Navvye/Olympiad-Problems-CSV huggingface dataset:
2015 Caucasus Mathematical Olympiad
Let's call a natural number a palindrome, the decimal notation of which is equally readable from left to right and right to left (decimal notation cannot start from zero; for example, the number $1221$ is a palindrome, but the numbers $1231, 1212$ and $1010$ are not). Which palindromes among the numbers from $10,000$ to $999,999$ have an odd sum of digits, which have an one even, and how many times are the ones with odd sum more than the ones with the even sum?
2015 Caucasus Mathematical Olympiad
Find the roots of the equation  $(x-a)(x-b)=(x-c)(x-d)$, if you know that $a+d=b+c=2015$ and $a \ne c$ (numbers $a, b, c, d$ are not given).
2015 Caucasus Mathematical Olympiad
Vasya chose a certain number $x$ and calculated the following:

$a_1=1+x^2+x^3, a_2=1+x^3+x^4, a_3=1+x^4+x^5, ..., a_n=1+x^{n+1}+x^{n+2} ,...$

It turned out that $a_2^2 = a_1a_3$.

Prove that for all $n\ge 3$, the equality $a_n^2 = a_{n-1}a_{n+1}$ holds.
2015 Caucasus Mathematical Olympiad
What is the smallest number of $3$-cell corners that you need to paint in a $5 \times5$ square so that you cannot paint more than one corner of one it?

(Shaded corners should not overlap.)
2015 Caucasus Mathematical Olympiad
We call a number greater than $25$,  semi-prime if it is the sum of some two different prime numbers. What is the greatest number of consecutive natural numbers that can be semi-prime?
2015 Caucasus Mathematical Olympiad
Let $AA_1$ and $CC_1$ be the altitudes of the acute-angled  triangle $ABC$. Let $K,L$ and $M$ be the midpoints of the sides $AB,BC$ and $CA$ respectively. Prove that if  $\angle C_1MA_1 =\angle ABC$, then $C_1 K = A_1L$.
2015 Caucasus Mathematical Olympiad
Is there an eight-digit number without zero digits, which when divided by the first digit gives the remainder $1$, when divided by the second digit will give the remainder $2$, ..., when divided by the eighth digit will give the remainder $8$?
2015 Caucasus Mathematical Olympiad
The equation $(x+a) (x+b) = 9$  has a root $a+b$. Prove that  $ab\le 1$.
2015 Caucasus Mathematical Olympiad
The workers laid a floor of size $n\times n$ ($10 <n <20$) with two types of tiles: $2 \times 2$ and $5\times 1$. It turned out that they were able to completely lay the floor so that the same number of tiles of each type was used. For which $n$ could this happen?

(You can’t cut tiles and also put them on top of each other.)
2015 Caucasus Mathematical Olympiad
The midpoint of the edge $SA$ of the triangular pyramid of $SABC$ has equal distances  from all the vertices of the pyramid. Let $SH$ be the height of the pyramid. Prove that $BA^2 + BH^2 = C A^2 + CH^2$.
2015 Caucasus Mathematical Olympiad
Are there natural $a, b >1000$ , such that for any $c$ that is a perfect square, the three numbers $a, b$ and $c$ are not the lengths of the sides of a triangle?
2021 Centroamerican and Caribbean Math Olympiad
An ordered triple $(p, q, r)$ of prime numbers is called parcera if $p$ divides $q^2-4$, $q$ divides $r^2-4$ and $r$ divides $p^2-4$. Find all parcera triples.
2021 Centroamerican and Caribbean Math Olympiad
Let $ABC$ be a triangle and let $\Gamma$ be its circumcircle. Let $D$ be a point on $AB$ such that $CD$ is parallel to the line tangent to $\Gamma$ at $A$. Let $E$ be the intersection of $CD$ with $\Gamma$ distinct from $C$, and $F$ the intersection of $BC$ with the circumcircle of $\bigtriangleup ADC$ distinct from $C$. Finally, let $G$ be the intersection of the line $AB$ and the internal bisector of $\angle DCF$. Show that $E,\ G,\ F$ and $C$ lie on the same circle.
2021 Centroamerican and Caribbean Math Olympiad
In a table consisting of $2021\times 2021$ unit squares, some unit squares are colored black in such a way that if we place a mouse in the center of any square on the table it can walk in a straight line (up, down, left or right along a column or row) and leave the table without walking on any black square (other than the initial one if it is black). What is the maximum number of squares that can be colored black?
2021 Centroamerican and Caribbean Math Olympiad
There are $2021$ people at a meeting. It is known that one person at the meeting doesn't have any friends there and another person has only one friend there. In addition, it is true that, given any $4$ people, at least $2$ of them are friends. Show that there are $2018$ people at the meeting that are all friends with each other.

