a) The real number $a$ and the continuous function $f : [a, \infty) \rightarrow [a, \infty)$ are such that $|f(x)-f(y)| < |x–y|$ for every two different $x, y \in [a, \infty)$. Is it always true that the equation $f(x)=x$ has only one solution in the interval $[a, \infty)$? b) The real numbers $a$ and $b$ and the continuous function $f : [a, b] \rightarrow [a, b]$ are such that $|f(x)-f(y)| < |x–y|$, for every two different $x, y \in [a, b]$. Is it always true that the equation $f(x)=x$ has only one solution in the interval $[a, b]$?

In triangle $ABC$, $M$ is the midpoint of side $BC$, the bisector of angle $BAC$ intersects $BC$ and $(ABC)$ at $K$ and $L$, respectively. If the circle with diameter $[BC]$ is tangent to the external angle bisector of angle $BAC$, prove that this circle is tangent to $(KLM)$ as well.

Plot points $A,B,C$ at coordinates $(0, 0)$, $(0, 1)$, and $(1,1)$ in the plane, respectively. Let $S$ denote the union of the two line segments $AB$ and $BC$. Let $X_1$ be the area swept out when Bobby rotates $S$ counterclockwise $45$ degrees about point $A$. Let $X_2$ be the area swept out when Calvin rotates $S$ clockwise $45$ degrees about point $A$. Find $\frac{X_1+X_2}{2}$ .

A house is in the shape of a triangle, perimeter $P$ metres and area $A$ square metres. The garden consists of all the land within 5 metres of the house. How much land do the garden and house together occupy?

Let $P$ be an interior point of a circle $\Gamma$ centered at $O$ where $P \ne O$. Let $A$ and $B$ be distinct points on $\Gamma$. Lines $AP$ and $BP$ meet $\Gamma$ again at $C$ and $D$, respectively. Let $S$ be any interior point on line segment $PC$. The circumcircle of $\vartriangle ABS$ intersects line segment $PD$ at $T$. The line through $S$ perpendicular to $AC$ intersects $\Gamma$ at $U$ and $V$ . The line through $T$ perpendicular to $BD$ intersects $\Gamma$ at $X$ and $Y$ . Let $M$ and $N$ be the midpoints of $UV$ and $XY$ , respectively. Let $AM$ and $BN$ meet at $Q$. Suppose that $AB$ is not parallel to $CD$. Show that $P, Q$, and $O$ are collinear if and only if $S$ is the midpoint of $PC$.

Let $n\geqslant 2$ be an integer. Prove that for any positive real numbers $a_1, a_2,\ldots, a_n$, \[\frac{1}{2\sqrt{2}}\sum_{i=1}^{n}2^{i}a_i^2 \geqslant\sum_{1 \leqslant i < j \leqslant n}a_i a_j.\][i]Proposed by Andrei Vila[/i]

For positive rational $x$, if $x$ is written in the form $p/q$ with $p, q$ positive relatively prime integers, define $f(x)=p+q$. For example, $f(1)=2$. a) Prove that if $f(x)=f(mx/n)$ for rational $x$ and positive integers $m, n$, then $f(x)$ divides $|m-n|$. b) Let $n$ be a positive integer. If all $x$ which satisfy $f(x)=f(2^nx)$ also satisfy $f(x)=2^n-1$, find all possible values of $n$. [i]Anderson Wang.[/i]

Let $x_0, x_1, \ldots, x_{n-1}, x_n=x_0$ be reals and let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function. The numbers $y_i$ for $i=0,1, \ldots, n-1$ are chosen such that $y_i$ is between $x_i$ and $x_{i+1}$. Prove that $\sum_{i=0}^{n-1}(x_{i+1}-x_i)f(y_i)$ can attain both positive and negative values, by varying the $y_i$.

Compute the largest prime factor of $3^{12}+3^9+3^5+1.$

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.

A rectangular city is exactly $m$ blocks long and $n$ blocks wide (see diagram). A woman lives in the southwest corner of the city and works in the northeast corner. She walks to work each day but, on any given trip, she makes sure that her path does not include any intersection twice. Show that the number $f(m,n)$ of different paths she can take to work satisfies $f(m,n) \le 2^{mn}$. [asy] unitsize(0.4 cm); for(int i = 0; i <= 11; ++i) { draw((i,0)--(i,7)); } for(int j = 0; j <= 7; ++j) { draw((0,j)--(11,j)); } label("$\underbrace{\hspace{4.4 cm}}$", (11/2,-0.5)); label("$\left. \begin{array}{c} \vspace{2.4 cm} \end{array} \right\}$", (11,7/2)); label("$m$ blocks", (11/2,-1.5)); label("$n$ blocks", (14,7/2)); [/asy]

Prove that there is no continuous function $ f: \mathbb{R} \to \mathbb{R} $ satisfying the condition $ f(f(x)) = - x $ for every $ x $.

