Navvye/Olympiad-Problems-CSV huggingface dataset:
1997 IMO Shortlist
Let $ X,Y,Z$ be the midpoints of the small arcs $ BC,CA,AB$ respectively (arcs of the circumcircle of $ ABC$). $ M$ is an arbitrary point on $ BC$, and the parallels through $ M$ to the internal bisectors of $ \angle B,\angle C$ cut the external bisectors of $ \angle C,\angle B$ in $ N,P$ respectively. Show that $ XM,YN,ZP$ concur.
1997 IMO Shortlist
For every integer $ n \geq 2$ determine the minimum value that the sum $ \sum^n_{i=0} a_i$ can take for nonnegative numbers $ a_0, a_1, \ldots, a_n$ satisfying the condition $ a_0 = 1,$ $ a_i \leq a_{i+1} + a_{i+2}$ for $ i = 0, \ldots, n - 2.$
1996 IMO Shortlist
Suppose that $a, b, c > 0$ such that $abc = 1$. Prove that  $$ \frac{ab}{ab + a^5 + b^5} + \frac{bc}{bc + b^5 + c^5} + \frac{ca}{ca + c^5 + a^5} \leq 1. $$
1996 IMO Shortlist
Let $ a_1 \geq a_2 \geq \ldots \geq a_n$ be real numbers such that for all integers $ k > 0,$



$$ a^k_1 + a^k_2 + \ldots + a^k_n \geq 0.$$



Let $ p =\max\{|a_1|, \ldots, |a_n|\}.$ Prove that $ p = a_1$ and that



$$ (x - a_1) \cdot (x - a_2) \cdots (x - a_n) \leq x^n - a^n_1$$ for all $ x > a_1.$
1996 IMO Shortlist
Let $ a > 2$ be given, and starting $ a_0 = 1, a_1 = a$ define recursively:



$$ a_{n+1} = \left(\frac{a^2_n}{a^2_{n-1}} - 2 \right) \cdot a_n.$$



Show that for all integers $ k > 0,$ we have: $ \sum^k_{i = 0} \frac{1}{a_i} < \frac12 \cdot (2 + a - \sqrt{a^2-4}).$
1996 IMO Shortlist
Let $ a_{1}, a_{2}...a_{n}$ be non-negative reals, not all zero. Show that that

(a) The polynomial $ p(x) = x^{n} - a_{1}x^{n - 1} + ... - a_{n - 1}x - a_{n}$ has preceisely 1 positive real root $ R$.

(b) let $ A = \sum_{i = 1}^n a_{i}$ and $ B = \sum_{i = 1}^n ia_{i}$. Show that $ A^{A} \leq R^{B}$.
1996 IMO Shortlist
Let $ P(x)$ be the real polynomial function, $ P(x) = ax^3 + bx^2 + cx + d.$ Prove that if $ |P(x)| \leq 1$ for all $ x$ such that $ |x| \leq 1,$ then



$$ |a| + |b| + |c| + |d| \leq 7.$$
1996 IMO Shortlist
Let $ n$ be an even positive integer. Prove that there exists a positive inter $ k$ such that



$$ k = f(x) \cdot (x+1)^n + g(x) \cdot (x^n + 1)$$



for some polynomials $ f(x), g(x)$ having integer coefficients. If $ k_0$ denotes the least such $ k,$ determine $ k_0$ as a function of $ n,$ i.e. show that $ k_0 = 2^q$ where $ q$ is the odd integer determined by $ n = q \cdot 2^r, r \in \mathbb{N}.$



Note: This is variant A6' of the three variants given for this problem.
1996 IMO Shortlist
Let $ f$ be a function from the set of real numbers $ \mathbb{R}$ into itself such for all $ x \in \mathbb{R},$ we have $ |f(x)| \leq 1$ and



$$ f \left( x + \frac{13}{42} \right) + f(x) = f \left( x + \frac{1}{6} \right) + f \left( x + \frac{1}{7} \right).$$



