Navvye/Olympiad-Problems-CSV huggingface dataset:
1990 IMO Longlists
Let S be a 1990-element set and P be a set of 100-ary sequences $(a_1,a_2,...,a_{100})$ ,where $a_i's$ are distinct elements of S.An ordered pair (x,y) of elements of S is said to appear in $(a_1,a_2,...,a_{100})$ if $x=a_i$ and $y=a_j$ for some i,j with $1\leq i<j\leq 100$.Assume that every ordered pair (x,y) of elements of S appears in at most one member in P.Show that $|P|\leq 800$.
1990 IMO Longlists
There are $n$ non-coplanar points in space. Prove that there exists a circle exactly passes through three points of them.
1990 IMO Longlists
Prove that if $|x| < 1$, then

$$ \frac{x}{(1-x)^2}+\frac{x^2}{(1+x^2)^2} + \frac{x^3}{(1-x^3)^2}+\cdots=\frac{x}{1-x}+\frac{2x^2}{1+x^2}+\frac{3x^3}{1-x^3}+\cdots$$
1990 IMO Longlists
Let $\alpha$ be the positive root of the quadratic equation $x^2 = 1990x + 1$. For any $m, n \in \mathbb N$, define the operation $m*n = mn + [\alpha m][ \alpha n]$, where $[x]$ is the largest integer no larger than $x$. Prove that $(p*q)*r = p*(q*r)$ holds for all $p, q, r \in \mathbb N.$
1990 IMO Longlists
Let $a, b, c$ be integers. Prove that there exist integers $p_1, q_1, r_1, p_2, q_2$ and $r_2$, satisfying $a = q_1r_2 - q_2r_1, b = r_1p_2 - r_2p_1$ and $c = p_1q_2 - p_2q_1.$
1990 IMO Longlists
Given three letters $X, Y, Z$, we can construct letter sequences arbitrarily, such as $XZ, ZZYXYY, XXYZX$, etc. For any given sequence, we can perform following operations:



$T_1$: If the right-most letter is $Y$, then we can add $YZ$ after it, for example, $T_1(XYZXXY) =

(XYZXXYYZ).$



$T_2$: If The sequence contains $YYY$, we can replace them by $Z$, for example, $T_2(XXYYZYYYX) =

(XXYYZZX).$



$T_3$: We can replace $Xp$ ($p$ is any sub-sequence) by $XpX$, for example, $T_3(XXYZ) = (XXYZX).$



$T_4$: In a sequence containing one or more $Z$, we can replace the first $Z$ by $XY$, for example,

$T_4(XXYYZZX) = (XXYYXYZX).$



$T_5$: We can replace any of $XX, YY, ZZ$ by $X$, for example, $T_5(ZZYXYY) = (XYXX)$ or $(XYXYY)$ or $(ZZYXX).$



Using above operations, can we get $XYZZ$ from $XYZ \ ?$
1990 IMO Longlists
Let $n$ be an arbitrary positive integer. Calculate $S_n = \sum_{r=0}^n 2^{r-2n} \binom{2n-r}{n}.$
1990 IMO Longlists
Find $n$ points $p_1, p_2, \ldots, p_n$ on the circumference of a unit circle, such that $\sum_{1\leq i< j \leq n} p_i p_j$ is maximal.
1990 IMO Longlists
Let $V$ be a finite set of points in three-dimensional space. Let $S_1, S_2, S_3$ be the sets consisting of the orthogonal projections of the points of $V$ onto the $yz$-plane, $zx$-plane, $xy$-plane, respectively. Prove that $| V|^2 \leq | S1|\cdot|S2|\cdot |S3|$, where $| A|$ denotes the number of elements in the finite set $A.$
1990 IMO Longlists
Prove that for any positive integer $n$, the number of odd integers among the binomial coefficients $\binom nh \ ( 0 \leq h \leq n)$ is a power of 2.
1990 IMO Longlists
The tourist on an island can play the "getting treasure" game. He has to open a series of doors, each door is colored with one of n colors, according to the following rules:



(i) The tourist has n keys, each key with a different color.