Let $n$ be a positive integer. There exist $n$ ordered triples$$(x_1, y_1, z_1), (x_2, y_2, z_2), \dots, (x_n, y_n, z_n)$$where each coordinate is an integer between $1$ and $100$ (inclusive), satisfying the following condition: [center] [i]For every infinite sequence $(a_1, a_2, a_3, \dots)$ of integers between $1$ and $100$, there exist a positive integer $i$ and an index $j$ (with $1 \leqslant j \leqslant n$) such that $(a_i, a_{i+1}, a_{i+2}) = (x_j, y_j, z_j)$.[/i] [/center] Determine the minimum possible value of $n$.

Α disk of radius $1$ is covered by seven identical disks. Prove that their radii are not less than $\frac12$ .

Let $\mathbb{N}$ be the set of positive integers. Determine all positive integers $k$ for which there exist functions $f:\mathbb{N} \to \mathbb{N}$ and $g: \mathbb{N}\to \mathbb{N}$ such that $g$ assumes infinitely many values and such that $$ f^{g(n)}(n)=f(n)+k$$ holds for every positive integer $n$. ([i]Remark.[/i] Here, $f^{i}$ denotes the function $f$ applied $i$ times i.e $f^{i}(j)=f(f(\dots f(j)\dots ))$.)

Find all integer solutions $(x,y)$ to the equation $\displaystyle \frac{x+y}{x^2-xy+y^2}=\frac{3}{7}$.

There are $1000$ balls and $500$ bins that can fit arbitrarily many balls. All of the balls are then placed independently and at random into the bins. Estimate how many bins, on average, are empty. (Estimate the expected number of empty bins. In other words, if this were done over and over again, how many bins would be empty on average?) Your estimate must be an integer, or you will receive a score of zero.

A particle moves so that its speed for the second and subsequent miles varies inversely as the integral number of miles already traveled. For each subsequent mile the speed is constant. If the second mile is traversed in $2$ hours, then the time, in hours, needed to traverse the $n$th mile is: $\textbf{(A) }\dfrac2{n-1}\qquad \textbf{(B) }\dfrac{n-1}2\qquad \textbf{(C) }\dfrac2n\qquad \textbf{(D) }2n\qquad \textbf{(E) }2(n-1)$

Let be three real numbers $ u,v,t $ under the condition $ u+v+t=0. $ Prove that for any positive real number $ a\neq 1 $ the following inequality is true with equality only and only if $ u=v=t=0: $ $$ a^u/a^v+a^v/a^t+a^{v+t}\ge a^u+a^v+1 $$ [i]Ion Tecu[/i]

In an isosceles triangle $ABC$ with $AB = AC$, $D$ is the midpoint of $AC$ and $E$ is the projection of $D$ onto $BC$. Let $F$ be the midpoint of $DE$. Prove that the lines $BF$ and $AE$ are perpendicular if and only if the triangle $ABC$ is equilateral.

Find all pairs of positive integers $(m, n)$ satisfying the equation $$m!+n!=m^n+1.$$

Let $n$ be a positive integer. Elfie the Elf travels in $\mathbb{R}^3$. She starts at the origin: $(0,0,0)$. In each turn she can teleport to any point with integer coordinates which lies at distance exactly $\sqrt{n}$ from her current location. However, teleportation is a complicated procedure: Elfie starts off [i]normal[/i] but she turns [i]strange[/i] with her first teleportation. Next time she teleports she turns [i]normal[/i] again, then [i]strange [/i]again... etc. For which $n$ can Elfie travel to any point with integer coordinates and be [i]normal [/i]when she gets there?

Prove that any finite set $H$ of lattice points on the plane has a subset $K$ with the following properties: [list] [*]any vertical or horizontal line in the plane cuts $K$ in at most $2$ points, [*]any point of $H\setminus K$ is contained by a segment with endpoints from $K$.[/list]

Let $f(x) = 15x - 2016$. If $f(f(f(f(f(x))))) = f(x)$, find the sum of all possible values of $x$.

Let $b_1$, $\dots$ , $b_n$ be nonnegative integers with sum $2$ and $a_0$, $a_1$, $\dots$ , $a_n$ be real numbers such that $a_0=a_n=0$ and $|a_i-a_{i-1}|\leq b_i$ for each $i=1$, $\dots$ , $n$. Prove that $$\sum_{i=1}^n(a_i+a_{i-1})b_i\leq 2$$ [hide]I believe that the original problem was for nonnegative real numbers and it was a typo on the version of the exam paper we had but I'm not sure the inequality would hold[/hide]