Note. If $A$ is friend of $B$ then $B$ is a friend of $A$.
2021 Centroamerican and Caribbean Math Olympiad
Let $n \geq 3$ be an integer and $a_1,a_2,...,a_n$ be positive real numbers such that $m$ is the smallest and $M$ is the largest of these numbers. It is known that for any distinct integers $1 \leq i,j,k \leq n$, if $a_i \leq a_j \leq a_k$ then $a_ia_k \leq a_j^2$. Show that



$$ a_1a_2 \cdots a_n \geq m^2M^{n-2} $$

and determine when equality holds
2021 Centroamerican and Caribbean Math Olympiad
Let $ABC$ be a triangle with $AB<AC$ and let $M$ be the midpoint of $AC$. A point $P$ (other than $B$) is chosen on the segment $BC$ in such a way that $AB=AP$. Let $D$ be the intersection of $AC$ with the circumcircle of $\bigtriangleup ABP$ distinct from $A$, and $E$ be the intersection of $PM$ with the circumcircle of $\bigtriangleup ABP$ distinct from $P$. Let $K$ be the intersection of lines $AP$ and $DE$. Let $F$ be a point on $BC$ (other than $P$) such that $KP=KF$. Show that $C,\ D,\ E$ and $F$ lie on the same circle.
2020 Centroamerican and Caribbean Math Olympiad
A four-digit positive integer is called virtual if it has the form $\overline{abab}$, where $a$ and $b$ are digits and $a \neq 0$. For example 2020, 2121 and 2222 are virtual numbers, while 2002 and 0202 are not. Find all virtual numbers of the form $n^2+1$, for some positive integer $n$.
2020 Centroamerican and Caribbean Math Olympiad
Suppose you have identical coins distributed in several piles with one or more coins in each pile. An action consists of taking two piles, which have an even total of coins among them, and redistribute their coins in two piles so that they end up with the same number of coins.



A distribution is levelable if it is possible, by means of 0 or more operations, to end up with all the piles having the same number of coins.



Determine all positive integers $n$ such that, for all positive integers $k$, any distribution of $nk$ coins in $n$ piles is levelable.
2020 Centroamerican and Caribbean Math Olympiad
Find all the functions $f: \mathbb{Z}\to \mathbb{Z}$ satisfying the following property: if $a$, $b$ and $c$ are integers such that $a+b+c=0$, then



$$f(a)+f(b)+f(c)=a^2+b^2+c^2.$$
2020 Centroamerican and Caribbean Math Olympiad
Consider a triangle $ABC$ with $BC>AC$. The circle with center $C$ and radius $AC$ intersects the segment $BC$ in $D$. Let $I$ be the incenter of triangle $ABC$ and $\Gamma$ be the circle that passes through $I$ and is tangent to the line $CA$ at $A$. The line $AB$ and $\Gamma$ intersect at a point $F$ with $F \neq A$. Prove that $BF=BD$.
2020 Centroamerican and Caribbean Math Olympiad
Let $P(x)$ be a polynomial with real non-negative coefficients. Let $k$ be a positive integer and $x_1, x_2, \dots, x_k$ positive real numbers such that $x_1x_2\cdots x_k=1$. Prove that



$$P(x_1)+P(x_2)+\cdots+P(x_k)\geq kP(1).$$
2020 Centroamerican and Caribbean Math Olympiad
A positive integer $N$ is interoceanic if its prime factorization



