Equilateral triangles with a parallelogram

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

9 minutes ago

Concurrent lines

BR1F1SZ 4

N 20 minutes ago by NicoN9

Source: 2025 CJMO P2

Let $ABCD$ be a trapezoid with parallel sides $AB$ and $CD$ , where $BC\neq DA$ . A circle passing through $C$ and $D$ intersects $AC, AD, BC, BD$ again at $W, X, Y, Z$ respectively. Prove that $WZ, XY, AB$ are concurrent.

4 replies

BR1F1SZ

Mar 7, 2025

NicoN9

20 minutes ago

Arithmetic progression

BR1F1SZ 2

N 31 minutes ago by NicoN9

Source: 2025 CJMO P1

Suppose an infinite non-constant arithmetic progression of integers contains $1$ in it. Prove that there are an infinite number of perfect cubes in this progression. (A perfect cube is an integer of the form $k^3$ , where $k$ is an integer. For example, $-8$ , $0$ and $1$ are perfect cubes.)

2 replies

BR1F1SZ

Mar 7, 2025

NicoN9

31 minutes ago

Number Theory Chain!

JetFire008 51

N 40 minutes ago by Primeniyazidayi

I will post a question and someone has to answer it. Then they have to post a question and someone else will answer it and so on. We can only post questions related to Number Theory and each problem should be more difficult than the previous. Let's start!

Question 1

Starting with the simplest
What is $2+3$ ?

51 replies

+1 w

JetFire008

Apr 7, 2025

Primeniyazidayi

40 minutes ago

Injective arithmetic comparison

adityaguharoy 1

N an hour ago by Mathzeus1024

Source: Own .. probably own

Show or refute :
For every injective function $f: \mathbb{N} \to \mathbb{N}$ there are elements $a,b,c$ in an arithmetic progression in the order $a<b<c$ such that $f(a)<f(b)<f(c)$ .

1 reply

adityaguharoy

Jan 16, 2017

Mathzeus1024

an hour ago

Abelkonkurransen 2025 1b

Lil_flip38 2

N an hour ago by Lil_flip38

Source: abelkonkurransen

In Duckville there is a perpetual trophy with the words “Best child of Duckville” engraved on it. Each inhabitant of Duckville has a non-empty list (which never changes) of other inhabitants of Duckville. Whoever receives the trophy
gets to keep it for one day, and then passes it on to someone on their list the next day. Gregers has previously received the trophy. It turns out that each time he does receive it, he is guaranteed to receive it again exactly $2025$ days later (but perhaps earlier, as well). Hedvig received the trophy today. Determine all integers $n>0$ for which we can be absolutely certain that she cannot receive the trophy again in $n$ days, given the above information.

2 replies

Lil_flip38

Mar 20, 2025

Lil_flip38

an hour ago

Symmetric inequalities under two constraints

ChrP 4

N an hour ago by ChrP

Let $a+b+c=0$ such that $a^2+b^2+c^2=1$ . Prove that $\sqrt{2-3a^2}+\sqrt{2-3b^2}+\sqrt{2-3c^2} \leq 2\sqrt{2}$
and

$a\sqrt{2-3a^2}+b\sqrt{2-3b^2}+c\sqrt{2-3c^2} \geq 0$
What about the lower bound in the first case and the upper bound in the second?

4 replies

ChrP

Apr 7, 2025

ChrP

an hour ago

Terms of a same AP

adityaguharoy 1

N 2 hours ago by Mathzeus1024

Given $p,q,r$ are positive integers pairwise distinct and $n$ is also a positive integer $n \ne 1$ .
Determine under which conditions can $\sqrt[n]{p},\sqrt[n]{q},\sqrt[n]{r}$ form terms of a same arithmetic progression.

