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Stage 7 · Mixed

10 problems · National Olympiad second round; IMO P1/P4 · mathsolympiadprep.com

Print or save as PDF The answer key prints on its own page at the end.

1. Find all integers $n \geq 3$ such that the following property holds: if we list the divisors of $n !$ in increasing order as $1=d_1<d_2<\cdots<d_k=n!$ , then we have $$d_2-d_1 \leq d_3-d_2 \leq \cdots \leq d_k-d_{k-1} .$$
   Contents 1 Solution (Explanation of Video) 2 Solution 2 3 Video Solution 4

   USAMO

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2. Let $a_1,a_2,\cdots,a_n$ be a permutation of $1,2,\cdots,n$. Among all possible permutations, find the minimum of $$\sum_{i=1}^n \min \{ a_i,2i-1 \}.$$

   China Team Selection Test

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3. Find all triples of positive integers $(x,y,z)$ that satisfy the equation
   $$2(x+y+z+2xyz)^2=(2xy+2yz+2zx+1)^2+2023.$$

   USAJMO

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4. Find all pairs $(p,q)$ of prime numbers which $p>q$ and
   $$\frac{(p+q)^{p+q}(p-q)^{p-q}-1}{(p+q)^{p-q}(p-q)^{p+q}-1}$$
   is an integer.

   IMO Shortlist

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5. Let $a_1,a_2,\cdots,a_{41}\in\mathbb{R},$ such that $a_{41}=a_1, \sum_{i=1}^{40}a_i=0,$ and for any $i=1,2,\cdots,40, |a_i-a_{i+1}|\leq 1.$ Determine the greatest possible value of
   $(1)a_{10}+a_{20}+a_{30}+a_{40};$
   $(2)a_{10}\cdot a_{20}+a_{30}\cdot a_{40}.$

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6. For which positive integers $m$ does there exist an infinite arithmetic sequence of integers $a_1, a_2, . . .$ and an infinite geometric sequence of integers $g_1, g_2, . . .$ satisfying the following properties?

   - $a_n - g_n$ is divisible by $m$ for all integers $n \ge 1$;

   - $a_2 - a_1$ is not divisible by $m$.

   Holden Mui

   USAJMO

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7. Find the smallest positive integer $n$ such that if $n$ squares of a $1000 \times 1000$ chessboard are colored, then there will exist three colored squares whose centers form a right triangle with sides parallel to the edges of the board.

   USAMO

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8. Given a acute triangle $PA_1B_1$ is inscribed in the circle $\Gamma$ with radius $1$. for all integers $n \ge 1$ are defined:
   $C_n$ the foot of the perpendicular from $P$ to $A_nB_n$
   $O_n$ is the center of $\odot (PA_nB_n)$
   $A_{n+1}$ is the foot of the perpendicular from $C_n$ to $PA_n$
   $B_{n+1} \equiv PB_n \cap O_nA_{n+1}$

   If $PC_1 =\sqrt{2}$, find the length of $PO_{2015}$

   Cono Sur Olympiad - 2015 - Day 1 - Problem 3

   Cono Sur Olympiad

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9. Let $n>5$ be an integer. There are $n$ points in the plane, no three of them collinear. Each day, Tom erases one of the points, until there are three points left. On the $i$-th day, for $1<i<n-3$, before erasing that day's point, Tom writes down the positive integer $v(i)$ such that the convex hull of the points at that moment has $v(i)$ vertices. Finally, he writes down $v(n-2) = 3$. Find the greatest possible value that the expression
   $$|v(1)-v(2)|+ |v(2)-v(3)| + \ldots + |v(n-3)-v(n-2)|$$
   can obtain among all possible initial configurations of $n$ points and all possible Tom's moves.

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10. Find all positive integers $n, k_1, \dots, k_n$ such that $k_1 + \cdots + k_n = 5n-4$ and $$\frac{1}{k_1} + \cdots + \frac{1}{k_n} = 1.$$

    Putnam

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Answer key — Stage 7 · Mixed

Worked solutions for every problem are on the site, one page per problem.

1. The integers $n \geq 3$ that satisfy the given property are $n = 3$ and $n = 4$. open
2. $\sum_{i=1}^n \min \{ n + 1 - i, 2i-1 \}$ open
3. $(2, 3, 3)$ open
4. $(3, 2)$ open
5. $10$ open
6. $\text{m is not squarefree}$ open
7. $\boxed{1999}$ open
8. $\frac{1}{2^{1007}}$ open
9. $2n - 8$ open
10. $n = 1, k_1 = 1; n = 3, (k_1,k_2,k_3) = (2,3,6); n = 4, (k_1,k_2,k_3,k_4) = (4,4,4,4)$ open

Problems belong to the competitions that set them. Each question on this sheet names the competition it came from, and the licences the material is used under are listed below. Free to copy for classroom use.

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