Navvye/Olympiad-Problems-CSV huggingface dataset:
2019 Peru IMO TST
A power is a positive integer of the form $a^k$, where $a$ and $k$ are positive integers with $k\geq 2$. Let $S$ be the set of positive integers which cannot be expressed as sum of two powers (for example, $4,\ 7,\ 15$ and $27$ are elements of $S$). Determine whether the set $S$ has a finite or infinite number of elements.
2019 Peru IMO TST
Let $I,\ O$ and $\Gamma$ be the incenter, circumcenter and the circumcircle of triangle $ABC$, respectively. Line $AI$ meets $\Gamma$ at $M$ $(M\neq A)$. The circumference $\omega$ is tangent internally to $\Gamma$ at $T$, and is tangent to the lines $AB$ and $AC$. The tangents through $A$ and $T$ to $\Gamma$ intersect at $P$. Lines $PI$ and $TM$ meet at $Q$. Prove that the lines $QA$ and $MO$ meet at a point on $\Gamma$.
2019 Peru IMO TST
Let $k\geq 0$ an integer. The sequence $a_0,\ a_1,\ a_2, \ a_3, \ldots$ is defined as follows:



$a_0=k$

For $n\geq 1$, we have that $a_n$ is the smallest integer greater than $a_{n-1}$ so that $a_n+a_{n-1}$ is a perfect square.



Prove that there are exactly $\left \lfloor{\sqrt{2k}} \right \rfloor$ positive integers that cannot be written as the difference of two elements of such a sequence.

Note. If $x$ is a real number, $\left \lfloor{x} \right \rfloor$ denotes the greatest integer smaller or equal than $x$.
2019 Peru IMO TST
Let $m$ and $n$ two given integers. Ana thinks of a pair of real numbers $x$, $y$ and then she tells Beto the values of $x^m+y^m$ and $x^n+y^n$, in this order. Beto's goal is to determine the value of $xy$ using that information. Find all values of $m$ and $n$ for which it is possible for Beto to fulfill his wish, whatever numbers that Ana had chosen.
2019 Peru IMO TST
Let $p$ and $q$ two positive integers. Determine the greatest value of $n$ for which there exists sets $A_1,\ A_2,\ldots,\ A_n$ and $B_1,\ B_2,\ldots,\ B_n$ such that:



The sets $A_1,\ A_2,\ldots,\ A_n$ have $p$ elements each one.

The sets  $B_1,\ B_2,\ldots,\ B_n$ have $q$ elements each one.

For all $1\leq i,\ j \leq n$, sets $A_i$ and $B_j$ are disjoint if and only if $i=j$.
2018 Peru IMO TST
Find all pairs $(p,q)$ of prime numbers which $p>q$ and

$$\frac{(p+q)^{p+q}(p-q)^{p-q}-1}{(p+q)^{p-q}(p-q)^{p+q}-1}$$is an integer.
2018 Peru IMO TST
Let $d$ be a positive integer.

The seqeunce $a_1, a_2, a_3,...$ of positive integers is defined by  $a_1 = 1$ and $a_{n + 1} = n\left \lfloor \frac{a_n}{n}  \right \rfloor+ d$ for $n = 1,2,3, ...$ .

Prove that there exists a positive integer $N$ so that the terms $a_N,a_{N + 1}, a_{N + 2},...$ form an arithmetic progression.



Note: If $x$ is a real number, $\left \lfloor  x \right \rfloor $ denotes the largest integer that is less than or equal to $x$.
2018 Peru IMO TST
Let $ABC$ be, with $AC>AB$, an acute-angled triangle with circumcircle $\Gamma$ and $M$ the midpoint of side $BC$. Let $N$ be a point in the interior of $\bigtriangleup ABC$. Let $D$ and $E$ be the feet of the perpendiculars from $N$ to $AB$ and $AC$, respectively. Suppose that $DE\perp AM$. The circumcircle of $\bigtriangleup ADE$ meets $\Gamma$ at $L$ ($L\neq A$), lines $AL$ and $DE$ intersects at $K$ and line $AN$ meets $\Gamma$ at $F$ ($F\neq A$). Prove that if $N$ is the midpoint of the segment $AF$ then $KA=KF$.
2018 Peru IMO TST
You want to paint some edges of a regular dodecahedron red so that each face has an even number of painted edges (which can be zero). Determine from How many ways this coloration can be done.