Prove that $ f$ is a periodic function (that is, there exists a non-zero real number $ c$ such $ f(x+c) = f(x)$ for all $ x \in \mathbb{R}$).
1996 IMO Shortlist
Let the sequence $ a(n), n = 1,2,3, \ldots$ be generated as follows with $ a(1) = 0,$ and for $ n > 1:$



$$ a(n) = a\left( \left \lfloor \frac{n}{2} \right \rfloor \right) + (-1)^{\frac{n(n+1)}{2}}.$$



1.) Determine the maximum and minimum value of $ a(n)$ over $ n \leq 1996$ and find all $ n \leq 1996$ for which these extreme values are attained.



2.) How many terms $ a(n), n \leq 1996,$ are equal to 0?
1996 IMO Shortlist
Let $ ABC$ be a triangle, and $ H$ its orthocenter. Let $ P$ be a point on the circumcircle of triangle $ ABC$ (distinct from the vertices $ A$, $ B$, $ C$), and let $ E$ be the foot of the altitude of triangle $ ABC$ from the vertex $ B$. Let the parallel to the line $ BP$ through the point $ A$ meet the parallel to the line $ AP$ through the point $ B$ at a point $ Q$. Let the parallel to the line $ CP$ through the point $ A$ meet the parallel to the line $ AP$ through the point $ C$ at a point $ R$. The lines $ HR$ and $ AQ$ intersect at some point $ X$. Prove that the lines $ EX$ and $ AP$ are parallel.
1996 IMO Shortlist
Let $ABC$ be an equilateral triangle and let $P$ be a point in its interior. Let the lines $AP$, $BP$, $CP$ meet the sides $BC$, $CA$, $AB$ at the points $A_1$, $B_1$, $C_1$, respectively. Prove that



$A_1B_1 \cdot B_1C_1 \cdot C_1A_1 \ge A_1B \cdot B_1C \cdot C_1A$.
1996 IMO Shortlist
Let the sides of two rectangles be $ \{a,b\}$ and $ \{c,d\},$ respectively, with $ a < c \leq d < b$ and $ ab < cd.$ Prove that the first rectangle can be placed within the second one if and only if



$$ \left(b^2 - a^2\right)^2 \leq \left(bc - ad \right)^2 + \left(bd - ac \right)^2.$$
1996 IMO Shortlist
Let $ABC$ be an acute triangle with circumcenter $O$ and circumradius $R$. $AO$ meets the circumcircle of $BOC$ at $A'$, $BO$ meets the circumcircle of $COA$ at $B'$ and $CO$ meets the circumcircle of $AOB$ at $C'$. Prove that $$OA'\cdot OB'\cdot OC'\geq 8R^{3}.$$ Sorry if this has been posted before since this is a very classical problem, but I failed to find it with the search-function.
1996 IMO Shortlist
Let $ ABCD$ be a convex quadrilateral, and let $ R_A, R_B, R_C, R_D$ denote the circumradii of the triangles  $ DAB, ABC, BCD, CDA,$ respectively. Prove that $ R_A + R_C > R_B + R_D$ if and only if  $ \angle A + \angle C > \angle B + \angle D.$
1996 IMO Shortlist
In the plane, consider a point $ X$ and a polygon $ \mathcal{F}$ (which is not necessarily convex). Let $ p$ denote the perimeter of $ \mathcal{F}$, let $ d$ be the sum of the distances from the point $ X$ to the vertices of $ \mathcal{F}$, and let $ h$ be the sum of the distances from the point $ X$ to the sidelines of $ \mathcal{F}$. Prove that $ d^2 - h^2\geq\frac {p^2}{4}.$
1996 IMO Shortlist
Four integers are marked on a circle. On each step we simultaneously replace each number by the difference between this number and next number on the circle, moving in a clockwise direction; that is, the numbers $ a,b,c,d$ are replaced by $ a-b,b-c,c-d,d-a.$ Is it possible after 1996 such to have numbers $ a,b,c,d$ such the numbers $ |bc-ad|, |ac - bd|, |ab - cd|$ are primes?
1996 IMO Shortlist
A finite sequence of integers $ a_0, a_1, \ldots, a_n$ is called quadratic if for each $ i$ in the set $ \{1,2 \ldots, n\}$ we have the equality $ |a_i - a_{i-1}| = i^2.$