$$N=p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}$$

satisfies



$$x_1+x_2+\dots +x_k=p_1+p_2+\cdots +p_k.$$

Find all interoceanic numbers less than 2020.
2019 Centroamerican and Caribbean Math Olympiad
Let $N=\overline{abcd}$ be a positive integer with four digits. We name plátano power to the smallest positive integer $p(N)=\overline{\alpha_1\alpha_2\ldots\alpha_k}$ that can be inserted between  the numbers $\overline{ab}$ and $\overline{cd}$ in such a way the new number $\overline{ab\alpha_1\alpha_2\ldots\alpha_kcd}$ is divisible by $N$. Determine the value of $p(2025)$.
2019 Centroamerican and Caribbean Math Olympiad
We have a regular polygon $P$ with 2019 vertices, and in each vertex there is a coin. Two players Azul and Rojo take turns alternately, beginning with Azul, in the following way: first, Azul chooses a triangle with vertices in $P$ and colors its interior with blue, then Rojo selects a triangle with vertices in $P$ and colors its interior with red, so that the triangles formed in each move don't intersect internally the previous colored triangles. They continue playing until it's not possible to choose another triangle to be colored. Then, a player wins the coin of a vertex if he colored the greater quantity of triangles incident to that vertex (if the quantities of triangles colored with blue or red incident to the vertex are the same, then no one wins that coin and the coin is deleted). The player with the greater quantity of coins wins the game.  Find a winning strategy for one of the players.



Note. Two triangles can share vertices or sides.
2019 Centroamerican and Caribbean Math Olympiad
Let $ABC$ be a triangle and $\Gamma$ its circumcircle. Let $D$ be the foot of the altitude from $A$ to the side $BC$, $M$ and $N$ the midpoints of $AB$ and $AC$, and $Q$ the point on $\Gamma$ diametrically opposite to $A$. Let $E$ be the midpoint of $DQ$. Show that the lines perpendicular to $EM$ and $EN$ passing through $M$ and $N$, respectively, meet on $AD$.
2019 Centroamerican and Caribbean Math Olympiad
Let $ABC$ be a triangle, $\Gamma$ its circumcircle and $l$ the tangent to $\Gamma$ through $A$. The altitudes from $B$ and $C$ are extended and meet $l$ at $D$ and $E$, respectively. The lines $DC$ and $EB$ meet $\Gamma$ again at $P$ and $Q$, respectively. Show that the triangle $APQ$ is isosceles.
2019 Centroamerican and Caribbean Math Olympiad
Let $a,\ b$ and $c$ be positive real numbers so that $a+b+c=1$. Show that

$$a\sqrt{a^2+6bc}+b\sqrt{b^2+6ac}+c\sqrt{c^2+6ab}\leq\frac{3\sqrt{2}}{4}$$
2019 Centroamerican and Caribbean Math Olympiad
A triminó is a rectangular tile of $1\times 3$. Is it possible to cover a $8\times8$ chessboard using $21$ triminós, in such a way there remains exactly one $1\times 1$ square without covering? In case the answer is in the affirmative, determine all the possible locations of such a unit square in the chessboard.
2018 Centroamerican and Caribbean Math Olympiad
There are 2018 cards numbered from 1 to 2018. The numbers of the cards are visible at all times. Tito and Pepe play a game. Starting with Tito, they take turns picking cards until they're finished. Then each player sums the numbers on his cards and whoever has an even sum wins. Determine which player has a winning strategy and describe it.







P.S. Proposed by yours truly  :-D
2018 Centroamerican and Caribbean Math Olympiad
Let $\Delta ABC$ be a triangle inscribed in the circumference $\omega$ of center $O$. Let $T$ be the symmetric of $C$ respect to $O$ and $T'$ be the reflection of $T$ respect to line $AB$. Line $BT'$ intersects $\omega$ again at $R$. The perpendicular to $CT$ through $O$ intersects line $AC$ at $L$. Let $N$ be the intersection of lines $TR$ and $AC$. Prove that $\overline{CN}=2\overline{AL}$.
2018 Centroamerican and Caribbean Math Olympiad
Let $x, y$ be real numbers such that $x-y, x^2-y^2, x^3-y^3$ are all prime numbers. Prove that $x-y=3$.



EDIT: Problem submitted by Leonel Castillo, Panama.
2018 Centroamerican and Caribbean Math Olympiad
Determine all triples $(p, q, r)$ of positive integers, where $p, q$ are also primes, such that $\frac{r^2-5q^2}{p^2-1}=2$.
2018 Centroamerican and Caribbean Math Olympiad
Let $n$ be a positive integer, $1<n<2018$. For each $i=1, 2, \ldots ,n$ we define the polynomial $S_i(x)=x^2-2018x+l_i$, where $l_1, l_2, \ldots, l_n$ are distinct positive integers. If the polynomial $S_1(x)+S_2(x)+\cdots+S_n(x)$ has at least an integer root, prove that at least one of the $l_i$ is greater or equal than $2018$.
2018 Centroamerican and Caribbean Math Olympiad
A dance with 2018 couples takes place in Havana. For the dance, 2018 distinct points labeled $0, 1,\ldots, 2017$ are marked in a circumference and each couple is placed on a different point. For $i\geq1$, let $s_i=i\ (\textrm{mod}\ 2018)$ and $r_i=2i\ (\textrm{mod}\ 2018)$. The dance begins at minute $0$. On the $i$-th minute, the couple at point $s_i$ (if there's any) moves to point $r_i$, the couple on point $r_i$ (if there's any) drops out, and the dance continues with the remaining couples. The dance ends after $2018^2$ minutes. Determine how many couples remain at the end.