Note: A regular dodecahedron has twelve pentagonal faces and in each vertex concur three edges. The edges of the dodecahedron are all different for the purpose of the coloring . In this way, two colorings are the same only if the painted edges they are the same.
2018 Peru IMO TST
A sequence of real numbers $a_1,a_2,\ldots$ satisfies the relation

$$a_n=-\max_{i+j=n}(a_i+a_j)\qquad\text{for all}\quad n>2017.$$Prove that the sequence is bounded, i.e., there is a constant $M$ such that $|a_n|\leq M$ for all positive integers $n$.
2018 Peru IMO TST
For each positive integer $m> 1$, let $P (m)$ be the product of all prime numbers that divide $m$.

Define the sequence $a_1, a_2, a_3,...$ as followed:

$a_1> 1$ is an arbitrary positive integer,

$a_{n + 1} = a_n  + P (a_n)$ for each positive integer $n$.

Prove that there exist positive integers $j$ and $k$ such that $a_j$ is the product of the first $k$  prime numbers.
2017 Peru IMO TST
Find all functions $f: \mathbb{R}\rightarrow \mathbb{R}$ such that



$$ f(xy-1) + f(x)f(y) = 2xy-1 $$for all x and y
2017 Peru IMO TST
Let $n\geq3$ an integer. Mario draws $20$ lines in the plane, such that there are not two parallel lines.

For each equilateral triangle formed by three of these lines, Mario receives three coins.

For each isosceles and non-equilateral triangle (at the same time) formed by three of these lines, Mario receives a coin. How is the maximum number of coins that can Mario receive?
2017 Peru IMO TST
The inscribed circle of the triangle $ABC$ is tangent to the sides $BC, AC$ and $AB$ at points $D, E$ and $F$, respectively. Let $M$ be the midpoint of $EF$. The circle circumscribed around the triangle $DMF$ intersects line $AB$ at $L$, the circle circumscribed around the triangle $DME$ intersects the line $AC$ at $K$. Prove that the circle circumscribed around the triangle  $AKL$ is tangent to the line $BC$.
2017 Peru IMO TST
The product $1\times 2\times  3\times ...\times  n$  is written on the board.  For what integers $n \ge 2$, we can add exclamation marks to some factors to convert them into factorials, in such a way that the final product can be a perfect square?
2017 Peru IMO TST
Let $I$ be the incentre of a non-equilateral triangle $ABC$, $I_A$ be the $A$-excentre, $I'_A$ be the reflection of $I_A$ in $BC$, and $l_A$ be the reflection of line $AI'_A$ in $AI$. Define points $I_B$, $I'_B$ and line $l_B$ analogously. Let $P$ be the intersection point of $l_A$ and $l_B$.



Prove that $P$ lies on line $OI$ where $O$ is the circumcentre of triangle $ABC$.

Let one of the tangents from $P$ to the incircle of triangle $ABC$ meet the circumcircle at points $X$ and $Y$. Show that $\angle XIY = 120^{\circ}$.
2017 Peru IMO TST
Let $ABCD$ be a cyclie quadrilateral, $\omega$ be it's circumcircle and $M$ be the midpoint of the arc  $AB$ of $\omega$  which does not contain the vertices $C$ and $D$. The line that passes through $M$ and the point of intersection of segments $AC$ and $BD$, intersects again $\omega$  in $N$. Let $P$ and $Q$ be points in the $CD$ segment such that $\angle AQD = \angle  DAP$ and $\angle  BPC = \angle CBQ$. Prove that the circumcircle of $NPQ$ and  $\omega$ are tangent to each other.
2017 Peru IMO TST
Let $P (n)$ and $Q (n)$ be two polynomials (not constant) whose coefficients are integers not negative. For each positive integer $n$, define $x_n = 2016^{P (n)} + Q (n)$. Prove that there exist infinite primes $p$ for which there is a positive integer $m$, squarefree, such that $p | x_m$.