a.) Prove that any two integers $ b$ and $ c,$ there exists a natural number $ n$ and a quadratic sequence with $ a_0 = b$ and $ a_n = c.$



b.) Find the smallest natural number $ n$ for which there exists a quadratic sequence with  $ a_0 = 0$ and $ a_n = 1996.$
1996 IMO Shortlist
Find all positive integers $ a$ and $ b$ for which



$$ \left \lfloor \frac{a^2}{b} \right \rfloor + \left \lfloor \frac{b^2}{a} \right \rfloor = \left \lfloor \frac{a^2 + b^2}{ab} \right \rfloor + ab.$$
1996 IMO Shortlist
Show that there exists a bijective function $ f: \mathbb{N}_{0}\to \mathbb{N}_{0}$ such that for all $ m,n\in \mathbb{N}_{0}$:

$$ f(3mn + m + n) = 4f(m)f(n) + f(m) + f(n).

$$
1996 IMO Shortlist
Let $ k,m,n$ be integers such that $ 1 < n \leq m - 1 \leq k.$ Determine the maximum size of a subset $ S$ of the set $ \{1,2,3, \ldots, k-1,k\}$ such that no $ n$ distinct elements of $ S$ add up to $ m.$
1996 IMO Shortlist
Determine whether or nor there exist two disjoint infinite sets $ A$ and $ B$ of points in the plane satisfying the following conditions:



a.) No three points in $ A \cup B$ are collinear, and the distance between any two points in $ A \cup B$ is at least 1.



b.) There is a point of $ A$ in any triangle whose vertices are in $ B,$ and there is a point of $ B$ in any triangle whose vertices are in $ A.$
1996 IMO Shortlist
A finite number of coins are placed on an infinite row of squares. A sequence of moves is performed as follows: at each stage a square containing more than one coin is chosen. Two coins are taken from this square; one of them is placed on the square immediately to the left while the other is placed on the square immediately to the right of the chosen square. The sequence terminates if at some point there is at most one coin on each square. Given some initial configuration, show that any legal sequence of moves will terminate after the same number of steps and with the same final configuration.
1996 IMO Shortlist
let $ V$ be a finitive  set and $ g$ and $ f$ be two injective surjective functions from $ V$to$ V$.let $ T$ and $ S$ be two sets such that they are defined as following"

$ S = \{w \in V: f(f(w)) = g(g(w))\}$

$ T = \{w \in V: f(g(w)) = g(f(w))\}$

we know that $ S \cup T = V$, prove:

for each $ w \in V : f(w) \in S$ if and only if $ g(w) \in S$
1995 IMO Shortlist
Let $ a$ and $ b$ be non-negative integers such that $ ab \geq c^2,$ where $ c$ is an integer. Prove that there is a number $ n$ and integers $ x_1, x_2, \ldots, x_n, y_1, y_2, \ldots, y_n$ such that



$$ \sum^n_{i=1} x^2_i = a, \sum^n_{i=1} y^2_i = b, \text{ and } \sum^n_{i=1} x_iy_i = c.$$
1995 IMO Shortlist
Let $ n$ be an integer, $ n \geq 3.$ Let $ a_1, a_2, \ldots, a_n$ be real numbers such that $ 2 \leq a_i \leq 3$ for $ i = 1, 2, \ldots, n.$ If $ s = a_1 + a_2 + \ldots + a_n,$ prove that



$$ \frac{a^2_1 + a^2_2 - a^2_3}{a_1 + a_2 - a_3} + \frac{a^2_2 + a^2_3 - a^2_4}{a_2 + a_3 - a_4} + \ldots + \frac{a^2_n + a^2_1 - a^2_2}{a_n + a_1 - a_2} \leq 2s - 2n.$$
1995 IMO Shortlist
Find all of the positive real numbers like $ x,y,z,$ such that :



