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jlacosta   0
May 1, 2025
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0 replies
jlacosta
May 1, 2025
0 replies
Functional equation of nonzero reals
proglote   8
N 24 minutes ago by jasperE3
Source: Brazil MO 2013, problem #3
Find all injective functions $f\colon \mathbb{R}^* \to \mathbb{R}^* $ from the non-zero reals to the non-zero reals, such that \[f(x+y) \left(f(x) + f(y)\right) = f(xy)\] for all non-zero reals $x, y$ such that $x+y \neq 0$.
8 replies
proglote
Oct 24, 2013
jasperE3
24 minutes ago
Interesting inequalities
sqing   1
N 36 minutes ago by sqing
Source: Own
Let $ a,b\geq  0 ,a^2-ab+b^2+a+b=3  $. Prove that
$$  \frac{39+\sqrt{13}}{78}\geq  \frac{1}{a^2+3}+ \frac{1}{b^2+ 3} \geq  \frac{1}{2}$$Let $ a,b\geq  0 ,a^2+ab+b^2+a+b=3  $. Prove that
$$  \frac{19+\sqrt{10}}{39}\geq  \frac{1}{a^2+3}+ \frac{1}{b^2+ 3} \geq    \frac{39+\sqrt{13}}{78}$$Let $ a,b\geq  0 ,a^2+ab+b^2+a+b=5  $. Prove that
$$  \frac{3}{5}> \frac{1}{a^2+3}+ \frac{1}{b^2+ 3} \geq     \frac{185+3\sqrt{21}}{402}$$
1 reply
+1 w
sqing
an hour ago
sqing
36 minutes ago
4-var inequality
sqing   2
N 2 hours ago by sqing
Source: SXTB (4)2025 Q2837
Let $ a,b,c,d> 0  $. Prove that
$$   \frac{1}{(3a+1)^4}+ \frac{1}{(3b+1)^4}+\frac{1}{(3c+1)^4}+\frac{1}{(3d+1)^4} \geq \frac{1}{16(3abcd+1)}$$
2 replies
sqing
Yesterday at 2:59 PM
sqing
2 hours ago
Inspired by Bet667
sqing   2
N 2 hours ago by sqing
Source: Own
Let $ a,b $ be a real numbers such that $a^2+kab+b^2\ge a^3+b^3.$Prove that$$a+b\leq k+2$$Where $ k\geq 0. $
2 replies
sqing
Yesterday at 2:46 PM
sqing
2 hours ago
A Collection of Good Problems from my end
SomeonecoolLovesMaths   24
N 4 hours ago by ReticulatedPython
This is a collection of good problems and my respective attempts to solve them. I would like to encourage everyone to post their solutions to these problems, if any. This will not only help others verify theirs but also perhaps bring forward a different approach to the problem. I will constantly try to update the pool of questions.

The difficulty level of these questions vary from AMC 10 to AIME. (Although the main pool of questions were prepared as a mock test for IOQM over the years)

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5
24 replies
SomeonecoolLovesMaths
May 4, 2025
ReticulatedPython
4 hours ago
n and n+100 have odd number of divisors (1995 Belarus MO Category D P2)
jasperE3   4
N Yesterday at 9:50 PM by KTYC
Find all positive integers $n$ so that both $n$ and $n + 100$ have odd numbers of divisors.
4 replies
jasperE3
Apr 6, 2021
KTYC
Yesterday at 9:50 PM
Closed form expression of 0.123456789101112....
ReticulatedPython   3
N Yesterday at 8:15 PM by ReticulatedPython
Is there a closed form expression for the decimal number $$0.123456789101112131415161718192021...$$which is defined as all the natural numbers listed in order, side by side, behind a decimal point, without commas? If so, what is it?
3 replies
ReticulatedPython
Yesterday at 8:05 PM
ReticulatedPython
Yesterday at 8:15 PM
primes and perfect squares
Bummer12345   5
N Yesterday at 8:08 PM by Shan3t
If $p$ and $q$ are primes, then can $2^p + 5^q + pq$ be a perfect square?
5 replies
Bummer12345
Monday at 5:08 PM
Shan3t
Yesterday at 8:08 PM
trapezoid
Darealzolt   1
N Yesterday at 7:38 PM by vanstraelen
Let \(ABCD\) be a trapezoid such that \(A, B, C, D\) lie on a circle with center \(O\), and side \(AB\) is parallel to side \(CD\). Diagonals \(AC\) and \(BD\) intersect at point \(M\), and \(\angle AMD = 60^\circ\). It is given that \(MO = 10\). It is also known that the difference in length between \(AB\) and \(CD\) can be expressed in the form \(m\sqrt{n}\), where \(m\) and \(n\) are positive integers and \(n\) is square-free. Compute the value of \(m + n\).
1 reply
Darealzolt
Yesterday at 2:03 AM
vanstraelen
Yesterday at 7:38 PM
Polynomial Minimization
ReticulatedPython   1
N Yesterday at 5:36 PM by clarkculus
Consider the polynomial $$p(x)=x^{n+1}-x^{n}$$, where $x, n \in \mathbb{R+}.$

