Navvye/Olympiad-Problems-CSV huggingface dataset:
2012 CentroAmerican
Trilandia is a very unusual city. The city has the shape of an equilateral triangle of side lenght 2012. The streets divide the city into several blocks that are shaped like equilateral triangles of side lenght 1. There are streets at the border of Trilandia too. There are 6036 streets in total. The mayor wants to put sentinel sites at some intersections of the city to monitor the streets. A sentinel site can monitor every street on which it is located. What is the smallest number of sentinel sites that are required to monitor every street of Trilandia?
2012 CentroAmerican
Alexander and Louise are a pair of burglars. Every morning, Louise steals one third of Alexander's money, but feels remorse later in the afternoon and gives him half of all the money she has. If Louise has no money at the beginning and starts stealing on the first day, what is the least positive integer amount of money Alexander must have so that at the end of the 2012th day they both have an integer amount of money?
2012 CentroAmerican
Let $ABC$ be a triangle with $AB < BC$, and let $E$ and $F$ be points in $AC$ and $AB$ such that $BF = BC = CE$, both on the same halfplane as $A$ with respect to  $BC$.



Let $G$ be the intersection of $BE$ and $CF$. Let $H$ be a point in the parallel through $G$ to $AC$ such that $HG = AF$ (with $H$ and $C$ in opposite halfplanes with respect to $BG$). Show that $\angle EHG = \frac{\angle BAC}{2}$.
2011 CentroAmerican
Consider a cube with a fly standing at each of its vertices. When a whistle blows, each fly moves to a vertex in the same face as the previous one but diagonally opposite to it. After the whistle blows, in how many ways can the flies change position so that there is no vertex with 2 or more flies?
2011 CentroAmerican
In a scalene triangle $ABC$, $D$ is the foot of the altitude through $A$, $E$ is the intersection of $AC$ with the bisector of $\angle ABC$ and $F$ is a point on $AB$. Let $O$ the circumcenter of $ABC$ and $X=AD\cap BE$, $Y=BE\cap CF$, $Z=CF \cap AD$. If $XYZ$ is an equilateral triangle, prove that one of the triangles $OXY$, $OYZ$, $OZX$ must be equilateral.
2011 CentroAmerican
A slip on an integer $n\geq 2$ is an operation that consists in choosing a prime divisor $p$ of $n$ and replacing $n$ by $\frac{n+p^2}{p}.$



Starting with an arbitrary integer $n\geq 5$, we successively apply the slip operation on it. Show that one eventually reaches $5$, no matter the slips applied.
2011 CentroAmerican
Find all positive integers $p$, $q$, $r$ such that $p$ and $q$ are prime numbers and  $\frac{1}{p+1}+\frac{1}{q+1}-\frac{1}{(p+1)(q+1)} = \frac{1}{r}.$
2011 CentroAmerican
If $x$, $y$, $z$ are positive numbers satisfying

$$x+\frac{y}{z}=y+\frac{z}{x}=z+\frac{x}{y}=2.$$

Find all the possible values of $x+y+z$.
2011 CentroAmerican
Let $ABC$ be an acute triangle and $D$, $E$, $F$ be the feet of the altitudes through $A$, $B$, $C$ respectively.  Call $Y$ and $Z$ the feet of the perpendicular lines from $B$ and $C$ to $FD$ and $DE$, respectively. Let $F_1$ be the symmetric of $F$ with respect to $E$ and $E_1$ be the symmetric of $E$ with respect to $F$. If $3EF=FD+DE$, prove that $\angle BZF_1=\angle CYE_1$.
2010 CentroAmerican
Denote by $S(n)$ the sum of the digits of the positive integer $n$. Find all the solutions of the equation



$n(S(n)-1)=2010.$
2010 CentroAmerican
Let $ABC$ be a triangle and $L$, $M$, $N$ be the midpoints of $BC$, $CA$ and $AB$, respectively. The tangent to the circumcircle of $ABC$ at $A$ intersects $LM$ and $LN$ at $P$ and $Q$, respectively. Show that $CP$ is parallel to $BQ$.
2010 CentroAmerican
A token is placed in one square of a $m\times n$ board, and is moved according to the following rules:



In each turn, the token can be moved to a square sharing a side with the one currently occupied.



The token cannot be placed in a square that has already been occupied.



Any two consecutive moves cannot have the same direction.





The game ends when the token cannot be moved. Determine the values of $m$ and $n$ for which, by placing the token in some square, all the squares of the board will have been occupied in the end of the game.
2010 CentroAmerican
Find all positive integers $N$ such that an $N\times N$ board can be tiled using tiles of size $5\times 5$ or $1\times 3$.