1.) $ x + y + z = a + b + c$



2.) $ 4xyz = a^2x + b^2y + c^2z + abc$



Proposed to Gazeta Matematica in the 80s by VASILE CÎRTOAJE and then by Titu Andreescu to IMO 1995.
1995 IMO Shortlist
Let $ \mathbb{R}$ be the set of real numbers. Does there exist a function $ f: \mathbb{R} \mapsto \mathbb{R}$ which simultaneously satisfies the following three conditions?



(a) There is a positive number $ M$ such that $ \forall x:$ $ - M \leq f(x) \leq M.$

(b) The value of $f(1)$ is $1$.

(c) If $ x \neq 0,$ then

$$ f \left(x + \frac {1}{x^2} \right) = f(x) + \left[ f \left(\frac {1}{x} \right) \right]^2

$$
1995 IMO Shortlist
Let $ n$ be an integer,$ n \geq 3.$ Let $ x_1, x_2, \ldots, x_n$ be real numbers such that $ x_i < x_{i+1}$ for $ 1 \leq i \leq n - 1$. Prove that



$$ \frac{n(n-1)}{2} \sum_{i < j} x_ix_j > \left(\sum^{n-1}_{i=1} (n-i)\cdot x_i \right) \cdot \left(\sum^{n}_{j=2} (j-1) \cdot x_j \right)$$
1995 IMO Shortlist
Let $ A, B$ and $ C$ be non-collinear points. Prove that there is a unique point $ X$ in the plane of $ ABC$ such that $$ XA^2 + XB^2 + AB^2 = XB^2 + XC^2 + BC^2 = XC^2 + XA^2 + CA^2.$$
1995 IMO Shortlist
The incircle of triangle $ \triangle ABC$ touches the sides $ BC$, $ CA$, $ AB$ at $ D, E, F$ respectively. $ X$ is a point inside triangle of $ \triangle ABC$ such that the incircle of triangle $ \triangle XBC$ touches $ BC$ at $ D$, and touches $ CX$ and $ XB$ at $ Y$ and $ Z$ respectively.

Show that $ E, F, Z, Y$ are concyclic.
1995 IMO Shortlist
An acute triangle $ ABC$ is given. Points $ A_1$ and $ A_2$ are taken on the side $ BC$ (with $ A_2$ between $ A_1$ and $ C$), $ B_1$ and $ B_2$ on the side $ AC$ (with $ B_2$ between $ B_1$ and $ A$), and $ C_1$ and $ C_2$ on the side $ AB$ (with $ C_2$ between $ C_1$ and $ B$) so that



$$ \angle AA_1A_2 = \angle AA_2A_1 = \angle BB_1B_2 = \angle BB_2B_1 = \angle CC_1C_2 = \angle CC_2C_1.$$



The lines $ AA_1,BB_1,$ and $ CC_1$ bound a triangle, and the lines $ AA_2,BB_2,$ and $ CC_2$ bound a second triangle. Prove that all six vertices of these two triangles lie on a single circle.
1995 IMO Shortlist
Let $ A_1A_2A_3A_4$ be a tetrahedron, $ G$ its centroid, and $ A'_1, A'_2, A'_3,$ and $ A'_4$ the points where the circumsphere of $ A_1A_2A_3A_4$ intersects $ GA_1,GA_2,GA_3,$ and $ GA_4,$ respectively. Prove that



$$ GA_1 \cdot GA_2 \cdot GA_3 \cdot GA_ \cdot4 \leq GA'_1 \cdot GA'_2 \cdot GA'_3 \cdot GA'_4$$



