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Stage 9 · Mixed

10 problems · IMO P2/P5; hard shortlist · mathsolympiadprep.com

The answer key prints on its own page at the end.

Let $n$ be a positive integer. A Nordic square is an $n \times n$ board containing all the integers from $1$ to $n^2$ so that each cell contains exactly one number. Two different cells are considered adjacent if they share a common side. Every cell that is adjacent only to cells containing larger numbers is called a valley. An uphill path is a sequence of one or more cells such that:
(i) the first cell in the sequence is a valley,
(ii) each subsequent cell in the sequence is adjacent to the previous cell, and
(iii) the numbers written in the cells in the sequence are in increasing order.
Find, as a function of $n$ , the smallest possible total number of uphill paths in a Nordic square.
Author: Nikola Petrovi?

Combinatorics Solution and answer checking →
Let $\mathcal{S}$ be a finite nonempty set of prime numbers. Let $1 = b_{1} < b_{2} < \cdots$ be the sequence of all positive integers whose prime divisors all belong to $\mathcal{S}$ . Prove that, for all but finitely many positive integers $n$ , there exist positive integers $a_{1}, a_{2}, \ldots, a_{n}$ such that
$\frac{a_{1}}{b_{1}} + \frac{a_{2}}{b_{2}} + \cdots + \frac{a_{n}}{b_{n}} = \left\lceil \frac{1}{b_{1}} + \frac{1}{b_{2}} + \cdots + \frac{1}{b_{n}} \right\rceil.$

Number theory Solution and answer checking →
Let $\overline{AH_1}$ , $\overline{BH_2}$ , and $\overline{CH_3}$ be the altitudes of an acute scalene triangle $ABC$ . The incircle of triangle $ABC$ is tangent to $\overline{BC}$ , $\overline{CA}$ , and $\overline{AB}$ at $T_1, T_2$ , and $T_3$ , respectively. For $k = 1, 2, 3$ , let $P_i$ be the point on line $H_iH_{i+1}$ (where $H_4 = H_1$ ) such that $H_iT_iP_i$ is an acute isosceles triangle with $H_iT_i = H_iP_i$ . Prove that the circumcircles of triangles $T_1P_1T_2$ , $T_2P_2T_3$ , $T_3P_3T_1$ pass through a common point.

Geometry Solution and answer checking →
Assume $a$ is a real number in $[\frac{1}{2}, \frac{2}{3}]$ . Consider two sequences $(u_n), (v_n), (n = 0, 1, \dots)$ , defined by:
$u_n = \frac{3}{2^{n+1}} \cdot (-1)^{\lfloor 2^{n+1}a \rfloor}, \quad v_n = \frac{3}{2^{n+1}} \cdot (-1)^{n+\lfloor 2^{n+1}a \rfloor}.$
a. Prove that
$\left(\sum_{i=0}^{2018} u_i\right)^2 + \left(\sum_{i=0}^{2018} v_i\right)^2 \le 72a^2 - 48a + 10 + \frac{2}{4^{2019}}.$
b. Find all value of $a$ for which equality occurs.

Algebra Solution and answer checking →
There are $2025$ people and $66$ given colors. Each person has $66$ balls, one of each color, with a total weight of $1$ for all $66$ balls.
Find the smallest real number $C$ such that, no matter how the balls are weighted, one can always select exactly one ball from each person so that for every color, the total weight of the selected balls of that color does not exceed $C$ .

Combinatorics Solution and answer checking →
Let $n$ be a positive integer with $b$ digits and $l, r$ be non-negative integers satisfying $l + r < b$ . We say that a positive integer number is a sub-divisor of $n$ , if it divides the number obtained by erasing the first $l$ and last $r$ digits of $n$ . (For example, sub-divisors of $143$ are $1$ , $2$ , $3$ , $4$ , $7$ , $11$ , $13$ , $14$ , $43$ and $143$ .) For any positive integer $d$ , let $A_d$ be the set of positive integers for which $d$ is not a sub-divisor. Find all positive integers $d$ for which the set $A_d$ is finite.

Number theory Solution and answer checking →
Let $ABC$ be an acute scalene triangle. The incircle of $ABC$ touches $BC$ , $CA$ , $AB$ at $D$ , $E$ , $F$ respectively. Let $X$ , $Y$ , $Z$ be feet of the altitudes from $A$ , $B$ , $C$ to the sides $BC$ , $CA$ , $AB$ respectively. Let $A'$ , $B'$ , $C'$ be the reflections of $X$ , $Y$ , $Z$ in $EF$ , $FD$ , $DE$ respectively. Prove that triangles $ABC$ and $A'B'C'$ are similar.

Geometry Solution and answer checking →
Determine all sequences $a_{1}, a_{2}, \ldots$ of positive integers such that, for any pair of positive integers $m \leqslant n$ , the arithmetic and geometric means
$\frac{a_{m}+a_{m+1}+\cdots+a_{n}}{n-m+1} \quad \text{ and } \quad \left(a_{m} a_{m+1} \cdots a_{n}\right)^{\frac{1}{n-m+1}}$
are both integers.

Algebra Solution and answer checking →
At a gala banquet, $12n + 6$ chairs, where $n \in \mathbb{N}$ , are equally arranged around a large round table. A seating will be called a proper seating of rank $n$ if a gathering of $6n + 3$ married couples sit around this table such that each seated person also has exactly one sibling (brother/sister) of the opposite gender present (siblings cannot be married to each other) and each man is seated closer to his wife than his sister. Among all proper seats of rank $n$ find the maximum possible number of women seated closer to their brother than their husband. (The maximum is taken not only across all possible seating arrangements for a given gathering, but also across all possible gatherings.)

Combinatorics Solution and answer checking →
Let a simple polynomial function be a polynomial function $P(x)$ whose coefficients belong to the set $\{-1, 0, 1\}$ . Let $n$ be a positive integer, $n > 1$ . Find the smallest possible number of non-zero coefficients in a simple polynomial function of $n$ th order whose values at all integral arguments are divisible by $n$ .
Answer: 2.

Number theory Solution and answer checking →

Answer key — Stage 9 · Mixed

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$2n(n - 1) + 1$ open
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