Navvye/Olympiad-Problems-CSV huggingface dataset:
2014 Dutch IMO TST
Let $a$, $b$ and $c$ be rational numbers for which $a+bc$, $b+ac$ and $a+b$ are all non-zero and for which we have

$$\frac{1}{a+bc}+\frac{1}{b+ac}=\frac{1}{a+b}.$$

Prove that $\sqrt{(c-3)(c+1)}$ is rational.
2014 Dutch IMO TST
Let $\triangle ABC$ be a triangle with $|AC|=2|AB|$ and let $O$ be its circumcenter. Let $D$ be the intersection of the bisector of $\angle A$ with $BC$. Let $E$ be the orthogonal projection of $O$ to $AD$ and let $F\ne D$ be the point on $AD$ satisfying $|CD|=|CF|$. Prove that $\angle EBF=\angle ECF$.
2014 Dutch IMO TST
On each of the $2014^2$ squares of a $2014 \times 2014$-board a light bulb is put. Light bulbs can be either on or off. In the starting situation a number of the light bulbs is on. A move consists of choosing a row or column in which at least $1007$ light bulbs are on and changing the state of all $2014$ light bulbs in this row or column (from on to off or from off to on). Find the smallest non-negative integer $k$ such that from each starting situation there is a finite sequence of moves to a situation in which at most $k$ light bulbs are on.
2014 Dutch IMO TST
Let $ f:\mathbb{Z}_{>0}\rightarrow\mathbb{R} $ be a function such that for all $n > 1$ there is a prime divisor $p$ of $n$ such that $$ f(n)=f\left(\frac{n}{p}\right)-f(p). $$

Furthermore, it is given that $ f(2^{2014})+f(3^{2015})+f(5^{2016})=2013 $. Determine $ f(2014^2)+f(2015^3)+f(2016^5) $.
2014 Dutch IMO TST
The sets $A$ and $B$ are subsets of the positive integers. The sum of any two distinct elements of $A$ is an element of $B$. The quotient of any two distinct elements of $B$ (where we divide the largest by the smallest of the two) is an element of $A$. Determine the maximum number of elements in $A\cup B$.
2014 Dutch IMO TST
Let $H$ be the orthocentre of an acute triangle $ABC$. The line through $A$ perpendicular to $AC$ and the line through $B$ perpendicular to $BC$ intersect in $D$. The circle with centre $C$ through $H$ intersects the circumcircle of triangle $ABC$ in the points $E$ and $F$. Prove that $|DE| = |DF| = |AB|$.
2014 Dutch IMO TST
Determine all pairs $(p, q)$ of primes for which $p^{q+1}+q^{p+1}$ is a perfect square.
2014 Dutch IMO TST
Let $P(x)$ be a polynomial of degree $n \le 10$ with integral coefficients such that for every $k \in \{1, 2, \dots, 10\}$ there is an integer $m$ with $P(m) = k$. Furthermore, it is given that $|P(10) - P(0)| < 1000$. Prove that for every integer $k$ there is an integer $m$ such that $P(m) = k.$
2013 Dutch IMO TST
Determine all 4-tuples ($a, b,c, d$) of real numbers satisfying the following four equations: $\begin{cases} ab + c + d = 3 \\
bc + d + a = 5 \\
cd + a + b = 2 \\
da + b + c = 6 \end{cases}$
2013 Dutch IMO TST
Determine all integers $n$ for which $\frac{4n-2}{n+5}$ is the square of a rational number.
2013 Dutch IMO TST
Fix a triangle $ABC$. Let $\Gamma_1$ the circle through $B$, tangent  to edge in $A$. Let  $\Gamma_2$ the circle through C tangent to edge $AB$ in $A$. The second intersection of $\Gamma_1$ and $\Gamma_2$ is denoted by $D$. The line $AD$ has second intersection $E$ with the circumcircle of  $\vartriangle ABC$. Show that $D$ is the midpoint of the segment $AE$.
2013 Dutch IMO TST
Let $n \ge 3$ be an integer, and consider a $n \times n$-board, divided into $n^2$ unit squares. For all $m \ge 1$, arbitrarily many $1\times m$-rectangles (type I) and arbitrarily many $m\times 1$-rectangles (type II) are available. We cover the board with $N$ such rectangles, without overlaps, and such that every rectangle lies entirely inside the board. We require that the number of type I rectangles used is equal to the number of type II rectangles used.(Note that a $1 \times 1$-rectangle has both types.)