1 reply

adityaguharoy

May 4, 2017

Mathzeus1024

2 hours ago

Inspired by old results

sqing 8

N 2 hours ago by sqing

Source: Own

Let $a,b\geq 0$ and $a^2+ab+b^2=2$ . Prove that
$(a+b-ab)\left( \frac{1}{a+1} + \frac{1}{b+1}\right)\leq 2$ $(a+b-ab)\left( \frac{a}{b+1} + \frac{2b}{a+2}\right)\leq 2$ $(a+b-ab)\left( \frac{a}{b+1} + \frac{2b}{a+1}\right)\leq 4$ Let $a,b$ be reals such that $a^2+b^2=2$ . Prove that
$(a+b)\left( \frac{1}{a+1} + \frac{1}{b+1}\right)= 2$ $(a+b)\left( \frac{a}{b+1} + \frac{b}{a+1}\right)=2$

8 replies

sqing

Today at 2:42 AM

sqing

2 hours ago

Transform the sequence

steven_zhang123 1

N 2 hours ago by vgtcross

Given a sequence of $n$ real numbers $a_1, a_2, \ldots, a_n$ , we can select a real number $\alpha$ and transform the sequence into $|a_1 - \alpha|, |a_2 - \alpha|, \ldots, |a_n - \alpha|$ . This transformation can be performed multiple times, with each chosen real number $\alpha$ potentially being different
(i) Prove that it is possible to transform the sequence into all zeros after a finite number of such transformations.
(ii) To ensure that the above result can be achieved for any given initial sequence, what is the minimum number of transformations required?

1 reply

steven_zhang123

Today at 3:57 AM

vgtcross

2 hours ago

NEPAL TST DAY 2 PROBLEM 2

Tony_stark0094 5

N 2 hours ago by ThatApollo777

Kritesh manages traffic on a $45 \times 45$ grid consisting of 2025 unit squares. Within each unit square is a car, facing either up, down, left, or right. If the square in front of a car in the direction it is facing is empty, it can choose to move forward. Each car wishes to exit the $45 \times 45$ grid.

Kritesh realizes that it may not always be possible for all the cars to leave the grid. Therefore, before the process begins, he will remove $k$ cars from the $45 \times 45$ grid in such a way that it becomes possible for all the remaining cars to eventually exit the grid.

What is the minimum value of $k$ that guarantees that Kritesh's job is possible?

$\textbf{Proposed by Shining Sun. USA}$

5 replies

Tony_stark0094

Yesterday at 8:37 AM

ThatApollo777

2 hours ago

Product of distinct integers in arithmetic progression -- ever a perfect power ?

adityaguharoy 1

N 2 hours ago by Mathzeus1024

Source: Well known (the gen. is more difficult, but may be not this one -- so this is here)

Let $a_1,a_2,a_3,a_4$ be four positive integers in arithmetic progression (that is, $a_1-a_2=a_2-a_3=a_3-a_4$ ) and with $a_1 \ne a_2$ . Can the product $a_1 \cdot a_2 \cdot a_3 \cdot a_4$ ever be a number of the form $n^k$ for some $n \in \mathbb{N}$ and some $k \in \mathbb{N}$ , with $k \ge 2$ ?

1 reply

adityaguharoy

Aug 31, 2019

Mathzeus1024

2 hours ago

Equilateral triangles with a parallelogram

kred9 1

N Apr 6, 2025 by bjump

Source: 2025 Utah Math Olympiad #5

Given parallelogram $ABCD$ , we construct equilateral triangle $ABP$ such that $P$ is on the same side of $\overline{AB}$ as $C$ and $D$ . It is given that $\overleftrightarrow{CP}$ intersects $\overleftrightarrow{DA}$ at $Q$ . Prove that there exists a point $R$ on $\overleftrightarrow{AB}$ such that $\triangle CQR$ is equilateral.

1 reply

kred9

Apr 6, 2025

bjump

Apr 6, 2025

Equilateral triangles with a parallelogram

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parallelogram

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Source: 2025 Utah Math Olympiad #5

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kred9

1021 posts

kred9

#1 h Apr 6, 2025, 12:06 AM

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bjump

999 posts

bjump

#2 h Apr 6, 2025, 2:19 AM • 1 Y

Y by kred9

Let $I$ be the midpoint of $QC$ , $J$ be the midpoint of $BA$ , let $G$ be where the perpendicular bisector of $CQ$ intersects $AB$ . Note that
$\measuredangle GJE = \measuredangle GIE = 90^\circ$ Therefore $IEJG$ is cyclic. Observe as $IJ \parallel BC$ . Converse of Reim with the previous two facts gives $GBEC$ cyclic. Therefore:
$\measuredangle GCE = \measuredangle GBE = 60^\circ$ Therefore $\triangle GFC$ is equilateral.

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