Clarification: A positive integer is squarefree if it is not divisible by the square of any prime number.
2017 Peru IMO TST
Let $ABC$ be an acute and scalene  of circumcircle $\Gamma$   and orthocenter $H$. Let  $A_1,B_1,C_1$ be the second points of intersection of the lines $AH, BH, CH$ with   $\Gamma$, respectively. The lines that pass through $A_1,B_1,C_1$ and are parallel to $BC,CA, AB$ intersect again to $\Gamma$ at $A_2,B_2,C_2$, respectively. Let $M$ be the point of intersection of $AC_2$ and $BC_1, N$ the intersection point of $BA_2$ and $CA_1$, and $P$ the point of intersection of $CB_2$ and $AB_1$. Prove that $\angle MNB = \angle  AMP$ .
2017 Peru IMO TST
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(0)\neq 0$ and for all $x,y\in\mathbb{R}$,

$$ f(x+y)^2 = 2f(x)f(y) + \max \left\{ f(x^2+y^2), f(x^2)+f(y^2) \right\}. $$
2017 Peru IMO TST
Let $n$ and $k$ be positive integers. A simple graph $G$ does not contain any cycle whose length be an odd number greater than $1$ and less than $ 2k + 1$. If $G$ has at most $n + \frac{(k-1) (n-1) (n+2)}{2}$  vertices, prove that the vertices of $G$ can be painted with $n$ colors in such a way that any edge of $G$ has its ends of different colors.
2016 Peru IMO TST
The real problem was:



The positive real numbers $a, b, c$ with $ab+bc+ca = 1$ Show that:

$\sqrt{a + \frac{1}{a}} + \sqrt{b + \frac{1}{b}} + \sqrt{c + \frac{1}{c}}$ ≥ $2(\sqrt{a} + \sqrt{b} + \sqrt{c})$
2016 Peru IMO TST
Determine how many $100$-positive integer sequences satisfy the two conditions following:

- At least one term of the sequence is equal to $4$ or $5$.

-  Any two adjacent terms differ as a maximum in $2$.
2016 Peru IMO TST
Let $ABCD$ a convex quadrilateral such that $AD$ and $BC$ are not parallel. Let $M$ and $N$ the midpoints of $AD$ and $BC$ respectively. The segment $MN$ intersects $AC$ and $BD$ in $K$ and $L$ respectively, Show that at least one point of the intersections of the circumcircles of $AKM$ and $BNL$ is in the line $AB$.
2016 Peru IMO TST
Let $n> 2$ be an integer. A child has $n^2$ candies, which are distributed in $n$ boxes. An operation consists in choosing two boxes that together contain an even number of candies and redistribute the candy from those boxes so that both contain the same amount of candy. Determine all the values of $n$ for which the child, after some operations, can get each box containing $n$ candies, no matter which the initial distribution of candies is.
2016 Peru IMO TST
Let $\mathbb{Z}_{>0}$ denote the set of positive integers. Consider a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$. For any $m, n \in \mathbb{Z}_{>0}$ we write $f^n(m) = \underbrace{f(f(\ldots f}_{n}(m)\ldots))$. Suppose that $f$ has the following two properties:



(i) if $m, n \in \mathbb{Z}_{>0}$, then $\frac{f^n(m) - m}{n} \in \mathbb{Z}_{>0}$;

(ii) The set $\mathbb{Z}_{>0} \setminus \{f(n) \mid n\in \mathbb{Z}_{>0}\}$ is finite.



Prove that the sequence $f(1) - 1, f(2) - 2, f(3) - 3, \ldots$ is periodic.