(a) Prove that the minimum value of $p(x)$ always occurs at $x=\frac{n}{n+1}.$
1 reply
ReticulatedPython
Yesterday at 5:07 PM
clarkculus
Yesterday at 5:36 PM
Easy one
irregular22104   0
Yesterday at 5:03 PM
Given two positive integers a,b written on the board. We apply the following rule: At each step, we will add all the numbers that are the sum of the two numbers on the board so that the sum does not appear on the board. For example, if the two initial numbers are 2.5, then the numbers on the board after step 1 are 2,5,7; after step 2 are 2,5,7,9,12;...
1) With a = 3; b = 12, prove that the number 2024 cannot appear on the board.
2) With a = 2; b = 3, prove that the number 2024 can appear on the board.
0 replies
irregular22104
Yesterday at 5:03 PM
0 replies
This shouldn't be a problem 15
derekli   2
N Yesterday at 4:09 PM by aarush.rachak11
Hey guys I was practicing AIME and came across this problem which is definitely misplaced. It asks for the surface area of a plane within a cylinder which we can easily find out using a projection that is easy to find. I think this should be placed in problem 10 or below. What do you guys think?
2 replies
derekli
Yesterday at 2:15 PM
aarush.rachak11
Yesterday at 4:09 PM
Regular tetrahedron
vanstraelen   7
N Yesterday at 3:46 PM by ReticulatedPython
Given the points $O(0,0,0),A(1,0,0),B(\frac{1}{2},\frac{\sqrt{3}}{2},0)$
a) Determine the point $C$, above the xy-plane, such that the pyramid $OABC$ is a regular tetrahedron.
b) Calculate the volume.
c) Calculate the radius of the inscribed sphere and the radius of the circumscribed sphere.
7 replies
vanstraelen
May 4, 2025
ReticulatedPython
Yesterday at 3:46 PM
[ABCD] = n [CDE], areas in trapezoid - IOQM 2020-21 p1
parmenides51   4
N Yesterday at 3:44 PM by Kizaruno
Let $ABCD$ be a trapezium in which $AB \parallel CD$ and $AB = 3CD$. Let $E$ be then midpoint of the diagonal $BD$. If $[ABCD] = n \times  [CDE]$, what is the value of $n$?

