,
denote the number of its positive divisors (including 1 and itself). Determine all positive integers 
for which there exists a positive integer 
.
let 
be the foot of the altitude from 
to the side 
and 
,
,
be the incenter, 
-
and 
the other intersection points of the circle 
with the lines 
and 
,
.
such that![\[(a^2-b^2)(b^2-c^2) = abc.\]](http://latex.artofproblemsolving.com/7/1/8/7188261c40a45f5e15533100432d36abbf515efd.png)
.
that 
is a prime.
is divided into rectangles with two cells. Two rectangles are called adjacent if they share a segment of length 1 or 2. In each rectangle the amount of adjacent with it rectangles is written.
such that 
for all 
.
,
is the greatest prime divisor of 
.
be a nonzero square matrix of order 
satisfying![\[
a_{ik} a_{jk} = a_{kk} a_{ij}, \quad \text{for all } i, j, k.
\]](http://latex.artofproblemsolving.com/7/5/2/752b16a277d9fd18bfda6c3be35d23d42d75ebaa.png)
Denote by 
the trace of 
,
.
be a ring. Let 
such that 
.
is nilpotent, so is 
.
,
,
.
id defined by![\[
f(x) = \frac{1}{\sqrt{2\pi} \sigma} e^{-\frac{(x-\mu)^2}{2\sigma^2}}
\]](http://latex.artofproblemsolving.com/0/2/c/02c7b4af3cac77da8e3cc102c79a7db2e95f437f.png)
now what is CDF or cumulative distribution function ![$F_X(x)=P[X\leq x]$](http://latex.artofproblemsolving.com/7/9/1/79150c7527581feec873c9c790f8aeddc8f8efb8.png)
![$f(x)\in C[0,3]$](http://latex.artofproblemsolving.com/8/e/2/8e2d8e1cf446975c3e3aa894610ac8b6371737b4.png)
satisfy 
for all 
and
Show that
Y by pog, mathman3880, MP8148, 606234, centslordm, Sugiyem, Nofancyname, amar_04, zuss77, GorgonMathDota, sotpidot, EpicNumberTheory, Aryan27, samrocksnature, judgefan99, mijail, Hyperbolic_, megarnie, Taco12, OlympusHero, EmilXM, brickmaster8, SerdarBozdag, Triangle_Center, BVKRB-, sargamsujit, MintTea, CyclicISLscelesTrapezoid, Jalil_Huseynov, Seicchi28, Miku_, GoldenBoy03, LoloChen, GeoKing, Lamboreghini, oolite, Mahdi_Mashayekhi, TheBatmanWhoLaughs, CalciumSodium01, nguyenducmanh2705, Paramizo_Dicrominique, UI_MathZ_25, Siddharth03, Rounak_iitr
In order all the olympiad geometry geometry mocks to be collected in one place I created Aops Geo Mocks post collection. After all a few people have spent a lot of time, preparing those mocks, testsolving and grading.
includes so far:
* USA Computational Geo Mocks - AMC, AIME, HMMT
* Geometry Mains Olympiad (GMO) (IMO level)
* 2024 MODS Geometry Contest
* 2024 Geo Mock - GPO by Eeightqx, ychk, xiaolu12, Wenzi, Mang, smoth_silver
* 2024 Geo Mock - SAGF (pdfs: part I, part II) by LoloChen, GeoKing , Inequality., nervy
* 2023 DGO III (pdfs: individual + team ) (Discord Geometry Olympiad ΙΙI 2023)
* 2023 DJGO II (pdfs: individual + team ) (Discord Junior Geometry Olympiad ΙIΙ 2023)
* 2023 Geo Mock - SAGF (pdfs: part I, part II) by LoloChen, GeoKing , Inequality., nervy
* 2023 Geo Mock - BMoEG - III (pdf inside zip) (Buratino's Mock of Euclidean Geometry - III 2023) by buratinogigle , parmenides51
* 2022 DGO II (pdfs: individual + team ) (Discord Geometry Olympiad ΙΙ 2022)
* 2022 DJGO I (pdfs: individual + team ) (Discord Junior Geometry Olympiad Ι 2022)
* 2022 Geo Mock - BMoEG - II (pdf) (Buratino's Mock of Euclidean Geometry - II 2022) by buratinogigle , parmenides51 (solutions)
* 2022 Geo Mock MGMA (pdf) (Make Geometry Great Again!) by LoloChen, Inequality., GeoKing , a22886 (solutions)
* 2021 GUSAMO (pdf) (Geometry Unsuccessfully Set like Ankan's: the Mock Olympiad) by i3435
* 2021 DGO (Discord Geometry Olympiad Ι 2021)
* 2021 Geo Mock - GOWACA (pdf) (Geometry Olympiad Without A Cool Acronym) by MP8148 (solutions)
* 2021 Geo Mock - BMoEG - I (pdf) (Buratino's Mock of Euclidean Geometry - I 2021) by buratinogigle , parmenides51 (solutions)
*2021 Geo Mock - AMASCIWLOFRIAA1PD (pdf) (A Mediocre and Silly Contest in Which Lack of Figures Results In An Automatic 1 Point Deduction) (solutions)
2021 Geo Mock - CSMGO (pdf) (Christmas Special Mock Geometry Olympiad) by amar_04
2020 Geo Mock - AQGO (pdf) (Aops Quarantine Geometry Olympiad) by GeoMetrix, Amar_04 , mueller.25
