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Largest Divisor
4everwise 19
N
by reni_wee
What is that largest positive integer 
for which 
is divisible by 
?
for which 
is divisible by 
?
19 replies
![\[
\sum_{0 \leq k \leq l} (l - k) \binom{m}{k} \binom{q + k}{n}
= \binom{l + q + 1}{m + n + 1},
\]](http://latex.artofproblemsolving.com/b/5/8/b58c5b1f0f9c95266d189593f02bae85c100c82a.png)
![\[
\text{integers } l, m \geq 0,\quad \text{integers } n \geq q \geq 0.
\]](http://latex.artofproblemsolving.com/8/e/2/8e25f8d7e35ac7813e40af7cc4806000e70999af.png)
[ABCD] = n [CDE], areas in trapezoid - IOQM 2020-21 p1
parmenides51 3
N
by iamahana008
Let 
be a trapezium in which 
and 
. Let 
be then midpoint of the diagonal 
. If ![$[ABCD] = n \times [CDE]$](//latex.artofproblemsolving.com/e/9/3/e9347ba2c8e9396adfe716159e711408a3b9dc69.png)
, what is the value of ?
(Here![$[t]$](//latex.artofproblemsolving.com/8/a/8/8a86702dc60b9db916798b1115a6b6822bf828b9.png)
denotes the area of the geometrical figure
.)
be a trapezium in which 
and 
. Let 
be then midpoint of the diagonal 
. If ![$[ABCD] = n \times [CDE]$](http://latex.artofproblemsolving.com/e/9/3/e9347ba2c8e9396adfe716159e711408a3b9dc69.png)
, what is the value of ?(Here
![$[t]$](http://latex.artofproblemsolving.com/8/a/8/8a86702dc60b9db916798b1115a6b6822bf828b9.png)
denotes the area of the geometrical figure
.)
3 replies
Looking for users and developers
derekli 5
N
by musicalpenguin
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Link: https://stellarlearning.app/competitive
5 replies
Sequences and GCD problem
BBNoDollar 0
Determine the general term of the sequence of non-zero natural numbers (a_n)n≥1, with the property that gcd(a_m, a_n, a_p) = gcd(m^2 ,n^2 ,p^2), for any distinct non-zero natural numbers m, n, p.
Note that gcd(a,b,c) denotes the greatest common divisor of the natural numbers a,b,c .
Note that gcd(a,b,c) denotes the greatest common divisor of the natural numbers a,b,c .
0 replies
Sum of digits is 18
Ecrin_eren 13
N
by NamelyOrange
How many 5 digit numbers are there such that sum of its digits is 18
13 replies
and 
Prove that![$$ a+b+kc^3\geq\sqrt[4]{\frac{4k} {27}}$$](http://latex.artofproblemsolving.com/5/6/1/561a990e5885863c77d454ffa3f1973e1bbf60b4.png)
![$$ a+b+kc^4\geq\frac{5} {8}\sqrt[5]{\frac{k} {2}}$$](http://latex.artofproblemsolving.com/4/b/1/4b14911672688250c909571d788aaf10724462eb.png)
Where 
)
Prove that
When to look at solutions - pre calc
omerrob13 1
N
by abartoha
Hey all.
I am doing the precalc book, and unfortunately, im getting into the habit of looking in the solutions quite fast on a problem I did not able to make any progress on.
My goal is mainly to develop problem solving and reasonning skills.
I divide the problems in AOPS to 2:
- Challenge problems at the end of the of each chapter.
- The problems that teach you the material itself, and the problems at the end of each section (1.1,1.2, etc...)
For non challenging problems, It takes around 20 mins of me not be able to solve a problem, and look at the solutions for it
Is it too little?
My goal is mainly to develop problem solving and reasoning skills.
I'm not sure if it's too little time to bring to a regular problem, or its ok to give 20 mins to a problem and continue if making no progress.
I am doing the precalc book, and unfortunately, im getting into the habit of looking in the solutions quite fast on a problem I did not able to make any progress on.
My goal is mainly to develop problem solving and reasonning skills.
I divide the problems in AOPS to 2:
- Challenge problems at the end of the of each chapter.
- The problems that teach you the material itself, and the problems at the end of each section (1.1,1.2, etc...)
For non challenging problems, It takes around 20 mins of me not be able to solve a problem, and look at the solutions for it
Is it too little?
My goal is mainly to develop problem solving and reasoning skills.
I'm not sure if it's too little time to bring to a regular problem, or its ok to give 20 mins to a problem and continue if making no progress.
1 reply
