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IMO ShortList 2008, Number Theory problem 3
April   24
N 44 minutes ago by sansgankrsngupta
Source: IMO ShortList 2008, Number Theory problem 3
Let $ a_0$, $ a_1$, $ a_2$, $ \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $ \gcd (a_i, a_{i + 1}) > a_{i - 1}$. Prove that $ a_n\ge 2^n$ for all $ n\ge 0$.

Proposed by Morteza Saghafian, Iran
24 replies
April
Jul 9, 2009
sansgankrsngupta
44 minutes ago
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Find points with sames integer distances as given
nAalniaOMliO   1
N an hour ago by Rohit-2006
Source: Belarus TST 2024
Points $A_1, \ldots A_n$ with rational coordinates lie on a plane. It turned out that the distance between every pair of points is an integer. Prove that there exist points $B_1, \ldots ,B_n$ with integer coordinates such that $A_iA_j=B_iB_j$ for every pair $1 \leq i \leq j \leq n$
N. Sheshko, D. Zmiaikou
1 reply
nAalniaOMliO
Jul 17, 2024
Rohit-2006
an hour ago
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Coincide
giangtruong13   2
N an hour ago by giangtruong13
Source: Hanoi Specialized School's Math Test (Round 2 - Phase 1)
Let $ABCD$ be a trapezoid inscribed in circle $(O)$, $AD||BC, AD < BC$. Let $P$ is the symmetric point of $A$ across $BC$, $AP$ intersects $BC$ at $K$. Let $M$ is midpoint of $BC$ and $H$ is orthocenter of triangle $ABC$. On $BD$ take a point $F$ so that $AF||HM$. Prove that: $ FK,AC,PD$ coincide
2 replies
giangtruong13
Sunday at 4:05 PM
giangtruong13
an hour ago
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Interesting number theory
giangtruong13   3
N an hour ago by giangtruong13
Source: Hanoi Specialized School’s Practical Math Entrance Exam (Round 2)
Let $a,b$ be integer numbers $\geq 3$ satisfy that:$a^2=b^3+ab$. Prove that:
a) $a,b$ are even
b) $4b+1$ is a perfect square number
c) $a$ can’t be any power $\geq 1$ of a positive integer number
3 replies
giangtruong13
Yesterday at 4:15 PM
giangtruong13
an hour ago
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Arbitrary point on BC and its relation with orthocenter
falantrng   22
N an hour ago by Rotten_
Source: Balkan MO 2025 P2
In an acute-angled triangle \(ABC\), \(H\) be the orthocenter of it and \(D\) be any point on the side \(BC\). The points \(E, F\) are on the segments \(AB, AC\), respectively, such that the points \(A, B, D, F\) and \(A, C, D, E\) are cyclic. The segments \(BF\) and \(CE\) intersect at \(P.\) \(L\) is a point on \(HA\) such that \(LC\) is tangent to the circumcircle of triangle \(PBC\) at \(C.\) \(BH\) and \(CP\) intersect at \(X\). Prove that the points \(D, X, \) and \(L\) lie on the same line.

Proposed by Theoklitos Parayiou, Cyprus
22 replies
falantrng
Sunday at 11:47 AM
Rotten_
an hour ago
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Hard Inequality Problem
Omerking   2
N an hour ago by surfstyle
$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=3$ is given where $a,b,c$ are positive reals. Prove that:
$$\frac{1}{\sqrt{a^3+1}}+\frac{1}{\sqrt{b^3+1}}+\frac{1}{\sqrt{c^3+1}} \le \frac{3}{\sqrt{2}}$$
2 replies
Omerking
Yesterday at 3:51 PM
surfstyle
an hour ago
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d | \overline{aabbcc} iff d | \overline{abc} where d is two digit number
parmenides51   1
N 2 hours ago by luphuc
Source: Czech-Polish-Slovak Junior Match 2013, Individual p4 CPSJ
Determine the largest two-digit number $d$ with the following property:
for any six-digit number $\overline{aabbcc}$ number $d$ is a divisor of the number $\overline{aabbcc}$ if and only if the number $d$ is a divisor of the corresponding three-digit number $\overline{abc}$.

