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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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0 replies
jlacosta
May 1, 2025
0 replies
\sqrt{a^2+b^2+2}+\sqrt{b^2+c^2+2 }+\sqrt{c^2+a^2+2}\ge 6
parmenides51   19
N an hour ago by NicoN9
Source: JBMO Shortlist 2017 A1
Let $a, b, c$ be positive real numbers such that $a + b + c + ab + bc + ca + abc = 7$. Prove
that $\sqrt{a^2 + b^2 + 2 }+\sqrt{b^2 + c^2 + 2 }+\sqrt{c^2 + a^2 + 2 } \ge 6$ .
19 replies
parmenides51
Jul 25, 2018
NicoN9
an hour ago
Inspired by Austria 2025
sqing   1
N an hour ago by sqing
Source: Own
Let $ a,b\geq 0 ,a,b\neq 1$ and $  a^2+b^2=1. $ Prove that$$   (a + b ) \left( \frac{a}{(b -1)^2} + \frac{b}{(a - 1)^2} \right) \geq 12+8\sqrt 2$$
1 reply
sqing
an hour ago
sqing
an hour ago
IMO Genre Predictions
ohiorizzler1434   51
N an hour ago by ethan2011
Everybody, with IMO upcoming, what are you predictions for the problem genres?


Personally I predict: predict
51 replies
1 viewing
ohiorizzler1434
May 3, 2025
ethan2011
an hour ago
Concurrency from isogonal Mittenpunkt configuration
MarkBcc168   17
N 3 hours ago by Ilikeminecraft
Source: Fake USAMO 2020 P3
Let $\triangle ABC$ be a scalene triangle with circumcenter $O$, incenter $I$, and incircle $\omega$. Let $\omega$ touch the sides $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ at points $D$, $E$, and $F$ respectively. Let $T$ be the projection of $D$ to $\overline{EF}$. The line $AT$ intersects the circumcircle of $\triangle ABC$ again at point $X\ne A$. The circumcircles of $\triangle AEX$ and $\triangle AFX$ intersect $\omega$ again at points $P\ne E$ and $Q\ne F$ respectively. Prove that the lines $EQ$, $FP$, and $OI$ are concurrent.

Proposed by MarkBcc168.
17 replies
MarkBcc168
Apr 28, 2020
Ilikeminecraft
3 hours ago
anyone who can help me this 2 problems?
auroracliang   2
N 3 hours ago by ReticulatedPython
1. Let r be the radius of the largest circle which is tangent to the parabola y=x^2 at x=0 and which lies entirely on or inside (that is, above) the parabola, find r.

2. Counting number n has the following property,: if we take any 50 different numbers from 1,2,3, ... n, there always are two numbers with the difference of 7. what is the largest value among all value of n?


thanks a lot
2 replies
auroracliang
Nov 3, 2024
ReticulatedPython
3 hours ago
What conic section is this? Is this even a conic section?
invincibleee   2
N 3 hours ago by ReticulatedPython
IMAGE

The points in this are given by
P = (sin2A, sin4A)∀A [0,2π]
Is this a conic section? what is this?
2 replies
invincibleee
Nov 15, 2024
ReticulatedPython
3 hours ago
Spheres, ellipses, and cones
ReticulatedPython   0
3 hours ago
A sphere is inscribed inside a cone with base radius $1$ and height $2.$ Another sphere of radius $r$ is internally tangent to the lateral surface of the cone, but does not intersect the larger inscribed sphere. A plane is tangent to both of these spheres, and passes through the inside of the cone. The intersection of the plane and the cone forms an ellipse. Find the maximum area of this ellipse.
0 replies
ReticulatedPython
3 hours ago
0 replies
Looking for users and developers
derekli   13
N 4 hours ago by DreamineYT
Guys I've been working on a web app that lets you grind high school lvl math. There's AMCs, AIME, BMT, HMMT, SMT etc. Also, it's infinite practice so you can keep grinding without worrying about finding new problems. Please consider helping me out by testing and also consider joining our developer team! :P :blush:

Link: https://stellarlearning.app/competitive
13 replies
derekli
May 4, 2025
DreamineYT
4 hours ago
trigonometric functions
VivaanKam   12
N 4 hours ago by aok
Hi could someone explain the basic trigonometric functions to me like sin, cos, tan etc.
Thank you!
12 replies
VivaanKam
Apr 29, 2025
aok
4 hours ago
find number of elements in H
Darealzolt   1
N Yesterday at 6:47 PM by alexheinis
If \( H \) is the set of positive real solutions to the system
\[
x^3 + y^3 + z^3 = x + y + z
\]\[
x^2 + y^2 + z^2 = xyz
\]then find the number of elements in \( H \).
1 reply
Darealzolt
Yesterday at 1:50 AM
alexheinis
Yesterday at 6:47 PM
primes and perfect squares
Bummer12345   0
Yesterday at 5:08 PM
If $p$ and $q$ are primes, then can $2^p + 5^q + pq$ be a perfect square?
0 replies
Bummer12345
Yesterday at 5:08 PM
0 replies
simple trapezoid
gggzul   0
Yesterday at 4:44 PM
Let $ABCD$ be a trapezoid. By $x$ we denote the angle bisector of angle $X$ . Let $P=a\cap c$ and $Q=b\cap d$. Prove that $ABPQ$ is cyclic.
0 replies
gggzul
Yesterday at 4:44 PM
0 replies
geometry
JetFire008   0
Yesterday at 4:14 PM
Given four concyclic points. For each subset of three points take the incenter. Show that the four incentres form a rectangle.
0 replies
JetFire008
Yesterday at 4:14 PM
0 replies
Inequalities
sqing   11
N Yesterday at 3:02 PM by sqing
Let $a,b,c> 0$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1.$ Prove that
$$  (1-abc) (1-a)(1-b)(1-c)  \ge 208 $$$$ (1+abc) (1-a)(1-b)(1-c)  \le -224 $$$$(1+a^2b^2c^2) (1-a)(1-b)(1-c)  \le -5840 $$
11 replies
1 viewing
sqing
Jul 12, 2024
sqing
Yesterday at 3:02 PM
Connecting chaos in a grid
Assassino9931   3
N Apr 25, 2025 by dgrozev
Source: Bulgaria National Olympiad 2025, Day 1, Problem 2
Exactly \( n \) cells of an \( n \times n \) square grid are colored black, and the remaining cells are white. The cost of such a coloring is the minimum number of white cells that need to be recolored black so that from any black cell \( c_0 \), one can reach any other black cell \( c_k \) through a sequence \( c_0, c_1, \ldots, c_k \) of black cells where each consecutive pair \( c_i, c_{i+1} \) are adjacent (sharing a common side) for every \( i = 0, 1, \ldots, k-1 \). Let \( f(n) \) denote the maximum possible cost over all initial colorings with exactly \( n \) black cells. Determine a constant $\alpha$ such that
\[
\frac{1}{3}n^{\alpha} \leq f(n) \leq 3n^{\alpha}
\]for any $n\geq 100$.
3 replies
Assassino9931
Apr 8, 2025
dgrozev
Apr 25, 2025
Connecting chaos in a grid
G H J
Source: Bulgaria National Olympiad 2025, Day 1, Problem 2
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Assassino9931
1320 posts
#1 • 1 Y
Y by cubres
Exactly \( n \) cells of an \( n \times n \) square grid are colored black, and the remaining cells are white. The cost of such a coloring is the minimum number of white cells that need to be recolored black so that from any black cell \( c_0 \), one can reach any other black cell \( c_k \) through a sequence \( c_0, c_1, \ldots, c_k \) of black cells where each consecutive pair \( c_i, c_{i+1} \) are adjacent (sharing a common side) for every \( i = 0, 1, \ldots, k-1 \). Let \( f(n) \) denote the maximum possible cost over all initial colorings with exactly \( n \) black cells. Determine a constant $\alpha$ such that
\[
\frac{1}{3}n^{\alpha} \leq f(n) \leq 3n^{\alpha}
\]for any $n\geq 100$.
This post has been edited 1 time. Last edited by Assassino9931, Apr 9, 2025, 8:54 AM
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internationalnick123456
134 posts
#2 • 1 Y
Y by cubres
Answer
Solution
This post has been edited 3 times. Last edited by internationalnick123456, Apr 9, 2025, 10:15 AM
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Assassino9931
1320 posts
#3 • 1 Y
Y by internationalnick123456
Answer

