Navvye/Olympiad-Problems-CSV huggingface dataset:
2008 Canada National Olympiad
$ ABCD$ is a convex quadrilateral for which $ AB$ is the longest side. Points $ M$ and $ N$ are located on sides $ AB$ and $ BC$ respectively, so that each of the segments $ AN$ and $ CM$ divides the quadrilateral into two parts of equal area. Prove that the segment $ MN$ bisects the diagonal $ BD$.
2008 Canada National Olympiad
Determine all functions $ f$ defined on the set of rational numbers that take rational values for which

$$ f(2f(x) + f(y)) = 2x + y,

$$

for each $ x$ and $ y$.
2008 Canada National Olympiad
Let $ a$, $ b$, $ c$ be positive real numbers for which $ a + b + c = 1$. Prove that

$$ {{a-bc}\over{a+bc}} + {{b-ca}\over{b+ca}} + {{c-ab}\over{c+ab}}

\leq {3 \over 2}.$$
2008 Canada National Olympiad
Determine all functions $ f$ defined on the natural numbers that take values among the natural numbers for which

$$ (f(n))^p \equiv n\quad {\rm mod}\; f(p)

$$

for all $ n \in {\bf N}$ and all prime numbers $ p$.
2008 Canada National Olympiad
A self-avoiding rook walk on a chessboard (a rectangular grid of unit squares) is a path traced by a sequence of moves parallel to an edge of the board from one unit square to another, such that each begins where the previous move ended and such that no move ever crosses a square that has previously been crossed, i.e., the rook's path is non-self-intersecting.



Let $ R(m, n)$ be the number of self-avoiding rook walks on an $ m \times n$ ($ m$ rows, $ n$ columns) chessboard which begin at the lower-left corner and end at the upper-left corner. For example, $ R(m, 1) = 1$ for all natural numbers $ m$; $ R(2, 2) = 2$; $ R(3, 2) = 4$; $ R(3, 3) = 11$. Find a formula for $ R(3, n)$ for each natural number $ n$.
2007 Canada National Olympiad
What is the maximum number of non-overlapping $ 2\times 1$ dominoes that can be placed on a $ 8\times 9$ checkerboard if six of them are placed as shown? Each domino must be placed horizontally or vertically so as to cover two adjacent squares of the board.
2007 Canada National Olympiad
You are given a pair of triangles for which two sides of one triangle are equal in length to two sides of the second triangle, and the triangles are similar, but not necessarily congruent. Prove that the ratio of the sides that correspond under the similarity is a number between $ \frac {1}{2}(\sqrt {5} - 1)$ and $ \frac {1}{2}(\sqrt {5} + 1)$.
2007 Canada National Olympiad
Suppose that $ f$ is a real-valued function for which $$ f(xy)+f(y-x)\geq f(y+x)$$for all real numbers $ x$ and $ y$.



a) Give a non-constant polynomial that satisfies the condition.

b) Prove that $ f(x)\geq 0$ for all real $ x$.
2007 Canada National Olympiad
For two real numbers $ a$, $ b$, with $ ab\neq 1$, define the $ \ast$ operation by

$$ a\ast b=\frac{a+b-2ab}{1-ab}.$$ Start with a list of $ n\geq 2$ real numbers whose entries $ x$ all satisfy $ 0<x<1$. Select any two numbers $ a$ and $ b$ in the list; remove them and put the number $ a\ast b$ at the end of the list, thereby reducing its length by one. Repeat this procedure until a single number remains.



$ a.$ Prove that this single number is the same regardless of the choice of pair at each stage.

$ b.$ Suppose that the condition on the numbers $ x$ is weakened to $ 0<x\leq 1$. What happens if the list contains exactly one $ 1$?
2007 Canada National Olympiad
Let the incircle of triangle $ ABC$ touch sides $ BC,\, CA$ and $ AB$ at $ D,\, E$ and $ F,$ respectively. Let $ \omega,\,\omega_{1},\,\omega_{2}$ and $ \omega_{3}$ denote the circumcircles of triangle $ ABC,\, AEF,\, BDF$ and $ CDE$ respectively.



