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order of a function greater than c*n-1
YLG_123   2
N an hour ago by SimplisticFormulas
Source: Brazil EGMO TST2 2024 #1
Let \( \mathbb{N} \) be the set of all positive integers. We say that a function \( f: \mathbb{N} \to \mathbb{N} \) is Georgian if \( f(1) = 1 \) and, for every positive integer \( n \), there exists a positive integer \( k \) such that
\[
f^{(k)}(n) = 1, \quad \text{where } f^{(k)} = f \circ f \cdots \circ f \quad \text{(applied } k \text{ times)}.
\]If \( f \) is a Georgian function, we define, for each positive integer \( n \), \( \text{ord}(n) \) as the smallest positive integer \( m \) such that \( f^{(m)}(n) = 1 \). Determine all positive real numbers \( c \) for which there exists a Georgian function such that, for every positive integer \( n \geq 2024 \), it holds that \( \text{ord}(n) \geq cn - 1 \).
2 replies
YLG_123
Oct 12, 2024
SimplisticFormulas
an hour ago
IMO Shortlist 2012, Algebra 2
lyukhson   26
N 2 hours ago by ezpotd
Source: IMO Shortlist 2012, Algebra 2
Let $\mathbb{Z}$ and $\mathbb{Q}$ be the sets of integers and rationals respectively.
a) Does there exist a partition of $\mathbb{Z}$ into three non-empty subsets $A,B,C$ such that the sets $A+B, B+C, C+A$ are disjoint?
b) Does there exist a partition of $\mathbb{Q}$ into three non-empty subsets $A,B,C$ such that the sets $A+B, B+C, C+A$ are disjoint?

Here $X+Y$ denotes the set $\{ x+y : x \in X, y \in Y \}$, for $X,Y \subseteq \mathbb{Z}$ and for $X,Y \subseteq \mathbb{Q}$.
26 replies
lyukhson
Jul 29, 2013
ezpotd
2 hours ago
Inequality with x+y+z=1.
FrancoGiosefAG   3
N 3 hours ago by JARP091
Let $x,y,z$ be positive real numbers such that $x+y+z=1$. Show that
\[ \frac{x^2-yz}{x^2+x}+\frac{y^2-zx}{y^2+y}+\frac{z^2-xy}{z^2+z}\leq 0. \]
3 replies
1 viewing
FrancoGiosefAG
Yesterday at 8:36 PM
JARP091
3 hours ago
Count the number of balanced colorings
TUAN2k8   4
N 3 hours ago by aidan0626
Source: A book
Given a $2n \times 2n$ grid ($n \in \mathbb{Z}^{+}$), we color some of its cells black.A coloring is called balanced if each row and each cell contains exactly $n$ black cells.Detemine the number of balanced colorings.
4 replies
TUAN2k8
4 hours ago
aidan0626
3 hours ago
Sneaky one
Sunjee   5
N 3 hours ago by TBazar
Find minimum and maximum value of following function.
$$f(x,y)=\frac{\sqrt{x^2+y^2}+\sqrt{(x-2)^2+(y-1)^2}}{\sqrt{x^2+(y-1)^2}+\sqrt{(x-2)^2+y^2}} $$
5 replies
Sunjee
May 16, 2025
TBazar
3 hours ago
Tangencies with cyclic quadrilateral
tapir1729   21
N 3 hours ago by Mathandski
Source: TSTST 2024, problem 4
Let $ABCD$ be a quadrilateral inscribed in a circle with center $O$ and $E$ be the intersection of segments $AC$ and $BD$. Let $\omega_1$ be the circumcircle of $ADE$ and $\omega_2$ be the circumcircle of $BCE$. The tangent to $\omega_1$ at $A$ and the tangent to $\omega_2$ at $C$ meet at $P$. The tangent to $\omega_1$ at $D$ and the tangent to $\omega_2$ at $B$ meet at $Q$. Show that $OP=OQ$.

Merlijn Staps
21 replies
+1 w
tapir1729
Jun 24, 2024
Mathandski
3 hours ago
Binary multiples of three
tapir1729   8
N 3 hours ago by Mathandski
Source: TSTST 2024, problem 5
For a positive integer $k$, let $s(k)$ denote the number of $1$s in the binary representation of $k$. Prove that for any positive integer $n$,
\[\sum_{i=1}^{n}(-1)^{s(3i)} > 0.\]Holden Mui
8 replies
+2 w
tapir1729
Jun 24, 2024
Mathandski
3 hours ago
Geometry
MathsII-enjoy   3
N 3 hours ago by MathsII-enjoy
Given triangle $ABC$ inscribed in $(O)$ with $M$ being the midpoint of $BC$. The tangents at $B, C$ of $(O)$ intersect at $D$. Let $N$ be the projection of $O$ onto $AD$. On the perpendicular bisector of $BC$, take a point $K$ that is not on $(O)$ and different from M. Circle $(KBC)$ intersects $AK$ at $F$. Lines $NF$ and $AM$ intersect at $E$. Prove that $AEF$ is an isosceles triangle.
3 replies
MathsII-enjoy
May 15, 2025
MathsII-enjoy
3 hours ago
IMO 2018 Problem 2
juckter   98
N 4 hours ago by ezpotd
Find all integers $n \geq 3$ for which there exist real numbers $a_1, a_2, \dots a_{n + 2}$ satisfying $a_{n + 1} = a_1$, $a_{n + 2} = a_2$ and
$$a_ia_{i + 1} + 1 = a_{i + 2},$$for $i = 1, 2, \dots, n$.

