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(help urgent) Classic Geo Problem / Angle Chasing?
orangesyrup   0
5 minutes ago
Source: own
In the given figure, ABC is an isosceles triangle with AB = AC and ∠BAC = 78°. Point D is chosen inside the triangle such that AD=DC. Find the measure of angle X (∠BDC).

ps: see the attachment for figure
0 replies
orangesyrup
5 minutes ago
0 replies
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Lord Evan the Reflector
whatshisbucket   21
N 25 minutes ago by ezpotd
Source: ELMO 2018 #3, 2018 ELMO SL G3
Let $A$ be a point in the plane, and $\ell$ a line not passing through $A$. Evan does not have a straightedge, but instead has a special compass which has the ability to draw a circle through three distinct noncollinear points. (The center of the circle is not marked in this process.) Additionally, Evan can mark the intersections between two objects drawn, and can mark an arbitrary point on a given object or on the plane.

(i) Can Evan construct* the reflection of $A$ over $\ell$?

(ii) Can Evan construct the foot of the altitude from $A$ to $\ell$?

*To construct a point, Evan must have an algorithm which marks the point in finitely many steps.

Proposed by Zack Chroman
21 replies
whatshisbucket
Jun 28, 2018
ezpotd
25 minutes ago
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Feet of perpendiculars to diagonal in cyclic quadrilateral
jl_   2
N 27 minutes ago by lw202277
Source: Malaysia IMONST 2 2023 (Primary) P6
Suppose $ABCD$ is a cyclic quadrilateral with $\angle ABC = \angle ADC = 90^{\circ}$. Let $E$ and $F$ be the feet of perpendiculars from $A$ and $C$ to $BD$ respectively. Prove that $BE = DF$.
2 replies
1 viewing
jl_
3 hours ago
lw202277
27 minutes ago
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The old one is gone.
EeEeRUT   9
N 30 minutes ago by Jupiterballs
Source: EGMO 2025 P2
An infinite increasing sequence $a_1 < a_2 < a_3 < \cdots$ of positive integers is called central if for every positive integer $n$ , the arithmetic mean of the first $a_n$ terms of the sequence is equal to $a_n$.

Show that there exists an infinite sequence $b_1, b_2, b_3, \dots$ of positive integers such that for every central sequence $a_1, a_2, a_3, \dots, $ there are infinitely many positive integers $n$ with $a_n = b_n$.
9 replies
EeEeRUT
Apr 16, 2025
Jupiterballs
30 minutes ago
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Inequalities
Humberto_Filho   2
N 30 minutes ago by damyan
Source: From the material : A brief introduction to inequalities.
Let a,b be nonnegative real numbers such that $a + b \leq 2$. Prove that :

$$(1+a^2)(1+b^2) \geq (1 + (\frac{a+b}{2})^2)^2$$
2 replies
1 viewing
Humberto_Filho
Apr 12, 2023
damyan
30 minutes ago
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3 var inequalities
sqing   1
N 37 minutes ago by sqing
Source: Own
Let $ a,b> 0 $ and $ a+b\leq 2ab . $ Prove that
$$ \frac{ a + b }{ a^2(1+ b^2)} \leq\frac{1 }{\sqrt 2}-\frac{1 }{2}$$$$ \frac{ a +ab+ b }{ a^2(1+ b^2)} \leq \sqrt 2-1$$$$ \frac{ a +a^2b^2+ b }{ a^2(1+ b^2)} \leq\frac{\sqrt5 }{2}$$
1 reply
sqing
43 minutes ago
sqing
37 minutes ago
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Divisibility holds for all naturals
XbenX   13
N 43 minutes ago by zRevenant
Source: 2018 Balkan MO Shortlist N5
Let $x,y$ be positive integers. If for each positive integer $n$ we have that $$(ny)^2+1\mid x^{\varphi(n)}-1.$$Prove that $x=1$.