(ii) Once a key is used, it is not permitted to change until it is destroyed.



(iii) Each key can open any door, and keeps intact when it opens the door having different color with it, but is destroyed when it opens the door having the same color with it.



Find the least number of doors to ensure that no tourist, no matter how he choose the order of the keys to use, can get the treasure.
1990 IMO Longlists
For each $P$ inside the triangle $ABC$, let $A(P), B(P)$, and $C(P)$ be the points of intersection of the lines $AP, BP$, and $CP$ with the sides opposite to $A, B$, and $C$, respectively. Determine $P$ in such a way that the area of the triangle $A(P)B(P)C(P)$ is as large as possible.
1990 IMO Longlists
Prove that $\sqrt 2 +\sqrt 3 +\sqrt{1990}$ is irrational.
1990 IMO Longlists
$AB, AC$ are two chords of the circle centered at $O$. The diameter, which is perpendicular to $BC$, intersects $AB, AC$ at $F, G$ respectively ($F$ is in the circle). The tangent from $G$ tangents the circle at $T$. Prove that $F$ is the projection of $T$ on $OG. $
1990 IMO Longlists
During the class interval, $n$ children sit in a circle and play the game described below. The teacher goes around the children clockwisely and hands out candies to them according to the following regulations: Select a child, give him a candy; and give the child next to the first child a candy too; then skip over one child and give next child a candy; then skip over two children; give the next child a candy; then skip over three children; give the next child a candy;...



Find the value of $n$ for which the teacher can ensure that every child get at least one candy eventually (maybe after many circles).
1990 IMO Longlists
Let real numbers $a_1, a_2, \ldots, a_n$ satisfy $0 < a_i \leq a, \  i = 1, 2, \ldots, n$. Prove that



(i) If $n = 4$, then

\[\frac 1a \sum_{i=1}^4 a_i - \frac{a_1a_2 + a_2a_3 + a_3 a_4 + a_4 a_1}{a^2}  \leq 2.\]



(ii) If $n = 6$, then

\[\[\frac 1a \sum_{i=1}^6 a_i - \frac{a_1a_2 + a_2a_3 + \cdots + a_5 a_6 + a_6 a_1}{a^2}  \leq 3.\]
1990 IMO Longlists
Find the real solution(s) for the system of equations

$$\begin{cases}x^3+y^3 &=1\\x^5+y^5 &=1\end{cases}$$
1990 IMO Longlists
Let $M = \{1, 2, \ldots, n\}$ and $\phi : M \to M$ be a bijection.



(i) Prove that there exist bijections $\phi_1, \phi_2 : M \to M$ such that $\phi_1 \cdot \phi_2 = \phi , \phi_1^2 =\phi_2^2=E$, where $E$ is the identity mapping.



(ii) Prove that the conclusion in (i) is also true if $M$ is the set of all positive integers.
1990 IMO Longlists
Given points $A, M, M_1$ and rational number $\lambda \neq -1$. Construct the triangle $ABC$, such that $M$ lies on $BC$ and $M_1$ lies on $B_1C_1$ ($B_1, C_1$ are the projections of $B, C$ on $AC, AB$ respectively), and $\frac{BM}{MC}=\frac{B_1M_1}{M_1C_1}=\lambda.$
1990 IMO Longlists
For positive integers $n, p$ with $n \geq p$, define real number $K_{n, p}$ as follows:

$K_{n, 0} = \frac{1}{n+1}$ and $K_{n, p} = K_{n-1, p-1} -K_{n, p-1}$ for $1 \leq p \leq n.$



(i) Define $S_n = \sum_{p=0}^n K_{n,p} , \ n = 0, 1, 2, \ldots$ . Find $\lim_{n \to \infty} S_n.$



