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this hAOpefully shoudn't BE weird
popop614   40
N an hour ago by peace09
Source: 2023 IMO Shortlist G1
Let $ABCDE$ be a convex pentagon such that $\angle ABC = \angle AED = 90^\circ$. Suppose that the midpoint of $CD$ is the circumcenter of triangle $ABE$. Let $O$ be the circumcenter of triangle $ACD$.

Prove that line $AO$ passes through the midpoint of segment $BE$.
40 replies
1 viewing
popop614
Jul 17, 2024
peace09
an hour ago
Geo with an angle condition and fixed angle
a_507_bc   7
N an hour ago by L13832
Source: CGMO 2024 P4
Let $ABC$ be a triangle with $AB<BC<CA$ and let $D$ be a variable point on $BC$. The point $E$ on the circumcircle of $ABC$ is such that $\angle BAD=\angle BED$. The line through $D$ perpendicular to $AB$ meets $AC$ at $F$. Show that the measure of $\angle BEF$ is constant as $D$ varies.
7 replies
a_507_bc
Aug 12, 2024
L13832
an hour ago
Three triangle center projections (OTIS MOCK AIME 2025 I #13)
dolphinday   3
N an hour ago by P_Groudon
Source: OTIS MOCK AIME 2025 I
Let $ABC$ be an acute triangle.
Suppose the distances from its circumcenter, incenter, and orthocenter to side $BC$ are $8$, $6$, and $4$, respectively.
Compute $BC^2$.


Tanishq Pauskar
3 replies
dolphinday
Today at 12:37 AM
P_Groudon
an hour ago
Fermat's prime numbers and divisibility
Megus   6
N an hour ago by zuat.e
Source: Taiwan 1997
Let $k = 2^{2^n} + 1$ for some $n \in N$. Show that k is prime if and only if k
divides $3^{\frac{k-1}{2}}+1$
6 replies
Megus
May 27, 2005
zuat.e
an hour ago
weird FE
tobiSALT   8
N an hour ago by ItsBesi
Source: Pan American Girls' Mathematical Olympiad 2024, P5
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
$f(f(x+y) - f(x)) + f(x)f(y) = f(x^2) - f(x+y),$

for all real numbers $x, y$.
8 replies
tobiSALT
Nov 27, 2024
ItsBesi
an hour ago
Coloring edges (OTIS MOCK AIME 2025 I #10)
dolphinday   2
N 2 hours ago by bachkieu
Source: OTIS MOCK AIME 2025 I
An $8 \times 8$ grid of unit squares is drawn; it thus has $144$ unit edges.
Let $N$ be the number of ways to color each of the $144$ unit edges one of six colors (red, orange, yellow, green, blue, or purple) such that every unit square is surrounded by exactly $3$ different colors.
Then $N$ can be written as a prime factorization $p_1^{e_1} \dots p_k^{e_k}$ where $p_1 < \dots < p_k$ are primes and $e_i$ are positive integers.
Compute $e_1 + \dots + e_k$.


Tanishq Pauskar
2 replies
dolphinday
Today at 12:30 AM
bachkieu
2 hours ago
AJ combo
codyj   33
N 2 hours ago by Ywgh1
Source: IMO 2015 #6
The sequence $a_1,a_2,\dots$ of integers satisfies the conditions:

(i) $1\le a_j\le2015$ for all $j\ge1$,
(ii) $k+a_k\neq \ell+a_\ell$ for all $1\le k<\ell$.

Prove that there exist two positive integers $b$ and $N$ for which\[\left\vert\sum_{j=m+1}^n(a_j-b)\right\vert\le1007^2\]for all integers $m$ and $n$ such that $n>m\ge N$.

Proposed by Ivan Guo and Ross Atkins, Australia
33 replies
codyj
Jul 11, 2015
Ywgh1
2 hours ago
MN _|_ PQ wanted, ABC , ADE right isosceles triangles, 4 midpoints
parmenides51   1
N 3 hours ago by vanstraelen
Source: 2015 Iberoamerican Shortlist G1 https://artofproblemsolving.com/community/c2533295_iberoamerican_shortlist_2011_geometry
Let $ABC$ and $ADE$ be two right isosceles triangles that share only point $A$, which is the vertex opposite the hypotenuse for both triangles. The order in which the vertices are given is that of clockwise, for both triangles. Let $M, N, P$ and $Q$ be the midpoints of the segments $BE$, $CD$, $BC$ and $DE$, respectively. Show that the segments $MN$ and $PQ$ intersect and are perpendicular.
1 reply
parmenides51
Nov 3, 2021
vanstraelen
3 hours ago
19 divides x^2 + 10x +1
Evo_Zarpic   1
N 3 hours ago by CatinoBarbaraCombinatoric
Find all integers $x$ such that $19 | x^2 + 10x + 1$.
Determine The Minimum and Maximum of Them
1 reply
Evo_Zarpic
3 hours ago
CatinoBarbaraCombinatoric
3 hours ago
Thanks u!
Ruji2018252   7
N 3 hours ago by youochange
$x,y,z\ge 0$ and $x+y+z=3$. Find minimum (and prove):
$$P=x^2+y^2+2z^2+2xyz$$
7 replies
Ruji2018252
Today at 1:08 PM
youochange
3 hours ago
What is the minimum number of friendships that must already exist so that every
Khalifakhalifa   12
N 3 hours ago by Khalifakhalifa


There are \( n \) users on a social network called Mathbook, and some of them are Mathbook-friends. (On Mathbook, friendship is always mutual and permanent.)
Starting now, Mathbook will only allow a new friendship to be formed between two users if they have at least \( k \) friends in common, where \( k \geq 2 \) is a fixed integer.
What is the minimum number of friendships that must already exist so that every user could eventually become friends with every other user?
12 replies
Khalifakhalifa
Dec 8, 2024
Khalifakhalifa
3 hours ago
thanks u!
Ruji2018252   0
3 hours ago
$a,b,c\ge 0$ and $a+b+c=1$. Prove:
$$\sqrt {7a^2-8a+3} +\sqrt {7b^2-8b+3} + \sqrt{7c^2-8c+3} \ge \sqrt {12(a^2+b^2+c^2)+6}$$
0 replies
Ruji2018252
3 hours ago
0 replies
Perspective Triangles
MNJ2357   2
N Jan 26, 2020 by Pathological
Source: Korea Winter Program Practice Test 2 P4
$\triangle ABC$ and $\triangle A_1B_1C_1$ are perspective triangles. $(ABB_1)$ and $(ACC_1)$ meet at $A_2 (\neq A)$. Define $B_2,C_2$ analogously. Prove that $AA_2, BB_2,CC_2$ are concurrent.
2 replies
MNJ2357
Jan 23, 2020
Pathological
Jan 26, 2020
Perspective Triangles
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G H BBookmark kLocked kLocked NReply
Source: Korea Winter Program Practice Test 2 P4
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MNJ2357
643 posts
#1 • 2 Y
Y by stroller, Adventure10
$\triangle ABC$ and $\triangle A_1B_1C_1$ are perspective triangles. $(ABB_1)$ and $(ACC_1)$ meet at $A_2 (\neq A)$. Define $B_2,C_2$ analogously. Prove that $AA_2, BB_2,CC_2$ are concurrent.
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stroller
894 posts
#2 • 3 Y
Y by MNJ2357, Adventure10, Mango247
mOvInG pOiNtS

why am I such a troll
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Pathological
578 posts
#4 • 3 Y
Y by stroller, Pluto1708, Adventure10
Easy Trig
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