5}{b^2 + 2c^3} + \frac{2b^5}{3c + a^6} + \frac{c^7}{a + b^4} \geq 2.
\]](http://latex.artofproblemsolving.com/f/f/5/ff5043705025592aeefc55a764a8b7170e13ff4b.png)
Easy geometry problem
dwrty 1
N
by Ianis
Let ABCD be a square with center O. Triangles BJC and CKD are constructed outward from the square such that BJ = CJ = CK = DK. Let M be the midpoint of CJ. Prove that the lines OM and BK are perpendicular.
1 reply
gcd of a^2+b^2-nab and a+b divides n+2 , when a,b are coprime positive
parmenides51 3
N
by FrancoGiosefAG
Source: Mexican Mathematical Olympiad 1988 OMM P5
If 
and 
are coprime positive integers and 
an integer, prove that the greatest common divisor of 
and 
divides 
.
and 
are coprime positive integers and 
an integer, prove that the greatest common divisor of 
and 
divides 
.
3 replies
no of ways of selecting 8 integers a_i such that 1<a_1<=...<=a_8<=8
parmenides51 12
N
by FrancoGiosefAG
Source: Mexican Mathematical Olympiad 1988 OMM P4
In how many ways can one select eight integers 
, not necesarily distinct, such that 
?
, not necesarily distinct, such that 
?
12 replies
China Northern MO 2009 p4 CNMO
parkjungmin 4
N
by exoticc
Source: China Northern MO 2009 p4 CNMO
China Northern MO 2009 p4 CNMO
The problem is too difficult.
Is there anyone who can help me?
The problem is too difficult.
Is there anyone who can help me?
4 replies
functional equation
pratyush 3
N
by soryn
For the functional equation 
, if f ' (0)=p and f ' (5)=q, then prove f ' (-5) = q
, if f ' (0)=p and f ' (5)=q, then prove f ' (-5) = q
3 replies
My Unsolved Problem
ZeltaQN2008 2
N
by ErTeeEs06
Let 
satisfy 
. The circumcircle 
and the incircle 
of are tangent to the sides 
at 
, respectively. The line 
meets 
at 
and intersects 
at 
, while the line 
meets 
at 
. Construct the common external tangent 
(different from 
) to the incircles of the triangles 
and 
. Show that is parallel to the line 
.
satisfy 
. The circumcircle 
and the incircle 
of are tangent to the sides 
at 
, respectively. The line 
meets 
at 
and intersects 
at 
, while the line 
meets 
at 
. Construct the common external tangent 
(different from 
) to the incircles of the triangles 
and 
. Show that is parallel to the line 
.2 replies
Computing functions
BBNoDollar 4
N
by ICE_CNME_4
Let 
, 
, with 
, 
. Prove that there exists 
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](//latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For and , the notation 
represents 
. )
, 
, with 
, 
. Prove that there exists 
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For and , the notation 
represents 
. )4 replies
Nice concurrency
navi_09220114 1
N
by bin_sherlo
Source: TASIMO 2025 Day 1 Problem 2
Four points 
, 
, 
, 
lie on a semicircle 
in this order with diameter 
, and is not parallel to . Points 
and 
lie on segments and 
respectively such that 
and 
. A circle 
passes through and is tangent to , and intersects again at 
. Prove that the lines 
, and 
are concurrent.
, 
, 
, 
lie on a semicircle 
in this order with diameter 
, and is not parallel to . Points 
and 
lie on segments and 
respectively such that 
and 
. A circle 
passes through and is tangent to , and intersects again at 
. Prove that the lines 
, and 
are concurrent.
1 reply
k-triangular sets
navi_09220114 0
Source: TASIMO 2025 Day 2 Problem 6
For an integer 
, we call a set 
of 
points in a plane 
-triangular if no three of them lie on the same line and whenever at most (possibly zero) points are removed from , the convex hull of the resulting set is a non-degenerate triangle. For given positive integer , find all integers such that there exists a -triangular set consisting of points.
Note. A set of points in a Euclidean plane is defined to be convex if it contains the line segments connecting each pair of its points. The convex hull of a shape is the smallest convex set that contains it.
, we call a set 
of 
points in a plane 
-triangular if no three of them lie on the same line and whenever at most (possibly zero) points are removed from , the convex hull of the resulting set is a non-degenerate triangle. For given positive integer , find all integers such that there exists a -triangular set consisting of points.Note. A set of points in a Euclidean plane is defined to be convex if it contains the line segments connecting each pair of its points. The convex hull of a shape is the smallest convex set that contains it.
0 replies
