maxarea of a quadrilateral with sidelengths 1, 4, 7,8

Old High School Olympiads Regional, National, and International Math Olympiads before 2001 mostly, recent years have been posted in order to make contest collections richer

Regional, National, and International Math Olympiads before 2001 mostly, recent years have been posted in order to make contest collections richer

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Old High School Olympiads Regional, National, and International Math Olympiads before 2001 mostly, recent years have been posted in order to make contest collections richer

Regional, National, and International Math Olympiads before 2001 mostly, recent years have been posted in order to make contest collections richer

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k a April Highlights and 2025 AoPS Online Class Information

jlacosta 0

2 hours ago

Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

Looking for summer camps in math and language arts? Be sure to check out the video-based summer camps offered at the Virtual Campus that are 2- to 4-weeks in duration. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following events:
[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
[*]April 8th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS State Discussion
April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]April 16th (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies

jlacosta

2 hours ago

0 replies

k a My Retirement & New Leadership at AoPS

rrusczyk 1571

N Mar 26, 2025 by SmartGroot

I write today to announce my retirement as CEO from Art of Problem Solving. When I founded AoPS 22 years ago, I never imagined that we would reach so many students and families, or that we would find so many channels through which we discover, inspire, and train the great problem solvers of the next generation. I am very proud of all we have accomplished and I’m thankful for the many supporters who provided inspiration and encouragement along the way. I'm particularly grateful to all of the wonderful members of the AoPS Community!

I’m delighted to introduce our new leaders - Ben Kornell and Andrew Sutherland. Ben has extensive experience in education and edtech prior to joining AoPS as my successor as CEO, including starting like I did as a classroom teacher. He has a deep understanding of the value of our work because he’s an AoPS parent! Meanwhile, Andrew and I have common roots as founders of education companies; he launched Quizlet at age 15! His journey from founder to MIT to technology and product leader as our Chief Product Officer traces a pathway many of our students will follow in the years to come.

Thank you again for your support for Art of Problem Solving and we look forward to working with millions more wonderful problem solvers in the years to come.

And special thanks to all of the amazing AoPS team members who have helped build AoPS. We’ve come a long way from here:IMAGE

1571 replies

rrusczyk

Mar 24, 2025

SmartGroot

Mar 26, 2025

k i testing latex post , please ignore

parmenides51 3

N Feb 23, 2025 by parmenides51

Source: used with https://artofproblemsolving.com/community/c3170239_latex_testing_for_post_collections