(Here $[t]$ denotes the area of the geometrical figure$ t$.)
4 replies
parmenides51
Jan 18, 2021
Kizaruno
Yesterday at 3:44 PM
Problem 4 (second day)
darij grinberg   93
N Apr 25, 2025 by Ilikeminecraft
Source: IMO 2004 Athens
Let $n \geq 3$ be an integer. Let $t_1$, $t_2$, ..., $t_n$ be positive real numbers such that \[n^2 + 1 > \left( t_1 + t_2 + \cdots + t_n \right) \left( \frac{1}{t_1} + \frac{1}{t_2} + \cdots + \frac{1}{t_n} \right).\] Show that $t_i$, $t_j$, $t_k$ are side lengths of a triangle for all $i$, $j$, $k$ with $1 \leq i < j < k \leq n$.
93 replies
darij grinberg
Jul 13, 2004
Ilikeminecraft
Apr 25, 2025
Problem 4 (second day)
G H J
Source: IMO 2004 Athens
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darij grinberg
6555 posts
#1 • 11 Y
Y by Adventure10, chessgocube, megarnie, fluff_E, Mango247, bjump, buddyram, cubres, and 3 other users
Let $n \geq 3$ be an integer. Let $t_1$, $t_2$, ..., $t_n$ be positive real numbers such that \[n^2 + 1 > \left( t_1 + t_2 + \cdots + t_n \right) \left( \frac{1}{t_1} + \frac{1}{t_2} + \cdots + \frac{1}{t_n} \right).\] Show that $t_i$, $t_j$, $t_k$ are side lengths of a triangle for all $i$, $j$, $k$ with $1 \leq i < j < k \leq n$.
Attachments:
This post has been edited 2 times. Last edited by djmathman, Jun 27, 2015, 11:54 PM
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Peter Scholze
644 posts
#2 • 5 Y
Y by chessgocube, Adventure10, Illuzion, Mango247, cubres
solution by induction: the hardest part is to show that it works for n=3. so assume $t_1\geq t_2\geq t_3$. it is easy to see that the right is strictly increasing in $t_3$. so if we show that the reverse inequality holds for $t_1=t_2+t_3$, we are done. a quick calculation is enough to show that(it is too boring to post). but if we know is true for n-1, then let's do the following computation:
$((t_1+...+t_{n-1})+t_n)(\frac{1}{t_1}+...+\frac{1}{t_{n-1}}+\frac{1}{t_n})\geq (t_1+...+t_{n-1})(\frac{1}{t_1}+...+\frac{1}{t_{n-1}})+2n-1$,
by AM-GM and AM-HM. so we get the inequality reduces to that one for n-1 variables.

Peter
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pbornsztein
3004 posts
#3 • 5 Y
Y by anantmudgal09, Adventure10, chessgocube, Mango247, sabkx
It reminds me the following problem from China 1987/88 :

Let $n \geq 3$ be an integer. Suppose the inequality
$(a_1^2 + a_2^2 + ... + a_n^2)^2 > (n-1)(a_1^4 + ... a_n^4)$
holds for some positive real numbers $a_1,...a_n$. Prove that $a_i,a_j,a_k$ are the three sides of some triangle for all $i<j<k$.

Even there is an elegant direct solution, the most common way is induction.

Pierre.
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jmerry
12096 posts
#4 • 16 Y
Y by GammaBetaAlpha, Polynom_Efendi, ETS1331, Adventure10, oneteen11, Illuzion, Mango247, aidan0626, EquationTracker, RobertRogo, and 6 other users
When I saw this, my immediate thought was- an elegant direct solution.

The inequality
$n^2 + 1 > \left( t_1 + t_2 + ... + t_n \right) \left( \frac{1}{t_1} + \frac{1}{t_2} + ... + \frac{1}{t_n} \right)$
reduces to $\displaystyle{n^2+1>\sum_{j=1}^n \sum_{k=1}^n{\frac{t_j}{t_k}}}$
$\displaystyle{n^2+1>n+\sum_{1 \le j<k \le n}{\frac{t_j}{t_k}+\frac{t_k}{t_j}}}$
Each of the terms in the last sum is at least 2, so if the sum of some $m$ terms is at least $2m+1$, the whole sum will be at least $n^2-n+1$ and the inequality will be violated.

Suppose $t_i+t_j \le t_k$. Then $\frac{t_i}{t_k}+\frac{t_j}{t_k} \le 1$
If we can show that $a+b+\frac{1}{a}+\frac{1}{b} \ge 5$ when $a+b \le 1$, we will be done by the argument above.
First, we have $a+\frac{1}{a}=\frac{a^2+1}{a} \ge \frac{(a-\frac{1}{2})^2+a+\frac{3}{4}}{a} \ge 1+\frac{3}{4a}$
and $b+\frac{1}{b} \ge 1+\frac{3}{4b}$
so $a+b+\frac{1}{a}+\frac{1}{b} \ge 2+\frac{3}{4}(\frac{1}{a}+\frac{1}{b})$
By AM-HM, $\frac{2}{\frac{1}{a}+\frac{1}{b}} \le \frac{a+b}{2} \le \frac{1}{2}$, so $\frac{1}{a}+\frac{1}{b} \ge 4$ and $a+b+\frac{1}{a}+\frac{1}{b} \ge 5$

In fact the case n=3 is sufficient to prove the desired fact for all n - we can simply pair off the other terms and only worry about the ones in our non-triangle.
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Kantor
21 posts
#5 • 3 Y
Y by GammaBetaAlpha, Adventure10, Mango247
it is, indeed, a very elegant solution jmerry!