Note: The tiles must completely cover all the board, with no overlappings.
2010 CentroAmerican
If $p$, $q$ and $r$ are nonzero rational numbers such that $\sqrt[3]{pq^2}+\sqrt[3]{qr^2}+\sqrt[3]{rp^2}$ is a nonzero rational number, prove that



$\frac{1}{\sqrt[3]{pq^2}}+\frac{1}{\sqrt[3]{qr^2}}+\frac{1}{\sqrt[3]{rp^2}}$



is also a rational number.
2010 CentroAmerican
Let $\Gamma$ and $\Gamma_1$ be two circles internally tangent at $A$, with centers $O$ and  $O_1$ and radii $r$ and $r_1$, respectively ($r>r_1$). $B$ is a point diametrically opposed to $A$ in $\Gamma$, and $C$ is a point on $\Gamma$ such that $BC$ is tangent to $\Gamma_1$ at $P$. Let $A'$ the midpoint of $BC$. Given that $O_1A'$ is parallel to $AP$, find the ratio $r/r_1$.
2009 CentroAmerican
Let $ P$ be the product of all non-zero digits of the positive integer $ n$. For example, $ P(4) = 4$, $ P(50) = 5$, $ P(123) = 6$, $ P(2009) = 18$.

Find the value of the sum: P(1) + P(2) + ... + P(2008) + P(2009).
2009 CentroAmerican
\item Two circles $ \Gamma_1$ and $ \Gamma_2$ intersect at points $ A$ and $ B$. Consider a circle $ \Gamma$ contained in $ \Gamma_1$ and $ \Gamma_2$, which is tangent to both of them at $ D$ and $ E$ respectively. Let $ C$ be one of the intersection points of line $ AB$ with $ \Gamma$, $ F$ be the intersection of line $ EC$ with $ \Gamma_2$ and $ G$ be the intersection of line $ DC$ with $ \Gamma_1$. Let $ H$ and $ I$ be the intersection points of line $ ED$ with $ \Gamma_1$ and $ \Gamma_2$ respectively. Prove that $ F$, $ G$, $ H$ and $ I$ are on the same circle.
2009 CentroAmerican
There are 2009 boxes numbered from 1 to 2009, some of which contain stones. Two players, $ A$ and $ B$, play alternately, starting with $ A$. A move consists in selecting a non-empty box $ i$, taking one or more stones from that box and putting them in box $ i + 1$. If $ i = 2009$, the selected stones are eliminated. The player who removes the last stone wins

a) If there are 2009 stones in the box 2 and the others are empty, find a winning strategy for either player.

b) If there is exactly one stone in each box, find a winning strategy for either player.
2009 CentroAmerican
We wish to place natural numbers around a circle such that the following property is satisfied: the absolute values of the differences of each pair of neighboring numbers are all different.

a) Is it possible to place the numbers from 1 to 2009 satisfying this property

b) Is it possible to suppress one of the numbers from 1 to 2009 in such a way that the remaining 2008 numbers can be placed satisfying the property
2009 CentroAmerican
Given an acute and scalene triangle $ ABC$, let $ H$ be its orthocenter, $ O$ its circumcenter, $ E$ and $ F$ the feet of the altitudes drawn from $ B$ and $ C$, respectively. Line $ AO$ intersects the circumcircle of the triangle again at point $ G$ and segments $ FE$ and $ BC$ at points $ X$ and $ Y$ respectively. Let $ Z$ be the point of intersection of line $ AH$ and the tangent line to the circumcircle at $ G$. Prove that $ HX$ is parallel to $ YZ$.
2009 CentroAmerican
Find all prime numbers $ p$ and $ q$ such that $ p^3 - q^5 = (p + q)^2$.
2008 CentroAmerican
Find the least positive integer $ N$ such that the sum of its digits is 100 and the sum of the digits of $ 2N$ is 110.
2008 CentroAmerican
Let $ ABCD$ be a convex cuadrilateral inscribed in a circumference centered at $ O$ such that $ AC$ is a diameter. Pararellograms $ DAOE$ and $ BCOF$ are constructed. Show that if $ E$ and $ F$ lie on the circumference then $ ABCD$ is a rectangle.
2008 CentroAmerican
There are 2008 bags numbered from 1 to 2008, with 2008 frogs in each one of them. Two people play in turns. A play consists in selecting  a bag and taking out of it any number of frongs (at least one), leaving $ x$ frogs in it ($ x\geq 0$). After each play, from each bag with a number higher than the selected one and having more than $ x$ frogs, some frogs scape until there are $ x$ frogs in the bag. The player that takes out the last frog from bag number 1 looses. Find and explain a winning strategy.
2008 CentroAmerican
Five girls have a little store that opens from Monday through Friday. Since two people are always enough for taking care of it, they decide to do a work plan for the week, specifying who will work each day, and fulfilling the following conditions:



a) Each girl will work exactly two days a week

b) The 5 assigned couples for the week must be different

In how many ways can the girls do the work plan?
2008 CentroAmerican
Find a polynomial $ p\left(x\right)$ with real coefficients such that