and



$$ \frac{1}{GA'_1} + \frac{1}{GA'_2} + \frac{1}{GA'_3} + \frac{1}{GA'_4} \leq \frac{1}{GA_1} + \frac{1}{GA_2} + \frac{1}{GA_3} + \frac{1}{GA_4}.$$
1995 IMO Shortlist
Let ABCD be a convex quadrilateral and O a point inside it. Let the parallels to the lines BC, AB, DA, CD through the point O meet the sides AB, BC, CD, DA of the quadrilateral ABCD at the points E, F, G, H, respectively. Then, prove that $ \sqrt {\left|AHOE\right|} + \sqrt {\left|CFOG\right|}\leq\sqrt {\left|ABCD\right|}$, where $ \left|P_1P_2...P_n\right|$ is an abbreviation for the non-directed area of an arbitrary polygon $ P_1P_2...P_n$.
1995 IMO Shortlist
Suppose that $ ABCD$ is a cyclic quadrilateral. Let $ E = AC\cap BD$ and $ F = AB\cap CD$. Denote by $ H_{1}$ and $ H_{2}$ the orthocenters of triangles $ EAD$ and $ EBC$, respectively. Prove that the points $ F$, $ H_{1}$, $ H_{2}$ are collinear.



Original formulation:



Let $ ABC$ be a triangle. A circle passing through $ B$ and $ C$ intersects the sides $ AB$ and $ AC$ again at $ C'$ and $ B',$ respectively. Prove that $ BB'$, $CC'$ and $ HH'$ are concurrent, where $ H$ and $ H'$ are the orthocentres of triangles $ ABC$ and $ AB'C'$ respectively.
1995 IMO Shortlist
Let $ k$ be a positive integer. Show that there are infinitely many perfect squares of the form $ n \cdot 2^k - 7$ where $ n$ is a positive integer.
1995 IMO Shortlist
Let $ \mathbb{Z}$ denote the set of all integers. Prove that for any integers $ A$ and $ B,$ one can find an integer $ C$ for which $ M_1 = \{x^2 + Ax + B : x \in \mathbb{Z}\}$ and $ M_2 = {2x^2 + 2x + C : x \in \mathbb{Z}}$ do not intersect.
1995 IMO Shortlist
Find all $ x,y$ and $ z$ in positive integer: $ z + y^{2} + x^{3} = xyz$ and $ x = \gcd(y,z)$.
1995 IMO Shortlist
At a meeting of $ 12k$ people, each person exchanges greetings with exactly $ 3k+6$ others. For any two people, the number who exchange greetings with both is the same. How many people are at the meeting?
1995 IMO Shortlist
Does there exist an integer $ n > 1$ which satisfies the following condition? The set of positive integers can be partitioned into $ n$ nonempty subsets, such that an arbitrary sum of $ n - 1$ integers, one taken from each of any $ n - 1$ of the subsets, lies in the remaining subset.
1995 IMO Shortlist
Let $ p$ be an odd prime. Determine positive integers $ x$ and $ y$ for which $ x \leq y$ and $ \sqrt{2p} - \sqrt{x} - \sqrt{y}$ is non-negative and as small as possible.
1995 IMO Shortlist
Does there exist a sequence $ F(1), F(2), F(3), \ldots$ of non-negative integers that simultaneously satisfies the following three conditions?



(a) Each of the integers $ 0, 1, 2, \ldots$ occurs in the sequence.

(b) Each positive integer occurs in the sequence infinitely often.