Note: If $r_i=s_i$, the couple on $s_i$ stays there and does not drop out.
2017 CentroAmerican
The figure below shows a hexagonal net formed by many congruent equilateral triangles. Taking turns, Gabriel and Arnaldo play a game as follows. On his turn, the player colors in a segment, including the endpoints, following these three rules:



i) The endpoints must coincide with vertices of the marked equilateral triangles.



ii) The segment must be made up of one or more of the sides of the triangles.



iii) The segment cannot contain any point (endpoints included) of a previously colored segment.



Gabriel plays first, and the player that cannot make a legal move loses. Find a winning strategy and describe it.
2017 CentroAmerican
We call a pair $(a,b)$ of positive integers, $a<391$, pupusa if



$$\textup{lcm}(a,b)>\textup{lcm}(a,391)$$

Find the minimum value of $b$ across all pupusa pairs.



Fun Fact: OMCC 2017 was held in El Salvador. Pupusa is their national dish. It is a corn tortilla filled with cheese, meat, etc.
2017 CentroAmerican
Let $ABC$ be a triangle and $D$ be the foot of the altitude from $A$. Let $l$ be the line that passes through the midpoints of $BC$ and $AC$. $E$ is the reflection of $D$ over $l$. Prove that the circumcentre of $\triangle ABC$ lies on the line $AE$.
2017 CentroAmerican
$ABC$ is a right-angled triangle, with $\angle ABC = 90^{\circ}$. $B'$ is the reflection of $B$ over $AC$. $M$ is the midpoint of $AC$. We choose $D$ on $\overrightarrow{BM}$, such that $BD = AC$. Prove that $B'C$ is the angle bisector of $\angle MB'D$.



NOTE: An important condition not mentioned in the original problem is $AB<BC$. Otherwise, $\angle MB'D$ is not defined or $B'C$ is the external bisector.
2017 CentroAmerican
Susana and Brenda play a game writing polynomials on the board. Susana starts and they play taking turns.



1) On the preparatory turn (turn 0), Susana choose a positive integer $n_0$ and writes the polynomial $P_0(x)=n_0$.



2) On turn 1, Brenda choose a positive integer $n_1$, different from $n_0$, and either writes the polynomial



$$P_1(x)=n_1x+P_0(x) \textup{  or  } P_1(x)=n_1x-P_0(x)$$

3) In general, on turn $k$, the respective player chooses an integer $n_k$, different from $n_0, n_1, \ldots, n_{k-1}$, and either writes the polynomial



