Olympiad problems

1988 IMO Shortlist, 28

The sequence $ \{a_n\}$ of integers is defined by \[ a_1 \equal{} 2, a_2 \equal{} 7 \] and \[ \minus{} \frac {1}{2} < a_{n \plus{} 1} \minus{} \frac {a^2_n}{a_{n \minus{} 1}} \leq \frac {}{}, n \geq 2. \] Prove that $ a_n$ is odd for all $ n > 1.$

1987 IMO Shortlist, 7

Tags: inequality system , algebra

Given five real numbers $u_0, u_1, u_2, u_3, u_4$, prove that it is always possible to find five real numbers $v0, v_1, v_2, v_3, v_4$ that satisfy the following conditions: $(i)$ $u_i-v_i \in \mathbb N, \quad 0 \leq i \leq 4$ $(ii)$ $\sum_{0 \leq i<j \leq 4} (v_i - v_j)^2 < 4.$ [i]Proposed by Netherlands.[/i]

1983 IMO Shortlist, 6

Tags: inequalities , inequality system , algebra

Suppose that ${x_1, x_2, \dots , x_n}$ are positive integers for which $x_1 + x_2 + \cdots+ x_n = 2(n + 1)$. Show that there exists an integer $r$ with $0 \leq r \leq n - 1$ for which the following $n - 1$ inequalities hold: \[x_{r+1} + \cdots + x_{r+i} \leq 2i+ 1, \qquad \qquad \forall i, 1 \leq i \leq n - r; \] \[x_{r+1} + \cdots + x_n + x_1 + \cdots+ x_i \leq 2(n - r + i) + 1, \qquad \qquad \forall i, 1 \leq i \leq r - 1.\] Prove that if all the inequalities are strict, then $r$ is unique and that otherwise there are exactly two such $r.$

1988 IMO Longlists, 74

Tags: linear recurrence , inequality system , inequality , sequence

Let $ \{a_k\}^{\infty}_1$ be a sequence of non-negative real numbers such that: \[ a_k \minus{} 2 a_{k \plus{} 1} \plus{} a_{k \plus{} 2} \geq 0 \] and $ \sum^k_{j \equal{} 1} a_j \leq 1$ for all $ k \equal{} 1,2, \ldots$. Prove that: \[ 0 \leq a_{k} \minus{} a_{k \plus{} 1} < \frac {2}{k^2} \] for all $ k \equal{} 1,2, \ldots$.

1972 IMO, 1

Tags: inequality system , algebra

Find all positive real solutions to: \begin{eqnarray*} (x_1^2-x_3x_5)(x_2^2-x_3x_5) &\le& 0 \\ (x_2^2-x_4x_1)(x_3^2-x_4x_1) &\le& 0 \\ (x_3^2-x_5x_2)(x_4^2-x_5x_2) &\le& 0 \\ (x_4^2-x_1x_3)(x_5^2-x_1x_3) &\le & 0 \\ (x_5^2-x_2x_4)(x_1^2-x_2x_4) &\le& 0 \\ \end{eqnarray*}

1987 IMO Shortlist, 15

Tags: inequalities , inequality system , linear algebra , pigeonhole principle

Let $x_1,x_2,\ldots,x_n$ be real numbers satisfying $x_1^2+x_2^2+\ldots+x_n^2=1$. Prove that for every integer $k\ge2$ there are integers $a_1,a_2,\ldots,a_n$, not all zero, such that $|a_i|\le k-1$ for all $i$, and $|a_1x_1+a_2x_2+\ldots+a_nx_n|\le{(k-1)\sqrt n\over k^n-1}$. [i](IMO Problem 3)[/i] [i]Proposed by Germany, FR[/i]

1980 IMO, 1

Tags: algebra , inequality system , sequence , induction

Given a sequence $\{a_n\}$ of real numbers such that $|a_{k+m} - a_k - a_m| \leq 1$ for all positive integers $k$ and $m$, prove that, for all positive integers $p$ and $q$, \[|\frac{a_p}{p} - \frac{a_q}{q}| < \frac{1}{p} + \frac{1}{q}.\]

2007 IMO Shortlist, 1

Tags: induction , combinatorics , sequence , inequality system

Let $ n > 1$ be an integer. Find all sequences $ a_1, a_2, \ldots a_{n^2 \plus{} n}$ satisfying the following conditions: \[ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 \plus{} n; \] \[ \text{ (b) } a_{i \plus{} 1} \plus{} a_{i \plus{} 2} \plus{} \ldots \plus{} a_{i \plus{} n} < a_{i \plus{} n \plus{} 1} \plus{} a_{i \plus{} n \plus{} 2} \plus{} \ldots \plus{} a_{i \plus{} 2n} \text{ for all } 0 \leq i \leq n^2 \minus{} n. \] [i]Author: Dusan Dukic, Serbia[/i]

1969 IMO Shortlist, 48

Tags: maximization , inequality system , algebra

$(NET 3)$ Let $x_1, x_2, x_3, x_4,$ and $x_5$ be positive integers satisfying \[x_1 +x_2 +x_3 +x_4 +x_5 = 1000,\] \[x_1 -x_2 +x_3 -x_4 +x_5 > 0,\] \[x_1 +x_2 -x_3 +x_4 -x_5 > 0,\] \[-x_1 +x_2 +x_3 -x_4 +x_5 > 0,\] \[x_1 -x_2 +x_3 +x_4 -x_5 > 0,\] \[-x_1 +x_2 -x_3 +x_4 +x_5 > 0\] $(a)$ Find the maximum of $(x_1 + x_3)^{x_2+x_4}$ $(b)$ In how many different ways can we choose $x_1, . . . , x_5$ to obtain the desired maximum?

