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Difference between revisions of "1996 IMO Problems"

(Created page with "Problems of the 1996 IMO. ==Day I== ===Problem 1=== We are given a positive integer <math>r</math> and a rectangular board <math>ABCD</math> with dimensions <math>|AB|=20...")
 
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Let <math>S</math> denote the set of nonnegative integers.  Find all functions <math>f</math> from <math>S</math> to itself such that
 
Let <math>S</math> denote the set of nonnegative integers.  Find all functions <math>f</math> from <math>S</math> to itself such that
  
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<math>f(m+f(n))=f(f(m))+f(n)</math>      <math>\forall m,n \in S</math>
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<math>f(m+f(n))=f(f(m))+f(n)</math>      <math>\forall m,n \in S.</math>
  
 
[[1996 IMO Problems/Problem 3|Solution]]
 
[[1996 IMO Problems/Problem 3|Solution]]
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Let <math>ABCDEF</math> be a convex hexagon such that <math>AB</math> is parallel to <math>DE</math>, <math>BD</math> is parallel to <math>EF</math>, and <math>CD</math> is parallel to <math>FA</math>.  Let <math>R_{A}</math>, <math>R_{C}</math>, <math>R_{E}</math> denote the circumradii of triangles <math>FAB</math>, <math>BCD</math>, <math>DEF</math>, respectively, and let <math>P</math> denote the perimeter of the hexagon.  Prove that
 
Let <math>ABCDEF</math> be a convex hexagon such that <math>AB</math> is parallel to <math>DE</math>, <math>BD</math> is parallel to <math>EF</math>, and <math>CD</math> is parallel to <math>FA</math>.  Let <math>R_{A}</math>, <math>R_{C}</math>, <math>R_{E}</math> denote the circumradii of triangles <math>FAB</math>, <math>BCD</math>, <math>DEF</math>, respectively, and let <math>P</math> denote the perimeter of the hexagon.  Prove that
  
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<math>R_{A}+R_{C}+R_{E} \ge \frac{P}{2}</math>
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<math>R_{A}+R_{C}+R_{E} \ge \frac{P}{2}.</math>
  
 
[[1996 IMO Problems/Problem 5|Solution]]
 
[[1996 IMO Problems/Problem 5|Solution]]

Latest revision as of 20:41, 4 July 2024

Problems of the 1996 IMO.

Day I

Problem 1

We are given a positive integer $r$ and a rectangular board $ABCD$ with dimensions $|AB|=20$, $|BC|=12$. The rectangle is divided into a grid of $20 \times 12$ unit squares. The following moves are permitted on the board: one can move from one square to another only if the distance between the centers of the two squares is $\sqrt{r}$. The task is to find a sequence of moves leading from the square with $A$ as a vertex to the square with $B$ as a vertex.

(a) Show that the task cannot be done if $r$ is divisible by $2$ or $3$.

(b) Prove that the task is possible when $r=73$.

(c) Can the task be done when $r=97$?

Solution

Problem 2

Let $P$ be a point inside triangle $ABC$ such that

\[\angle APB-\angle ACB = \angle APC-\angle ABC\]

Let $D$, $E$ be the incenters of triangles $APB$, $APC$, respectively. Show that $AP$, $BD$, $CE$ meet at a point.

Solution

Problem 3

Let $S$ denote the set of nonnegative integers. Find all functions $f$ from $S$ to itself such that

$f(m+f(n))=f(f(m))+f(n)$ $\forall m,n \in S.$

Solution

Day II

Problem 4

The positive integers $a$ and $b$ are such that the numbers $15a+16b$ and $16a-15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?

Solution

Problem 5

Let $ABCDEF$ be a convex hexagon such that $AB$ is parallel to $DE$, $BD$ is parallel to $EF$, and $CD$ is parallel to $FA$. Let $R_{A}$, $R_{C}$, $R_{E}$ denote the circumradii of triangles $FAB$, $BCD$, $DEF$, respectively, and let $P$ denote the perimeter of the hexagon. Prove that

$R_{A}+R_{C}+R_{E} \ge \frac{P}{2}.$

Solution

Problem 6

Let $p, q, n$ be three positive integers with $p+q<n$. Let $(x_0,x_1,\cdots ,x_n)$ be an $(n+1)$-tuple of integers satisfying the following conditions:

(i) $x_0=x_n=0$;

(ii) For each $i$ with $1 \le i \le n$, either $x_i-x_{i-1}=p$ or $x_i-x_{i-1}=-q$.

Show that there exists indices $i<j$ with $(i,j) \ne (0,n)$, such that $x_i=x_j$.

Solution

See Also

1996 IMO (Problems) • Resources
Preceded by
1995 IMO 		1 • 2 • 3 • 4 • 5 • 6 Followed by
1997 IMO
All IMO Problems and Solutions
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