(ii) Find $T_n = \sum_{p=0}^n (-1)^p K_{n,p} , \ n = 0, 1, 2, \ldots$.
1990 IMO Longlists
The sequence $\{u_n\}$ is defined by $u_1 = 1, u_2 = 1, u_n = u_{n-1} + 2u_{n-2} for n \geq 3$. Prove that for any positive integers $n, p \  (p > 1), u_{n+p} = u_{n+1}u_{p} + 2u_nu_{p-1}$. Also find the greatest common divisor of $u_n$ and $u_{n+3}.$
1990 IMO Longlists
Given eight real numbers $a_1 \leq a_2 \leq  \cdots \leq a_7 \leq a_8$. Let $x = \frac{ a_1 + a_2 +  \cdots + a_7 + a_8}{8}$, $y = \frac{ a_1^2 + a_2^2 +  \cdots + a_7^2 + a_8^2}{8}$. Prove that

$$2 \sqrt{y-x^2} \leq a_8  - a_1 \leq 4 \sqrt{y-x^2}.$$
1990 IMO Longlists
Prove that we can fill in the three dimensional space with regular tetrahedrons and regular octahedrons, all of which have the same edge-lengths. Also find the ratio of the number of the regular tetrahedrons used and the number of the regular octahedrons used.
1990 IMO Longlists
Given an $m$-element set $M$ and a $k$-element subset $K \subset M$. We call a function $f: K \to M$ has "path", if there exists an element $x_0  \in K$ such that $f(x_0) = x_0$, or there exists a chain $x_0, x_1, \ldots, x_j = x_0 \in K$ such that $_xi = f(x_{i-1})$ for $i = 1, 2, \ldots, j$. Find the number of functions $f: K \to M$ which have path.
1990 IMO Longlists
Find all the continuous bounded functions $f: \mathbb R \to \mathbb R$ such that

$$(f(x))^2 -(f(y))^2  = f(x + y)f(x - y) \text{  for all } x, y \in \mathbb R.$$
1990 IMO Longlists
Let $a + bi$ and $c + di$ be two roots of the equation $x^n = 1990$, where $n \geq 3$ is an integer and $a,b,c,d \in \mathbb R$. Under the linear transformation $f =\left(\begin{array}{cc}a&c\\b &d\end{array}\right)$, we have $(2, 1) \to  (1, 2)$. Denote $r$ to be the distance from the image of $(2, 2)$ to the origin. Find the range of $r.$
1990 IMO Longlists
In coordinate plane, a variable point $M$, starting from the origin $O(0, 0)$, moves on the line $l$ with slope $k$, where $k$ is an irrational number.



(i) Prove that point $O(0, 0)$ is the only rational point (namely, the coordinates of which are both rationals) on the line $l.$



(ii) Prove that for any number $\varepsilon > 0$, there exist integers $m, n$ such that the distance between $l$ and the point $(m, n)$ is less than $\varepsilon.$
1990 IMO Longlists
Consider the set of cuboids: the three edges $a, b, c$ from a common vertex satisfy the condition

$$\frac ab = \frac{a^2}{c^5}$$

(i) Prove that there are $100$ pairs of cuboids in this set with equal volumes in each pair.



(ii) For each pair of the above cuboids, find the ratio of the sum of their edges.
1990 IMO Longlists
$BC$ is a segment, $M$ is point on $BC$, $A$ is a point such that $A, B, C$ are non-collinear.



(i) Prove that if $M$ is the midpoint of $BC$, then $AB^2 + AC^2 = 2(AM^2 + BM^2).$