Let $ \omega$ and $ \omega_{1}$ intersect at $ A$ and $ P,\,\omega$ and $ \omega_{2}$ intersect at $ B$ and $ Q,\,\omega$ and $ \omega_{3}$ intersect at $ C$ and $ R.$



$ a.$ Prove that $ \omega_{1},\,\omega_{2}$ and $ \omega_{3}$ intersect in a common point.



$ b.$ Show that $ PD,\, QE$ and $ RF$ are concurrent.
2006 Canada National Olympiad
Let $ f(n,k)$ be the number of ways of distributing $ k$ candies to $ n$ children so that each child receives at most $ 2$ candies. For example $ f(3,7) = 0,f(3,6) = 1,f(3,4) = 6$. Determine the value of $ f(2006,1) + f(2006,4) + \ldots + f(2006,1000) + f(2006,1003) + \ldots + f(2006,4012)$.
2006 Canada National Olympiad
Let $ABC$ be acute triangle. Inscribe a rectangle $DEFG$ in this triangle such that $D\in AB,E\in AC,F\in BC,G\in BC$. Describe the locus of (i.e., the curve occupied by) the intersections of the diagonals of all possible rectangles $DEFG$.
2006 Canada National Olympiad
In a rectangular array of nonnegative reals with $m$ rows and $n$ columns, each row and each column contains at least one positive element. Moreover, if a row and a column intersect in a positive element, then the sums of their elements are the same. Prove that $m=n$.
2006 Canada National Olympiad
Consider a round-robin tournament with $2n+1$ teams, where each team plays each other team exactly one. We say that three teams $X,Y$ and $Z$, form a cycle triplet  if $X$ beats $Y$, $Y$ beats $Z$ and $Z$ beats $X$. There are no ties.

a)Determine the minimum number of cycle triplets possible.

b)Determine the maximum number of cycle triplets possible.
2006 Canada National Olympiad
The vertices of a right triangle $ABC$ inscribed in a circle divide the circumference into three arcs. The right angle is at $A$, so that the opposite arc $BC$ is a semicircle while arc $BC$ and arc $AC$ are supplementary. To each of three arcs, we draw a tangent such that its point of tangency is the mid point of that portion of the tangent intercepted by the extended lines $AB,AC$. More precisely, the point $D$ on arc $BC$ is the midpoint of the segment joining the points $D'$ and $D''$ where tangent at $D$ intersects the extended lines $AB,AC$. Similarly for $E$ on arc $AC$ and $F$ on arc $AB$. Prove that triangle $DEF$ is equilateral.
2005 Canada National Olympiad
An equilateral triangle of side length $ n$ is divided into unit triangles. Let $ f(n)$ be the number of paths from the triangle in the top row to the middle triangle in the bottom row, such that adjacent triangles in a path share a common edge and the path never travels up (from a lower row to a higher row) or revisits a triangle. An example is shown on the picture for $ n = 5$. Determine the value of $ f(2005)$.
2005 Canada National Olympiad
Let $(a,b,c)$ be a Pythagorean triple, i.e. a triplet of positive integers with $ a^2+b^2=c^2$.



$a)$ Prove that $\left(\frac{c}{a}+\frac{c}{b}\right)^2>8$.

$b)$ Prove that there are no integer $n$ and Pythagorean triple $(a,b,c)$ satisfying $\left(\frac{c}{a}+\frac{c}{b}\right)^2=n$.
2005 Canada National Olympiad
Let $S$ be a set of $n\ge 3$ points in the interior of a circle.

$a)$ Show that there are three distinct points $a,b,c\in S$ and three distinct points $A,B,C$ on the circle such that $a$ is (strictly) closer to $A$ than any other point in $S$, $b$ is closer to $B$ than any other point in $S$ and $c$ is closer to $C$ than any other point in $S$.

$b)$ Show that for no value of $n$ can four such points in $S$ (and corresponding points on the circle) be guaranteed.
2005 Canada National Olympiad
Let $ ABC$ be a triangle with circumradius $ R$, perimeter $ P$ and area $ K$. Determine the maximum value of: $ \frac{KP}{R^3}$.
2005 Canada National Olympiad
Let's say that an ordered triple of positive integers $(a,b,c)$ is $n$-powerful if $a\le b\le c,\gcd (a,b,c)=1$ and $a^n+b^n+c^n$ is divisible by $a+b+c$. For example, $(1,2,2)$ is $5$-powerful.