(Silouanos Brazitikos, Greece)
13 replies
XbenX
May 22, 2019
zRevenant
43 minutes ago
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Sum and product of 5 numbers
jl_   1
N an hour ago by jl_
Source: Malaysia IMONST 2 2023 (Primary) P2
Ivan bought $50$ cats consisting of five different breeds. He records the number of cats of each breed and after multiplying these five numbers he obtains the number $100000$. How many cats of each breed does he have?
1 reply
jl_
4 hours ago
jl_
an hour ago
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Interesting inequalities
sqing   3
N an hour ago by sqing
Source: Own
Let $ a,b> 0 $ and $ a+b\leq 2ab . $ Prove that
$$\frac{ 9a^2- ab +9b^2 }{ a^2(1+b^4)}\leq\frac{17 }{2}$$$$\frac{a- ab+b }{ a^2(1+b^4)}\leq\frac{1 }{2}$$$$\frac{2a- 3ab+2b }{ a^2(1+b^4)}\leq\frac{1 }{2}$$
3 replies
sqing
5 hours ago
sqing
an hour ago
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a+b+c=2 ine
KhuongTrang   30
N an hour ago by KhuongTrang
Source: own
Problem. Given non-negative real numbers $a,b,c: ab+bc+ca>0$ satisfying $a+b+c=2.$ Prove that $$\color{blue}{\frac{a}{\sqrt{2a+3bc}}+\frac{b}{\sqrt{2b+3ca}}+\frac{c}{\sqrt{2c+3ab}} \le \sqrt{\frac{2}{ab+bc+ca}}. }$$
30 replies
KhuongTrang
Jun 25, 2024
KhuongTrang
an hour ago
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2021 EGMO P1: {m, 2m+1, 3m} is fantabulous
anser   55
N 2 hours ago by NicoN9
Source: 2021 EGMO P1
The number 2021 is fantabulous. For any positive integer $m$, if any element of the set $\{m, 2m+1, 3m\}$ is fantabulous, then all the elements are fantabulous. Does it follow that the number $2021^{2021}$ is fantabulous?
55 replies
anser
Apr 13, 2021
NicoN9
2 hours ago
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Complicated FE
XAN4   0
2 hours ago
Source: own
Find all solutions for the functional equation $f(xyz)+\sum_{cyc}f(\frac{yz}x)=f(x)\cdot f(y)\cdot f(z)$, in which $f$: $\mathbb R^+\rightarrow\mathbb R^+$
Note: the solution is actually quite obvious - $f(x)=x^n+\frac1{x^n}$, but the proof is important.
Note 2: it is likely that the result can be generalized into a more advanced questions, potentially involving more bash.
0 replies
XAN4
2 hours ago
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Maximum area of the triangle
adityaguharoy   1
N Apr 13, 2025 by Mathzeus1024
If in some triangle $\triangle ABC$ we are given :
$\sqrt{3} \cdot \sin(C)=\frac{2- \sin A}{\cos A}$ and one side length of the triangle equals $2$, then under these conditions find the maximum area of the triangle $ABC$.
1 reply
adityaguharoy
Jan 19, 2017
Mathzeus1024
Apr 13, 2025
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Maximum area of the triangle
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adityaguharoy
4655 posts
adityaguharoy
#1 h  Jan 19, 2017, 11:42 AM • 1 Y
Y by Adventure10
If in some triangle $\triangle ABC$ we are given :
$\sqrt{3} \cdot \sin(C)=\frac{2- \sin A}{\cos A}$ and one side length of the triangle equals $2$, then under these conditions find the maximum area of the triangle $ABC$.
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Mathzeus1024
826 posts
Mathzeus1024
#2 h  Apr 13, 2025, 11:57 AM
Y by
Let us take $\sin(\angle{C}) = \frac{2-\sin(\angle{A})}{\sqrt{3}\cos(\angle{A})},$ and since $\angle{C} \in (0, \pi) \Rightarrow \sin(\angle{C}) \in (0,1]$ we require:

$0 < \frac{2-\sin(\angle{A})}{\sqrt{3}\cos(\angle{A})} \le 1$;

or $2-\sin(\angle{A}) \le \sqrt{3}\cos(\angle{A})$;

or $4-4\sin(\angle{A})+\sin^{2}(\angle{A}) \le 3[1-\sin^{2}(\angle{A})]$;

or $4\sin^{2}(\angle{A}) -4\sin(\angle{A})+1 \le 0$;

or $[2\sin(\angle{A})-1]^2 \le 0$;

or $\angle{A} = \frac{\pi}{6}, \frac{5\pi}{6}$ (i).

Checking both values of (i) for $\sin(\angle{C}) = \frac{2-\sin(\angle{A})}{\sqrt{3}\cos(\angle{A})}$ yields $\pm 1$, or $\angle{C} = \frac{\pi}{2}, \frac{3\pi}{2}$ respectively, which only the former angle is admissible $\Rightarrow \angle{A} = \frac{\pi}{6}, \angle{B}=\frac{\pi}{3}, \angle{C}=\frac{\pi}{2}$. If one of the sides of $\Delta ABC$ has length $2$, then we have the following triplets $(a,b,c) = (t,\sqrt{3}t,2t)$:

$(a,b,c) = (2,2\sqrt{3},4); (1,\sqrt{3},2); \left(\frac{2}{\sqrt{3}},2,\frac{4}{\sqrt{3}}\right)$ (ii).

If $A=\frac{1}{2}ab$ is the area, then (ii) yields: $\frac{\sqrt{3}}{2} < \frac{2\sqrt{3}}{3} < 2\sqrt{3} \Rightarrow \textcolor{red}{A_{MAX} = 2\sqrt{3}}$.
This post has been edited 3 times. Last edited by Mathzeus1024, Apr 13, 2025, 12:02 PM
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