$ \left(x+10\right)p\left(2x\right)=\left(8x-32\right)p\left(x+6\right)$



for all real $ x$ and $ p\left(1\right)=210$.
2008 CentroAmerican
Let $ ABC$ be an acute triangle. Take points $ P$ and $ Q$ inside $ AB$ and $ AC$, respectively, such that $ BPQC$ is cyclic. The circumcircle of $ ABQ$ intersects $ BC$ again in $ S$ and the circumcircle of $ APC$ intersects $ BC$ again in $ R$, $ PR$ and $ QS$ intersect again in $ L$. Prove that the intersection of $ AL$ and $ BC$ does not depend on the selection of $ P$ and $ Q$.
2007 CentroAmerican
The Central American Olympiad is an annual competition. The ninth Olympiad is held in 2007. Find all the positive integers $n$ such that $n$ divides the number of the year in which the $n$-th Olympiad takes place.
2007 CentroAmerican
In a triangle $ABC$, the angle bisector of $A$ and the cevians $BD$ and $CE$ concur at a point $P$ inside the triangle. Show that the quadrilateral $ADPE$ has an incircle if and only if $AB=AC$.
2007 CentroAmerican
Let $S$ be a finite set of integers. Suppose that for every two different elements of $S$, $p$ and $q$, there exist not necessarily distinct integers $a \neq 0$, $b$, $c$ belonging to $S$, such that $p$ and $q$ are the roots of the polynomial $ax^{2}+bx+c$. Determine the maximum number of elements that $S$ can have.
2007 CentroAmerican
In a remote island, a language in which every word can be written using only the letters $a$, $b$, $c$, $d$, $e$, $f$, $g$ is spoken. Let's say two words are synonymous if we can transform one into the other according to the following rules:



i) Change a letter by another two in the following way: $$a \rightarrow bc,\ b \rightarrow cd,\ c \rightarrow de,\ d \rightarrow ef,\ e \rightarrow fg,\ f\rightarrow ga,\ g\rightarrow ab$$

ii) If a letter is between other two equal letters, these can be removed. For example, $dfd \rightarrow f$.



Show that all words in this language are synonymous.
2007 CentroAmerican
Given two non-negative integers $m>n$, let's say that $m$ ends in $n$ if we can get $n$ by erasing some digits (from left to right) in the decimal representation of $m$. For example, 329 ends in 29, and also in 9.



Determine how many three-digit numbers end in the product of their digits.
2007 CentroAmerican
Consider a circle $S$, and a point $P$ outside it. The tangent lines from $P$ meet $S$ at $A$ and $B$, respectively. Let $M$ be the midpoint of $AB$. The perpendicular bisector of $AM$ meets $S$ in a point $C$ lying inside the triangle $ABP$. $AC$ intersects $PM$ at $G$, and $PM$ meets $S$ in a point $D$ lying outside the triangle $ABP$. If $BD$ is parallel to $AC$, show that $G$ is the centroid of the triangle $ABP$.



Arnoldo Aguilar (El Salvador)
2006 CentroAmerican
For $0 \leq d \leq 9$, we define the numbers $$S_{d}=1+d+d^{2}+\cdots+d^{2006}$$Find the last digit of the number $$S_{0}+S_{1}+\cdots+S_{9}.$$
2006 CentroAmerican
Let $\Gamma$ and $\Gamma'$ be two congruent circles centered at $O$ and $O'$, respectively, and let $A$ be one of their two points of intersection. $B$ is a point on $\Gamma$, $C$ is the second point of intersection of $AB$ and $\Gamma'$, and $D$ is a point on $\Gamma'$ such that $OBDO'$ is a parallelogram. Show that the length of $CD$ does not depend on the position of $B$.
2006 CentroAmerican
For every natural number $n$ we define $$f(n)=\left\lfloor n+\sqrt{n}+\frac{1}{2}\right\rfloor$$ Show that for every integer $k \geq 1$ the equation $$f(f(n))-f(n)=k$$ has exactly $2k-1$ solutions.
2006 CentroAmerican
The product of several distinct positive integers is divisible by ${2006}^{2}$. Determine the minimum value the sum of such numbers can take.
2006 CentroAmerican
The Olimpia country is formed by $n$ islands. The most populated one is called Panacenter, and every island has a different number of inhabitants. We want to build bridges between these islands, which we'll be able to travel in both directions, under the following conditions:



a) No pair of islands is joined by more than one bridge.



b) Using the bridges we can reach every island from Panacenter.



c) If we want to travel from Panacenter to every other island, in such a way that we use each bridge at most once, the number of inhabitants of the islands we visit is strictly decreasing.