(c) For any $ n \geq 2,$

$$ F(F(n^{163})) = F(F(n)) + F(F(361)).

$$
1995 IMO Shortlist
For an integer $x \geq 1$, let $p(x)$ be the least prime that does not divide $x$, and define $q(x)$ to be the product of all primes less than $p(x)$. In particular, $p(1) = 2.$ For $x$ having $p(x) = 2$, define $q(x) = 1$. Consider the sequence $x_0, x_1, x_2, \ldots$ defined by $x_0 = 1$ and $$ x_{n+1} = \frac{x_n p(x_n)}{q(x_n)} $$ for $n \geq 0$. Find all $n$ such that $x_n = 1995$.
1995 IMO Shortlist
Suppose that $ x_1, x_2, x_3, \ldots$ are positive real numbers for which $$ x^n_n = \sum^{n-1}_{j=0} x^j_n$$ for $ n = 1, 2, 3, \ldots$ Prove that $ \forall n,$ $$ 2 - \frac{1}{2^{n-1}} \leq x_n < 2 - \frac{1}{2^n}.$$
1995 IMO Shortlist
For positive integers $ n,$ the numbers $ f(n)$ are defined inductively as follows: $ f(1) = 1,$ and for every positive integer $ n,$ $ f(n+1)$ is the greatest integer $ m$ such that there is an arithmetic progression of positive integers $ a_1 < a_2 < \ldots < a_m = n$ for which



$$ f(a_1) = f(a_2) = \ldots = f(a_m).$$



Prove that there are positive integers $ a$ and $ b$ such that $ f(an+b) = n+2$ for every positive integer $ n.$
1995 IMO Shortlist
Let $ \mathbb{N}$ denote the set of all positive integers. Prove that there exists a unique function $ f: \mathbb{N} \mapsto \mathbb{N}$ satisfying

$$ f(m + f(n)) = n + f(m + 95)

$$

for all $ m$ and $ n$ in $ \mathbb{N}.$ What is the value of $ \sum^{19}_{k = 1} f(k)?$
1994 IMO Shortlist
Let $ a_{0} = 1994$ and $ a_{n + 1} = \frac {a_{n}^{2}}{a_{n} + 1}$ for each nonnegative integer $ n$. Prove that $ 1994 - n$ is the greatest integer less than or equal to $ a_{n}$, $ 0 \leq n \leq 998$
1994 IMO Shortlist
Let $ \mathbb{R}$ denote the set of all real numbers and $ \mathbb{R}^+$ the subset of all positive ones. Let $ \alpha$ and $ \beta$ be given elements in $ \mathbb{R},$ not necessarily distinct. Find all functions $ f: \mathbb{R}^+ \mapsto \mathbb{R}$ such that



$$ f(x)f(y) = y^{\alpha} f \left( \frac{x}{2} \right) + x^{\beta} f \left( \frac{y}{2} \right) \forall x,y \in \mathbb{R}^+.$$
1994 IMO Shortlist
Let $ f(x) = \frac{x^2+1}{2x}$ for $ x \neq 0.$ Define $ f^{(0)}(x) = x$ and $ f^{(n)}(x) = f(f^{(n-1)}(x))$ for all positive integers $ n$ and $ x \neq 0.$ Prove that for all non-negative integers $ n$ and $ x \neq \{-1,0,1\}$



