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Cyclic ine
m4thbl3nd3r   7
N 16 minutes ago by Victoria_Discalceata1
Let $a,b,c>0$ such that $a+b+c=3$. Prove that $$a^3b+b^3c+c^3a+9abc\le 12$$
7 replies
m4thbl3nd3r
Yesterday at 3:17 PM
Victoria_Discalceata1
16 minutes ago
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Guessing with intervals
navi_09220114   1
N 35 minutes ago by ja.
Source: Malaysian IMO TST 2025 P2
Let $n\ge 4$ be a positive integer. Megavan and Minivan are playing a game, where Megavan secretly chooses a real number $x$ in $[0, 1]$. At the start of the game, the only information Minivan has about $x$ is $x$ in $[0, 1]$. He needs to now learn about $x$ based on the following protocols: at each turn of his, Minivan chooses a number $y$ and submits to Megavan, where Megavan replies immediately with one of $y > x$, $y < x$, or $y\simeq x$, subject to two rules:

$\bullet$ The answers in the form of $y > x$ and $y < x$ must be truthful;

$\bullet$ Define the score of a round, known only to Megavan, as follows: $0$ if the answer is in the form $y > x$ and $y < x$, and $|x - y|$ if in the form $y\simeq x$. Then for every positive integer $k$ and every $k$ consecutive rounds, at least one round has score no more than $\frac{1}{k + 1}$.

Minivan's goal is to produce numbers $a, b$ such that $a\le x\le b$ and $b - a\le \frac 1n$. Let $f(n)$ be the minimum number of queries that Minivan needs in order to guarantee success, regardless of Megavan's strategy. Prove that $$n\le f(n) \le 4n$$
Proposed by Anzo Teh Zhao Yang
1 reply
navi_09220114
Yesterday at 12:55 PM
ja.
35 minutes ago
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permutations of sets
cloventeen   1
N an hour ago by alexheinis
Find the number of permutations of the set \( A = (1, 2, \dots, n) \) with the set \( B = (1, 1, 2, 3, \dots, n) \) such that each element in the permutations has at most one immediate neighbor greater than itself.
1 reply
cloventeen
Today at 2:36 AM
alexheinis
an hour ago
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EM // AC wanted, isosceles trapezoid related
parmenides51   5
N an hour ago by Tsikaloudakis
Source: 2020 Austrian Mathematical Olympiad Junior Regional Competition , Problem 3
Given is an isosceles trapezoid $ABCD$ with $AB \parallel CD$ and $AB> CD$. The projection from $D$ on $ AB$ is $E$. The midpoint of the diagonal $BD$ is $M$. Prove that $EM$ is parallel to $AC$.

(Karl Czakler)
5 replies
parmenides51
Dec 18, 2020
Tsikaloudakis
an hour ago
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triangles with equal areas
mathuz   7
N an hour ago by Tsikaloudakis
Source: SRMC 2023, P1
Let $ABCD$ be a trapezoid with $AD\parallel BC$. A point $M $ is chosen inside the trapezoid, and a point $N$ is chosen inside the triangle $BMC$ such that $AM\parallel CN$, $BM\parallel DN$. Prove that triangles $ABN$ and $CDM$ have equal areas.
7 replies
1 viewing
mathuz
Dec 28, 2023
Tsikaloudakis
an hour ago
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number theory
karimeow   0
an hour ago
Prove that there exist infinitely many positive integers m such that the equation (xz+1)(yz+1) = mz^3 + 1 has infinitely many positive integer solutions.
0 replies
karimeow
an hour ago
0 replies
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Eventually constant sequence with condition
PerfectPlayer   3
N 2 hours ago by egxa
Source: Turkey TST 2025 Day 3 P8
A positive real number sequence $a_1, a_2, a_3,\dots $ and a positive integer \(s\) is given.
Let $f_n(0) = \frac{a_n+\dots+a_1}{n}$ and for each $0<k<n$
\[f_n(k)=\frac{a_n+\dots+a_{k+1}}{n-k}-\frac{a_k+\dots+a_1}{k}\]Then for every integer $n\geq s,$ the condition
\[a_{n+1}=\max_{0\leq k<n}(f_n(k))\]is satisfied. Prove that this sequence must be eventually constant.
3 replies
PerfectPlayer
Mar 18, 2025
egxa
2 hours ago
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A lot of tangent circle
ItzsleepyXD   2
N 2 hours ago by WLOGQED1729
Source: Own
Let \( \triangle ABC \) be a triangle with circumcircle \( \omega \) and circumcenter \( O \). Let \( \omega_A \) and \( I_A \) represent the \( A \)-excircle and \( A \)-excenter, respectively. Denote by \( \omega_B \) the circle tangent to \( AB \), \( BC \), and \( \omega \) on the arc \( BC \) not containing \( A \), and similarly for \( \omega_C \). Let the tangency points of \( \omega_A, \omega_B, \omega_C \) with line \( BC \) be \( X, Y, Z \), respectively. Let \( P \neq A \) be the intersection point of \( (AYZ) \) and \( \omega \). Define \( Q \) as the point on segment \( OI_A \) such that \( 2 \cdot OQ = QI_A \). Suppose that \( XP \) intersects \( \omega \) again at \( R \). Let \( T \) be the touch point of the \( A \)-mixtilinear incircle and \( \omega \), and let \( A' \) be the antipode of \( A \) with respect to \( \omega \). Let \( S \) be the intersection of \( A'Q \) and \( I_AT \).