Kantor
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harazi
5526 posts
#6 • 3 Y
Y by Adventure10, Mango247, jolynefag
Well, I just adore this problem. So easy, beautiful and elegant. But try to think about the following problem, which I think it's much more difficult and interesting:
For iven n>2, find the greatest constant k (which may depend on n) such that if positive reals $ x_1,...,x_n$ verify $ k>(x_1+...+x_n)(\frac{1}{x_1}+...+\frac{1}{x_n}) $ then any three of them are sides of a triangle. I think I've managed to solve this problem and if my solution is correct, then it's a really hard and nice problem.
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Anh Cuong
431 posts
#7 • 2 Y
Y by Adventure10, Mango247
If my computation is not wrong then it should be
n^2+(2\sqrt10-6)n+19-6\sqrt{10}
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harazi
5526 posts
#8 • 3 Y
Y by Adventure10, Mango247, jolynefag
Yes, it is $ (n+\sqrt{10}-3)^2  $. I think this one is much more interesting.
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Valentin Vornicu
7301 posts
#9 • 4 Y
Y by adityaguharoy, Adventure10, Mango247, jolynefag
btw, the creator of this problem is Hojoo Lee :)
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billzhao
827 posts
#10 • 13 Y
Y by threehandsnal, Mateescu Constantin, Wizard0001, Adventure10, Periwinkle27, Jasurbek, Illuzion, Mango247, ehuseyinyigit, Bluecloud123, FairyBlade, and 2 other users
Another solution:

Note that

\[ (t_1+ \cdots + t_n) ( \frac{1}{t_1} + \cdots  + \frac{1}{t_n}) = n^2 + \sum_{i<j} (\sqrt{\frac{t_i}{t_j}} - \sqrt{\frac{t_j}{t_i}} )^2 \]

And so if three of the terms satisfy $a \geq b + c$ then

\[{{ (\sqrt{a/b} - \sqrt{b/a})^2 + (\sqrt{a/c} - \sqrt{c/a})^2 =
\frac{ (a-b)^2}{ab} } + \frac{ (a-c)^2}{ac} } 
\geq \frac{(a-b)c}{ab} + \frac{(a-c)b}{ac}
= c/b + b/c -(b+c)/a \geq 2 - 1 = 1 \]

And the result follows immediately.

(During the IMO, I accidentally forgot the square root, costing me a huge chunk of marks).
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vinoth_90_2004
301 posts
#11 • 2 Y
Y by Adventure10, Mango247
just wanted to say that this is *really* nice ...
btw, anh cuong/harazi could you please post the solution to the harder version (where n^2+1 is replaced by ( n + :sqrt: 10 - 3 ) ^ 2 )? thanks.
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harazi
5526 posts
#12 • 2 Y
Y by Adventure10, Mango247
Of course, Hojoo Lee is the official proposer. But I knew the case n=3 some four months ago, when I tried to create a problem for the junior TST in Romania.
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Sung-yoon Kim
324 posts
#13 • 2 Y
Y by Adventure10, Mango247
It's my solution in the 2004 IMO.