What is the minimal value of $N$ for which this is possible?
2013 Dutch IMO TST
Let $a, b$, and $c$ be positive real numbers satisfying $abc = 1$.

Show that $a + b + c \ge \sqrt{\frac13  (a + 2)(b + 2)(c + 2)}$
2013 Dutch IMO TST
Show that $\sum_{n=0}^{2013}\frac{4026!}{(n!(2013-n)!)^2}$ is a perfect square.
2013 Dutch IMO TST
Let $P$ be the point of intersection of the diagonals of a convex quadrilateral $ABCD$.Let $X,Y,Z$ be points on the interior of $AB,BC,CD$ respectively such that $\frac{AX}{XB}=\frac{BY}{YC}=\frac{CZ}{ZD}=2$. Suppose that $XY$ is tangent to the circumcircle of $\triangle CYZ$ and that $Y Z$ is tangent to the circumcircle of $\triangle BXY$.Show that $\angle APD=\angle XYZ$.
2013 Dutch IMO TST
Fix a sequence $a_1,a_2,a_3\ldots$ of integers satisfying the following condition:for all prime numbers $p$ and all positive integers $k$,we have $a_{pk+1}=pa_k-3a_p+13$.Determine all possible values of $a_{2013}$.
2013 Dutch IMO TST
Determine all positive integers $n\ge 2$ satisfying $i+j\equiv\binom ni +\binom nj \pmod{2}$ for all $i$ and $j$ with $0\le i\le j\le n$.
2013 Dutch IMO TST
Let $ABCDEF$ be a cyclic hexagon satisfying $AB\perp BD$ and $BC=EF$.Let $P$ be the intersection of lines $BC$ and $AD$ and let $Q$ be the intersection of lines $EF$ and $AD$.Assume that $P$ and $Q$ are on the same side of $D$ and $A$ is on the opposite side.Let $S$ be the midpoint of $AD$.Let $K$ and $L$ be the incentres of $\triangle BPS$ and $\triangle EQS$ respectively.Prove that $\angle KDL=90^0$.
2012 Dutch IMO TST
Let $a, b, c$ and $d$ be positive real numbers. Prove that

$$\frac{a - b}{b + c}+\frac{b - c}{c + d}+\frac{c - d}{d + a} +\frac{d - a}{a + b } \ge 0 $$
2012 Dutch IMO TST
Determine all positive integers that cannot be written as $\frac{a}{b} + \frac{a+1}{b+1}$ where $a$ and $b$ are positive integers.
2012 Dutch IMO TST
Let $n$ be a positive integer divisible by $4$. We consider the permutations $(a_1, a_2,...,a_n)$ of $(1,2,..., n)$ having the following property: for each j we have $a_i + j = n + 1$ where $i = a_j$ . Prove that there are exactly $\frac{ (\frac12 n)!}{(\frac14 n)!}$ such permutations.
2012 Dutch IMO TST
Let $\Gamma$ be the circumcircle of the acute triangle $ABC$. The angle bisector of angle $ABC$ intersects $AC$ in the point $B_1$ and the short arc $AC$ of $\Gamma$  in the point $P$. The line through $B_1$ perpendicular to $BC$ intersects the short arc $BC$ of $\Gamma$  in $K$. The line through $B$ perpendicular to $AK$ intersects $AC$ in $L$. Prove that $K, L$ and $P$ lie on a line.
2012 Dutch IMO TST
For all positive integers $a$ and $b$, we dene $a @ b = \frac{a - b}{gcd(a, b)}$ .