(ii) If there exists another point (except $M$) on segment $BC$ satisfying (i), find the region of point $A$ might occupy.
1990 IMO Longlists
Given a  point $P = (p_1, p_2, \ldots, p_n)$ in $n$-dimensional space . Find point $X = (x_1, x_2, \ldots, x_n)$, such that $x_1 \leq x_2 \leq\cdots \leq x_n$ and $\sqrt{(x_1-p_1)^2 + (x_2-p_2)^2+\cdots+(x_n-p_n)^2}$ is minimal.
1990 IMO Longlists
Let $n \geq 5$ be a positive integer. $a_1, b_1, a_2, b_2, \ldots, a_n, b_n$ are integers. $( a_i, b_i)$ are pairwisely distinct for $i = 1, 2, \ldots, n$, and $|a_1b_2 - a_2b_1| = |a_2b_3 -a_3b_2| = \cdots = |a_{n-1}b_n -a_nb_{n-1}| = 1$. Prove that there exists a pair of indexes $i, j$ satisfying $2 \leq |i - j| \leq n - 2$ and $|a_ib_j -a_jb_i| = 1.$
1990 IMO Longlists
Let $\mathbb Q$ be the set of all rational numbers and $\mathbb R$ be the set of real numbers. Function $f: \mathbb Q \to \mathbb R$ satisfies the following conditions:



(i) $f(0) = 0$, and for any nonzero $a \in Q, f(a) > 0.$

(ii) $f(x + y) = f(x)f(y) \qquad \forall x,y \in \mathbb Q.$

(iii) $f(x + y) \leq \max\{f(x), f(y)\} \qquad \forall x,y \in \mathbb Q , x,y \neq 0.$



Let $x$ be an integer and $f(x) \neq 1$. Prove that $f(1 + x + x^2+ \cdots + x^n) = 1$ for any positive integer $n.$
1990 IMO Longlists
Let $L$ be a subset in the coordinate plane defined by $L = \{(41x + 2y, 59x + 15y) | x, y \in \mathbb Z \}$, where $\mathbb Z$ is set of integers. Prove that for any parallelogram with center in the origin of coordinate and area $1990$, there exist at least two points of $L$ located in it.
1990 IMO Longlists
Prove that there exist at least two non-congruent quadrilaterals, both having a circumcircle, such that they have equal perimeters and areas.
1990 IMO Longlists
Let $a, b, c \in \mathbb R$. Prove that

$$(a^2 + ab + b^2)(b^2 + bc + c^2)(c^2 + ca + a^2) \geq (ab + bc + ca)^3.$$

When does the equality hold?
1990 IMO Longlists
Function $f(x, y): \mathbb N \times \mathbb N \to \mathbb Q$ satisfies the conditions:



(i) $f(1, 1) =1$,



(ii) $f(p + 1, q) + f(p, q + 1) = f(p, q)$ for all $p, q \in  \mathbb N$, and



(iii) $qf(p + 1, q) = pf(p, q + 1)$ for all $p, q \in  \mathbb N$.



Find $f(1990, 31).$
1990 IMO Longlists
Point $D$ is on the hypotenuse $BC$ of right-angled triangle $ABC$. The inradii of triangles $ADB$ and $ADC$ are equal. Prove that $S_{ABC} = AD^2$, where $S$ is the area function.
1990 IMO Longlists
Let $n \geq 4$ be an integer. $a_1, a_2, \ldots, a_n \in (0, 2n)$ are $n$ distinct integers. Prove that there exists a subset of the set $\{a_1, a_2, \ldots, a_n \}$ such that the sum of its elements is divisible by $2n.$
1990 IMO Longlists
Let $A_1, A_2, \ldots, A_n (n \geq 4)$ be $n$ convex sets in plane. Knowing that every three convex sets have a common point. Prove that there exists a point belonging to all the sets.
1990 IMO Longlists
Given function $f(x) = \sin x + \sin \pi x$ and positive number $d$. Prove that there exists real number $p$ such that $|f(x + p) - f(x)| < d$ holds for all real numbers $x$, and the value of $p$ can be arbitrarily large.
1990 IMO Longlists
Let $m$ be an positive odd integer not divisible by $3$. Prove that $\left[4^m -(2+\sqrt 2)^m\right]$ is divisible by $112.$
1990 IMO Longlists
Let $n$ be a positive integer. $S_1, S_2, \ldots, S_n$ are pairwise non-intersecting sets, and $S_k $ has exactly $k$ elements $(k = 1, 2, \ldots, n)$. Define $S = S_1\cup S_2\cup\cdots \cup S_n$. The function $f: S \to S $ maps all elements in $S_k$ to a fixed element of $S_k$, $k = 1, 2, \ldots, n$. Find the number of functions $g: S \to S$ satisfying $f(g(f(x))) = f(x).$
1990 IMO Longlists
Let $P$ be a variable point on the circumference of a quarter-circle with radii $OA, OB$ and $\angle AOB = 90^\circ$. H is the projection of $P$ on $OA$. Find the locus of the incenter of the right-angled triangle $HPO.$
1990 IMO Longlists
Quadrilateral $ABCD$ has an inscribed circle with center $O$. Knowing that $AB = CD$, and $M, K$ are the midpoints of $BC, AD$ respectively. Prove that $OM = OK.$
1990 IMO Longlists
Let $n$ be a positive integer and $m = \frac{(n+1)(n+2)}{2}$. In coordinate plane, there are $n$ distinct lines $L_1, L_2, \ldots, L_n$ and $m$ distinct points $A_1, A_2, \ldots, A_m$, satisfying the following conditions:



