Y by
Given a convex quadrilateral 
, define
![\[
\alpha = |\angle ADB - \angle ACB| = |\angle DAC - \angle DBC|
\quad\text{and}\quad
\beta = |\angle ABD - \angle ACD| = |\angle BAC - \angle BDC|.
\]](//latex.artofproblemsolving.com/3/e/e/3eef0425824a79ba8615288dcfe2df79fca048b1.png)
Prove that
![\[
AC \cdot BD = AD \cdot BC \cos\alpha + AB \cdot CD \cos\beta.
\]](//latex.artofproblemsolving.com/d/4/a/d4adc5d510093da3520b26cbce172cf791b2847a.png)
, define![\[
\alpha = |\angle ADB - \angle ACB| = |\angle DAC - \angle DBC|
\quad\text{and}\quad
\beta = |\angle ABD - \angle ACD| = |\angle BAC - \angle BDC|.
\]](http://latex.artofproblemsolving.com/3/e/e/3eef0425824a79ba8615288dcfe2df79fca048b1.png)
Prove that![\[
AC \cdot BD = AD \cdot BC \cos\alpha + AB \cdot CD \cos\beta.
\]](http://latex.artofproblemsolving.com/d/4/a/d4adc5d510093da3520b26cbce172cf791b2847a.png)
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|a^2-b^2-2abc|<2c implies abc EVEN!
tom-nowy 1
N
by Tkn
Source: Own
Prove that if integers 
and 
satisfy 
, then 
is an even number.
and 
satisfy 
, then 
is an even number.
1 reply
Tricky inequality
Orestis_Lignos 28
N
by MR.1
Source: JBMO 2023 Problem 2
Prove that for all non-negative real numbers 
, not all equal to 
, the following inequality holds
Determine all the triples
for which the equality holds.
Milan Mitreski, Serbia
, not all equal to 
, the following inequality holds
Determine all the triples
for which the equality holds.Milan Mitreski, Serbia
28 replies
JBMO Shortlist 2023 A1
Orestis_Lignos 5
N
by MR.1
Source: JBMO Shortlist 2023, A1
Prove that for all positive real numbers 
,
and determine when equality occurs.
,
and determine when equality occurs.
5 replies
find angle
TBazar 5
N
by TBazar
Given 
triangle with 
. We take 
, 
point on AC, AB respectively such that 
, 
. 
, 
lines intersect at point 
. If 
, find 
triangle with 
. We take 
, 
point on AC, AB respectively such that 
, 
. 
, 
lines intersect at point 
. If 
, find 
5 replies
2 var inquality
Iveela 19
N
by sqing
Source: Izho 2025 P1
Let be positive reals such that 
. Prove that ![\[(a-b)^2 + a + b \geq 2\]](//latex.artofproblemsolving.com/c/f/a/cfa01a5b54a72afe6746f547390e65dbbb82ea64.png)
be positive reals such that 
. Prove that ![\[(a-b)^2 + a + b \geq 2\]](http://latex.artofproblemsolving.com/c/f/a/cfa01a5b54a72afe6746f547390e65dbbb82ea64.png)
19 replies
Set of perfect powers is irreducible
Assassino9931 0
Source: Al-Khwarizmi International Junior Olympiad 2025 P4
For two sets of integers 
and 
we define 
as the set of all products of an element of and an element of . For example, if 
and 
then 
We call a set 
of positive integers good if there do not exist sets 
of positive integers, each with at least two elements and such that the sets 
and are the same. Prove that the set of perfect powers greater than or equal to 
is good.
(In any of the sets
, 
, no two elements are equal, but any two or three of these sets may have common elements. A perfect power is an integer of the form 
, where 
and 
are integers.)
Lajos Hajdu and Andras Sarkozy, Hungary
and 
we define 
as the set of all products of an element of and an element of . For example, if 
and 
then 
We call a set 
of positive integers good if there do not exist sets 
of positive integers, each with at least two elements and such that the sets 
and are the same. Prove that the set of perfect powers greater than or equal to 
is good.(In any of the sets
, 
, no two elements are equal, but any two or three of these sets may have common elements. A perfect power is an integer of the form 
, where 
and 
are integers.)Lajos Hajdu and Andras Sarkozy, Hungary
0 replies
Al-Khwarizmi birth year in a combi process
Assassino9931 0
Source: Al-Khwarizmi International Junior Olympiad 2025 P3
On a circle are arranged 
baskets, each containing at least one candy. The total number of candies is 
. Asad and Sevinch make moves alternatingly, with Asad going first. On one move, Asad takes all the candies from 
consecutive non-empty baskets, while Sevinch takes all the candies from a single non-empty basket that has at least one empty neighboring basket. Prove that Asad can take overall at least 
candies, regardless of the initial distribution of candies and Sevinch's actions.
Shubin Yakov, Russia
baskets, each containing at least one candy. The total number of candies is 
. Asad and Sevinch make moves alternatingly, with Asad going first. On one move, Asad takes all the candies from 
consecutive non-empty baskets, while Sevinch takes all the candies from a single non-empty basket that has at least one empty neighboring basket. Prove that Asad can take overall at least 
candies, regardless of the initial distribution of candies and Sevinch's actions.Shubin Yakov, Russia
0 replies
Grouping angles in a pentagon with bisectors
Assassino9931 0
Source: Al-Khwarizmi International Junior Olympiad 2025 P2
Let be a convex quadrilateral with ![\[\angle ADC = 90^\circ, \ \ \angle BCD = \angle ABC > 90^\circ, \mbox{ and } AB = 2CD.\]](//latex.artofproblemsolving.com/b/4/1/b416d5c0b29c3a69a3e87cd22b6000aa8e70d456.png)
The line through 
, parallel to 
, intersects the external angle bisector of 
at point 
. Prove that the angles 
, 
, 
, 
, 
can be divided into two groups, so that the angles in each group have a sum of 
.
Miroslav Marinov, Bulgaria
be a convex quadrilateral with ![\[\angle ADC = 90^\circ, \ \ \angle BCD = \angle ABC > 90^\circ, \mbox{ and } AB = 2CD.\]](http://latex.artofproblemsolving.com/b/4/1/b416d5c0b29c3a69a3e87cd22b6000aa8e70d456.png)
The line through 
, parallel to 
, intersects the external angle bisector of 
at point 
. Prove that the angles 
, 
, 
, 
, 
can be divided into two groups, so that the angles in each group have a sum of 
.Miroslav Marinov, Bulgaria
0 replies
Nice Functional Equations (ISI 2021)
integrated_JRC 11
N
by lakshya2009
Source: ISI 2021 P2
Let 
be a function satisfying 
. Assume also that 
satisfies equations (A) and (B) below. 
for all integers 
.
(i) Determine explicitly the set
.
(ii) Assuming that there is a non-zero integer
such that 
, prove that the set 
is infinite.
be a function satisfying 
. Assume also that 
satisfies equations (A) and (B) below. 
for all integers 
.(i) Determine explicitly the set
.(ii) Assuming that there is a non-zero integer
such that 
, prove that the set 
is infinite.
11 replies
A Generalization of Ptolemy's Theorem
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H
G
H
Source: VN Math Olympiad For High School Students P7 - 2025
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