maximum minimum of function

For $A,B,C$ non-negative angles such that $A+B+C = \pi / 2$, find the maximum and minimum of: $$ f = \sin A + \sin B + \sin C + \sin^2 A + \sin^2 B + \sin^2 C$$ and $$g = \cos A (\sin A -1) + \cos B (\sin B - 1) + \cos C (\sin C - 1)$$

Teacher distributing coupons

Let $n$ be a positive integer, and $N = n(n+1)(2n+1)$. In a class with $N$ students, each student has an ID number from $1$ to $N$. The teacher then gives each student red, green, and blue coupons as follows:

A student with id $k$ will receive $x$ red coupons, $y$ green groupons, $z$ blue coupons where
$1 \leq x \leq n, n | k-x$
$1 \leq y \leq n+1, (n+1) | k-y$
$1 \leq z \leq 2n+1, (2n+1) | k-z$.

For each student then the teacher gives him a penny based on the median number of coupons he receives. For example, if a student receives 2 red, 4 green, and 5 blue coupons, he gets 4 cents. If another student receives 2 red, 5 green, and 4 blue, he also gets 4 cents. If yet another student get 3 red, 5 green, 5 blue, he gets 5 cents.

Find the total number of money that the teacher has to pay to the entire class.

Solution

Note that $n, n+1, 2n+1$ are all pairwise prime, and $N$ is their LCM. By Chinese Remainder Theorem, we find that for each triplet $(x,y,z)$ such that $1 \leq x \leq n, 1 \leq y \leq n+1, 1 \leq z \leq 2n+1$, there is exactly one $k$ such that $1 \leq k \leq N, k \equiv x \mod N, k \equiv y \mod N, k \equiv z \mod N$. In other words, for each triplet of $(x,y,z)$ within the allowable coupon range, there is exactly one student who receives $x$ red, $y$ green, and $z$ blue coupons.

Solution 1

We divide the cases based on the amount of money a particular student receives. Suppose a student receives $k$ cents.

Case 1: $1 \leq k \leq n$.

Let $x,y,z$ be the number of red, green, and blue coupons he receives.

Case 1a: $x,y,z$ are all distinct.

If $x < y < z$, then $k = y$. For a student to be paid $k$ cents in this manner, $x$ must be from 1 to $k-1$, and $z$ must be from $k$ to $2n+1$, inclusive. Conversely, as long as $x$ is any value from $1$ to $k-1$, and $z$ is any value from $k+1$ to $2n+1$, then $(x,y,z)$ form a triple where the median is $k$. For each of these triples, there is exactly one student who gets paid $k$ cents. So the number of students who gets paid $k$ cents in this manner is $(k-1)(2n+1-k)$, so the total money spent on them is $k(k-1)(2n+1-k)$

Since $k$ is allowed to take range from $1$ to $n$, then the number of students who gets paid via $x < y < z$ configuration is
$$\sum_{k=1}^{n} k(k-1)(2n+1-k)$$

Now, that's just for $x < y < z$. There are 6 permutations of order, and we can repeat the same analysis for each. The total number or $x,y,z$ all distinct is:

$$2 \sum_{k=1}^{n} k(k-1)(n-k) + 2 \sum_{k=1}^{n} k(k-1)(n+1-k) + 2 \sum_{k=1}^{n} k(k-1)(2n+1-k)$$
$$ = 2 \sum_{k=1}^{n} k(k-1)(4n+2-3k) $$

Case 1b: two of them are the same, the third higher.

Suppose $x=y < z$, then $k = x = y$. Similar analysis as above, the number of students who gets paid this way is $(2n+1-k)$, and the total money spent on them is $k(2n+1-k)$. Summing from $k=1$ to $n$, and also taking into account cases of $x=z < y, y=z < x$, we have:

$$\sum_{k=1}^{n} k (4n+2-3k)$$

Case 1c: two of them are the same, the third lower.

Suppose $x < y= z$, then $k = y = z$. Again, the number of students is $k-1$, so total amount of money, including cases of $y < x= z, z < x=y$ is:
$$ 3 \sum_{k=1}^{n} k(k-1)$$

Case 1d: $x=y=z$

$$\sum_{k=1}^{n} k$$

Case 2: $k = n+1$.

Since $x \leq n$, then $x$ cannot be the median nor the top, and $y$ cannot be on top, so the only configuration left is $x < y \leq z$. There are $n$ values for $x$ and $n+1$ values for $z$ (from $n+1$ to $2n+1$). So the number of students is $n(n+1)$ and total amount paid is $n(n+1)^2$

These are all the cases, since the median cannot be greater than $n+1$. Summing over all the sums, we have:

(do the algebra here)

Solution 2:

We divide cases based on the range of $x,y,z$.

Case 1: $1 \leq x,y,z \leq n$

Recall that each triplet $(x,y,z)$ corresponds to exactly one student. We form student pairs as follows: student who receives $(x,y,z)$ is paired with the student who receives $(n+1-x, n+1-y, n+1-z)$. Note that if $n$ is even, then each student gets a pair, but if $n$ is odd, there is a student $((n+1)/2, (n+1)/2, (n+1)/2$ who is paired with himself.

Suppose student $A$ and student $B$ is a pair. Note that if $x < y < z$, then $n+1-x > n+1-y > n+1-z$, so if $A$ gets paid $y$, then $B$ gets paid $n+1-y$. The sum of payment from this pair is $n+1$. It's easy to check that this rule holds for all order configurations ($x = y < z, x < y = z$, etc). In other words, the average income of this pair is $(n+1)/2$. This holds for all pairs, even in the case where a student is paired to himself. The total amount of income for all students in this group is then $n^3 (n+1)/2$.