$a)$ Determine all ordered triples (if any) which are $n$-powerful for all $n\ge 1$.

$b)$ Determine all ordered triples (if any) which are $2004$-powerful and $2005$-powerful, but not $2007$-powerful.
2004 Canada National Olympiad
Find all ordered triples $ (x,y,z)$ of real numbers which satisfy the following system of equations:

$$ \left\{\begin{array}{rcl} xy & = & z - x - y \\
xz & = & y - x - z \\
yz & = & x - y - z \end{array} \right.

$$
2004 Canada National Olympiad
How many ways can $ 8$ mutually non-attacking rooks be placed on the $ 9\times9$ chessboard (shown here) so that all $ 8$ rooks are on squares of the same color?



(Two rooks are said to be attacking each other if they are placed in the same row or column of the board.)



[asy]unitsize(3mm);

defaultpen(white);

fill(scale(9)*unitsquare,black);

fill(shift(1,0)*unitsquare);

fill(shift(3,0)*unitsquare);

fill(shift(5,0)*unitsquare);

fill(shift(7,0)*unitsquare);



fill(shift(0,1)*unitsquare);

fill(shift(2,1)*unitsquare);

fill(shift(4,1)*unitsquare);

fill(shift(6,1)*unitsquare);

fill(shift(8,1)*unitsquare);



fill(shift(1,2)*unitsquare);

fill(shift(3,2)*unitsquare);

fill(shift(5,2)*unitsquare);

fill(shift(7,2)*unitsquare);



fill(shift(0,3)*unitsquare);

fill(shift(2,3)*unitsquare);

fill(shift(4,3)*unitsquare);

fill(shift(6,3)*unitsquare);

fill(shift(8,3)*unitsquare);



fill(shift(1,4)*unitsquare);

fill(shift(3,4)*unitsquare);

fill(shift(5,4)*unitsquare);

fill(shift(7,4)*unitsquare);



fill(shift(0,5)*unitsquare);

fill(shift(2,5)*unitsquare);

fill(shift(4,5)*unitsquare);

fill(shift(6,5)*unitsquare);

fill(shift(8,5)*unitsquare);



fill(shift(1,6)*unitsquare);

fill(shift(3,6)*unitsquare);

fill(shift(5,6)*unitsquare);

fill(shift(7,6)*unitsquare);



fill(shift(0,7)*unitsquare);

fill(shift(2,7)*unitsquare);

fill(shift(4,7)*unitsquare);

fill(shift(6,7)*unitsquare);

fill(shift(8,7)*unitsquare);



fill(shift(1,8)*unitsquare);

fill(shift(3,8)*unitsquare);

fill(shift(5,8)*unitsquare);

fill(shift(7,8)*unitsquare);



draw(scale(9)*unitsquare,black);[/asy]
2004 Canada National Olympiad
Let $ A,B,C,D$ be four points on a circle (occurring in clockwise order), with $ AB<AD$ and $ BC>CD$. The bisectors of angles $ BAD$ and $ BCD$ meet the circle at $ X$ and $ Y$, respectively. Consider the hexagon formed by these six points on the circle. If four of the six sides of the hexagon have equal length, prove that $ BD$ must be a diameter of the circle.
2004 Canada National Olympiad
Let $p$ be an odd prime. Prove that:

$$\displaystyle\sum_{k=1}^{p-1}k^{2p-1} \equiv \frac{p(p+1)}{2} \pmod{p^2}$$
2004 Canada National Olympiad
Let $ T$ be the set of all positive integer divisors of $ 2004^{100}$. What is the largest possible number of elements of a subset $ S$ of $ T$ such that no element in $ S$ divides any other element in $ S$?
2003 Canada National Olympiad
Consider a standard twelve-hour clock whose hour and minute hands move continuously. Let $m$ be an integer, with $1 \leq m \leq 720$. At precisely $m$ minutes after 12:00, the angle made by the hour hand and minute hand is exactly $1^\circ$.