i) Any two lines are non-parallel.



ii) Any three lines are non-concurrent.



iii) Only $A_1$ does not lies on any line $L_k$, and there are exactly $k + 1$ points $A_j$'s that lie on line $L_k$ $(k = 1, 2, \ldots, n).$



Prove that there exist a unique polynomial $p(x, y)$ with degree $n$ satisfying $p(A_1) = 1$ and $p(A_j) = 0$ for $j = 2, 3, \ldots, m.$
1990 IMO Longlists
Given integer $n > 1$ and real number $t \geq 1$. $P$ is a parallelogram with four vertices $(0, 0), (0, t), (tF_{2n+1}, tF_{2n}), (tF_{2n+1}, tF_{2n} + t)$. Here, ${F_n}$ is the $n$-th term of Fibonacci sequence defined by $F_0 = 0, F_1 = 1$ and $F_{m+1} = F_m + F_{m-1}$. Let $L$ be the number of integral points (whose coordinates are integers) interior to $P$, and $M$ be the area of $P$, which is $t^2F_{2n+1}.$



i) Prove that for any integral point $(a, b)$, there exists a unique pair of integers $(j, k)$ such that$ j(F_{n+1}, F_n) + k(F_n, F_{n-1}) = (a, b)$, that is,$ jF_{n+1} + kF_n = a$ and $jF_n + kF_{n-1} = b.$



ii) Using i) or not, prove that $|\sqrt L-\sqrt M| \leq \sqrt 2.$
1990 IMO Longlists
In convex hexagon $ABCDEF$, we know that $\angle BCA = \angle DEC = \angle AFB = \angle CBD = \angle EDF.$ Prove that $AB = CD = EF.$
1990 IMO Longlists
Given a $10 \times 10$ chessboard colored as black-and-white alternately. Prove that for any $46$ unit squares without common edges, there are at least $30$ unit squares with the same color.
1990 IMO Longlists
The side-lengths of two equilaterals $ABC$ and $KLM$ are $1$ and $1/4$, respectively. And triangle $KLM$ located inside triangle $ABC$. Denote by $\Sigma$ the sum of the distances from $A$ to lines $KL, LM$ and $MK$. Find the location of triangle $KLM$ when $\Sigma$ is maximal.
1990 IMO Longlists
In coordinate plane, we call a point $(x, y)$ "lattice point" if both $x$ and $y$ are integers. Knowing that the vertices of triangle $ABC$ are all lattice points, and there exists exactly one lattice point interior to triangle $ABC$ (there might exist lattice points on the sides of $ABC$). Prove that the area of triangle $ABC$ is no larger than $\frac 92.$
1990 IMO Longlists
Find the minimal value of the function