Proposed by Ang Jie Jun, Singapore
2016 Peru IMO TST
Find all pairs $ (m, n)$ of positive integers that have the following property:

For every polynomial $P (x)$ of real coefficients and degree $m$, there exists a polynomial $Q (x)$ of real coefficients and degree $n$ such that $Q (P (x))$ is divisible by $Q (x)$.
2015 Peru IMO TST
Find all positive integers $n$ for which there exist real numbers $x_1, x_2,. . . , x_n$ satisfying all of the following conditions:

(i) $-1 <x_i <1,$ for all $1\leq  i \leq  n.$

(ii) $ x_1 + x_2 + ... + x_n = 0.$

(iii) $\sqrt{1 - x_1^2} +\sqrt{1 - x^2_2} + ... +\sqrt{1 - x^2_n} = 1.$
2015 Peru IMO TST
Ana chose some unit squares of a  $50 \times  50$ board and placed a chip on each of them. Prove that  Beto can always choose at most $99$ empty unit squares and place a chip on each so that each row and each column of the board contains an even number of chips.
2015 Peru IMO TST
Let $M$ be the midpoint of the arc $BAC$ of the circumcircle of the triangle $ABC,$ $I$ the incenter of the triangle $ABC$ and $L$ a point on the side $BC$ such that $AL$ is bisector. The line $MI$ cuts the circumcircle again at $K.$ The circumcircle of the triangle $AKL$ cuts the line $BC$ again at $P.$ Prove that $\angle AIP = 90^{\circ}.$
2015 Peru IMO TST
Let $n\geq  2$ be an integer. The permutation $a_1,a_2,..., a_n$ of the numbers $1, 2,...,n$ is called quadratic if $a_ia_{i +1} + 1$ is a perfect square for all $1\leq  i \leq  n-1.$ The permutation $a_1,a_2,..., a_n$ of the numbers $1, 2,...,n$ is called cubic if $a_ia_{i + 1} + 1$ is a perfect cube for all $1\leq  i \leq  n - 1.$

a) Prove that for infinitely many values of $n$ is there at least one quadratic permutation of the numbers $1, 2,...,n.$

b) Prove that for no value of $n$  is there a cubic permutation of the numbers $1, 2,..., n.$
2015 Peru IMO TST
For a sequence $x_1,x_2,\ldots,x_n$ of real numbers, we define its $\textit{price}$ as $$\max_{1\le i\le n}|x_1+\cdots +x_i|.$$ Given $n$ real numbers, Dave and George want to arrange them into a sequence with a low price. Diligent Dave checks all possible ways and finds the minimum possible price $D$. Greedy George, on the other hand, chooses $x_1$ such that $|x_1 |$ is as small as possible; among the remaining numbers, he chooses $x_2$ such that $|x_1 + x_2 |$ is as small as possible, and so on. Thus, in the $i$-th step he chooses $x_i$ among the remaining numbers so as to minimise the value of $|x_1 + x_2 + \cdots  x_i |$. In each step, if several numbers provide the same value, George chooses one at random. Finally he gets a sequence with price $G$.



Find the least possible constant $c$ such that for every positive integer $n$, for every collection of $n$ real numbers, and for every possible sequence that George might obtain, the resulting values satisfy the inequality $G\le cD$.



Proposed by Georgia
2015 Peru IMO TST
Let $I$ be the incenter of the $ABC$ triangle. The circumference that passes through $I$ and has center

in $A$ intersects the circumscribed circumference of the $ABC$ triangle at points $M$ and

$N$. Prove that the line $MN$ is tangent to the inscribed circle of the $ABC$ triangle.
2015 Peru IMO TST
Let $A$ be a finite set of functions $f: \Bbb{R}\to \Bbb{R.}$  It is known that:



If $f, g\in A$ then $f (g (x)) \in A.$

For all $f \in A$ there exists $g \in A$ such that $f (f (x) + y) = 2x + g (g (y) - x),$ for all $x, y\in \Bbb{R}.$



Let $i:\Bbb{R}\to \Bbb{R}$ be the identity function, ie, $i (x) = x$ for all $x\in \Bbb{R}.$ Prove that $i \in A.$
2015 Peru IMO TST
A card deck consists of $1024$ cards. On each card, a set of distinct decimal digits is written in such a way that no two of these sets coincide (thus, one of the cards is empty). Two players alternately take cards from the deck, one card per turn. After the deck is empty, each player checks if he can throw out one of his cards so that each of the ten digits occurs on an even number of his remaining cards. If one player can do this but the other one cannot, the one who can is the winner; otherwise a draw is declared.