Determine all possible values of $m$.
2003 Canada National Olympiad
Find the last three digits of the number $2003^{{2002}^{2001}}$.
2003 Canada National Olympiad
Find all real positive solutions (if any) to

\begin{align*}

x^3+y^3+z^3 &= x+y+z, \mbox{ and}  \\
x^2+y^2+z^2 &= xyz. 

\end{align*}
2003 Canada National Olympiad
Prove that when three circles share the same chord $AB$, every line through $A$ different from $AB$ determines the same ratio $X Y : Y Z$, where $X$ is an arbitrary point different from $B$ on the ﬁrst circle while $Y$ and $Z$ are the points where AX intersects the other two circles (labeled so that $Y$ is between $X$ and $Z$).
2003 Canada National Olympiad
Let $S$ be a set of $n$ points in the plane such that any two points of $S$ are at least $1$ unit apart.

Prove there is a subset $T$ of $S$ with at least $\frac{n}{7}$ points such that any two points of $T$ are at least $\sqrt{3}$ units apart.
2002 Canada National Olympiad
Let $S$ be a subset of $\{1, 2, \dots, 9\}$, such that the sums formed by adding each unordered pair of distinct numbers from $S$ are all different.  For example, the subset $\{1, 2, 3, 5\}$ has this property, but $\{1, 2, 3, 4, 5\}$ does not, since the pairs $\{1, 4\}$ and $\{2, 3\}$ have the same sum, namely 5.



What is the maximum number of elements that $S$ can contain?
2002 Canada National Olympiad
Call a positive integer $n$ practical if every positive integer less than or equal to $n$ can be written as the sum of distinct divisors of $n$.



For example, the divisors of 6 are 1, 2, 3, and 6. Since

$$ \centerline{1={\bf 1}, ~~ 2={\bf 2}, ~~ 3={\bf 3}, ~~ 4={\bf 1}+{\bf 3}, ~~ 5={\bf 2}+ {\bf 3}, ~~ 6={\bf 6},} $$

we see that 6 is practical.



Prove that the product of two practical numbers is also practical.
2002 Canada National Olympiad
Prove that for all positive real numbers $a$, $b$, and $c$,

$$ \frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ab} \geq a+b+c $$

and determine when equality occurs.
2002 Canada National Olympiad
Let $\Gamma$ be a circle with radius $r$.  Let $A$ and $B$ be distinct points on $\Gamma$ such that $AB < \sqrt{3}r$.  Let the circle with centre $B$ and radius $AB$ meet $\Gamma$ again at $C$.  Let $P$ be the point inside $\Gamma$ such that triangle $ABP$ is equilateral.  Finally, let the line $CP$ meet $\Gamma$ again at $Q$.



Prove that $PQ = r$.
2002 Canada National Olympiad
Let $\mathbb N = \{0,1,2,\ldots\}$.  Determine all functions $f: \mathbb N \to \mathbb N$ such that

$$ xf(y) + yf(x) = (x+y) f(x^2+y^2)  $$

for all $x$ and $y$ in $\mathbb N$.
2001 Canada National Olympiad
Randy: "Hi Rachel, that's an interesting quadratic equation you have written down.  What are its roots?''



Rachel: "The roots are two positive integers.  One of the roots is my age, and the other root is the age of my younger brother, Jimmy.''



Randy: "That is very neat!  Let me see if I can figure out how old you and Jimmy are.  That shouldn't be too difficult since all of your coefficients are integers.  By the way, I notice that the sum of the three coefficients is a prime number.''



Rachel: "Interesting.  Now figure out how old I am.''



Randy: "Instead, I will guess your age and substitute it for $x$ in your quadratic equation $\dots$ darn, that gives me $-55$, and not $0$.''



Rachel: "Oh, leave me alone!''



(1) Prove that Jimmy is two years old.