$$P_k(x)=n_kx^k+P_{k-1}(x) \textup{  or  } P_k(x)=n_kx^k-P_{k-1}(x)$$

The first player to write a polynomial with at least one whole whole number root wins. Find and describe a winning strategy.
2017 CentroAmerican
Tita the Frog sits on the number line. She is initially on the integer number $k>1$. If she is sitting on the number $n$, she hops to the number $f(n)+g(n)$, where $f(n)$ and $g(n)$ are, respectively, the biggest and smallest positive prime numbers that divide $n$. Find all values of $k$ such that Tita can hop to infinitely many distinct integers.
2016 CentroAmerican
Find all positive integers $n$ that have 4 digits, all of them perfect squares, and such that $n$ is divisible by 2, 3, 5 and 7.
2016 CentroAmerican
Let $ABC$ be an acute-angled triangle, $\Gamma$ its circumcircle and $M$ the midpoint of $BC$. Let $N$ be a point in the arc $BC$ of $\Gamma$ not containing $A$ such that $\angle NAC= \angle BAM$. Let $R$ be the midpoint of $AM$, $S$ the midpoint of $AN$ and $T$ the foot of the altitude through $A$. Prove that $R$, $S$ and $T$ are collinear.
2016 CentroAmerican
The polynomial $Q(x)=x^3-21x+35$ has three different real roots. Find real numbers $a$ and $b$ such that the polynomial $x^2+ax+b$ cyclically permutes the roots of $Q$, that is, if $r$, $s$ and $t$ are the roots of $Q$ (in some order) then $P(r)=s$, $P(s)=t$ and $P(t)=r$.
2016 CentroAmerican
The number "3" is written on a board. Ana and Bernardo take turns, starting with Ana, to play the following game. If the number written on the board is $n$, the player in his/her turn must replace it by an integer $m$ coprime with $n$ and such that $n<m<n^2$. The first player that reaches a number greater or equal than 2016 loses. Determine which of the players has a winning strategy and describe it.
2016 CentroAmerican
We say a number is irie if it can be written in the form $1+\dfrac{1}{k}$ for some positive integer $k$. Prove that every integer $n \geq 2$ can be written as the product of $r$ distinct irie numbers for every integer $r \geq n-1$.
2016 CentroAmerican
Let $\triangle ABC$ be triangle with incenter $I$ and circumcircle $\Gamma$. Let $M=BI\cap \Gamma$ and $N=CI\cap \Gamma$, the line parallel to $MN$ through $I$ cuts $AB$, $AC$ in $P$ and $Q$. Prove that the circumradius of $\odot (BNP)$ and $\odot (CMQ)$ are equal.
2015 CentroAmerican
We wish to write $n$ distinct real numbers $(n\geq3)$ on the circumference of a circle in such a way that each number is equal to the product of its immediate neighbors to the left and right. Determine all of the values of $n$ such that this is possible.
2015 CentroAmerican
A sequence $(a_n)$ of real numbers is defined by $a_0=1$, $a_1=2015$ and for all $n\geq1$, we have

$$a_{n+1}=\frac{n-1}{n+1}a_n-\frac{n-2}{n^2+n}a_{n-1}.$$

Calculate the value of $\frac{a_1}{a_2}-\frac{a_2}{a_3}+\frac{a_3}{a_4}-\frac{a_4}{a_5}+\ldots+\frac{a_{2013}}{a_{2014}}-\frac{a_{2014}}{a_{2015}}$.
2015 CentroAmerican
Let $ABCD$ be a cyclic quadrilateral with $AB<CD$, and let $P$ be the point of intersection of the lines $AD$ and $BC$.The circumcircle of the triangle $PCD$ intersects the line $AB$ at the points $Q$ and $R$. Let $S$ and $T$ be the points where the tangents from $P$ to the circumcircle of $ABCD$ touch that circle.



(a) Prove that $PQ=PR$.



(b) Prove that $QRST$ is a cyclic quadrilateral.
2015 CentroAmerican
Anselmo and Bonifacio start a game where they alternatively substitute a number written on a board. In each turn, a player can substitute the written number by either the number of divisors of the written number or by the difference between the written number and the number of divisors it has. Anselmo is the first player to play, and whichever player is the first player to write the number $0$ is the winner. Given that the initial number is $1036$, determine which player has a winning strategy and describe that strategy.



Note: For example, the number of divisors of $14$ is $4$, since its divisors are $1$, $2$, $7$, and $14$.
2015 CentroAmerican
Let $ABC$ be a triangle such that $AC=2AB$. Let $D$ be the point of intersection of the angle bisector of the angle $CAB$ with $BC$. Let $F$ be the point of intersection of the line parallel to $AB$ passing through $C$ with the perpendicular line to $AD$ passing through $A$. Prove that $FD$ passes through the midpoint of $AC$.
2015 CentroAmerican
$39$ students participated in a math competition. The exam consisted of $6$ problems and each problem was worth $1$ point for a correct solution and $0$ points for an incorrect solution. For any $3$ students, there is at most $1$ problem that was not solved by any of the three. Let $B$ be the sum of all of the scores of the $39$ students. Find the smallest possible value of $B$.
2014 CentroAmerican
A positive integer is called tico if it is the product of three different prime numbers that add up to 74. Verify that 2014 is tico. Which year will be the next tico year? Which one will be the last tico year in history?
2014 CentroAmerican
Let $ABCD$ be a trapezoid with bases $AB$ and $CD$, inscribed in a circle of center $O$. Let $P$ be the intersection of the lines $BC$ and $AD$. A circle through $O$ and $P$ intersects the segments $BC$ and $AD$ at interior points $F$ and $G$, respectively. Show that $BF=DG$.
2014 CentroAmerican
Let $a$, $b$, $c$ and $d$ be real numbers such that no two of them are equal,



$$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=4$$ and $ac=bd$. Find the maximum possible value of



$$\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}.$$
2014 CentroAmerican
Using squares of side 1, a stair-like figure is formed in stages following the pattern of the drawing.