issue 1 ..,
Click to reveal hidden text

Γραμμικά Συστήματα 2x2 στα σχολικά βιβλία μαθηματικών
ΒΓ 1 Δώδεκα μικρά λεωφορεία των 8 και 14 ατόμων μεταφέρουν συνολικά 126 επιβάτες. Πόσα λεωφορεία είναι των 8 και πόσα των 14 ατόμων;
2 .Ένας πατέρας είναι 44 ετών και ο γιος του είναι 8 ετών. Μετά από πόσα έτη η ηλικία του πατέρα θα είναι τριπλάσια της ηλικίας του γιου;
ΓΓ 3. Ο κερματοδέκτης ενός μηχανήματος πώλησης αναψυκτικών δέχεται κέρματα των 50 λεπτών και του 1 ευρώ. Όταν ανοίχτηκε, διαπιστώθηκε ότι περιείχε 126 κέρματα συνολικής αξίας 90 ευρώ. Πόσα κέρματα υπήρχαν από κάθε είδος;
4. Συσκευάσαμε 2,5 τόνους ελαιόλαδου σε 800 δοχεία των 2 και 5 κιλών. Να βρείτε πόσα δοχεία χρησιμοποιήσαμε από κάθε είδος.
5. O μέσος όρος της βαθμολογίας ενός μαθητή στη Φυσική και τη Χημεία κατά το πρώτο τρίμηνο ήταν 16. Στο δεύτερο τρίμηνο ο βαθμός της Φυσικής μειώθηκε κατά 2 μονάδες, ο βαθμός της Χημείας αυξήθηκε κατά 4 μονάδες με αποτέλεσμα οι δύο βαθμοί να γίνουν ίσοι. Ποιους βαθμούς είχε ο μαθητής σε καθένα από τα δύο μαθήματα κατά το πρώτο τρίμηνο;
6. Αν οι μαθητές ενός τμήματος καθίσουν ανά ένας σε κάθε θρανίο, τότε θα μείνουν όρθιοι 8 μαθητές, ενώ αν καθίσουν ανά δύο θα μείνουν κενά 4 θρανία. Να βρείτε πόσοι είναι οι μαθητές και πόσα τα θρανία.
7. Μια ποτοποιία παρασκεύασε 400 λίτρα ούζο περιεκτικότητας 38% vol, αναμειγνύοντας δύο ποιότητες με περιεκτικότητες 32% vol και 48% vol αντίστοιχα. Πόσα λίτρα από κάθε ποιότητα χρησιμοποίησε;
8. Από ένα σταθμό διοδίων πέρασαν 945 αυτοκίνητα και μοτοσικλέτες και εισπράχτηκαν 1810 €. Αν ο οδηγός κάθε αυτοκινήτου πλήρωσε 2 € και ο οδηγός κάθε μοτοσικλέτας πλήρωσε 1,2 €, να βρείτε πόσα ήταν τα αυτοκίνητα και πόσες οι μοτοσικλέτες.
9. Σ' ένα τηλεοπτικό παιχνίδι σε κάθε παίκτη υποβάλλονται 10 ερωτήσεις και για κάθε σωστή απάντηση προστίθενται βαθμοί, ενώ για κάθε λανθασμένη απάντηση αφαιρούνται βαθμοί. Ένας παίκτης έδωσε 7 σωστές απαντήσεις και συγκέντρωσε 64 βαθμούς, ενώ ένας άλλος έδωσε 4 σωστές απαντήσεις και συγκέντρωσε 28 βαθμούς. Πόσους βαθμούς παίρνει ένας παίκτης για κάθε σωστή απάντηση και πόσοι βαθμοί τού αφαιρούνται για κάθε λανθασμένη απάντηση;
10. Να βρείτε τις ηλικίες δύο αδελφών, αν σήμερα διαφέρουν κατά 5 χρόνια, ενώ μετά από 11 χρόνια οι ηλικίες τους θα έχουν λόγο 4/3.
11. Σ ´ ένα ταξίδι με πλοίο, το εισιτήριο της Α΄ θέσης κοστίζει 18 € και της Β΄ θέσης κοστίζει 6 € λιγότερα. Αν σ ´ ένα ταξίδι κόπηκαν 350 εισιτήρια συνολικής αξίας 4500 €, να βρείτε πόσα εισιτήρια κόπηκαν από κάθε κατηγορία.
12. Να βρείτε ένα διψήφιο αριθμό, που το άθροισμα των ψηφίων του είναι ίσο με 10 και αν εναλλάξουμε τα ψηφία του, τότε θα προκύψει αριθμός κατά 18 μικρότερος.
ΑΛ. 13. Από κεφάλαιο 4000 € ένα μέρος του κατατέθηκε σε μια τράπεζα προς 5% και το υπόλοιπο σε μια άλλη τράπεζα προς 3%. Ύστερα από 1 χρόνο εισπράχθηκαν συνολικά 175€ τόκοι. Ποιο ποσό τοκίστηκε προς 5% και ποιο προς 3%;
14 Πόσο καθαρό οινόπνευμα πρέπει να προσθέσει ένας φαρμακοποιός σε 200ml διάλυμα οινοπνεύματος περιεκτικότητας 15%, για να πάρει διάλυμα οινοπνεύματος περιεκτικότητας 32%;
BΛ. 15. Ένα ξενοδοχείο έχει 26 δωμάτια, άλλα δίκλινα και άλλα τρίκλινα και συνολικά 68 κρεβάτια. Πόσα είναι τα δίκλινα και πόσα τα τρίκλινα δωμάτια;
16. Σε έναν αγώνα το παιδικό εισιτήριο κοστίζει 1,5 € και το εισιτήριο ενός ενήλικα 4€. Τον αγώνα παρακολούθησαν 2200 άτομα και εισπράχτηκαν 5050 €. Να βρείτε πόσα ήταν τα παιδιά και πόσοι οι
ενήλικες που παρακολούθησαν τον αγώνα.
17. Ένας χημικός έχει δύο διαλύματα υδροχλωρικού οξέως, το πρώτο έχει περιεκτικότητα 50% σε υδροχλωρικό οξύ και το δεύτερο έχει περιεκτικότητα 80% σε υδροχλωρικό οξύ. Ποια ποσότητα από κάθε διάλυμα πρέπει να αναμείξει ώστε να πάρει 100 ml διάλυμα περιεκτικότητας 68% σε υδροχλωρικό οξύ;