$$\begin{array}{c}\ f(x) =\sqrt{15 - 12 \cos x} + \sqrt{4 -2 \sqrt 3 \sin x}+\sqrt{7-4\sqrt 3 \sin x} +\sqrt{10-4 \sqrt 3 \sin x - 6 \cos x}\end{array}$$
1990 IMO Longlists
Let $x,y,z$ be positive reals and $x \geq y \geq z$. Prove that

$$\frac{x^2y}{z}+\frac{y^2z}{x}+\frac{z^2x}{y} \geq x^2+y^2+z^2$$
1990 IMO Longlists
Let $S, T$ be the circumcenter and centroid of triangle $ABC$, respectively. $M$ is a point in the plane of triangle $ABC$ such that $90^\circ  \leq  \angle SMT < 180^\circ$. $A_1, B_1, C_1$ are the intersections of $AM, BM, CM$ with the circumcircle of triangle $ABC$ respectively. Prove that $MA_1 + MB_1 + MC_1 \geq MA + MB + MC.$
1990 IMO Longlists
Let $S$ be the incenter of triangle $ABC$. $A_1, B_1, C_1$ are the intersections of $AS, BS, CS$ with the circumcircle of triangle $ABC$ respectively. Prove that $SA_1 + SB_1 + SC_1 \geq SA + SB + SC.$
1990 IMO Longlists
Let $a, b, c$ be the side lengths and $P$ be area of a triangle, respectively. Prove that

$$(a^2+b^2+c^2-4\sqrt 3 P) (a^2+b^2+c^2) \geq 2 \left(a^2(b - c)^2 + b^2(c - a)^2 + c^2(a - b)^2\right).$$
1990 IMO Longlists
Let $\{ a_1, a_2, \ldots, a_n\} = \{1, 2, \ldots, n\}$. Prove that

$$\frac 12 +\frac 23 +\cdots+\frac{n-1}{n} \leq \frac{a_1}{a_2} + \frac{a_2}{a_3} +\cdots+\frac{a_{n-1}}{a_n}.$$
1989 IMO Longlists
In the set $ S_n = \{1, 2,\ldots ,n\}$ a new multiplication $ a*b$ is defined with the following properties:



(i) $ c = a * b$ is in $ S_n$ for any $ a \in S_n, b \in S_n.$

(ii) If the ordinary product $ a \cdot b$ is less than or equal to $ n,$ then $ a*b = a \cdot b.$

(iii) The ordinary rules of multiplication hold for $ *,$ i.e.:



(1) $ a * b = b * a$ (commutativity)

(2) $ (a * b) * c = a * (b * c)$ (associativity)

(3) If $ a * b = a * c$ then $ b = c$ (cancellation law).



Find a suitable multiplication table for the new product for $ n = 11$ and $ n = 12.$
1989 IMO Longlists
The sequences $ a_0, a_1, \ldots$ and $ b_0, b_1, \ldots$ are defined for $ n = 0, 1, 2, \ldots$ by the equalities

$$ a_0 = \frac {\sqrt {2}}{2}, \quad a_{n + 1} = \frac {\sqrt {2}}{2} \cdot \sqrt {1 - \sqrt {1 - a^2_n}}

$$

and

$$ b_0 = 1, \quad b_{n + 1} = \frac {\sqrt {1 + b^2_n} - 1}{b_n}

$$

Prove the inequalities for every $ n = 0, 1, 2, \ldots$

$$ 2^{n + 2} a_n < \pi < 2^{n + 2} b_n.

$$
1989 IMO Longlists
The circles $ c_1$ and $ c_2$ are tangent at the point $ A.$ A straight line $ l$ through $ A$ intersects $ c_1$ and $ c_2$  at points $ C_1$ and $ C_2$ respectively. A circle $ c,$ which contains $ C_1$ and $ C_2,$ meets $ c_1$ and $ c_2$ at points $ B_1$ and $ B_2$ respectively. Let $ \omega$ be the circle circumscribed around triangle $ AB_1B_2.$ The circle $ k$ tangent to $ \omega$ at the point $ A$ meets $ c_1$ and $ c_2$ at the points $ D_1$ and $ D_2$ respectively. Prove that