Show that for every integer $n > 1$, the following holds:

$n$ is a prime power if and only if for all positive integers $m$ such that $m < n$, it holds that $gcd(n, n @m) = 1$.
2012 Dutch IMO TST
There are two boxes containing balls. One of them contains $m$ balls, and the other contains $n$ balls, where $m, n > 0$. Two actions are permitted:

(i) Remove an equal number of balls from both boxes.

(ii) Increase the number of balls in one of the boxes by a factor $k$.

Is it possible to remove all of the balls from both boxes with just these two actions,

1. if $k = 2$?

2. if $k = 3$?
2012 Dutch IMO TST
Determine all pairs $(x, y)$ of positive integers satisfying

$x + y + 1 | 2xy$ and $ x + y - 1 | x^2 + y^2 - 1$.
2012 Dutch IMO TST
Let $\vartriangle ABC$ be a triangle. The angle bisector of $\angle CAB$ intersects$ BC$ at $L$. On the interior of line segments $AC$ and $AB$, two points, $M$ and $N$, respectively, are chosen in such a way that the lines $AL, BM$ and $CN$ are concurrent, and such that $\angle AMN = \angle ALB$. Prove that $\angle NML = 90^o$.
2012 Dutch IMO TST
Find all functions $f : R \to R$ satisfying $f(x + xy + f(y))=(f(x) + \frac12)(f(y) + \frac12 )$ for all $x, y \in  R$.
2011 Dutch IMO TST
Find all pairs $(x, y)$ of integers that satisfy $x^2 + y^2 + 3^3 = 456\sqrt{x - y}$.
2011 Dutch IMO TST
We consider tilings of a rectangular $m \times n$-board with $1\times2$-tiles. The tiles can be placed either horizontally, or vertically, but they aren't allowed to overlap and to be placed partially outside of the board. All squares on theboard must be covered by a tile.

(a) Prove that for every tiling of a $4 \times 2010$-board with $1\times2$-tiles there is a straight line cutting the board into two pieces such that every tile completely lies within one of the pieces.

(b) Prove that there exists a tiling of a $5 \times  2010$-board with $1\times 2$-tiles such that there is no straight line cutting the board into two pieces such that every tile completely lies within one of the pieces.
2011 Dutch IMO TST
The circles $\Gamma_1$ and $\Gamma_2$ intersect at $D$ and $P$. The common tangent line of the two circles closest to point $D$ touches $\Gamma_1$ in A and $\Gamma_2$ in $B$. The line $AD$ intersects $\Gamma_2$ for the second time in $C$. Let $M$ be the middle of line segment $BC$. Prove that $\angle DPM = \angle BDC$.
2011 Dutch IMO TST
Determine all integers $n$ for which the polynomial $P(x) = 3x^3-nx-n-2$ can be written as the product of two non-constant polynomials with integer coeffcients.
2011 Dutch IMO TST
Let $ABC$ be a triangle with $|AB|> |BC|$. Let $D$ be the midpoint of $AC$. Let $E$ be the intersection of the angular bisector of $\angle ABC$ and the line $AC$. Let $F$ be the point on $BE$ such that $CF$ is perpendicular to $BE$. Finally, let $G$ be the intersection of $CF$ and $BD$. Prove that $DF$ divides the line segment $EG$ into two equal parts.
2011 Dutch IMO TST
Let $n \ge 2$ and $k  \ge1$ be positive integers. In a country there are $n$ cities and between each pair of cities there is a bus connection in both directions. Let $A$ and $B$ be two different cities. Prove that the number of ways in which you can travel from $A$ to $B$ by using exactly $k$ buses is equal to $\frac{(n - 1)^k - (-1)^k}{n}$