(2) Determine Rachel's age.
2001 Canada National Olympiad
There is a board numbered $-10$ to $10$. Each square is coloured either red or white, and the sum of the numbers on the red squares is $n$. Maureen starts with a token on the square labeled $0$. She then tosses a fair coin ten times. Every time she ﬂips heads, she moves the token one square to the right. Every time she ﬂips tails, she moves the token one square to the left.  At the end of the ten ﬂips, the probability that the token ﬁnishes on a red square is a rational number of the form $\frac a b$. Given that $a + b = 2001$, determine the largest possible value for $n$.
2001 Canada National Olympiad
Let $ABC$ be a triangle with $AC > AB$.  Let $P$ be the intersection point of the perpendicular bisector of $BC$ and the internal angle bisector of $\angle{A}$.  Construct points $X$ on $AB$ (extended) and $Y$ on $AC$ such that $PX$ is perpendicular to $AB$ and $PY$ is perpendicular to $AC$.  Let $Z$ be the intersection point of $XY$ and $BC$.

Determine the value of $\frac{BZ}{ZC}$.
2001 Canada National Olympiad
Let $n$ be a positive integer. Nancy is given a rectangular table in which each entry is a positive integer.  She is permitted to make either of the following two moves:



(1)  select a row and multiply each entry in this row by $n$;



(2)  select a column and subtract $n$ from each entry in this column.



Find all possible values of $n$ for which the following statement is true:



Given any rectangular table, it is possible for Nancy to perform a finite sequence of moves to create a table in which each entry is $0$.
2001 Canada National Olympiad
Let $P_0$, $P_1$, $P_2$ be three points on the circumference of a circle with radius $1$, where $P_1P_2 = t < 2$. For each $i \ge 3$, define $P_i$ to be the centre of the circumcircle of $\triangle P_{i-1} P_{i-2} P_{i-3}$.



(1)  Prove that the points $P_1, P_5, P_9, P_{13},\cdots$ are collinear.



(2)  Let $x$ be the distance from $P_1$ to $P_{1001}$, and let $y$ be the distance from $P_{1001}$ to $P_{2001}$. Determine all values of $t$ for which $\sqrt[500]{ \frac xy}$ is an integer.
2000 Canada National Olympiad
At 12:00 noon, Anne, Beth and Carmen begin running laps around a circular track of length $300$ meters, all starting from the same point on the track. Each jogger maintains a constant speed in one of the two possible directions for an indefinite period of time. Show that if Anne's speed is different from the other two speeds, then at some later time Anne will be at least $100$ meters from each of the other runners.  (Here, distance is measured along the  shorter of the two arcs separating two runners.)
2000 Canada National Olympiad
A permutation of the integers $1901, 1902, \cdots, 2000$ is a sequence  $a_1, a_2, \cdots, a_{100}$ in which each of those integers appears exactly once. Given such a permutation, we form the sequence of partial sums

$$s_1 = a_1,\;\;s_2 = a_1 + a_2,\;\;s_3 = a_1 + a_2 + a_3, \; \ldots\;, \; s_{100} = a_1 + a_2 + \cdots + a_{100}.$$How many of these permutations will have no terms of the sequence $s_1, \ldots, s_{100}$ divisible by three?
2000 Canada National Olympiad
Let $A = (a_1, a_2, \cdots ,a_{2000})$ be a sequence of integers each lying in the interval $[-1000,1000]$. Suppose that the entries in A sum to $1$. Show that some nonempty subsequence of $A$ sums to zero.
2000 Canada National Olympiad
Let $ABCD$ be a convex quadrilateral with $\angle CBD = 2 \angle ADB$, $\angle ABD = 2 \angle CDB$ and  $AB = CB$.