(a) the points $ C_1,C_2,D_1,D_2$ are concyclic or collinear,

(b) the points $ B_1,B_2,D_1,D_2$ are concyclic if and only if $ AC_1$ and $ AC_2$ are diameters of $ c_1$ and $ c_2.$
1989 IMO Longlists
Let $ m$ be a positive integer and define $ f(m)$ to be the number of factors of $ 2$ in $ m!$ (that is, the greatest positive integer $ k$ such that $ 2^k|m!$). Prove that there are infinitely many positive integers $ m$ such that $ m - f(m) = 1989.$
1989 IMO Longlists
Given the equation $$ 4x^3 + 4x^2y - 15xy^2 - 18y^3 - 12x^2 + 6xy + 36y^2 + 5x - 10y = 0,$$ find all positive integer solutions.
1989 IMO Longlists
Given the equation $$ y^4 + 4y^2x - 11y^2 + 4xy - 8y + 8x^2 - 40x + 52 = 0,$$ find all real solutions.
1989 IMO Longlists
Let $ P(x)$ be a polynomial such that the following inequalities are satisfied:

$$ P(0) > 0;$$$$ P(1) > P(0);$$$$ P(2) > 2P(1) - P(0);$$$$ P(3) > 3P(2) - 3P(1) + P(0);$$

and also for every natural number $ n,$ $$ P(n+4) > 4P(n+3) - 6P(n+2)+4P(n + 1) - P(n).$$

Prove that for every positive natural number $ n,$ $ P(n)$ is positive.
1989 IMO Longlists
Let $ n \leq 44, n \in \mathbb{N}.$ Prove that for any function $ f$ defined over $ \mathbb{N}^2$ whose images are in the set $ \{1, 2, \ldots , n\},$ there are four ordered pairs $ (i, j), (i, k), (l, j),$ and $ (l, k)$ such that  $$ f(i, j) = f(i, k) = f(l, j) = f(l, k),$$  in which $ i, j, k, l$ are chosen in such a way that there are natural numbers $ m, p$ that satisfy $$ 1989m \leq i < l  < 1989 + 1989m$$ and $$ 1989p \leq j < k  < 1989 + 1989p.$$
1989 IMO Longlists
A sequence $ a_1, a_2, a_3, \ldots$ is defined recursively by $ a_1 = 1$ and $ a_{2^k+j} = -a_j$ $ (j = 1, 2, \ldots, 2^k).$ Prove that this sequence is not periodic.
1989 IMO Longlists
Let $ a \in \mathbb{R}, 0 < a < 1,$ and $ f$ a continuous function on $ [0, 1]$ satisfying $ f(0) = 0, f(1) = 1,$ and



$$ f \left( \frac{x+y}{2} \right) = (1-a) f(x) + a f(y) \quad \forall x,y \in [0,1] \text{ with } x \leq y.$$



Determine $ f \left( \frac{1}{7} \right).$
1989 IMO Longlists
There are some boys and girls sitting in an $ n \times n$ quadratic array. We know the number of girls in every column and row and every line parallel to the diagonals of the array. For which $ n$ is this information sufficient to determine the exact positions of the girls in the array? For which seats can we say for sure that a girl sits there or not?
1989 IMO Longlists
Let $ a_1, \ldots, a_n$ be distinct positive integers that do not contain a $ 9$ in their decimal representations. Prove that the following inequality holds



$$ \sum^n_{i=1} \frac{1}{a_i} \leq 30.$$
1989 IMO Longlists
Let $ ABC$ be an equilateral triangle with side length equal to $ N \in \mathbb{N}.$ Consider the set $ S$ of all points $ M$ inside the triangle $ ABC$ satisfying



$$ \overrightarrow{AM} = \frac{1}{N} \cdot \left(n \cdot \overrightarrow{AB} + m \cdot \overrightarrow{AC} \right)$$



with $ m, n$ integers, $ 0 \leq n \leq N,$ $ 0 \leq m \leq N$ and $ n + m \leq N.$



