![$f : [0,1] \longrightarrow \mathbb{R}$](http://latex.artofproblemsolving.com/9/a/4/9a4b7997c35990f839742d29866122791ade8b7c.png)
is differentiable with 
.
for all ![$x \in [0,1]$](http://latex.artofproblemsolving.com/d/3/4/d3423a3200d0ad7660fc4265d59d5598a53d551d.png)
,
for all 
.
increasing with
?
?
2. 
3. 
Bracket 2
2. 
3. 
Bracket 3![$$\int_0^{1}\frac{dx}{1-\sqrt[3]{1-\sqrt{x}}}$$](http://latex.artofproblemsolving.com/d/5/e/d5e8db00985944b2eca341e37a94e34661cb5c86.png)
2. 
3. 
Bracket 4
2. 
3. 
Bracket 5
2. 
3. 
Bracket 6
2. 
3. 
Bracket 7
2. 
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Bracket 8
2. 
3. 
![$$\int_0^{2\pi}\sqrt[4]{\cos^4(x)-\cos(2x)}dx$$](http://latex.artofproblemsolving.com/7/e/e/7eebb4a488fec702e0130a253ef87f0f6dfedb2d.png)
2. ![$$\int_0^1 \frac{x^5}{\sqrt[3]{1-x^3}}dx$$](http://latex.artofproblemsolving.com/6/0/5/605bb9d81d0ccff54a276c9f7bf1c1726c136398.png)
3. 
Bracket 2
2. 
3. 
Bracket 3
2. ![$$\int_0^1 \frac{dx}{(\sqrt{x}+\sqrt[3]{x})^2}$$](http://latex.artofproblemsolving.com/6/9/8/69869c0ed754c0dfacf412ac6411742ffa164a6d.png)
3. 
Bracket 4
2. 
3. 
2. 
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Bracket 2
2. 
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2. 
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2. 
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5. 
Y by
Aman and Berdi, two biologists, they invented a new type of bacteria such that they can control the division of bacteria into several parts. They are also participants of 
with the aim to train for the first problem of . They play the following game.
Initially, they take
bacteria and choose a natural number 
. On each move, the player chooses any 
number from to . Then the player divides each bacterium into pants. Once chosen, the number cannot be chosen twice. If after any player's move the number of bacteria population is divisible by then that player loses. Determine who has the winning strategy depending on the given number if it's known that Amman starts first.
with the aim to train for the first problem of . They play the following game.Initially, they take
bacteria and choose a natural number 
. On each move, the player chooses any 
number from to . Then the player divides each bacterium into pants. Once chosen, the number cannot be chosen twice. If after any player's move the number of bacteria population is divisible by then that player loses. Determine who has the winning strategy depending on the given number if it's known that Amman starts first.
is the number of bacteria population, after a move, the number will be 
.
,
.
,
.
,
is a prime. If someone choose 
and lose. If B doesn't choose 
numbers less than 
,
,
.
(odd number), so B necessarily has to choose a multiple of 
,
and 