Determine all possible first moves of the first player after which he has a winning strategy.



Proposed by Ilya Bogdanov & Vladimir Bragin, Russia
2015 Peru IMO TST
Find the least positive real number $\alpha$  with the following property: if the weight of a finite number of pumpkins is $1$ ton and the weight of each pumpkin is not greater than $\alpha$   tons then the pumpkins can be distributed in $50$ boxes (some boxes can be empty) so that there is no more than $\alpha$ tons of pumpkins in each box.
2015 Peru IMO TST
Let $ABC$ be a triangle with circumcircle $\Omega$ and incentre $I$. Let the line passing through $I$ and perpendicular to $CI$ intersect the segment $BC$ and the arc $BC$ (not containing $A$) of $\Omega$ at points $U$ and $V$ , respectively. Let the line passing through $U$ and parallel to $AI$ intersect $AV$ at $X$, and let the line passing through $V$ and parallel to $AI$ intersect $AB$ at $Y$ . Let $W$ and $Z$ be the midpoints of $AX$ and $BC$, respectively. Prove that if the points $I, X,$ and $Y$ are collinear, then the points $I, W ,$ and $Z$ are also collinear.



Proposed by David B. Rush, USA
2015 Peru IMO TST
Let $ n$ be a positive integer and let $ a_1,a_2,\ldots,a_n$ be positive real numbers  such that:

$$ \sum^n_{i = 1} a_i = \sum^n_{i = 1} \frac {1}{a_i^2}.

$$

Prove that for every $ i = 1,2,\ldots,n$ we can find $ i$ numbers with sum at least $ i$.
2015 Peru IMO TST
Let $c \ge 1$ be an integer. Define a sequence of positive integers by $a_1 = c$ and $$a_{n+1}=a_n^3-4c\cdot a_n^2+5c^2\cdot a_n+c$$ for all $n\ge 1$. Prove that for each integer $n \ge 2$ there exists a prime number $p$ dividing $a_n$ but none of the numbers $a_1 , \ldots , a_{n -1}$ .



Proposed by Austria
2014 Peru IMO TST
a) Find at least two functions $f: \mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that  $$\displaystyle{2f(x^2)\geq xf(x) + x,}$$for all $x \in  \mathbb{R}^+.$



b) Let $f: \mathbb{R}^+ \rightarrow \mathbb{R}^+$ be a function such that $$\displaystyle{2f(x^2)\geq xf(x) + x,}$$for all $x \in  \mathbb{R}^+.$ Show that $ f(x^3)\geq x^2,$ for all $x \in  \mathbb{R}^+.$



Can we find the best constant $a\in \Bbb{R}$ such that $f(x)\geq x^a,$ for all $x \in  \mathbb{R}^+?$
2014 Peru IMO TST
Let $n$ be a positive integer. There is an infinite number of cards, each one of them having a non-negative integer written on it, such that for each integer $l \geq 0$, there are exactly $n$ cards that have the number $l$ written on them. A move consists of picking $100$ cards from the infinite set of cards and discarding them. Find the least possible value of $n$ for which there is an infinitely long series of moves such that for each positive integer $k$, the sum of the numbers written on the $100$ chosen cards during the $k$-th move is equal to $k$.
2014 Peru IMO TST
Let $ABC$ be an acuteangled triangle with $AB> BC$ inscribed in a circle. The perpendicular bisector of the side $AC$ cuts arc $AC,$ containing $B,$ in $Q.$ Let $M$ be a point on the segment $AB$ such that $AM = MB + BC.$ Prove that the circumcircle of the triangle $BMC$ cuts $BQ$ in its midpoint.
2014 Peru IMO TST
A positive integer is called lonely  if the sum of the reciprocals of its positive divisors (including 1 and itself) is different from the sum of the reciprocals of the positive divisors of any positive integer.