Determine the number of ways we can build the bridges.
2006 CentroAmerican
Let $ABCD$ be a convex quadrilateral. $I=AC\cap BD$, and $E$, $H$, $F$ and $G$ are points on $AB$, $BC$, $CD$ and $DA$ respectively, such that $EF \cap GH= I$. If $M=EG \cap AC$, $N=HF \cap AC$, show that $$\frac{AM}{IM}\cdot \frac{IN}{CN}=\frac{IA}{IC}.$$
2005 CentroAmerican
Among the positive integers that can be expressed as the sum of 2005 consecutive integers, which occupies the 2005th position when arranged in order?



Roland Hablutzel, Venezuela

RemarkThe original question was: Among the positive integers that can be expressed as the sum of 2004 consecutive integers, and also as the sum of 2005 consecutive integers, which occupies the 2005th position when arranged in order?
2005 CentroAmerican
Show that the equation $a^{2}b^{2}+b^{2}c^{2}+3b^{2}-c^{2}-a^{2}=2005$ has no integer solutions.



Arnoldo Aguilar, El Salvador
2005 CentroAmerican
Let $ABC$ be a triangle. $P$, $Q$ and $R$ are the points of contact of the incircle with sides $AB$, $BC$ and $CA$, respectively. Let $L$, $M$ and $N$ be the feet of the altitudes of the triangle $PQR$ from $R$, $P$ and $Q$, respectively.



a) Show that the lines $AN$, $BL$ and $CM$ meet at a point.



b) Prove that this points belongs to the line joining the orthocenter and the circumcenter of triangle $PQR$.



Aarón Ramírez, El Salvador
2005 CentroAmerican
Two players, Red and Blue, play in alternating turns on a 10x10 board. Blue goes first. In his turn, a player picks a row or column (not chosen by any player yet) and color all its squares with his own color. If any of these squares was already colored, the new color substitutes the old one.



The game ends after 20 turns, when all rows and column were chosen. Red wins if the number of red squares in the board exceeds at least by 10 the number of blue squares; otherwise Blue wins.



Determine which player has a winning strategy and describe this strategy.
2005 CentroAmerican
Let $ABC$ be a triangle, $H$ the orthocenter and $M$ the midpoint of $AC$. Let $\ell$ be the parallel through $M$ to the bisector of $\angle AHC$. Prove that $\ell$ divides the triangle in two parts of equal perimeters.



Pedro Marrone, Panamá
2005 CentroAmerican
Let $n$ be a positive integer and $p$ a fixed prime. We have a deck of $n$ cards, numbered $1,\ 2,\ldots,\ n$ and $p$ boxes for put the cards on them. Determine all posible integers $n$ for which is possible to distribute the cards in the boxes in such a way the sum of the numbers of the cards in each box is the same.
2004 CentroAmerican
On a whiteboard, the numbers $1$ to $9$ are written. Players $A$ and $B$ take turns, and $A$ is first. Each player in turn chooses one of the numbers on the whiteboard and removes it, along with all multiples (if any). The player who removes the last number loses.

Determine whether any of the players has a winning strategy, and explain why.
2004 CentroAmerican
Define the sequence $(a_n)$ as follows: $a_0=a_1=1$ and for $k\ge 2$, $a_k=a_{k-1}+a_{k-2}+1$.

Determine how many integers between $1$ and $2004$ inclusive can be expressed as $a_m+a_n$ with $m$ and $n$ positive integers and $m\not= n$.
2004 CentroAmerican
$ABC$ is a triangle, and $E$ and $F$ are points on the segments $BC$ and $CA$ respectively, such that $\frac{CE}{CB}+\frac{CF}{CA}=1$ and $\angle CEF=\angle CAB$. Suppose that $M$ is the midpoint of $EF$ and $G$ is the point of intersection between $CM$ and $AB$. Prove that triangle $FEG$ is similar to triangle $ABC$.
2004 CentroAmerican
In a $10\times 10$ square board, half of the squares are coloured white and half black. One side common to two squares on the board side is called a border if the two squares have different colours. Determine the minimum and maximum possible number of borders that can be on the board.
2004 CentroAmerican
Let $ABCD$ be a trapezium such that $AB||CD$ and $AB+CD=AD$. Let $P$ be the point on $AD$ such that $AP=AB$ and $PD=CD$.