Every point of S is colored in one of the three colors blue, white, red such that



(i) no point of $ S \cap [AB]$ is coloured blue

(ii) no point of $ S \cap [AC]$ is coloured white

(iii) no point of $ S \cap [BC]$ is coloured red



Prove that there exists an equilateral triangle the following properties:



(1) the three vertices of the triangle are points of $ S$ and coloured blue, white and red, respectively.

(2) the length of the sides of the triangle is equal to 1.



Variant: Same problem but with a regular tetrahedron and four different colors used.
1989 IMO Longlists
Let $ ABC$ be a triangle. Prove that there is a unique point $ U$ in the plane of $ ABC$ such that there exist real numbers $ \alpha, \beta, \gamma, \delta$ not all zero, such that



$$ \alpha PL^2 + \beta PM^2 + \gamma PN^2 + \delta UP^2$$



is constant for all points $ P$ of the plane, where $ L,M,N$ are the feet of the perpendiculars from $ P$ to $ BC,CA,AB$ respectively. Identify $ U.$
1989 IMO Longlists
Let $ a, b, c, d$ be positive integers such that $ ab = cd$ and $ a+b = c - d.$ Prove that there exists a right-angled triangle the measure of whose sides (in some unit) are integers and whose area measure is $ ab$ square units.
1989 IMO Longlists
The integers $ c_{m,n}$ with $ m \geq 0, \geq 0$ are defined by



$$ c_{m,0} = 1 \quad \forall m \geq 0, c_{0,n} = 1 \quad \forall n \geq 0,$$



and



$$ c_{m,n} = c_{m-1,n} - n \cdot c_{m-1,n-1} \quad \forall m > 0, n > 0.$$



Prove that $$ c_{m,n} = c_{n,m} \quad \forall m > 0, n > 0.$$
1989 IMO Longlists
Let $ b_1, b_2, \ldots, b_{1989}$ be positive real numbers such that the equations



$$ x_{r-1} - 2x_r + x_{r+1} + b_rx_r = 0 \quad (1 \leq r \leq 1989)$$



have a solution with $ x_0 = x_{1989} = 0$ but not all of $ x_1, \ldots, x_{1989}$ are equal to zero. Prove that



$$ \sum^{1989}_{k=1} b_k \geq \frac{2}{995}.$$
1989 IMO Longlists
Let $ L$ denote the set of all lattice points of the plane (points with integral coordinates). Show that for any three points $ A,B,C$ of $ L$ there is a fourth point $ D,$ different from $ A,B,C,$ such that the interiors of the segments $ AD,BD,CD$ contain no points of $ L.$ Is the statement true if one considers four points of $ L$ instead of three?
1989 IMO Longlists
In a triangle $ ABC$ for which $ 6(a+b+c)r^2 = abc$ holds and where $ r$ denotes the inradius of $ ABC,$ we consider a point M on the inscribed circle and the projections $ D,E, F$ of $ M$ on the sides $ BC=a, AC=b,$ and $ AB=c$ respectively.  Let $ S, S_1$ denote the areas of the triangles $ ABC$ and $ DEF$ respectively. Find the maximum and minimum values of the quotient $ \frac{S}{S_1}$
1989 IMO Longlists
Let $ ABC$ be an equilateral triangle. Let $ D,E, F,M,N,$ and $ P$ be the mid-points of $ BC, CA, AB, FD, FB,$ and $ DC$ respectively.



(a) Show that the line segments $ AM,EN,$ and $ FP$ are concurrent.

(b) Let $ O$ be the point of intersection of $ AM,EN,$ and $ FP.$ Find $ OM : OF : ON : OE : OP : OA.$