3 replies

parmenides51

Oct 10, 2022

parmenides51

Feb 23, 2025

i Geo Collections + Forums + Greek Aops Index

parmenides51 57

N Oct 12, 2024 by parmenides51

57 replies

parmenides51

Mar 7, 2020

parmenides51

Oct 12, 2024

i Forum's Purpose - Index of posted Contest Collections

parmenides51 29

N Mar 30, 2023 by huashiliao2020

I created this forum in order to post High School Math Olympiads before 2001 (not present in aops at the moment) in order to add more Contests to the existing Aops Contest Collections (without spamming the existing forums by posting too many posts in so little time). I also post recent problems not available at the moment at contest collections, in oder to create those post collections.
Anyone is welcome to post problems from a national competition (final round or TST) of his / her country not already posted in Aops Contest Collections, to get those links (just not post random problem but all from each year's competition). There is no limit to how many problems anyone may post in this forum, as it serves as a new problem collection source for more (older usually) posts collection (e.g. anyone may post in a day 40 different problems from 10 years of his \ her 4 problem's olympiad in 40 different posts - one problem at each post).

Forum Contests Collected:
- Mathcenter Contest (Thai)
- Mathlinks Contests (old name of aops)
- OIFMAT (FMAT - Chilean)
- Oliforum Contest (Italian)
- VMEO (VMF - Vietnamese)