$$ \frac{f^{(n)}(x)}{f^{(n+1)}(x)} = 1 + \frac{1}{f \left( \left( \frac{x+1}{x-1} \right)^{2n} \right)}.$$
1994 IMO Shortlist
$ C$ and $ D$ are points on a semicircle. The tangent at $ C$ meets the extended diameter of the semicircle at $ B$, and the tangent at $ D$ meets it at $ A$, so that $ A$ and $ B$ are on opposite sides of the center. The lines $ AC$ and $ BD$ meet at $ E$. $ F$ is the foot of the perpendicular from $ E$ to $ AB$. Show that $ EF$ bisects angle $ CFD$
1994 IMO Shortlist
$ ABCD$ is a quadrilateral with $ BC$ parallel to $ AD$. $ M$ is the midpoint of $ CD$, $ P$ is the midpoint of $ MA$ and $ Q$ is the midpoint of $ MB$. The lines $ DP$ and $ CQ$ meet at $ N$. Prove that $ N$ is inside the quadrilateral $ ABCD$.
1994 IMO Shortlist
A circle $ C$ has two parallel tangents $ L'$ and$ L"$. A circle $ C'$ touches $ L'$ at $ A$ and $ C$ at $ X$. A circle $ C"$ touches $ L"$ at $ B$, $ C$ at $ Y$ and $ C'$ at $ Z$. The lines $ AY$ and $ BX$ meet at $ Q$. Show that $ Q$ is the circumcenter of $ XYZ$
1994 IMO Shortlist
A circle $ C$ with center $ O.$ and a line $ L$ which does not touch circle $ C.$ $ OQ$ is perpendicular to $ L,$ $ Q$ is on $ L.$ $ P$ is on $ L,$ draw two tangents $ L_1, L_2$ to circle $ C.$ $ QA, QB$ are perpendicular to $ L_1, L_2$ respectively. ($ A$ on $ L_1,$ $ B$ on $ L_2$). Prove that, line $ AB$ intersect $ QO$ at a fixed point.



Original formulation:



A line $ l$ does not meet a circle $ \omega$ with center $ O.$ $ E$ is the point on $ l$ such that $ OE$ is perpendicular to $ l.$ $ M$ is any point on $ l$ other than $ E.$ The tangents from $ M$ to $ \omega$ touch it at $ A$ and $ B.$ $ C$ is the point on $ MA$ such that $ EC$ is perpendicular to $ MA.$ $ D$ is the point on $ MB$ such that $ ED$ is perpendicular to $ MB.$ The line $ CD$ cuts $ OE$ at $ F.$ Prove that the location of $ F$ is independent of that of $ M.$
1994 IMO Shortlist
$ M$ is a subset of $ \{1, 2, 3, \ldots, 15\}$ such that the product of any three distinct elements of $ M$ is not a square. Determine the maximum number of elements in $ M.$
1994 IMO Shortlist
Define the sequences $ a_n, b_n, c_n$ as follows. $ a_0 = k, b_0 = 4, c_0 = 1$.

If $ a_n$ is even then $ a_{n + 1} = \frac {a_n}{2}$, $ b_{n + 1} = 2b_n$, $ c_{n + 1} = c_n$.

If $ a_n$ is odd, then $ a_{n + 1} = a_n - \frac {b_n}{2} - c_n$, $ b_{n + 1} = b_n$, $ c_{n + 1} = b_n + c_n$.