.
2011 Dutch IMO TST
Find all functions $f : R\to R$ satisfying $xf(x + xy) = xf(x) + f(x^2)f(y)$ for all $x, y \in R$.
2011 Dutch IMO TST
Let $\Gamma_1$ and $\Gamma_2$ be two intersecting circles with midpoints respectively $O_1$ and $O_2$, such that $\Gamma_2$ intersects the line segment $O_1O_2$ in a point $A$. The intersection points of $\Gamma_1$ and $\Gamma_2$ are $C$ and $D$. The line $AD$ intersects $\Gamma_1$ a second time in $S$. The line $CS$ intersects $O_1O_2$ in $F$. Let $\Gamma_3$ be the circumcircle of triangle $AD$. Let $E$ be the second intersection point of $\Gamma_1$ and $\Gamma_3$. Prove that $O_1E$ is tangent to $\Gamma_3$.
2011 Dutch IMO TST
Prove that there exists no innite sequence of prime numbers $p_0, p_1, p_2,...$ such that for all positive integers $k$:

$p_k = 2p_{k-1} + 1$ or $p_k = 2p_{k-1} - 1$.
2011 Dutch IMO TST
Find all triples $(a, b, c)$ of positive integers with $a+b+c = 10$ such that there are $a$ red, $b$ blue and $c$ green points (all different) in the plane satisfying the following properties:

$\bullet$ for each red point and each blue point we consider the distance between these two points, the sum of these distances is $37$,

$\bullet$ for each green point and each red point we consider the distance between these two points, the sum of these distances is $30$,

$\bullet$ for each blue point and each green point we consider the distance between these two points, the sum of these distances is $1$.
2010 Dutch IMO TST
Let $ABC$ be an acute triangle such that $\angle BAC = 45^o$. Let $D$ a point on $AB$ such that $CD \perp AB$. Let $P$ be an internal point of the segment $CD$. Prove that $AP\perp BC$ if and only if $|AP| = |BC|$.
2010 Dutch IMO TST
Let $A$ and $B$ be positive integers. Define the arithmetic sequence $a_0, a_1, a_2, ...$ by $a_n = A_n + B$. Suppose that there exists an $n\ge 0$ such that $a_n$ is a square. Let $M$ be a positive integer such that $M^2$ is the smallest square in the sequence. Prove that $M < A +\sqrt{B}$.
2010 Dutch IMO TST
Let $n\ge  2$ be a positive integer and $p $ a prime such that $n|p-1$ and $p | n^3-1$. Show $ 4p-3$ is a square.
2010 Dutch IMO TST
Let $ABCD$ be a cyclic quadrilateral satisfying $\angle ABD = \angle DBC$. Let $E$ be the intersection of the diagonals $AC$ and $BD$. Let $M$ be the midpoint of $AE$, and $N$ be the midpoint of $DC$. Show that $MBCN$ is a cyclic quadrilateral.
2010 Dutch IMO TST
Find all triples $(x,y, z)$ of real (but not necessarily positive) numbers satisfying

$3(x^2 + y^2 + z^2) = 1$ , $x^2y^2 + y^2z^2 + z^2x^2 = xyz(x + y + z)^3$.
2010 Dutch IMO TST
Consider sequences $a_1, a_2, a_3,...$ of positive integers. Determine the smallest possible value of $a_{2010}$ if

(i) $a_n < a_{n+1}$ for all $n\ge 1$,

(ii) $a_i + a_l > a_j + a_k$ for all quadruples $ (i, j, k, l)$ which satisfy $1 \le i < j \le k < l$.
2010 Dutch IMO TST
Find all functions $f : R \to R$ which satisfy $f(x) = max_{y\in R} (2xy - f(y))$ for all $x \in  R$.
2010 Dutch IMO TST
(a) Let $a$ and $b$ be positive integers such that $M(a, b) = a - \frac1b +b(b + \frac3a)$ is an integer.

Prove that $M(a,b)$ is a square.

(b) Find nonzero integers $a$ and $b$ such that $M(a,b)$ is a positive integer, but not a square.
2010 Dutch IMO TST
Let $ABCD$ be a square with circumcircle $\Gamma_1$. Let $P$ be a point on the arc $AC$ that also contains $B$. A circle $\Gamma_2$ touches $\Gamma_1$ in $P$ and also touches the diagonal $AC$ in $Q$. Let $R$ be a point on $\Gamma_2$ such that the line $DR$ touches $\Gamma_2$. Proof that $|DR| = |DA|$.
2010 Dutch IMO TST
The polynomial $A(x) = x^2 + ax + b$ with integer coefficients has the following property:

for each prime $p$ there is an integer $k$ such that $A(k)$ and $A(k + 1)$ are both divisible by $p$.