Prove that $AD = CD$.
2000 Canada National Olympiad
Suppose that the real numbers $a_1, a_2, \ldots, a_{100}$ satisfy

\begin{eqnarray*} 0 \leq a_{100} \leq a_{99} \leq \cdots \leq a_2 &\leq& a_1 , \\ a_1+a_2 & \leq & 100 \\ a_3+a_4+\cdots+a_{100} &\leq & 100. \end{eqnarray*}

Determine the maximum possible value of $a_1^2 + a_2^2 + \cdots + a_{100}^2$, and find all possible sequences $a_1, a_2, \ldots , a_{100}$ which achieve this maximum.
1999 Canada National Olympiad
Find all real solutions to the equation $4x^2 - 40 \lfloor x \rfloor + 51 = 0$.
1999 Canada National Olympiad
Let $ABC$ be an equilateral triangle of altitude 1.  A circle with radius 1 and center on the same side of $AB$ as $C$ rolls along the segment $AB$.  Prove that the arc of the circle that is inside the triangle always has the same length.
1999 Canada National Olympiad
Determine all positive integers $n$ with the property that  $n = (d(n))^2$. Here $d(n)$ denotes the number of positive divisors of $n$.
1999 Canada National Olympiad
Suppose $a_1,a_2,\cdots,a_8$ are eight distinct integers from $\{1,2,\cdots,16,17\}$.  Show that there is an integer $k > 0$ such that the equation $a_i - a_j = k$ has at least three different solutions.

Also, find a specific set of 7 distinct integers from $\{1,2,\ldots,16,17\}$ such that the equation $a_i - a_j = k$ does not have three distinct solutions for any $k > 0$.
1999 Canada National Olympiad
Let $ x$, $ y$, and $ z$ be non-negative real numbers satisfying $ x + y + z = 1$.  Show that

$$ x^2 y + y^2 z + z^2 x \leq \frac {4}{27}

$$

and find when equality occurs.
1998 Canada National Olympiad
Determine the number of real solutions $a$ to the equation:

$$ \left[\,\frac{1}{2}\;a\,\right]+\left[\,\frac{1}{3}\;a\,\right]+\left[\,\frac{1}{5}\;a\,\right] = a. $$

Here, if $x$ is a real number, then $[\,x\,]$ denotes the greatest integer that is less than or equal to $x$.
1998 Canada National Olympiad
Find all real numbers $x$ such that: $$ x = \sqrt{ x - \frac{1}{x} } + \sqrt{ 1 - \frac{1}{x} }  $$
1998 Canada National Olympiad
Let $ n$ be a natural number such that $ n \geq 2$.  Show that

$$ \frac {1}{n + 1} \left( 1 + \frac {1}{3} + \cdot \cdot \cdot + \frac {1}{2n - 1} \right) > \frac {1}{n} \left( \frac {1}{2} + \frac {1}{4} + \cdot \cdot \cdot + \frac {1}{2n} \right).

$$
1998 Canada National Olympiad
Let $ABC$ be a triangle with $\angle{BAC} = 40^{\circ}$ and $\angle{ABC}=60^{\circ}$.  Let $D$ and $E$ be the points lying on the sides $AC$ and $AB$, respectively, such that $\angle{CBD} = 40^{\circ}$ and $\angle{BCE} = 70^{\circ}$.  Let $F$ be the point of intersection of the lines $BD$ and $CE$.  Show that the line $AF$ is perpendicular to the line $BC$.
1998 Canada National Olympiad
Let $m$ be a positive integer. Define the sequence $a_0, a_1, a_2, \cdots$ by $a_0 = 0,\; a_1 = m,$ and $a_{n+1} = m^2a_n - a_{n-1}$ for $n = 1,2,3,\cdots$.

Prove that an ordered pair $(a,b)$ of non-negative integers, with $a \leq b$, gives a solution to the equation

$$ {\displaystyle \frac{a^2 + b^2}{ab + 1} = m^2} $$

if and only if $(a,b)$ is of the form $(a_n,a_{n+1})$ for some $n \geq 0$.
1997 Canada National Olympiad
Determine the number of pairs of positive integers $x,y$ such that $x\le y$, $\gcd (x,y)=5!$ and $\text{lcm}(x,y)=50!$.
1997 Canada National Olympiad
The closed interval $A = [0, 50]$ is the union of a finite number of closed intervals, each of length $1$. Prove that some of the intervals can be removed so that those remaining are mutually disjoint and have total length greater than $25$.