Show that the line \( RS \) is the radical axis of \( \omega_B \) and \( \omega_C \).
2 replies
1 viewing
ItzsleepyXD
2 hours ago
WLOGQED1729
2 hours ago
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deduction from function
MetaphysicalWukong   3
N 2 hours ago by pco
can we then deduce that h has exactly 1 zero?
3 replies
MetaphysicalWukong
2 hours ago
pco
2 hours ago
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number theory question?
jag11   3
N 2 hours ago by Anabcde
Find the smallest positive integer n such that n is a multiple of 11, n +1 is a multiple of 10, n + 2 is a
multiple of 9, n + 3 is a multiple of 8, n +4 is a multiple of 7, n +5 is a multiple of 6, n +6 is a multiple of
5, n + 7 is a multiple of 4, n + 8 is a multiple of 3, and n + 9 is a multiple of 2.

I tried doing the mods and simplifying it but I'm kinda confused.
3 replies
jag11
Yesterday at 10:41 PM
Anabcde
2 hours ago
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Circles and Chords
steven_zhang123   0
2 hours ago
(1) Let \( A \) , \( B \) and \( C \) be points on circle \( O \) divided into three equal parts. Construct three equal circles \( O_1 \), \( O_2 \), and \( O_3 \) tangent to \( O \) internally at points \( A \), \( B \), and \( C \) respectively. Let \( P \) be any point on arc \( AC \), and draw tangents \( PD \), \( PE \), and \( PF \) to circles \( O_1 \), \( O_2 \), and \( O_3 \) respectively. Prove that \( PE = PD + PF \).

(2) Let \( A_1 \), \( A_2 \), \( \cdots \), \( A_n \) be points on circle \( O \) divided into \( n \) equal parts. Construct \( n \) equal circles \( O_1 \), \( O_2 \), \( \cdots \), \( O_n \) tangent to \( O \) internally at \( A_1 \), \( A_2 \), \( \cdots \), \( A_n \). Let \( P \) be any point on circle \( O \), and draw tangents \( PB_1 \), \( PB_2 \), \( \cdots \), \( PB_n \) to circles \( O_1 \), \( O_2 \), \( \cdots \), \( O_n \). If the sum of \( k \) of \( PB_1 \), \( PB_2 \), \( \cdots \), \( PB_n \) equals the sum of the remaining \( n-k \) (where \( n \geq k \geq 1 \)), find all such \( n \).
0 replies
steven_zhang123
2 hours ago
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Find min
hunghd8   7
N Yesterday at 9:14 AM by imnotgoodatmathsorry
Let $a,b,c$ be nonnegative real numbers such that $ a+b+c\geq 2+abc $. Find min
$$P=a^2+b^2+c^2.$$
7 replies
hunghd8
Friday at 12:10 PM
imnotgoodatmathsorry
Yesterday at 9:14 AM
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Find min
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hunghd8
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#1 h  Friday at 12:10 PM
Y by
Let $a,b,c$ be nonnegative real numbers such that $ a+b+c\geq 2+abc $. Find min
$$P=a^2+b^2+c^2.$$
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Mathzeus1024
742 posts
Mathzeus1024
#2 h  Friday at 2:06 PM
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Withdrawn
This post has been edited 3 times. Last edited by Mathzeus1024, Yesterday at 8:22 AM
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sqing
41181 posts
sqing
#5 h  Friday at 3:20 PM
Y by
Let $a,b,c$ be nonnegative real numbers such that $ a+b+c\geq 2+abc $. Prove that
$$a^2+b^2+c^2\geq 2$$*
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imnotgoodatmathsorry
60 posts
imnotgoodatmathsorry
#6 h  Friday at 4:19 PM
Y by
hunghd8 wrote:
Let $a,b,c$ be nonnegative real numbers such that $ a+b+c\geq 2+abc $. Find min
$$P=a^2+b^2+c^2.$$