Proof that there is an integer $m$ such that $A(m) = A(m + 1) = 0$.
2009 Dutch IMO TST
For a positive integer $n$ let $S(n)$ be the sum of digits in the decimal representation of $n$.   Any positive integer obtained by removing several (at least one) digits from the right-hand end of the decimal representation of $n$ is called a stump of $n$. Let $T(n)$ be the sum of all stumps of $n$. Prove that $n=S(n)+9T(n)$.
2009 Dutch IMO TST
Let $ABC$ be a triangle, $P$ the midpoint of $BC$, and $Q$ a point on segment $CA$ such that $|CQ| = 2|QA|$. Let $S$ be the intersection of $BQ$ and $AP$. Prove that $|AS| = |SP|$.
2009 Dutch IMO TST
Let $a, b$ and $c$ be positive reals such that $a + b + c \ge abc$. Prove that $a^2 + b^2 + c^2 \ge \sqrt3 abc$.
2009 Dutch IMO TST
Find all functions $f : Z \to Z$ satisfying $f(m + n) + f(mn -1) = f(m)f(n) + 2$ for all $m, n \in Z$.
2009 Dutch IMO TST
Suppose that we are given an $n$-gon of which all sides have the same length, and of which all the vertices have rational coordinates. Prove that $n$ is even.
2008 Dutch IMO TST
Find all funtions $f : Z_{>0}  \to Z_{>0}$ that satisfy $f(f(f(n))) + f(f(n)) + f(n) = 3n$ for all $n \in Z_{>0}$ .
2008 Dutch IMO TST
Julian and Johan are playing a game with an even number of cards, say $2n$ cards, ($n \in Z_{>0}$). Every card is marked with a positive integer. The cards are shuffled and are arranged in a row, in such a way that the numbers are visible. The two players take turns picking cards. During a turn, a player can pick either the rightmost or the leftmost card. Johan is the first player to pick a card (meaning Julian will have to take the last card). Now, a player’s score is the sum of the numbers on the cards that player acquired during the game.

Prove that Johan can always get a score that is at least as high as Julian’s.
2008 Dutch IMO TST
Let $m, n$ be positive integers. Consider a sequence of positive integers $a_1, a_2, ... , a_n$ that satisfies $m = a_1 \ge a_2\ge ... \ge a_n \ge 1$. Then define, for $1\le  i\le  m$, $b_i =$ # $\{ j \in \{1, 2, ... , n\}: a_j \ge i\}$,

so $b_i$ is the number of terms $a_j $ of the given sequence for which $a_j  \ge i$.

Similarly, we define, for $1\le   j \le  n$, $c_j=$ # $\{ i \in \{1, 2, ... , m\}: b_i \ge j\}$ , thus $c_j$ is the number of terms bi in the given sequence for which $b_i \ge j$.

E.g.: If $a$ is the sequence $5, 3, 3, 2, 1, 1$ then $b$ is the sequence $6, 4, 3, 1, 1$.

(a) Prove that $a_j = c_j $ for $1  \le j  \le n$.

(b) Prove that for $1\le  k \le m$: $\sum_{i=1}^{k}  b_i = k \cdot b_k + \sum_{j=b_{k+1}}^{n}  a_j$.
2008 Dutch IMO TST
Let $n$ be positive integer such that $\sqrt{1 + 12n^2}$ is an integer.

Prove that $2 + 2\sqrt{1 + 12n^2}$ is the square of an integer.
2008 Dutch IMO TST
Let $\vartriangle ABC$ be a right triangle with $\angle B = 90^o$ and $|AB| > |BC|$, and let $\Gamma$ be the semicircle with diameter $AB$ that lies on the same side as $C$. Let $P$ be a point on $\Gamma$ such that $|BP| = |BC|$ and let $Q$ be on $AB$ such that $|AP| = |AQ|$. Prove that the midpoint of $CQ$ lies on $\Gamma$.
2020 ELMO Problems
Let $\mathbb{N}$ be the set of all positive integers. Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that $$f^{f^{f(x)}(y)}(z)=x+y+z+1$$for all $x,y,z \in \mathbb{N}$.