1972 IMO Longlists, 26

Tags: algebra , inequality system

Find all positive real solutions to: \begin{eqnarray*} (x_1^2-x_3x_5)(x_2^2-x_3x_5) &\le& 0 \\ (x_2^2-x_4x_1)(x_3^2-x_4x_1) &\le& 0 \\ (x_3^2-x_5x_2)(x_4^2-x_5x_2) &\le& 0 \\ (x_4^2-x_1x_3)(x_5^2-x_1x_3) &\le & 0 \\ (x_5^2-x_2x_4)(x_1^2-x_2x_4) &\le& 0 \\ \end{eqnarray*}

1983 IMO Longlists, 16

Tags: algebra , inequalities , inequality system

Suppose that ${x_1, x_2, \dots , x_n}$ are positive integers for which $x_1 + x_2 + \cdots+ x_n = 2(n + 1)$. Show that there exists an integer $r$ with $0 \leq r \leq n - 1$ for which the following $n - 1$ inequalities hold: \[x_{r+1} + \cdots + x_{r+i} \leq 2i+ 1, \qquad \qquad \forall i, 1 \leq i \leq n - r; \] \[x_{r+1} + \cdots + x_n + x_1 + \cdots+ x_i \leq 2(n - r + i) + 1, \qquad \qquad \forall i, 1 \leq i \leq r - 1.\] Prove that if all the inequalities are strict, then $r$ is unique and that otherwise there are exactly two such $r.$

1980 IMO Shortlist, 18

Tags: induction , algebra , inequality system , sequence

Given a sequence $\{a_n\}$ of real numbers such that $|a_{k+m} - a_k - a_m| \leq 1$ for all positive integers $k$ and $m$, prove that, for all positive integers $p$ and $q$, \[|\frac{a_p}{p} - \frac{a_q}{q}| < \frac{1}{p} + \frac{1}{q}.\]

1977 IMO Shortlist, 6

Tags: algebra , counting , inequality system , combinatorics

Let $n$ be a positive integer. How many integer solutions $(i, j, k, l) , \ 1 \leq i, j, k, l \leq n$, does the following system of inequalities have: \[1 \leq -j + k + l \leq n\]\[1 \leq i - k + l \leq n\]\[1 \leq i - j + l \leq n\]\[1 \leq i + j - k \leq n \ ?\]

1987 IMO Longlists, 20

Tags: linear algebra , inequality system , pigeonhole principle , inequalities

Let $x_1,x_2,\ldots,x_n$ be real numbers satisfying $x_1^2+x_2^2+\ldots+x_n^2=1$. Prove that for every integer $k\ge2$ there are integers $a_1,a_2,\ldots,a_n$, not all zero, such that $|a_i|\le k-1$ for all $i$, and $|a_1x_1+a_2x_2+\ldots+a_nx_n|\le{(k-1)\sqrt n\over k^n-1}$. [i](IMO Problem 3)[/i] [i]Proposed by Germany, FR[/i]

1977 IMO Longlists, 16

Tags: combinatorics , counting , inequality system , algebra

Let $n$ be a positive integer. How many integer solutions $(i, j, k, l) , \ 1 \leq i, j, k, l \leq n$, does the following system of inequalities have: \[1 \leq -j + k + l \leq n\]\[1 \leq i - k + l \leq n\]\[1 \leq i - j + l \leq n\]\[1 \leq i + j - k \leq n \ ?\]

1969 IMO Longlists, 48

Tags: algebra , inequality system , maximization

$(NET 3)$ Let $x_1, x_2, x_3, x_4,$ and $x_5$ be positive integers satisfying \[x_1 +x_2 +x_3 +x_4 +x_5 = 1000,\] \[x_1 -x_2 +x_3 -x_4 +x_5 > 0,\] \[x_1 +x_2 -x_3 +x_4 -x_5 > 0,\] \[-x_1 +x_2 +x_3 -x_4 +x_5 > 0,\] \[x_1 -x_2 +x_3 +x_4 -x_5 > 0,\] \[-x_1 +x_2 -x_3 +x_4 +x_5 > 0\] $(a)$ Find the maximum of $(x_1 + x_3)^{x_2+x_4}$ $(b)$ In how many different ways can we choose $x_1, . . . , x_5$ to obtain the desired maximum?