Note: For reals $a\le b$, the closed interval $[a, b] := \{x\in \mathbb{R}:a\le x\le b\}$ has length $b-a$; disjoint intervals have empty intersection.
1997 Canada National Olympiad
Prove that $\frac{1}{1999}< \prod_{i=1}^{999}{\frac{2i-1}{2i}}<\frac{1}{44}$.
1997 Canada National Olympiad
The point $O$ is situated inside the parallelogram $ABCD$ such that $\angle AOB+\angle COD=180^{\circ}$. Prove that $\angle OBC=\angle ODC$.
1997 Canada National Olympiad
Write the sum $\sum_{i=0}^{n}{\frac{(-1)^i\cdot\binom{n}{i}}{i^3 +9i^2 +26i +24}}$ as the ratio of two explicitly defined polynomials with integer coefficients.
1996 Canada National Olympiad
If $\alpha$, $\beta$, and $\gamma$ are the roots of $x^3 - x - 1 = 0$, compute $\frac{1+\alpha}{1-\alpha} + \frac{1+\beta}{1-\beta} + \frac{1+\gamma}{1-\gamma}$.
1996 Canada National Olympiad
Find all real solutions to the following system of equations.  Carefully justify your answer.

$$ \left\{ \begin{array}{c} \displaystyle\frac{4x^2}{1+4x^2} = y \\ \\ \displaystyle\frac{4y^2}{1+4y^2} = z \\ \\ \displaystyle\frac{4z^2}{1+4z^2} = x \end{array} \right. $$
1996 Canada National Olympiad
We denote an arbitrary permutation of the integers $1$, $2$, $\ldots$, $n$ by $a_1$, $a_2$, $\ldots$, $a_n$.  Let $f(n)$ denote the number of these permutations such that:



(1) $a_1 = 1$;



(2):$|a_i - a_{i+1}| \leq 2$, $i = 1, \ldots, n - 1$.



Determine whether $f(1996)$ is divisible by 3.
1996 Canada National Olympiad
Let triangle $ABC$ be an isosceles triangle with $AB = AC$. Suppose that the angle bisector of its angle $\angle B$ meets the side $AC$ at a point $D$ and that $BC = BD+AD$.

Determine $\angle A$.
1996 Canada National Olympiad
Let $r_1$, $r_2$, $\ldots$, $r_m$ be a given set of $m$ positive rational numbers such that $\sum_{k=1}^m r_k = 1$.  Define the function $f$ by $f(n)= n-\sum_{k=1}^m \: [r_k n]$ for each positive integer $n$.  Determine the minimum and maximum values of $f(n)$.  Here ${\ [ x ]}$ denotes the greatest integer less than or equal to $x$.
1995 Canada National Olympiad
Let $f(x)=\frac{9^x}{9^x + 3}$. Evaluate $\sum_{i=1}^{1995}{f\left(\frac{i}{1996}\right)}$.
1995 Canada National Olympiad
Let $\{a,b,c\}\in \mathbb{R}^{+}$. Prove that $a^a b^b c^c \ge (abc)^{\frac{a+b+c}{3}}$.
1995 Canada National Olympiad
Define a boomerang as a quadrilateral whose opposite sides do not intersect and one of whose internal angles is greater than $180^{\circ}$. Let $C$ be a convex polygon with $s$ sides. The interior region of $C$ is the union of $q$ quadrilaterals, none of whose interiors overlap each other. $b$ of these quadrilaterals are boomerangs. Show that $q\ge b+\frac{s-2}{2}$.
1995 Canada National Olympiad
Let $n$ be a constant positive integer. Show that for only non-negative integers $k$, the Diophantine equation $\sum_{i=1 }^{n}{ x_i ^3}=y^{3k+2}$ has infinitely many solutions in the positive integers $x_i, y$.
1995 Canada National Olympiad
$u$ is a real parameter such that $0<u<1$.

For $0\le x \le u$, $f(x)=0$.

For $u\le x \le n$, $f(x)=1-\left(\sqrt{ux}+\sqrt{(1-u)(1-x)}\right)^2$.

The sequence $\{u_n\}$ is define recursively as follows: $u_1=f(1)$ and $u_n=f(u_{n-1})$ $\forall n\in \mathbb{N}, n\neq 1$.

Show that there exists a positive integer $k$ for which $u_k=0$.