2020 Geo Mock - GGG4 (pdf) by AndrewWu (wu2481632)
2020 Geo Mock - POGCHAMP (pdf) (Pretty Original Geometry Contest: HAs Mediocre Problems) by tworigami , mathman3880
* 2020 Geo Mock - Computational Contest (pdf) by Tovi Wen (two origami)
2019 Geo Mock - India Forum (pdf) by GeoMetrix, Pluto1708, Naruto.D.Luffy, amar_04
2019 Geo Mock - GOTEEM (pdf) (Geometry Olympiad: Tovi and Ethan's Exam of Mathematics) by Tovi Wen, Ethan Joo
* 2019 Geo Mock - Olympiad (pdf) by Tovi Wen (two origami)
* 2018 Geo Mock - AIME (pdf) by AndrewWu (wu2481632)
2017 Geo Mock - GGG3 (pdf) by AndrewWu (wu2481632)
2017 Geo Mock - GGG1 (pdf), GG2 (pdf) by AndrewWu (wu2481632)
2015 Geolympiad Fall Contest (pdf)
2015 Geolympiad Summer Contest (pdf)
2015 Geolympiad Spring Contest (pdf)
* 2014 Geo Mock - USAJMO (pdf) by Mike Ren
* 2012 Geo Mock - USAMO (pdf) by Jinduckey, cwein3
* 2012 Geo Mock (set 1 pdf, set 2 pdf) by David Stoner
Enjoy and suggest any geo mock that I might have missed. Outside aops I collect them here
* latest entries
includes so far:
* USA Computational Geo Mocks - AMC, AIME, HMMT
* Geometry Mains Olympiad (GMO) (IMO level)
* 2024 MODS Geometry Contest
* 2024 Geo Mock - GPO by Eeightqx, ychk, xiaolu12, Wenzi, Mang, smoth_silver
* 2024 Geo Mock - SAGF (pdfs: part I, part II) by LoloChen, GeoKing , Inequality., nervy
* 2023 DGO III (pdfs: individual + team ) (Discord Geometry Olympiad ΙΙI 2023)
* 2023 DJGO II (pdfs: individual + team ) (Discord Junior Geometry Olympiad ΙIΙ 2023)
* 2023 Geo Mock - SAGF (pdfs: part I, part II) by LoloChen, GeoKing , Inequality., nervy
* 2023 Geo Mock - BMoEG - III (pdf inside zip) (Buratino's Mock of Euclidean Geometry - III 2023) by buratinogigle , parmenides51
* 2022 DGO II (pdfs: individual + team ) (Discord Geometry Olympiad ΙΙ 2022)
* 2022 DJGO I (pdfs: individual + team ) (Discord Junior Geometry Olympiad Ι 2022)
* 2022 Geo Mock - BMoEG - II (pdf) (Buratino's Mock of Euclidean Geometry - II 2022) by buratinogigle , parmenides51 (solutions)
* 2022 Geo Mock MGMA (pdf) (Make Geometry Great Again!) by LoloChen, Inequality., GeoKing , a22886 (solutions)
* 2021 GUSAMO (pdf) (Geometry Unsuccessfully Set like Ankan's: the Mock Olympiad) by i3435
* 2021 DGO (Discord Geometry Olympiad Ι 2021)
* 2021 Geo Mock - GOWACA (pdf) (Geometry Olympiad Without A Cool Acronym) by MP8148 (solutions)
* 2021 Geo Mock - BMoEG - I (pdf) (Buratino's Mock of Euclidean Geometry - I 2021) by buratinogigle , parmenides51 (solutions)
*2021 Geo Mock - AMASCIWLOFRIAA1PD (pdf) (A Mediocre and Silly Contest in Which Lack of Figures Results In An Automatic 1 Point Deduction) (solutions)
2021 Geo Mock - CSMGO (pdf) (Christmas Special Mock Geometry Olympiad) by amar_04
2020 Geo Mock - AQGO (pdf) (Aops Quarantine Geometry Olympiad) by GeoMetrix, Amar_04 , mueller.25
2020 Geo Mock - GGG4 (pdf) by AndrewWu (wu2481632)
2020 Geo Mock - POGCHAMP (pdf) (Pretty Original Geometry Contest: HAs Mediocre Problems) by tworigami , mathman3880
* 2020 Geo Mock - Computational Contest (pdf) by Tovi Wen (two origami)
2019 Geo Mock - India Forum (pdf) by GeoMetrix, Pluto1708, Naruto.D.Luffy, amar_04
2019 Geo Mock - GOTEEM (pdf) (Geometry Olympiad: Tovi and Ethan's Exam of Mathematics) by Tovi Wen, Ethan Joo
* 2019 Geo Mock - Olympiad (pdf) by Tovi Wen (two origami)
* 2018 Geo Mock - AIME (pdf) by AndrewWu (wu2481632)
2017 Geo Mock - GGG3 (pdf) by AndrewWu (wu2481632)
2017 Geo Mock - GGG1 (pdf), GG2 (pdf) by AndrewWu (wu2481632)
2015 Geolympiad Fall Contest (pdf)
2015 Geolympiad Summer Contest (pdf)
2015 Geolympiad Spring Contest (pdf)
* 2014 Geo Mock - USAJMO (pdf) by Mike Ren
* 2012 Geo Mock - USAMO (pdf) by Jinduckey, cwein3
* 2012 Geo Mock (set 1 pdf, set 2 pdf) by David Stoner
Enjoy and suggest any geo mock that I might have missed. Outside aops I collect them here
* latest entries
This post has been edited 68 times. Last edited by parmenides51, Jan 17, 2025, 3:49 PM
:
January 
(GMT+3)
:
January 
(GMT+3)