Construction for upper bound

Argument for lower bound

By accident, the constants actually matter!
This post has been edited 1 time. Last edited by Assassino9931, Apr 9, 2025, 8:46 PM
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dgrozev
2463 posts
#4 • 1 Y
Y by Miquel-point
Here is an argument using a spannong tree for the lower bound.

We may think of the configuration as a graph $G$ with $n^2$ vertices (cells) and edges that connect each two adjacent cells (sharing a side). We want to choose $m$ such that a grid with distance $m$ between two consecutive columns/rows contains at least $n$ cells. So $m$ must satisfy:
$$\left(\left\lfloor \frac{n-1}{m}\right\rfloor+1\right)^2\ge n.$$An easy calculation shows that it's enough to take $m:=\left\lfloor\sqrt{n}-\frac{1}{\sqrt{n}}\right\rfloor$. Let $V'$ be the set of $n$ cells, any two of them of distance at least $m$ between them. Let $T(V)$ be a connected subgraph of $G$ with a vertex-set $V$ such that $V'\subset V$. In what follows below, given an edge $e=uv$, by |uv| we denote its length, that is, the number of hops needed to reach $v$ from $u$, or, in other words, the so called manhattan distance between $u$ and $v$. |T| denotes the total length of $T$, that is, the sum of lengths of its edges.

Clearly $T$ is a tree, because we can remove the redundant edges until $T$ is still connected. Note that the claim is trivial in case $V'=V$, because in this case $|V|=n$ and each edge of $T$ has a length at least $m$, hence $|T|\ge (n-1)m$. (There was a glitch in the originally proposed solution, since it took for granted that $V'=V$, which cannot be guaranteed).

So, a bit more care is needed in case $V'\subsetneq V$. Let us refer to the vertices in $V\setminus V'$ as to the white vertices and to the vertices in $V'$ as to the black vertices. Let $v_0$ be a white vertex of $T$. We refer further to it as the root of $T$. We apply the following

Procedure. We delete all white leaves of $T$ (except of $v_0$). Take a leaf $v$ of $T$ with the largest number of vertices between $v$ and $v_0$. Let $v'$ be the neighbor of $v$. If $v'$ is black, we delete the edge $v'v$ whose length is at least $m$ and start the procedure from the beginning.

Assume now, $v'$ is white. If $\deg(v')=2$, we can assume that there is no need of $v'$ and just prolong the two edges that $v'$ is incident with. Let $v_1=v, v_2,\ldots,v_k$ be all the leaves that $v'$ is connected to, and $k\ge 2$. We have:
$$|v'v_i|+|v'v_{i+1}|\ge |v_iv_{i+1}|, i=1,2,\ldots,k,$$where we assume $v_{k+1}=v_1$. Note that $|v_iv_{i+1}|\ge m$ since all $v_i$ are black. Summing up these inequalities, we get:
$$\sum_{i=1}^k|v'v_i|\ge \frac{1}{2}\sum_{i=1}^k \left |v_i v_{i+1}\right|\ge \frac{km}{2}.$$Now, we delete the vertex $v'$. With this, we delete edges with length at least $mk/2$.
Repeat the procedure till we are left with the root $v_0$, which is white.

Note that each time we delete a set of $k$ black vertices, the corresponding deleted edges have total length at least $mk/2$, that is, the number of cells not in $V'$ that cover these edges is at least $mk/2-k=k(m-2)/2$. Therefore, the total number of cells not in $V'$ that cover $T$ is at least:
$$|V'|\cdot (m-2)/2=\frac{1}{2}n \cdot (m-2) \ge \frac{n(\sqrt{n}-3-1/\sqrt{n})}{2}\ge \frac{n^{3/2}}{3},$$as $n\ge 100$.
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