Case 2: $1 \leq x,y \leq n, z = n+1$.

In this case, since $z$ is the largest of all three, then the median is $\max(x,y)$, summed over all $1 \leq x,y \leq n$.

Case 2a: $x > y$.

Then $\max (x,y) = x$. For each $x$, there are $x-1$ possible values of $y$, meaning there are $x-1$ students, each of which are getting paid $x$ cents. So the total income here is:

$$\sum_{k=1}^{n} k(k-1)$$

Case 2b: $x < y$

By symmetry, it's identical to 2a.

Case 2c: $x = y$.

The total income here is $\sum_{k=1}^{n} k$

Case 3: $1 \leq x,z \leq n, y = n+1$.

By symmetry, it's identical to case 2

Case 4: $1 \leq x \leq n, y = z = n+1$.

The median here is $n+1$, and there are $n$ students, so total income here is $n(n+1)$.

Case 5: $1 \leq x \leq n, y = n+1, z > n+1$

The median here is $n+1$, and there are $n^2$ students, so total income is $n^2(n+1)$.

Case 6: $1 \leq x,y \leq n, z > n+1$

This case is similar to case 2. In fact, EACH value of $z$ is identical to case 2 in that we're summing $\max (x,y)$ over all $1 \leq x,y \leq n$. Case 6 is thus like $n$ instances of case 2.

Largest k for arbitrary function

Find the largest number $k$ that satisfies the following property:

For every real-valued function $f$ defined over $[0,1]$, there exist $a,b,c \in [0,1]$ such that:
$$|f(ab) + f(bc) + f(ca) - abc | \geq k$$

Solution

First we show that $k = 1/6$ satisfies the condition of the problem.

For $k=1/6$, suppose to the contrary that for all $a,b,c$ chosen in the interval $[0,1]$ we have:
$$|f(ab) + f(bc) + f(ca) - abc | < \frac{1}{6}$$

Let $y = f(0),y=f(1)$.
Plugging in $a=b=c=0$ we have $|3x| < 1/6 \iff |x| < 1/18$
Plugging in $a=b=c=1$ we have $|3y-1| < 1/6$
Plugging in $a=b=1, c=0$ we have $2x+y| < 1/6$

That means $y = (2x+y) - 2x < 1/6 + 2/18 = 5/18$
But also $3y-1 > -1/6 \iff y > 5/18$, a contradiction.

Thus $k= \frac{1}{6}$ satisfies the condition of the problem.

Now we show that it is the largest such constant. We choose a function $f(x) = \frac{6x^\frac{3}{2} - 1}{18}$ and prove that
$$ -\frac{1}{6} \leq f(ab)+f(bc)+f(ca) - abc \leq \frac{1}{6}$$
for all $a,b,c \in [0,1]$

Indeed, the first inequality is equivalent to
$$(ab)^\frac{3}{2} + (bc)^\frac{3}{2} + (ca)^\frac{3}{2} \geq 3abc$$ which is true by AM-GM.

The second inequality is equivalent to
$$(ab)^\frac{3}{2} + (bc)^\frac{3}{2} + (ca)^\frac{3}{2} - 3abc \leq 1$$

Now consider the LHS, and try to find its maximum value over $a,b,c \in [0,1]$
First, we try a substitution $x = \sqrt{ab}, y = \sqrt{ac}, z = \sqrt{bc}, x,y,z \in [0,1]$ and WLOG, we may assume that $x \geq y \geq z$ (which corresponds to $a \geq b \geq c$)
So now our objective function becomes $x^3 + y^3 + z^3 - 3xyz$.

Let $g(x) = x^3 + y^3 + z^3 - 3xyz$, and fix $y,z$.
$$g(1) - g(x) = (1-x^3) - 3yx(1-x) = (1-x)(1 + x + x^2 - 3yz)$$
Because $x \leq 1$, then $1-x \geq 0$
$1+x+x^2-3yz \geq 1 + 2x^2-3yz = (1-yz) + 2(x^2 - yz) \geq 0$
So $g(1) \geq g(x)$.

So the maximum happens when the largest of 3 variables equals to 1. But this means that $ab = 1$ which implies $a=b=1$. Then $y = z = \sqrt{c}$. Our objective function now becomes:
$$x^3 + y^3 + z^3 - 3xyz = 2y^3 - 3y^2 +1$$
Let $h(y) = 2y^3 - 3y^2 + 1$, then $h(y) - h(0) = y^2(2y-3) < 0$ because $y \geq 1$. So the maximum happens when $y = 0$

In conclusion, our maximum happens when $x = 1, y=z=0$ and that's when $a=b=1, c= 0$. At this point, the maximum value is 1, thus proving our assertion.

Another way to determine the maximum value of the function above is by observing that it is a convex function on each of its variable, and all three variables are freely chosen from the interval $[0,1]$. Thus, the maximum must happen when all of its variables are zero or one. Plugging in all permutations of $(0,0,0), (0,0,1), (0,1,1), (1,1,1)$, we find that the maximum is indeed 1.

3 arbitrary functions

Suppose $f,g,h$ are functions that are defined in the closed interval $0 \leq x \leq 1$. Show that we can always find $a,b,c \in [0,1]$ such that:

$$|f(a)+g(b)+h(c) - (1-a)(1-b)(1-c)| \geq \frac{1}{3}$$

Also show that the constant $\frac{1}{3}$ cannot be replaced by a larger constant.