Let $ABCD$ be a convex quadrilateral. A circle passing through the points $A$ and $D$ and a circle passing through the points $B$ and $C$ are externally tangent at a point $P$ inside the quadrilateral. Suppose that
$$
\angle PAB + \angle PDC \leq 90^{\circ} \quad \text{ and } \quad \angle PBA + \angle PCD \leq 90^{\circ}.
$$
Prove that $AB + CD \geq BC + AD$.
We have $n \geq 2$ lamps $L_{1}, \ldots, L_{n}$ in a row, each of them being either on or off. Every second we simultaneously modify the state of each lamp as follows:
- if the lamp $L_{i}$ and its neighbours (only one neighbour for $i=1$ or $i=n$, two neighbours for other $i$) are in the same state, then $L_{i}$ is switched off;
- otherwise, $L_{i}$ is switched on.
Initially all the lamps are off except the leftmost one which is on.
a. Prove that there are infinitely many integers $n$ for which all the lamps will eventually be off.
b. Prove that there are infinitely many integers $n$ for which the lamps will never be all off.
The sequence of real numbers $a_{0}, a_{1}, a_{2}, \ldots$ is defined recursively by
$$
a_{0}=-1, \quad \sum_{k=0}^{n} \frac{a_{n-k}}{k+1}=0 \quad \text{ for } \quad n \geq 1
$$
Show that $a_{n}\gt 0$ for $n \geq 1$.
A sequence of real numbers $a_{0}, a_{1}, a_{2}, \ldots$ is defined by the formula
$$
a_{i+1}=\left\lfloor a_{i}\right\rfloor \cdot\left\langle a_{i}\right\rangle \quad \text{ for } \quad i \geq 0
$$
here $a_{0}$ is an arbitrary real number, $\left\lfloor a_{i}\right\rfloor$ denotes the greatest integer not exceeding $a_{i}$, and $\left\langle a_{i}\right\rangle=a_{i}-\left\lfloor a_{i}\right\rfloor$. Prove that $a_{i}=a_{i+2}$ for $i$ sufficiently large.
Let $ABC$ be a triangle with incentre $I$. A point $P$ in the interior of the triangle satisfies
$$
\angle PBA + \angle PCA = \angle PBC + \angle PCB.
$$
Show that $AP \geq AI$ and that equality holds if and only if $P$ coincides with $I$.
In a triangle $A B C$, let $M_{a}, M_{b}, M_{c}$ be respectively the midpoints of the sides $B C, C A, A B$ and $T_{a}, T_{b}, T_{c}$ be the midpoints of the arcs $B C, C A, A B$ of the circumcircle of $A B C$, not containing the opposite vertices. For $i \in \{a, b, c\}$, let $\omega_{i}$ be the circle with $M_{i} T_{i}$ as diameter. Let $p_{i}$ be the common external tangent to $\omega_{j}, \omega_{k}(\{i, j, k\}=\{a, b, c\})$ such that $\omega_{i}$ lies on the opposite side of $p_{i}$ than $\omega_{j}, \omega_{k}$ do. Prove that the lines $p_{a}, p_{b}, p_{c}$ form a triangle similar to $A B C$ and find the ratio of similitude.
A point $D$ is chosen on the side $AC$ of a triangle $ABC$ with $\angle C \lt \angle A \lt 90^{\circ}$ in such a way that $BD = BA$. The incircle of $ABC$ is tangent to $AB$ and $AC$ at points $K$ and $L$, respectively. Let $J$ be the incentre of triangle $BCD$. Prove that the line $KL$ intersects the line segment $AJ$ at its midpoint.
(Russia)
Let $S$ be a finite set of points in the plane such that no three of them are on a line. For each convex polygon $P$ whose vertices are in $S$, let $a(P)$ be the number of vertices of $P$, and let $b(P)$ be the number of points of $S$ which are outside $P$. Prove that for every real number $x$
$$
\sum_{P} x^{a(P)}(1-x)^{b(P)}=1,
$$
where the sum is taken over all convex polygons with vertices in $S$.
NB. A line segment, a point and the empty set are considered as convex polygons of 2, 1 and 0 vertices, respectively.
A holey triangle is an upward equilateral triangle of side length $n$ with $n$ upward unit triangular holes cut out. A diamond is a $60^{\circ}$-$120^{\circ}$ unit rhombus. Prove that a holey triangle $T$ can be tiled with diamonds if and only if the following condition holds: Every upward equilateral triangle of side length $k$ in $T$ contains at most $k$ holes, for $1 \leq k \leq n$.