a) Prove that every prime number is lonely.

b) Prove that there are infinitely many positive integers that are not lonely.
2014 Peru IMO TST
$n$ vertices from a regular polygon with $2n$ sides are chosen and coloured red. The other $n$ vertices are coloured blue. Afterwards, the $\binom{n}{2}$ lengths of the segments formed with all pairs of red vertices are ordered in a non-decreasing sequence, and the same procedure is done with the $\binom{n}{2}$ lengths of the segments formed with all pairs of blue vertices. Prove that both sequences are identical.
2014 Peru IMO TST
Let $ABC$ be a triangle where $AB > BC$, and $D$ and $E$ be points on sides $AB$ and $AC$ respectively, such that $DE$ and $AC$ are parallel. Consider the circumscribed circumference of triangle $ABC$. A circumference that passes through points $D$ and $E$ is tangent to the arc $AC$ that does not contain $B$ at point $P$. Let $Q$ be the reflection of point $P$ with respect to the perpendicular bisector of $AC$. The segments $BQ$ and $DE$ intersect at $X$. Prove that $AX = XC$.
2014 Peru IMO TST
Let $n$ be a positive integer. Mariano divides a rectangle into $n^2$ smaller rectangles by drawing $n-1$ vertical lines and $n-1$ horizontal lines, parallel to the sides of the larger rectangle. On every step, Emilio picks one of the smaller rectangles and Mariano tells him its area. Find the least positive integer $k$ for which it is possible that Emilio can do $k$ conveniently thought steps in such a way that with the received information, he can determine the area of each one of the $n^2$ smaller rectangles.
2014 Peru IMO TST
Let $x, y, z$ be real numbers such that $$\displaystyle{\begin{cases} x^2+y^2+z^2+(x+y+z)^2=9 \\ xyz \leq \frac{15}{32} \end{cases}} $$Find the maximum possible value of $x.$
2014 Peru IMO TST
Prove that for every positive integer $n$ there exist integers $a$ and $b,$ both greater than $1,$ such that $a ^ 2 + 1 = 2b ^ 2$ and $a - b$ is a multiple of $n.$
2014 Peru IMO TST
Let $ABCDEF$ be a convex hexagon that does not have two parallel sides, such that $\angle AF B = \angle F DE,  \angle DF E = \angle BDC$  and $\angle BFC = \angle ADF.$ Prove that the lines $ AB, FC$  and $DE$  are concurrent if and only if the lines $ AF, BE$  and $CD$ are concurrent.
2014 Peru IMO TST
Let $ABC$ be a triangle, and $P$ be a variable point inside $ABC$ such that $AP$ and $CP$ intersect sides $BC$ and $AB$ at $D$ and $E$ respectively, and the area of the triangle $APC$ is equal to the area of quadrilateral $BDPE$. Prove that the circumscribed circumference of triangle $BDE$ passes through a fixed point different from $B$.
2014 Peru IMO TST
Every single point on the plane with integer coordinates is coloured either red, green or blue. Find the least possible positive integer $n$ with the following property: no matter how the points are coloured, there is always a triangle with area $n$ that has its $3$ vertices with the same colour.
2014 Peru IMO TST
Let $r$ be a positive integer and let $N$ be the smallest positive integer such that the numbers $\frac{N}{n+r}\binom{2n}{n}$,