Note The numbers $a \ne 0, b$ and $c$ need not be different.
1 reply
parmenides51
Mar 14, 2020
luphuc
2 hours ago
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Hard inequality
JK1603JK   1
N 2 hours ago by xytunghoanh
Source: unknown?
Let $a,b,c>0$ and $a^2+b^2+c^2=2(a+b+c).$ Find the minimum $$P=(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$$
1 reply
JK1603JK
4 hours ago
xytunghoanh
2 hours ago
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Functional Equation
JSGandora   13
N 2 hours ago by ray66
Source: 2006 Red MOP Homework Algebra 1.2
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfying
\[f(x+f(y))=x+f(f(y))\]
for all real numbers $x$ and $y$, with the additional constraint $f(2004)=2005$.
13 replies
JSGandora
Mar 17, 2013
ray66
2 hours ago
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Impossible divisibility
pohoatza   35
N 2 hours ago by cursed_tangent1434
Source: Romanian TST 3 2008, Problem 3
Let $ m,\ n \geq 3$ be positive odd integers. Prove that $ 2^{m}-1$ doesn't divide $ 3^{n}-1$.
35 replies
pohoatza
Jun 7, 2008
cursed_tangent1434
2 hours ago
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comp. geo starting with a 90-75-15 triangle. <APB =<CPQ, <BQA =<CQP.
parmenides51   1
N Apr 6, 2025 by Mathzeus1024
Source: 2013 Cuba 2.9
Let ABC be a triangle with $\angle A = 90^o$, $\angle B = 75^o$, and $AB = 2$. Points $P$ and $Q$ of the sides $AC$ and $BC$ respectively, are such that $\angle APB = \angle CPQ$ and $\angle BQA = \angle CQP$. Calculate the lenght of $QA$.
1 reply
parmenides51
Sep 20, 2024
Mathzeus1024
Apr 6, 2025
J
comp. geo starting with a 90-75-15 triangle. <APB =<CPQ, <BQA =<CQP.
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Source: 2013 Cuba 2.9
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parmenides51
30650 posts
parmenides51
#1 h  Sep 20, 2024, 9:25 PM
Y by
Let ABC be a triangle with $\angle A = 90^o$, $\angle B = 75^o$, and $AB = 2$. Points $P$ and $Q$ of the sides $AC$ and $BC$ respectively, are such that $\angle APB = \angle CPQ$ and $\angle BQA = \angle CQP$. Calculate the lenght of $QA$.
This post has been edited 2 times. Last edited by parmenides51, Sep 20, 2024, 9:37 PM
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Mathzeus1024
847 posts
Mathzeus1024
#2 h  Apr 6, 2025, 11:17 AM
Y by
Given right $\Delta ABC$ described above, let us examine the triangles $\Delta ABQ, \Delta APQ,$ and $\Delta CPQ$. If $\angle{APB}=\angle{CPQ}=\alpha$ and $\angle{BQA}=\angle{CQP}=\beta$, then we obtain $\alpha + \beta = 165^{\circ}$ (i) from $\Delta CPQ$. Next, an application of Law of Sines to $\Delta ABQ$ and $\Delta APQ$ yields the following system of equations in $AQ$ and $\alpha$:

$\frac{AB}{\sin(\beta)} = \frac{AQ}{\sin(75^{\circ})} \Rightarrow \frac{2}{\sin(165^{\circ}-\alpha)} = \frac{AQ}{\sin(75^{\circ})}$ (ii);

$\frac{AP}{\sin(180^{\circ}-2\beta)} = \frac{AQ}{\sin(180^{\circ}-\alpha)} \Rightarrow \frac{2\cot(\alpha)}{\sin(2\alpha-150^{\circ})} = \frac{AQ}{\sin(\alpha)}$ (iii)

of which we ultimately obtain $\textcolor{red}{AQ = \frac{\sqrt{19+9\sqrt{3}}}{3}}$.
This post has been edited 1 time. Last edited by Mathzeus1024, Apr 6, 2025, 11:54 AM
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