Below are the contest collections I have created using this forum :
- Arab Math Olympiad
- Argentina NMO Finals: 2008
- Argentina NMO Round 4 / Regional Geo, L1, L2, L3, years 1995-2022
- Argentina Provincional Geo: L1, L2, L3 , 1995-2022
- Argentina TST / geometry : Ibero , IMO, Cono Sur
- Austrian - Polish Competition : 1979-2000
- Austria Beginners' Competition: 2001-08
- Belarus National MO: 1995, 1997 missing problems , 1998 , 2001, 2002, 2003,2006 , 2014
- Belarus TST : 1998 , 2000 , 2009 , 2010, 2011, 2012, 2014, 2015, 2016
- Bosnia and Herzegovina JBMO TST: 2003-07
- Brazil TSTs: EGMO, Geometry : Cono Sur, Cono Sur Training List , ΙΜΟ + Ibero ,
- Brazil IMO TST: 2007-16
- Bulgaria National MO: 1993-96
- Chile Contests / Chile NMO 1989 -2015
- Czech and Slovak Olympiad III A: so far 1986, 1991- 2001
- Czech-Polish-Slovak Junior Match: all
- ~~Croatia National MO~~: 1996-2004
- Croatia TST: so far 2001, 2002, 2003
- Estonia Open Junior: ~~2001-2005~~, 2006, 2007, ~~2008-2019~~
- Estonia Open Senior: ~~2001-2005~~, 2006, 2007, ~~2008-2019~~
- Estonia National MO: so far 1996-2007, (soon 2008-2021)
- Estonia TST: ~~1992-95, 1998-2000~~, 2001-19
- ~~French National MO~~: 1986, 1988-2003
- Final Mathematical Cup
- Germany / Bundeswettbewerb Mathematik : 1974-1982 , 1990
- Germany / National MO: so far 1965, 1996- 2001, 2009
- Germany / QUEDMO (11 /15 editions so far)
- Greece Junior: 1996 - 2020
- Hungary / Kürschák: 1947-84
- Hungary / Dürer / Finals: 2019-21
- India / LIMIT : 2020
- India / Postal Coaching: 2007-09 , 2012
- Israel / National (Gillis) : 1995-98
- Israel / Grosman: so far 1995, 1997, 1999, 2001
- Italy TST: so far 1993-94, 1998
- Kyrgystan TST: 2010, 2018
- Lithuania / Grand Duchy of Lithuania 2009-20
- Macedonia North National MO: so far 1994, 1995, 1996, 1997, 1998, 1999, 2002, 2003, 2004, 2005
- May Olympiad all
- Moldova Contests /JBMO TST: 1999- 2020
- ~~Moldova National MO~~
- ~~Mongolia National MO~~: 1999-2002, 2007
- Netherlands / Dutch ΙΜΟ TST: 2021
- New Zealand / NZMOC Camp Selection Problems, New Zealand MO
- Norway / Abels Math Contest (Final Round): 1993- 2006
- Polish MO Finals: 1966-84
- Rioplatense L3 1990-99
- Romania MOP: ROMOP 2011 Juniors
- Romania JBMO TST : 2002-19
- Romania TST: so far 1989, 1990 , 1991 (missing), 1992, 1993, 1995
- Saudi Arabia / BMO TST: 2010-11, 2013, 2015-19
/ GMO TST: 2013-16, 2018
/ IMO TST: 2011.2013, 2015-19
/ Pre-TST + Training Tests: 2010-11, 2021
- Saudi Arabia JBMO TST: 2017, 2019
- ~~Serbia National MO~~: 1974 - 2006 (also known as Yugoslavia before 1990)
- Serbia TST:
- Singapore Junior : 2006-2019 collected
- Singapore Open: 1995 - 2004, 2014, 2016-17
- Singapore Senior : 1996 - 2008, 2011-13, 2015-19,
- Singapore TST: 1995 -2003
- ~~Slovenia National MO~~: 1997-2001, 2005
- Slovenia TST: so far 1997, 1998, 2005
- Soros Olympiad
- Spanish MO: 1966-83
- Sweden National MO: 1961 - 2020
- Switzerland TST : so far 1997 - 2004, 2015 .
- Thailand MO: 2004- 15
- Ukraine National MO: 1997, 1998, 1999, 2005
- Ukraine TST: 1999, 2009-18
- USA / iΤest: 2005

Basic Source: Imomath

PS. My posting ability in public forums is as everyone’s else limited. Therefore when I post in public forums, I post only the geometry problems. When my source is written not in English, only the geometry ones are posted as I spend time translating them by google translate. When my source is written in English, I post the whole problem set in aops, posting the non-geometry problems in this forum and adding all the problems in contest collections.
In this forum’s index, anyone may find which contests’ years have been posted inside this forum.
In the future, each year's post collections shall be available in the Contests Collections, till then anyone may find them above.

Related forums:
Old Russian Math Olympiads
China High School Contests
KöMaL (Hungarian Magazine)

For the friends of geometry:
Olympiad Geometry Collections + Forums
A Beautiful Journey Through Olympiad Geometry - solutions forum
Evan Chen's EGMO study group
Lemmas in Olympiad Geometry - active forum
Tran Quang Hung's geometry group

uc = Under Construction

PS. It is not permitted to create contest collections of each year inside aops from National Contests of UK and Australia.

PS.2 Test

29 replies

parmenides51

Feb 11, 2020

huashiliao2020

Mar 30, 2023

n^2 + 24n + 35 is a perfect square

parmenides51 5

N Mar 31, 2025 by lightsynth123

Source: Singapore Junior Math Olympiad 2016 2nd Round p1 SMO

Find all integers $n$ such that $n^2 + 24n + 35$ is a square.