$n=0,1,2,\ldots $, are all integer. Show that $N=\frac{r}{2}\binom{2r}{r}$.
2014 Peru IMO TST
Let $n$ be a positive integer, and consider a sequence $a_1 , a_2 , \cdots , a_n $ of positive integers. Extend it periodically to an infinite sequence $a_1 , a_2 , \cdots $ by defining $a_{n+i} = a_i $ for all $i \ge 1$. If $$a_1 \le a_2 \le \cdots \le a_n \le a_1 +n  $$ and $$a_{a_i } \le n+i-1 \quad\text{for}\quad i=1,2,\cdots, n, $$ prove that $$a_1 + \cdots +a_n \le n^2. $$
2014 Peru IMO TST
Let $n$ be a positive integer, and let $A$ be a subset of $\{ 1,\cdots ,n\}$. An $A$-partition of $n$ into $k$ parts is a representation of n as a sum $n = a_1 + \cdots + a_k$, where the parts $a_1 , \cdots , a_k $ belong to $A$ and are not necessarily distinct. The number of different parts in such a partition is the number of (distinct) elements in the set $\{ a_1 , a_2 , \cdots , a_k \} $.

We say that an $A$-partition of $n$ into $k$ parts is optimal if there is no $A$-partition of $n$ into $r$ parts with $r<k$. Prove that any optimal $A$-partition of $n$ contains at most $\sqrt[3]{6n}$ different parts.
2013 Peru IMO TST
Several positive integers are written in a row. Iteratively, Alice chooses two adjacent numbers $x$ and $y$ such that $x>y$ and $x$ is to the left of $y$, and replaces the pair $(x,y)$ by either $(y+1,x)$ or $(x-1,x)$. Prove that she can perform only finitely many such iterations.



Proposed by Warut Suksompong, Thailand
2013 Peru IMO TST
Let  $a \geq 3$ be a real number, and $P$ a polynomial of degree $n$ and having real coefficients. Prove that at least one of the following numbers is greater than or equal to $1:$ $$|a^0- P(0)|, \ |a^1-P(1)| , \ |a^2-P(2)|, \cdots, |a^{n + 1}-P(n + 1)|.$$
2013 Peru IMO TST
A point $P$ lies on side $AB$ of a convex quadrilateral $ABCD$. Let $\omega$ be the inscribed circumference of triangle $CPD$ and $I$ the centre of $\omega$. It is known that $\omega$ is tangent to the inscribed circumferences of triangles $APD$ and $BPC$ at points $K$ and $L$ respectively. Let $E$ be the point where the lines $AC$ and $BD$ intersect, and $F$ the point where the lines $AK$ and $BL$ intersect. Prove that the points $E, I, F$ are collinear.
2013 Peru IMO TST
Let $A$ be a point outside of a circumference $\omega$. Through $A$, two lines are drawn that intersect $\omega$, the first one cuts $\omega$ at $B$ and $C$, while the other one cuts $\omega$ at $D$ and $E$ ($D$ is between $A$ and $E$). The line that passes through $D$ and is parallel to $BC$ intersects $\omega$ at point $F \neq D$, and the line $AF$ intersects $\omega$ at $T \neq F$. Let $M$ be the intersection point of lines $BC$ and $ET$, $N$ the point symmetrical to $A$ with respect to $M$, and $K$ be the midpoint of $BC$. Prove that the quadrilateral $DEKN$ is cyclic.
2013 Peru IMO TST
Determine all integers $m \geq 2$ such that every $n$ with $\frac{m}{3} \leq n \leq \frac{m}{2}$ divides the binomial coefficient $\binom{n}{m-2n}$.
2012 Peru IMO TST
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $$\displaystyle{f(f(x)) = \frac{x^2 - x}{2}\cdot f(x) + 2-x,}$$for all $x \in  \mathbb{R}.$ Find all possible values of $f(2).$
2012 Peru IMO TST
Let $a, b, c$ be the lengths of the sides of a triangle, and $h_a, h_b, h_c$ the lengths of the heights corresponding to the sides $a, b, c,$ respectively. If $t \geq \frac{1} {2}$ is a real number, show that there is a triangle with sidelengths $$ t\cdot a + h_a, \  t\cdot b + h_b , \  t\cdot c + h_c.$$