Proposed by William Wang.
2020 ELMO Problems
Define the Fibonacci numbers by $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n\geq 3$. Let $k$ be a positive integer. Suppose that for every positive integer $m$ there exists a positive integer $n$ such that $m \mid F_n-k$. Must $k$ be a Fibonacci number?



Proposed by Fedir Yudin.
2020 ELMO Problems
Janabel has a device that, when given two distinct points $U$ and $V$ in the plane, draws the perpendicular bisector of $UV$. Show that if three lines forming a triangle are drawn, Janabel can mark the orthocenter of the triangle using this device, a pencil, and no other tools.



Proposed by Fedir Yudin.
2020 ELMO Problems
Let acute scalene triangle $ABC$ have orthocenter $H$ and altitude $AD$ with $D$ on side $BC$. Let $M$ be the midpoint of side $BC$, and let $D'$ be the reflection of $D$ over $M$. Let $P$ be a point on line $D'H$ such that lines $AP$ and $BC$ are parallel, and let the circumcircles of  $\triangle AHP$ and $\triangle BHC$ meet again at $G \neq H$. Prove that $\angle MHG = 90^\circ$.



Proposed by Daniel Hu.
2020 ELMO Problems
Let $m$ and $n$ be positive integers. Find the smallest positive integer $s$ for which there exists an $m \times n$ rectangular array of positive integers such that



each row contains $n$ distinct consecutive integers in some order,

each column contains $m$ distinct consecutive integers in some order, and

each entry is less than or equal to $s$.





Proposed by Ankan Bhattacharya.
2020 ELMO Problems
For any positive integer $n$, let



$\tau(n)$ denote the number of positive integer divisors of $n$,

$\sigma(n)$ denote the sum of the positive integer divisors of $n$, and

$\varphi(n)$ denote the number of positive integers less than or equal to $n$ that are relatively prime to $n$.



Let $a,b > 1$ be integers. Brandon has a calculator with three buttons that replace the integer $n$ currently displayed with $\tau(n)$, $\sigma(n)$, or $\varphi(n)$, respectively. Prove that if the calculator currently displays $a$, then Brandon can make the calculator display $b$ after a finite (possibly empty) sequence of button presses.



Proposed by Jaedon Whyte.
2019 ELMO Problems
Let $P(x)$ be a polynomial with integer coefficients such that $P(0)=1$, and let $c > 1$ be an integer. Define $x_0=0$ and $x_{i+1} = P(x_i)$ for all integers $i \ge 0$. Show that there are infinitely many positive integers $n$ such that $\gcd (x_n, n+c)=1$.



Proposed by Milan Haiman and Carl Schildkraut
2019 ELMO Problems
Let $m, n \ge 2$ be integers. Carl is given $n$ marked points in the plane and wishes to mark their centroid.* He has no standard compass or straightedge. Instead, he has a device which, given marked points $A$ and $B$, marks the $m-1$ points that divide segment $\overline{AB}$ into $m$ congruent parts (but does not draw the segment).



For which pairs $(m,n)$ can Carl necessarily accomplish his task, regardless of which $n$ points he is given?



*Here, the centroid of $n$ points with coordinates $(x_1, y_1), \dots, (x_n, y_n)$ is the point with coordinates $\left( \frac{x_1 + \dots + x_n}{n}, \frac{y_1 + \dots + y_n}{n}\right)$.



Proposed by Holden Mui and Carl Schildkraut
2019 ELMO Problems
Let $n \ge 3$ be a fixed integer. A game is played by $n$ players sitting in a circle. Initially, each player draws three cards from a shuffled deck of $3n$ cards numbered $1, 2, \dots, 3n$. Then, on each turn, every player simultaneously passes the smallest-numbered card in their hand one place clockwise and the largest-numbered card in their hand one place counterclockwise, while keeping the middle card.