Find the number of positive integers $ k < 1995$ such that some $ a_n = 0$.
1994 IMO Shortlist
Define the sequence $ a_1, a_2, a_3, ...$ as follows. $ a_1$ and $ a_2$ are coprime positive integers and $ a_{n + 2} = a_{n + 1}a_n + 1$. Show that for every $ m > 1$ there is an $ n > m$ such that $ a_m^m$ divides $ a_n^n$. Is it true that $ a_1$ must divide $ a_n^n$ for some $ n > 1$?
1994 IMO Shortlist
A wobbly number is a positive integer whose digits are alternately zero and non-zero with the last digit non-zero (for example, 201). Find all positive integers which do not divide any wobbly number.
1994 IMO Shortlist
Two players play alternately on a $ 5 \times 5$ board. The first player always enters a $ 1$ into an empty square and the second player always enters a $ 0$ into an empty square. When the board is full, the sum of the numbers in each of the nine $ 3 \times 3$ squares is calculated and the first player's score is the largest such sum. What is the largest score the first player can make, regardless of the responses of the second player?
1994 IMO Shortlist
In a certain city, age is reckoned in terms of real numbers rather than integers. Every two citizens $x$ and $x'$ either know each other or do not know each other. Moreover, if they do not, then there exists a chain of citizens $x = x_0, x_1, \ldots, x_n = x'$ for some integer $n \geq 2$ such that $ x_{i-1}$ and $x_i$ know each other. In a census, all male citizens declare their ages, and there is at least one male citizen. Each female citizen provides only the information that her age is the average of the ages of all the citizens she knows. Prove that this is enough to determine uniquely the ages of all the female citizens.
1994 IMO Shortlist
Peter has three accounts in a bank, each with an integral number of dollars. He is only allowed to transfer money from one account to another so that the amount of money in the latter is doubled. Prove that Peter can always transfer all his money into two accounts. Can Peter always transfer all his money into one account?
1994 IMO Shortlist
There are $ n + 1$ cells in a row labeled from $ 0$ to $ n$ and $ n + 1$ cards labeled from $ 0$ to $ n$. The cards are arbitrarily placed in the cells, one per cell. The objective is to get card $ i$ into cell $ i$ for each $ i$. The allowed move is to find the smallest $ h$ such that cell $ h$ has a card with a label $ k > h$, pick up that card, slide the cards in cells $ h + 1$, $ h + 2$, ... , $ k$ one cell to the left and to place card $ k$ in cell $ k$. Show that at most $ 2^n - 1$ moves are required to get every card into the correct cell and that there is a unique starting position which requires $ 2^n - 1$ moves. [For example, if $ n = 2$ and the initial position is 210, then we get 102, then 012, a total of 2 moves.]
1994 IMO Shortlist
$ 1994$ girls are seated at a round table. Initially one girl holds $ n$ tokens. Each turn a girl who is holding more than one token passes one token to each of her neighbours.



a.) Show that if $ n < 1994$, the game must terminate.

b.) Show that if $ n = 1994$ it cannot terminate.
1994 IMO Shortlist
Two players play alternatively on an infinite square grid. The first player puts an $X$ in an empty cell and the second player puts an $O$ in an empty cell. The first player wins if he gets $11$ adjacent $X$'s in a line - horizontally, vertically or diagonally. Show that the second player can always prevent the first player from winning.
1994 IMO Shortlist
Let  $ n > 2$.    Show that there is a set of $ 2^{n-1}$ points in the plane, no three collinear such that no $ 2n$ form a convex $ 2n$-gon.
1993 IMO Shortlist
Define a sequence $\langle f(n)\rangle^{\infty}_{n=1}$ of positive integers by $f(1) = 1$ and $$f(n) =  \begin{cases} f(n-1) - n & \text{ if } f(n-1) > n;\\ f(n-1) + n & \text{ if } f(n-1) \leq n, \end{cases}$$for $n \geq 2.$  Let $S = \{n \in \mathbb{N} \;\mid\; f(n) = 1993\}.$



(i) Prove that $S$ is an infinite set.

(ii) Find the least positive integer in $S.$

(iii) If all the elements of $S$ are written in ascending order as $$ n_1 < n_2 < n_3 < \ldots ,  $$show that $$ \lim_{i\rightarrow\infty} \frac{n_{i+1}}{n_i} = 3.  $$
1993 IMO Shortlist
Show that there exists a finite set $A \subset \mathbb{R}^2$ such that for every $X \in A$ there are points $Y_1, Y_2, \ldots, Y_{1993}$ in $A$ such that the distance between $X$ and $Y_i$ is equal to 1, for every $i.$
1993 IMO Shortlist
Prove that $$ \frac{a}{b+2c+3d} +\frac{b}{c+2d+3a} +\frac{c}{d+2a+3b}+ \frac{d}{a+2b+3c} \geq \frac{2}{3} $$ for all positive real numbers $a,b,c,d$.
1993 IMO Shortlist
Solve the following system of equations, in which $a$ is a given number satisfying $|a| > 1$:



$\begin{matrix} x_{1}^2 = ax_2 + 1 \\ x_{2}^2 = ax_3 + 1 \\ \ldots \\ x_{999}^2 = ax_{1000} + 1 \\ x_{1000}^2 = ax_1 + 1 \\ \end{matrix}$