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Stage 10 · Mixed

10 problems · Hardest shortlist tier · mathsolympiadprep.com

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1. Given an integer $n > 1$ and an integer $a$ that is coprime with $n$. There is a country consisting of $n$ islands $D_1, D_2, \dots, D_n$. For any two different islands $D_i$ and $D_j$, there is a one-way ferry from $D_i$ to $D_j$ if and only if $ij \equiv ia \pmod{n}$. A tourist hopes to visit as many islands as possible. He can first fly to any island he chooses to start the tour, and afterwards can only use the one-way ferry to tour freely between islands in this country. Find the maximum possible number of different islands that the tourist can visit.

   Number theory Solution and answer checking →

2. Let $\mathcal{V}$ be a finite set of points in the plane. We say that $\mathcal{V}$ is balanced if for any two distinct points $A, B \in \mathcal{V}$, there exists a point $C \in \mathcal{V}$ such that $AC = BC$. We say that $\mathcal{V}$ is center-free if for any distinct points $A, B, C \in \mathcal{V}$, there does not exist a point $P \in \mathcal{V}$ such that $PA = PB = PC$.

   a. Show that for all $n \geqslant 3$, there exists a balanced set consisting of $n$ points.

   b. For which $n \geqslant 3$ does there exist a balanced, center-free set consisting of $n$ points?

   Geometry Solution and answer checking →

3. Find all positive integers $n \ge 2$ for which there exist $n$ real numbers
   $$a_1 < a_2 < \dots < a_n$$
   and a real number $r > 0$ such that the $\frac{n(n-1)}{2}$ differences $a_j - a_i$ for $1 \le i < j \le n$ are equal, in some order, to the numbers
   $$r^1, r^2, \dots, r^{\frac{n(n-1)}{2}}.$$

   Algebra Solution and answer checking →

4. Let $A_{0} = (a_{1}, \ldots, a_{n})$ be a finite sequence of real numbers. For each $k \geq 0$, from the sequence $A_{k} = (x_{1}, \ldots, x_{n})$ we construct a new sequence $A_{k+1}$ in the following way.
   1. We choose a partition $\{1, \ldots, n\} = I \cup J$, where $I$ and $J$ are two disjoint sets, such that the expression
   $$\left|\sum_{i \in I} x_{i} - \sum_{j \in J} x_{j}\right|$$
   attains the smallest possible value. (We allow the sets $I$ or $J$ to be empty; in this case the corresponding sum is $0$.) If there are several such partitions, one is chosen arbitrarily.
   2. We set $A_{k+1} = (y_{1}, \ldots, y_{n})$, where $y_{i} = x_{i} + 1$ if $i \in I$, and $y_{i} = x_{i} - 1$ if $i \in J$.
   Prove that for some $k$, the sequence $A_{k}$ contains an element $x$ such that $|x| \geq n / 2$.

   Combinatorics Solution and answer checking →

5. Let $n$ be a positive integer. We say that a polynomial $P$ with integer coefficients is $n$-good if there exists a polynomial $Q$ of degree 2 with integer coefficients such that $Q(k)(P(k)+Q(k))$ is never divisible by $n$ for any integer $k$.
   Determine all integers $n$ such that every polynomial with integer coefficients is an $n$-good polynomial.

   Number theory Solution and answer checking →

6. Let $A B C D E$ be a convex pentagon with $C D = D E$ and $\angle E D C \neq 2 \cdot \angle A D B$. Suppose that a point $P$ is located in the interior of the pentagon such that $A P = A E$ and $B P = B C$. Prove that $P$ lies on the diagonal $C E$ if and only if $\operatorname{area}(B C D) + \operatorname{area}(A D E) = \operatorname{area}(A B D) + \operatorname{area}(A B P)$.

   Geometry Solution and answer checking →

7. Let $\mathbb{Z}_{\geqslant 0}$ be the set of all nonnegative integers. Find all the functions $f: \mathbb{Z}_{\geqslant 0} \rightarrow \mathbb{Z}_{\geqslant 0}$ satisfying the relation
   $$f(f(f(n)))=f(n+1)+1$$
   for all $n \in \mathbb{Z}_{\geqslant 0}$.

   Algebra Solution and answer checking →

8. Suppose there are 101 persons sitting around a round table in an arbitrary order. The $k$th person possesses $k$ pieces of cards, $k = 1, \dots, 101$. We call it a transition if one transits one of his cards to one of his adjacent persons. Find the minimum positive number $k$, such that whatever the order of the seating, there is a way of no more than $k$ transitions so that each person possesses 51 cards.

   Combinatorics Solution and answer checking →

9. Given a positive integer $n$, let $D$ be the set of positive divisors of $n$, and let $f: D \to \mathbb{Z}$ be a function. Prove that the following are equivalent:
   (A) for any positive divisor $m$ of $n$,
   $$n \mid \sum_{d|m} f(d) \binom{n/d}{m/d};$$
   (B) for any positive divisor $k$ of $n$,
   $$k \mid \sum_{d|k} f(d).$$

   Number theory Solution and answer checking →

10. Suppose there are beetles on a chessboard consisting of $2012 \times 2012$ unit squares. Each unit square can accommodate at most one beetle. At a moment, all beetles fly and land on the chessboard again. For a beetle, we call the vector from its flying unit to its landing unit the beetle's "displacement vector". We call the sum of all beetle's "displacement vectors" the "total displacement vectors".
    Find the maximum length of "total displacement vector" considering the number of beetles and all possible positions of flying and landing. (posed by Qu Zhenhua)

    Geometry Solution and answer checking →

Answer key — Stage 10 · Mixed

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