Proposed by Patrik Bak, Slovakia
98 replies
juckter
Jul 9, 2018
ezpotd
4 hours ago
Element sum of k others
akasht   19
N 4 hours ago by ezpotd
Source: ISL 2022 A2
Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.
19 replies
akasht
Jul 9, 2023
ezpotd
4 hours ago
Least swaps to get any labeling of a regular 99-gon
Photaesthesia   9
N 4 hours ago by Blast_S1
Source: 2024 China MO, Day 2, Problem 6
Let $P$ be a regular $99$-gon. Assign integers between $1$ and $99$ to the vertices of $P$ such that each integer appears exactly once. (If two assignments coincide under rotation, treat them as the same. ) An operation is a swap of the integers assigned to a pair of adjacent vertices of $P$. Find the smallest integer $n$ such that one can achieve every other assignment from a given one with no more than $n$ operations.

Proposed by Zhenhua Qu
9 replies
Photaesthesia
Nov 29, 2023
Blast_S1
4 hours ago
Equation with powers
a_507_bc   6
N Apr 3, 2025 by EVKV
Source: Serbia JBMO TST 2024 P1
Find all non-negative integers $x, y$ and primes $p$ such that $$3^x+p^2=7 \cdot 2^y.$$
6 replies
a_507_bc
May 25, 2024
EVKV
Apr 3, 2025
Equation with powers
G H J
G H BBookmark kLocked kLocked NReply
Source: Serbia JBMO TST 2024 P1
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a_507_bc
678 posts
#1
Y by
Find all non-negative integers $x, y$ and primes $p$ such that $$3^x+p^2=7 \cdot 2^y.$$
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NO_SQUARES
1133 posts
#2
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a_507_bc wrote:
Find all non-negative integers $x, y$ and primes $p$ such that $$3^x+p^2=7 \cdot 2^y.$$
If $y=0$ then $x=1, p=2$. If $y=1$ then by mod 3 $x=0$ and there are no solutions. Now let $y>1$, so $4 | RHS$.
Note that since $4|7-3$ we have $2 \not | x$. After this look at mod 8 to get $y<3$ ($p \not = 2$).
This post has been edited 2 times. Last edited by NO_SQUARES, May 25, 2024, 7:30 PM
Reason: was wrong
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RagvaloD
4918 posts
#3 • 2 Y
Y by NO_SQUARES, ehuseyinyigit
$p=2 \to y=0,x=1$
$p$ is odd $\to 3^x+p^2 \equiv 2,4 \pmod {8} \to y<3$
for $y=1$ there are no solutions
For $y=2: x=1,p=5$
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Assassino9931
1361 posts
#4
Y by
We only need that $p$ is odd if $p \geq 3$. Indeed, mod 8 we have $3^x \equiv 3,1$ and $p^2\equiv 1$ and so $3^x + p^2 \equiv 2, 4$ while $7 \cdot 2^y \equiv 0$ for $y\geq 3$, contradiction. Hence either $p=2$ or $y\leq 2$.

If $p=2$, then parity insists on $y=0$, so $x=1$. If $y=2$, then only $p=5$ and $x=1$ works. If $y=1$, then there are no solutions. If $y=0$, then $p=2$ and $x=1$ works.

Hence all solutions are $(x,y,p) = (1,0,2), (1,2,5)$.
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THE_SOLVER
1 post
#5 • 1 Y
Y by JelaByteEngineer
Simply by applying mod 8 which restricts or bounds the value of y i.e y<3
By checking manually which gives us 2 solutions i.e (1,2,0);(1,2,5)
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ali123456
52 posts
#6
Y by
sketch of my solution
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EVKV
71 posts
#7
Y by
p is odd for $y \neq 0$
and clearly x= 1 , y= 0, p=2 satisfies
$For y \geq 3$
$either 2,4 \equiv 0$ $mod$ $8$ nonsense
now checking remaining
all solutions are $(x,y,p) = (1,0,2), (1,2,5)$.
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