Let $T_r$ denote the configuration after $r$ turns (so $T_0$ is the initial configuration). Show that $T_r$ is eventually periodic with period $n$, and find the smallest integer $m$ for which, regardless of the initial configuration, $T_m=T_{m+n}$.



Proposed by Carl Schildkraut and Colin Tang
2019 ELMO Problems
Carl is given three distinct non-parallel lines $\ell_1, \ell_2, \ell_3$ and a circle $\omega$ in the plane. In addition to a normal straightedge, Carl has a special straightedge which, given a line $\ell$ and a point $P$, constructs a new line passing through $P$ parallel to $\ell$. (Carl does not have a compass.) Show that Carl can construct a triangle with circumcircle $\omega$ whose sides are parallel to $\ell_1,\ell_2,\ell_3$ in some order.



Proposed by Vincent Huang
2019 ELMO Problems
Let $S$ be a nonempty set of positive integers such that, for any (not necessarily distinct) integers $a$ and $b$ in $S$, the number $ab+1$ is also in $S$. Show that the set of primes that do not divide any element of $S$ is finite.



Proposed by Carl Schildkraut
2019 ELMO Problems
Carl chooses a functional expression* $E$ which is a finite nonempty string formed from a set $x_1, x_2, \dots$ of variables and applications of a function $f$, together with addition, subtraction, multiplication (but not division), and fixed real constants. He then considers the equation $E = 0$, and lets $S$ denote the set of functions $f \colon \mathbb R \to \mathbb R$ such that the equation holds for any choices of real numbers $x_1, x_2, \dots$. (For example, if Carl chooses the functional equation



$$ f(2f(x_1)+x_2) - 2f(x_1)-x_2 = 0, $$

then $S$ consists of one function, the identity function.



(a) Let $X$ denote the set of functions with domain $\mathbb R$ and image exactly $\mathbb Z$. Show that Carl can choose his functional equation such that $S$ is nonempty but $S \subseteq X$.



(b) Can Carl choose his functional equation such that $|S|=1$ and $S \subseteq X$?



*These can be defined formally in the following way: the set of functional expressions is the minimal one (by inclusion) such that (i) any fixed real constant is a functional expression, (ii) for any positive integer $i$, the variable $x_i$ is a functional expression, and (iii) if $V$ and $W$ are functional expressions, then so are $f(V)$, $V+W$, $V-W$, and $V \cdot W$.



Proposed by Carl Schildkraut
2018 ELMO Problems
Let $n$ be a positive integer. There are $2018n+1$ cities in the Kingdom of Sellke Arabia. King Mark wants to build two-way roads that connect certain pairs of cities such that for each city $C$ and integer $1\le i\le 2018,$ there are exactly $n$ cities that are a distance $i$ away from $C.$ (The distance between two cities is the least number of roads on any path between the two cities.)



For which $n$ is it possible for Mark to achieve this?



Proposed by Michael Ren
2018 ELMO Problems
Consider infinite sequences $a_1,a_2,\dots$ of positive integers satisfying $a_1=1$ and $$a_n \mid a_k+a_{k+1}+\dots+a_{k+n-1}$$for all positive integers $k$ and $n.$ For a given positive integer $m,$ find the maximum possible value of $a_{2m}.$



Proposed by Krit Boonsiriseth
2018 ELMO Problems
Let $A$ be a point in the plane, and $\ell$ a line not passing through $A$. Evan does not have a straightedge, but instead has a special compass which has the ability to draw a circle through three distinct noncollinear points. (The center of the circle is not marked in this process.) Additionally, Evan can mark the intersections between two objects drawn, and can mark an arbitrary point on a given object or on the plane.



(i) Can Evan construct* the reflection of $A$ over $\ell$?



(ii) Can Evan construct the foot of the altitude from $A$ to $\ell$?



*To construct a point, Evan must have an algorithm which marks the point in finitely many steps.



Proposed by Zack Chroman