Geometry angle chasing olympiads

by Foxellar, May 28, 2025, 10:32 PM

Let \( \triangle ABC \) be a triangle such that \( \angle ABC = 120^\circ \). Points \( X, Y, Z \) lie on segments \( BC, CA, AB \), respectively, such that lines \( AX, BY, \) and \( CZ \) are the angle bisectors of triangle \( ABC \). Find the measure of angle \( \angle XYZ \).

A sharp one with 3 var (3)

by mihaig, May 27, 2025, 5:17 PM

Let $a,b,c\geq0$ satisfying
$$\left(a+b+c-2\right)^2+8\leq3\left(ab+bc+ca\right).$$Prove
$$a^2+b^2+c^2+5abc\geq8.$$

Brilliant Problem

by M11100111001Y1R, May 27, 2025, 7:28 AM

Find all sequences \( (a_n) \) of natural numbers such that for every pair of natural numbers \( r \) and \( s \), the following inequality holds:
\[
\frac{1}{2} < \frac{\gcd(a_r, a_s)}{\gcd(r, s)} < 2
\]

Cup of Combinatorics

by M11100111001Y1R, May 27, 2025, 7:24 AM

There are \( n \) cups labeled \( 1, 2, \dots, n \), where the \( i \)-th cup has capacity \( i \) liters. In total, there are \( n \) liters of water distributed among these cups such that each cup contains an integer amount of water. In each step, we may transfer water from one cup to another. The process continues until either the source cup becomes empty or the destination cup becomes full.

$a)$ Prove that from any configuration where each cup contains an integer amount of water, it is possible to reach a configuration in which each cup contains exactly 1 liter of water in at most \( \frac{4n}{3} \) steps.

$b)$ Prove that in at most \( \frac{5n}{3} \) steps, one can go from any configuration with integer water amounts to any other configuration with the same property.
This post has been edited 1 time. Last edited by M11100111001Y1R, Tuesday at 7:26 AM

Bulgaria National Olympiad 1996

by Jjesus, Jun 10, 2020, 7:34 PM

Find all prime numbers $p,q$ for which $pq$ divides $(5^p-2^p)(5^q-2^q)$.

Problem 4

by codyj, Jul 11, 2015, 6:30 AM

Triangle $ABC$ has circumcircle $\Omega$ and circumcenter $O$. A circle $\Gamma$ with center $A$ intersects the segment $BC$ at points $D$ and $E$, such that $B$, $D$, $E$, and $C$ are all different and lie on line $BC$ in this order. Let $F$ and $G$ be the points of intersection of $\Gamma$ and $\Omega$, such that $A$, $F$, $B$, $C$, and $G$ lie on $\Omega$ in this order. Let $K$ be the second point of intersection of the circumcircle of triangle $BDF$ and the segment $AB$. Let $L$ be the second point of intersection of the circumcircle of triangle $CGE$ and the segment $CA$.

Suppose that the lines $FK$ and $GL$ are different and intersect at the point $X$. Prove that $X$ lies on the line $AO$.

Proposed by Greece
This post has been edited 3 times. Last edited by djmathman, Jun 14, 2018, 4:21 PM
Reason: italicized authors

Iran Inequality

by mathmatecS, Jun 11, 2015, 9:05 AM

When $x(\ge1),$ $y(\ge1),$ $z(\ge1)$ satisfy $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2,$ prove in equality.
$$\sqrt{x+y+z}\ge\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}$$
This post has been edited 1 time. Last edited by mathmatecS, Jun 11, 2015, 9:31 AM

Fractional Inequality

by sqing, Aug 10, 2012, 10:07 AM

Let $ a_1, a_2,\ldots, a_n$ be non-negative real numbers. Prove that
$\frac{1}{1+ a_1}+\frac{ a_1}{(1+ a_1)(1+ a_2)}+\frac{ a_1 a_2}{(1+ a_1)(1+ a_2)(1+ a_3)}+$ $\cdots+\frac{ a_1 a_2\cdots a_{n-1}}{(1+ a_1)(1+ a_2)\cdots (1+ a_n)} \le 1.$

Can't be power of 2

by shobber, Mar 17, 2006, 3:26 PM

Show that for any positive integers $a$ and $b$, $(36a+b)(a+36b)$ cannot be a power of $2$.

IMO96/2 [the lines AP, BD, CE meet at a point]

by Arne, Sep 30, 2003, 6:06 PM

Let $ P$ be a point inside a triangle $ ABC$ such that
\[ \angle APB - \angle ACB = \angle APC - \angle ABC.
\]
Let $ D$, $ E$ be the incenters of triangles $ APB$, $ APC$, respectively. Show that the lines $ AP$, $ BD$, $ CE$ meet at a point.

"Where wisdom and valor fail, all that remains is faith. . . And it can overcome all." -Toa Mata Tahu

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