# 2023-23 Don’t be negative!

Consider a function $$f: \{1,2,\dots, n\}\rightarrow \mathbb{R}$$ satisfying the following for all $$1\leq a,b,c \leq n-2$$ with $$a+b+c\leq n$$.

$f(a+b)+f(a+c)+f(b+c) – f(a)-f(b)-f(c)-f(a+b+c) \geq 0 \text{ and } f(1)=f(n)=0.$

Prove or disprove this: all such functions $$f$$ always have only nonnegative values on its domain.

Acknowledgement: This problem arises during a research discussion between June Huh, Jaehoon Kim and Matt Larson.

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# 2023-22 Simultaneously diagonalizable matrices

Does there exist a nontrivial subgroup $$G$$ of $$GL(10, \mathbb{C})$$ such that each element in $$G$$ is diagonalizable but the set of all the elements of $$G$$ is not simultaneously diagonalizable?

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# 2023-21 A limit

Find the following limit:

$\lim_{n \to \infty} \left( \frac{\sum_{k=1}^{n+2} k^k}{\sum_{k=1}^{n+1} k^k} – \frac{\sum_{k=1}^{n+1} k^k}{\sum_{k=1}^{n} k^k} \right)$

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# 2023-20 A sequence with small tail

Can we find a sequence $$a_i, i=0,1,2,…$$ with the following property: for each given integer $$n\geq 0$$, we have $\lim_{L\to +\infty}\sum_{i=0}^L 2^{ni} |a_i|\leq 23^{(n+11)^{10}} \quad \text{ and }\quad \lim_{L\to +\infty}\sum_{i=0}^L 2^{ni} a_i = (-1)^n ?$

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# 2023-19 Counting the number of solutions

Let $$N$$ be the number of ordered tuples of positive integers $$(a_1,a_2,\ldots, a_{27} )$$ such that $$\frac{1}{a_1} + \frac{1}{a_2} + \cdots +\frac{1}{a_{27}} = 1$$. Compute the remainder of $$N$$ when $$N$$ is divided by $$3$$.

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# 2023-18 Degrees of a graph

Find all integers $$n \geq 8$$ such that there exists a simple graph with $$n$$ vertices whose degrees are as follows:

(i) $$(n-4)$$ vertices of the graph are with degrees $$4, 5, 6, \dots, n-2, n-1$$, respectively.

(ii) The other $$4$$ vertices are with degrees $$n-2, n-2, n-1, n-1$$, respectively.

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# 2023-17 Comparing area of triangles

Let $$f(x) = x^4 + (2-a)x^3 – (2a+1)x^2 + (a-2)x + 2a$$ for some $$a \geq 2$$. Draw two tangent lines of its graph at the point $$(-1,0)$$ and $$(1,0)$$ and let $$P$$ be the intersection point. Denote by $$T$$ the area of the triangle whose vertices are $$(-1,0), (1,0)$$ and $$P$$. Let $$A$$ be the area of domain enclosed by the interval $$[-1,1]$$ and the graph of the function on this interval. Show that $$T \leq 3A/2.$$

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# 2023-16 Zeros in a sequence

Define the sequence $$x_n$$ by $$x_1 = 0$$ and
$x_n = x_{\lfloor n/2 \rfloor} + (-1)^{n(n+1)/2}$
for $$n \geq 2$$. Find the number of $$n \leq 2023$$ such that $$x_n = 0$$.

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Let $$p(z), q(z)$$and $$r(z)$$ be polynomials with complex coefficients in the complex plane. Suppose that $$|p(z)| + |q(z)| \leq |r(z)|$$ for every $$z$$. Show that there exist two complex numbers $$a,b$$ such that $$|a|^2 +|b|^2 =1$$ and $$a p(z) + bq(z) =0$$ for every $$z$$.
Let $$f(t)=(t^{pq}-1)(t-1)$$ and $$g(t)=(t^{p}-1)(t^q-1)$$ where $$p$$ and $$q$$ are relatively prime positive integers. Prove that $$\frac{f(t)}{g(t)}$$ can be written as a polynomial where it has just $$1$$ or $$-1$$ as coefficients. (For example, when $$p=2$$ and $$q=3$$, we have that $$\frac{f(t)}{g(t)} = t^2-t+1$$.)