WLOG, we have $(a-1)(b-1) \ge 0$ so we have: $a+b+c \ge 2 +abc \ge 2-c+c(a+b)$
so: $(a+b-2)(1-c)>=0$
Now you need to solve 2 cases: $a+b \ge 2$ and $c \le 1$ or $a+b \le 2$ and $c \ge 1$ which is kinda ez
So, you can prove $P \ge 2$ with equality occurs when $(a;b;c)=(2;0;0)$
This post has been edited 2 times. Last edited by imnotgoodatmathsorry, Friday at 4:20 PM
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hunghd8
40 posts
hunghd8
#7 h  Yesterday at 1:57 AM
Y by
imnotgoodatmathsorry wrote:
hunghd8 wrote:
Let $a,b,c$ be nonnegative real numbers such that $ a+b+c\geq 2+abc $. Find min
$$P=a^2+b^2+c^2.$$

WLOG, we have $(a-1)(b-1) \ge 0$ so we have: $a+b+c \ge 2 +abc \ge 2-c+c(a+b)$
so: $(a+b-2)(1-c)>=0$
Now you need to solve 2 cases: $a+b \ge 2$ and $c \le 1$ or $a+b \le 2$ and $c \ge 1$ which is kinda ez
So, you can prove $P \ge 2$ with equality occurs when $(a;b;c)=(2;0;0)$
$a+b \le 2$ and $c \ge 1$ which is kinda ez??
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imnotgoodatmathsorry
60 posts
imnotgoodatmathsorry
#8 h  Yesterday at 4:10 AM
Y by
hunghd8 wrote:
imnotgoodatmathsorry wrote:
hunghd8 wrote:
Let $a,b,c$ be nonnegative real numbers such that $ a+b+c\geq 2+abc $. Find min
$$P=a^2+b^2+c^2.$$

WLOG, we have $(a-1)(b-1) \ge 0$ so we have: $a+b+c \ge 2 +abc \ge 2-c+c(a+b)$
so: $(a+b-2)(1-c)>=0$
Now you need to solve 2 cases: $a+b \ge 2$ and $c \le 1$ or $a+b \le 2$ and $c \ge 1$ which is kinda ez
So, you can prove $P \ge 2$ with equality occurs when $(a;b;c)=(2;0;0)$
$a+b \le 2$ and $c \ge 1$ which is kinda ez??

Well, you should kind of present $P$ by $c$ and that's all right?
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hunghd8
40 posts
hunghd8
#9 h  Yesterday at 6:32 AM
Y by
imnotgoodatmathsorry wrote:
hunghd8 wrote:
Let $a,b,c$ be nonnegative real numbers such that $ a+b+c\geq 2+abc $. Find min
$$P=a^2+b^2+c^2.$$

WLOG, we have $(a-1)(b-1) \ge 0$ so we have: $a+b+c \ge 2 +abc \ge 2-c+c(a+b)$
so: $(a+b-2)(1-c)>=0$
Now you need to solve 2 cases: $a+b \ge 2$ and $c \le 1$ or $a+b \le 2$ and $c \ge 1$ which is kinda ez
So, you can prove $P \ge 2$ with equality occurs when $(a;b;c)=(2;0;0)$
You're wrong of course
Z K Y
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imnotgoodatmathsorry
60 posts
imnotgoodatmathsorry
#10 h  Yesterday at 9:14 AM
Y by
hunghd8 wrote:
imnotgoodatmathsorry wrote:
hunghd8 wrote:
Let $a,b,c$ be nonnegative real numbers such that $ a+b+c\geq 2+abc $. Find min
$$P=a^2+b^2+c^2.$$

WLOG, we have $(a-1)(b-1) \ge 0$ so we have: $a+b+c \ge 2 +abc \ge 2-c+c(a+b)$
so: $(a+b-2)(1-c)>=0$
Now you need to solve 2 cases: $a+b \ge 2$ and $c \le 1$ or $a+b \le 2$ and $c \ge 1$ which is kinda ez
So, you can prove $P \ge 2$ with equality occurs when $(a;b;c)=(2;0;0)$
You're wrong of course

oh, well, thanks for noticing btw
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