By Cauchy-Schwartz inequality,

(t_1+t_2+...+t_n)(1/t_1+1/t_2+...+1/t_n)=(3\times(t_1+t_2+t_3)/3 +t_4+...+t_n)(3\times(1/t_1+1/t_2+1/t_3)/3 +1/t_4+...+1/t_n)\geq(3\times \sqrt((t_1+t_2+t_3)/3\times(1/t_1+1/t_2+1/t_3)/3) +1+...+1)^2


So
n^2+1>(\sqrt((t_1+t_2+t_3)(1/t_1+1/t_2+1/t_3)) +n-3)^2


=>
(t_1+t_2+t_3)(1/t_1+1/t_2+1/t_3)<(\sqrt(n^2+1) -(n-3))^2




As
n\geq3
, it follows

(\sqrt(n^2+1) -(n-3))^2<10  <=>  2n^2 -6n<(2n-6) \sqrt(n^2+1)


So
(t_1+t_2+t_3)(1/t_1+1/t_2+1/t_3)<10




Let
t_1 \geq t_2 \geq t_3
. Then
t_1+t_2 \geq t_3, t_1+t_3 \geq t_2


Assume that
t_1>t_2+t_3
. Then
(t_1+t_2+t_3)(1/t_1+1/t_2+1/t_3)-((t_2+t_3)+t_2+t_3)(1/(t_2+t_3) +1/t_2+1/t_3) =(t_1-t_2-t_3)(1/t_2+1/t_3-1/t_1) \geq 0


so by Cauchy-Schwartz inequality,

10>(t_1+t_2+t_3)(1/t_1+1/t_2+1/t_3) \geq 2(t_2+t_3)(1/(t_2+t_3)+1/t_2+1/t_3)   <=>   4>(t_2+t_3)(1/t_2+1/t_3) \geq 4


and contradiction.
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heartwork
308 posts
#14 • 2 Y
Y by Adventure10, Mango247
harazi wrote:
...find the greatest constant k (which may depend on n) such that if positive reals
x_1,x_2,...,x_n
verify
k(n)>(x_1+x_2+...+x_n)(1/x_1+1/x_2+...+1/x_n)
then any three of them are sides of a triangle.
harazi wrote:
Yes, it is
(n+\sqrt{10}-3)^2
. I think this one is much more interesting.
Consider
f(x_1,x_2,...,x_n)=(x_1+x_2+...+x_n)(1/x_1+1/x_2+...+1/x_n)
where
x_1<=x_2<=...<=x_n
.

Greatest constant
k(n)=(n+\sqrt{10}-3)^2
might be found elementary. From the solution of Sung-yoon Kim:
Sung-yoon Kim wrote:
...By Cauchy-Schwartz inequality,
..........
So
n^2+1>(\sqrt((x_1+x_2+x_3)(1/x_1+1/x_2+1/x_3))+(n-3)^2


=>
(x_1+x_2+x_3)(1/x_1+1/x_2+1/x_3)<(\sqrt(n^2+1) -(n-3))^2

instead of
(n^2+1)
, use that
k(n)
and instead of
x_3
, use
x_n
:
(x_1+x_2+x_n)(1/x_1+1/x_2+1/x_n)<(\sqrt(k(n)) -(n-3))^2

Assume
x_1+x_2<=x_n
then:
(x_1+x_2+x_n)(1/x_1+1/x_2+1/x_n)>=10
, so we must have
10>=(\sqrt(k(n)) -(n-3))^2
,
which leads to the wanted answer. Indeed if:
(x_1+x_2+x_n)(1/x_1+1/x_2+1/x_n)<(\sqrt(k(n)) -(n-3))^2
, then:
x_1+x_2>x_n
, so for any
1<= i, j, k <= n
:
x_i+x_j>x_k
.

Of course, there is a quite simple (but not elementary) "forged" solution, using optimization theory:
k(n)=min f(x_1,x_2,...,x_n)
,
when
x_1<=x_2<=...<=x_n
and
x_1+x_2<=x_n
, we may get:
k(n)=f(1,1,2\sqrt(2/5),...,2\sqrt(2/5),2)
, where
2\sqrt(2/5)
appears
(n-3)
times.

I think it's pretty interesting.
This post has been edited 2 times. Last edited by heartwork, Aug 3, 2004, 6:36 PM
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heartwork
308 posts
#15 • 2 Y
Y by Adventure10, Mango247
Problem may be generalised as it follows:
Instead of a triangle, use a polygon of "given"
M
edges ,
M>=3
so if the same inequality holds for any
n>=M
positive reals (of course with something like
k(n,M)
in the RHS) then any
M
from our
n
real numbers can be the side lengths of a polygon with
M
edges. Solution is basically the same.
Z K Y
G
H
=
a