1987 IMO Longlists, 46

Tags: inequality system , algebra

Given five real numbers $u_0, u_1, u_2, u_3, u_4$, prove that it is always possible to find five real numbers $v0, v_1, v_2, v_3, v_4$ that satisfy the following conditions: $(i)$ $u_i-v_i \in \mathbb N, \quad 0 \leq i \leq 4$ $(ii)$ $\sum_{0 \leq i<j \leq 4} (v_i - v_j)^2 < 4.$ [i]Proposed by Netherlands.[/i]

1991 IMO Shortlist, 25

Tags: algebra , sequence , inequality , inequality system

Suppose that $ n \geq 2$ and $ x_1, x_2, \ldots, x_n$ are real numbers between 0 and 1 (inclusive). Prove that for some index $ i$ between $ 1$ and $ n \minus{} 1$ the inequality \[ x_i (1 \minus{} x_{i\plus{}1}) \geq \frac{1}{4} x_1 (1 \minus{} x_{n})\]

1980 IMO Longlists, 18

Tags: induction , inequality system , sequence , algebra

Given a sequence $\{a_n\}$ of real numbers such that $|a_{k+m} - a_k - a_m| \leq 1$ for all positive integers $k$ and $m$, prove that, for all positive integers $p$ and $q$, \[|\frac{a_p}{p} - \frac{a_q}{q}| < \frac{1}{p} + \frac{1}{q}.\]

2008 Germany Team Selection Test, 1

Tags: induction , sequence , inequality system , combinatorics

Let $ n > 1$ be an integer. Find all sequences $ a_1, a_2, \ldots a_{n^2 \plus{} n}$ satisfying the following conditions: \[ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 \plus{} n; \] \[ \text{ (b) } a_{i \plus{} 1} \plus{} a_{i \plus{} 2} \plus{} \ldots \plus{} a_{i \plus{} n} < a_{i \plus{} n \plus{} 1} \plus{} a_{i \plus{} n \plus{} 2} \plus{} \ldots \plus{} a_{i \plus{} 2n} \text{ for all } 0 \leq i \leq n^2 \minus{} n. \] [i]Author: Dusan Dukic, Serbia[/i]

2008 Germany Team Selection Test, 1

Tags: sequence , inequality system , induction , combinatorics

Let $ n > 1$ be an integer. Find all sequences $ a_1, a_2, \ldots a_{n^2 \plus{} n}$ satisfying the following conditions: \[ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 \plus{} n; \] \[ \text{ (b) } a_{i \plus{} 1} \plus{} a_{i \plus{} 2} \plus{} \ldots \plus{} a_{i \plus{} n} < a_{i \plus{} n \plus{} 1} \plus{} a_{i \plus{} n \plus{} 2} \plus{} \ldots \plus{} a_{i \plus{} 2n} \text{ for all } 0 \leq i \leq n^2 \minus{} n. \] [i]Author: Dusan Dukic, Serbia[/i]

1988 IMO Longlists, 80

Tags: sequence , inequality system , induction

The sequence $ \{a_n\}$ of integers is defined by \[ a_1 \equal{} 2, a_2 \equal{} 7 \] and \[ \minus{} \frac {1}{2} < a_{n \plus{} 1} \minus{} \frac {a^2_n}{a_{n \minus{} 1}} \leq \frac {}{}, n \geq 2. \] Prove that $ a_n$ is odd for all $ n > 1.$

1975 IMO Shortlist, 4

Tags: algebra , inequality system , sequence , convexity

Let $a_1, a_2, \ldots , a_n, \ldots $ be a sequence of real numbers such that $0 \leq a_n \leq 1$ and $a_n - 2a_{n+1} + a_{n+2} \geq 0$ for $n = 1, 2, 3, \ldots$. Prove that \[0 \leq (n + 1)(a_n - a_{n+1}) \leq 2 \qquad \text{ for } n = 1, 2, 3, \ldots\]

1972 IMO Shortlist, 9

Tags: inequality system , algebra

Find all positive real solutions to: \begin{eqnarray*} (x_1^2-x_3x_5)(x_2^2-x_3x_5) &\le& 0 \\ (x_2^2-x_4x_1)(x_3^2-x_4x_1) &\le& 0 \\ (x_3^2-x_5x_2)(x_4^2-x_5x_2) &\le& 0 \\ (x_4^2-x_1x_3)(x_5^2-x_1x_3) &\le & 0 \\ (x_5^2-x_2x_4)(x_1^2-x_2x_4) &\le& 0 \\ \end{eqnarray*}

1988 IMO Shortlist, 24

Tags: linear recurrence , inequality system , inequality , sequence

Let $ \{a_k\}^{\infty}_1$ be a sequence of non-negative real numbers such that: \[ a_k \minus{} 2 a_{k \plus{} 1} \plus{} a_{k \plus{} 2} \geq 0 \] and $ \sum^k_{j \equal{} 1} a_j \leq 1$ for all $ k \equal{} 1,2, \ldots$. Prove that: \[ 0 \leq a_{k} \minus{} a_{k \plus{} 1} < \frac {2}{k^2} \] for all $ k \equal{} 1,2, \ldots$.