For teachers / Printable sets /Stage 7 · Algebra

Stage 7 · Algebra

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. 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

   Algebra Solution and answer checking →

2. 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

   Algebra Solution and answer checking →

3. 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
   $(1)a_{10}+a_{20}+a_{30}+a_{40};$
   $(2)a_{10}\cdot a_{20}+a_{30}\cdot a_{40}.$

   China National Olympiad

   Algebra Solution and answer checking →

4. 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

   Algebra Solution and answer checking →

5. 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

   Algebra Solution and answer checking →

6. The function $f(n)$ is defined on the positive integers and takes non-negative integer values. $f(2)=0,f(3)>0,f(9999)=3333$ and for all $m,n:$ $$f(m+n)-f(m)-f(n)=0 \text{ or } 1.$$ Determine $f(1982)$.

   International Mathematical Olympiad

   Algebra Solution and answer checking →

7. Determine all functions $f:\mathbb{R}\to\mathbb{R}$ such that for every pair of real numbers $x$ and $y$,
   $$f(x+y^2)=f(x)+|yf(y)|.$$

   USA Team Selection Test

   Algebra Solution and answer checking →

8. Express
   $$\sum_{k=0}^{n} (-1)^k (n-k)!(n+k)!$$
   in closed form.

   USA IMO

   Algebra Solution and answer checking →

9. Determine (with proof) whether there is a subset $X$ of the integers with the following property: for any integer $n$ there is exactly one solution of $a + 2b = n$ with $a,b \in X$ .

   USAMO

   Algebra Solution and answer checking →

10. Let $n > 1$ be an integer. Find, with proof, all sequences $x_1, x_2, \ldots, x_{n-1}$ of positive integers with the following
    three properties:
    (a). ; (b). for all ; (c). given any two indices and (not necessarily distinct)
    for which , there is an index such
    that .

    USAJMO

    Algebra Solution and answer checking →

Answer key — Stage 7 · Algebra

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

1. $\sum_{i=1}^n \min \{ n + 1 - i, 2i-1 \}$ open
2. $(2, 3, 3)$ open
3. $10$ open
4. $\text{m is not squarefree}$ open
5. $n = 1, k_1 = 1; n = 3, (k_1,k_2,k_3) = (2,3,6); n = 4, (k_1,k_2,k_3,k_4) = (4,4,4,4)$ open
6. $660$ open
7. $f(x) = ax \text{ for } a \geq 0$ open
8. $\sum_{k=0}^{n} (-1)^k (n-k)!(n+k)! = \frac{(n!)^2}{2} + \frac{(-1)^n (2n+1)!}{2(n+1)}$ open
9. Yes, there does exist a subset $X$ of the integers such that for any integer $n$ there is exactly one solution to $a + 2b = n$ with $a, b \in X$. open
10. $\text{The sequence is (2, 4, 6, , 2n-2 ).}$ 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.

Apache-2.0 · CC-BY-4.0