5 replies

parmenides51

Mar 26, 2020

lightsynth123

Mar 31, 2025

integer p(x), p (1) = p (2) = 0, p (n) is a prime ,

parmenides51 2

N Mar 28, 2025 by AshAuktober

Source: 2010 Swedish Mathematical Competition p3

Find all natural numbers $n \ge 1$ such that there is a polynomial $p(x)$ with integer coefficients for which $p (1) = p (2) = 0$ and where $p (n)$ is a prime number .

2 replies

parmenides51

Apr 30, 2021

AshAuktober

Mar 28, 2025

no of words of the maximum length in a language with n letters

parmenides51 2

N Mar 28, 2025 by NicoN9

Source: Czech And Slovak Mathematical Olympiad, Round III, Category A 2001 p4 / Croatia TST 2002 p1

In a certain language there are $n$ letters. A sequence of letters is a word, if there are no two equal letters between two other equal letters. Find the number of words of the maximum length.

2 replies

parmenides51

Feb 11, 2020

NicoN9

Mar 28, 2025

f(xy) f( f(y)/x) = 1 , monotonic

parmenides51 8

N Mar 28, 2025 by navier3072

Source: 1990 Swedish Mathematical Competition p5

Find all monotonic positive functions $f(x)$ defined on the positive reals such that $f(xy) f\left( \frac{f(y)}{x}\right) = 1$ for all $x, y$ .

8 replies

parmenides51

Apr 2, 2021

navier3072

Mar 28, 2025

5^p + 4 p^4 is a perfect square

parmenides51 7

N Mar 27, 2025 by Chanome

Source: Singapore Math Olympiad 2008 2nd Round SMO, Senior p2, Junior p5

Determine all primes $p$ such that $5^p + 4 p^4$ is a perfect square, i.e., the square of an integer.

7 replies

parmenides51

Mar 25, 2020

Chanome

Mar 27, 2025

largest multiple of 55 using once digits 0,1,2,3,4,5,6

parmenides51 1

N Mar 27, 2025 by lightsynth123

Source: Singapore Junior Math Olympiad 2014 2nd Round p1 SMO

Consider the integers formed using the digits $0,1,2,3,4,5,6$ , without repetition. Find the largest multiple of $55$ . Justify your answer.

1 reply

parmenides51

Mar 26, 2020

lightsynth123

Mar 27, 2025

f(f(x)) +f(x) = 12x , for all x>=0

parmenides51 2

N Mar 27, 2025 by jasperE3

Source: 2002 Singapore TST 2.3

Find all functions $f : [0,\infty) \to [0,\infty)$ such that $f(f(x)) +f(x) = 12x$ , for all $x \ge 0$ .

2 replies

parmenides51

Dec 29, 2020

jasperE3

Mar 27, 2025

a\sqrt{b}=a+c, b\sqrt{c}=b+a, c\sqrt{a}=c+b , system

parmenides51 2

N Mar 26, 2025 by lightsynth123

Source: Singapore Junior Math Olympiad 2014 2nd Round p4 SMO

Find, with justification, all positive real numbers $a,b,c$ satisfying the system of equations:
$\begin{cases} a\sqrt{b}=a+c \\ b\sqrt{c}=b+a \\ c\sqrt{a}=c+b \end{cases}$

2 replies

parmenides51

Mar 26, 2020

lightsynth123

Mar 26, 2025

square is cut into rectangles with // sides, sum of ratio of sides >=1

parmenides51 1

N Mar 26, 2025 by lightsynth123

Source: Singapore Junior Math Olympiad 2017 2nd Round p1 SMO

A square is cut into several rectangles, none of which is a square, so that the sides of each rectangle are parallel to the sides of the square. For each rectangle with sides $a, b,a<b$ , compute the ratio $a/b$ . Prove that sum of these ratios is at least $1$ .