(Colombia)
The sequence $f(1), f(2), f(3), \ldots$ is defined by
$$
f(n)=\frac{1}{n}\left(\left\lfloor\frac{n}{1}\right\rfloor+\left\lfloor\frac{n}{2}\right\rfloor+\cdots+\left\lfloor\frac{n}{n}\right\rfloor\right),
$$
where $\lfloor x\rfloor$ denotes the integer part of $x$.
Let $X$ denote the set of all triples $(a, b, c)$ of integers. Define a function $f: X \rightarrow X$ by
$$
f(a, b, c) = (a + b + c, ab + bc + ca, abc)
$$
Find all triples $(a, b, c)$ in $X$ such that $f(f(a, b, c)) = (a, b, c)$.
1. In a non-equilateral triangle $A B C$, the sides $a, b, c$ form an arithmetic progression. Let $I$ and $O$ denote the incentre and circumcentre of the triangle respectively.
a. Prove that $I O$ is perpendicular to $B I$.
b. Suppose $B I$ extended meets $A C$ in $K$, and $D, E$ are the midpoints of $B C, B A$ respectively. Prove that $I$ is the circumcentre of triangle $D K E$.
Let $A_1, A_2, A_3, \dots, A_n$ be $n$ subsets of a finite set $S$ such that $|A_j| = 8$ for each $j$, $1 \le j \le n$. For a subset $B$ of $S$, let $F(B) = \{j : 1 \le j \le n \text{ and } A_j \subset B\}$. Suppose for each subset $B$ of $S$, at least one of the following conditions holds:
(i) $|B| \gt 25$;
(ii) $F(B) = \emptyset$;
(iii) $\cap_{j \in F(B)} A_j \ne \emptyset$.
Prove that for every positive integer $n$ there exists a unique ordered pair $(a, b)$ of positive integers such that
$$
n = \frac{1}{2}(a + b - 1)(a + b - 2) + a
$$
The positive divisors $d_1, d_2, \dots, d_l$ of a natural number $n$ are arranged in the form
$$1 = d_1 \lt d_2 \lt \dots \lt d_l = n.$$
Suppose it is known that $d_1^2 + d_{15}^2 = d_{16}^2$. Find all possible values of $d_{17}$.
There are $n$ markers, each with one side white and the other side black, aligned in a row so that their white sides are all up in the beginning. In each step, if possible, we choose a marker with the white side up (but not the one of outermost markers), remove it, and reverse the closest marker to the left and the closest marker to the right of it. Prove that one can achieve the state with only two markers remaining if and only if $(n-1)$ is not divisible by $3$. (Short-list, IMO-2005)
Let $(u_{jk})$, $1 \le j \le 3$, $1 \le k \le 2$ be real numbers and let $N$ be an integer such that
$$
\max_{1 \le k \le 2} \sum_{j=1}^{3} |u_{jk}| \le N.
$$
Suppose $M$ and $l$ are positive integers such that $l^2 \lt (M+1)^3$. Prove that there exist integers $\xi_1, \xi_2, \xi_3$, not all zero, such that
$$
\max_{1 \le j \le 3} |\xi_j| \le M \quad \text{and} \quad \left| \sum_{j=1}^{3} u_{jk} \xi_j \right| \le \frac{MN}{l}, \quad \text{for } k = 1, 2.
$$
a. Prove that if $n$ is a positive integer such that $n \geq 4011^{2}$, then there exists an integer $l$ such that $n \lt l^{2} \lt \left(1 + \frac{1}{2005}\right) n$.
b. Find the smallest positive integer $M$ for which whenever an integer $n$ is such that $n \geq M$, there exists an integer $l$, such that $n \lt l^{2} \lt \left(1 + \frac{1}{2005}\right) n$.
Let $A_1, A_2, \dots, A_n$ be arithmetic progressions of integers, each of $k$ terms, such that any two of these arithmetic progressions have at least two common elements. Suppose $b$ of these arithmetic progressions have common difference $d_1$ and the remaining arithmetic progressions have common difference $d_2$, where $0 \lt b \lt n$. Prove that
$$
b \le 2 \left( k - \frac{d_2}{\text{gcd}(d_1, d_2)} \right) - 1.
$$