For example, the first stage uses 1 square, the second uses 5, etc. Determine the last stage for which the corresponding figure uses less than 2014 squares.
2014 CentroAmerican
Points $A$, $B$, $C$ and $D$ are chosen on a line in that order, with $AB$ and $CD$ greater than $BC$. Equilateral triangles $APB$, $BCQ$ and $CDR$ are constructed so that $P$, $Q$ and $R$ are on the same side with respect to $AD$. If $\angle PQR=120^\circ$, show that



$$\frac{1}{AB}+\frac{1}{CD}=\frac{1}{BC}.$$
2014 CentroAmerican
A positive integer $n$ is funny if for all positive divisors $d$ of $n$, $d+2$ is a prime number. Find all funny numbers with the largest possible number of divisors.
2013 CentroAmerican
Juan writes the list of pairs $(n, 3^n)$, with $n=1, 2, 3,...$ on a chalkboard. As he writes the list, he underlines the pairs $(n, 3^n)$ when $n$ and $3^n$ have the same units digit. What is the $2013^{th}$ underlined pair?
2013 CentroAmerican
Around a round table the people $P_1, P_2,..., P_{2013}$ are seated in a clockwise order. Each person starts with a certain amount of coins (possibly none); there are a total of $10000$ coins. Starting with $P_1$ and proceeding in clockwise order, each person does the following on their turn:



If they have an even number of coins, they give all of their coins to their neighbor to the left.



If they have an odd number of coins, they give their neighbor to the left an odd number of coins (at least $1$ and at most all of their coins) and keep the rest.





Prove that, repeating this procedure, there will necessarily be a point where one person has all of the coins.
2013 CentroAmerican
Let $ABCD$ be a convex quadrilateral and let $M$ be the midpoint of side $AB$. The circle passing through $D$ and tangent to $AB$ at $A$ intersects the segment $DM$ at $E$. The circle passing through $C$ and tangent to $AB$ at $B$ intersects the segment $CM$ at $F$. Suppose that the lines $AF$ and $BE$ intersect at a point which belongs to the perpendicular bisector of side $AB$. Prove that $A$, $E$, and $C$ are collinear if and only if $B$, $F$, and $D$ are collinear.
2013 CentroAmerican
Ana and Beatriz take turns in a game that starts with a square of side $1$ drawn on an infinite grid. Each turn consists of drawing a square that does not overlap with the rectangle already drawn, in such a way that one of its sides is a (complete) side of the figure already drawn. A player wins if she completes a rectangle whose area is a multiple of $5$. If Ana goes first, does either player have a winning strategy?
2013 CentroAmerican
Let $ABC$ be an acute triangle and let $\Gamma$ be its circumcircle. The bisector of $\angle{A}$ intersects $BC$ at $D$, $\Gamma$ at $K$ (different from $A$), and the line through $B$ tangent to $\Gamma$ at $X$. Show that $K$ is the midpoint of $AX$ if and only if  $\frac{AD}{DC}=\sqrt{2}$.
2013 CentroAmerican
Determine all pairs of non-constant polynomials $p(x)$ and $q(x)$, each with leading coefficient $1$, degree $n$, and $n$ roots which are non-negative integers, that satisfy $p(x)-q(x)=1$.
2012 CentroAmerican
Find all positive integers that are equal to $700$ times the sum of its digits.
2012 CentroAmerican
Let $\gamma$ be the circumcircle of the acute triangle $ABC$. Let $P$ be the midpoint of the minor arc $BC$. The parallel to $AB$ through $P$ cuts $BC, AC$ and $\gamma$ at points $R,S$ and $T$, respectively. Let $K \equiv AP \cap BT$ and $L \equiv BS \cap AR$.  Show that $KL$ passes through the midpoint of $AB$ if and only if $CS = PR$.
2012 CentroAmerican
Let $a,b,c$ be real numbers that satisfy $\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{a+c} =1$ and $ab+bc+ac >0$.



Show that $$ a+b+c - \frac{abc}{ab+bc+ac} \ge 4 $$