$a)$ Prove that $\angle BPC=90^{\circ}$.

$b)$ $Q$ is the midpoint of $BC$ and $R$ is the point of intersection between the line $AD$ and the circle passing through the points $B,A$ and $Q$. Show that the points $B,P,R$ and $C$ are concyclic.
2004 CentroAmerican
With pearls of different colours form necklaces, it is said that a necklace is prime if it cannot be decomposed into strings of pearls of the same length, and equal to each other.

Let $n$ and $q$ be positive integers. Prove that the number of prime necklaces with $n$ beads, each of which has one of the $q^n$ possible colours, is equal to $n$ times the number of prime necklaces with $n^2$ pearls, each of which has one of $q$ possible colours.



Note: two necklaces are considered equal if they have the same number of pearls and you can get the same colour on both necklaces, rotating one of them to match it to the other.
2003 CentroAmerican
Two players $A$ and $B$ take turns playing the following game: There is a pile of $2003$ stones. In his first turn, $A$ selects a divisor of $2003$ and removes this number of stones from the pile. $B$ then chooses a divisor of the number of remaining stones, and removes that number of stones from the new pile, and so on. The player who has to remove the last stone loses. Show that one of the two players has a winning strategy and describe the strategy.
2003 CentroAmerican
$S$ is a circle with $AB$ a diameter and $t$ is the tangent line to $S$ at $B$. Consider the two points $C$ and $D$ on $t$ such that $B$ is between $C$ and $D$. Suppose $E$ and $F$ are the intersections of $S$ with $AC$ and $AD$ and $G$ and $H$ are the intersections of $S$ with $CF$ and $DE$. Show that $AH=AG$.
2003 CentroAmerican
Let $a$ and $b$ be positive integers with $a>1$ and $b>2$. Prove that $a^b+1\ge b(a+1)$ and determine when there is inequality.
2003 CentroAmerican
$S_1$ and $S_2$ are two circles that intersect at two different points $P$ and $Q$. Let $\ell_1$ and $\ell_2$ be two parallel lines such that $\ell_1$ passes through the point $P$ and intersects $S_1,S_2$ at $A_1,A_2$ respectively (both distinct from $P$), and $\ell_2$ passes through the point $Q$ and intersects $S_1,S_2$ at $B_1,B_2$ respectively (both distinct from $Q$).

Show that the triangles $A_1QA_2$ and $B_1PB_2$ have the same perimeter.
2003 CentroAmerican
A square board with $8\text{cm}$ sides is divided into $64$ squares square with each side $1\text{cm}$. Each box can be painted white or black. Find the total number of ways to colour the board so that each square of side $2\text{cm}$ formed by four squares with a common vertex contains two white and two black squares.
2002 CentroAmerican
For what integers $ n\ge 3$ is it possible to accommodate, in some order, the numbers $ 1,2,\cdots, n$ in a circular form such that every number divides the sum of the next two numbers, in a clockwise direction?
2002 CentroAmerican
Let $ ABC$ be an acute triangle, and let $ D$ and $ E$ be the feet of the altitudes drawn from vertexes $ A$ and $ B$, respectively. Show that if,

$$ Area[BDE]\le Area[DEA]\le Area[EAB]\le Area[ABD]$$

then, the triangle is isosceles.
2002 CentroAmerican
For every integer $ a>1$ an infinite list of integers is constructed $ L(a)$, as follows:





$ a$ is the first number in the list $ L(a)$.



Given a number

$ b$ in $ L(a)$, the next number in the list is $ b+c$, where $ c$ is the largest integer that divides $ b$ and is smaller than $ b$.

Find all the integers $ a>1$ such that $ 2002$ is in the list $ L(a)$.
2002 CentroAmerican
Let $ ABC$ be a triangle, $ D$ be the midpoint of $ BC$, $ E$ be a point on segment $ AC$ such that $ BE=2AD$ and $ F$ is the intersection point of $ AD$ with $ BE$. If $ \angle DAC=60^{\circ}$, find the measure of the angle $ FEA$.
2002 CentroAmerican
Find a set of infinite positive integers $ S$ such that for every $ n\ge 1$ and whichever $ n$ distinct elements $ x_1,x_2,\cdots, x_n$ of S, the number $ x_1+x_2+\cdots +x_n$ is not a perfect square.
2002 CentroAmerican
A path from $ (0,0)$ to $ (n,n)$ on the lattice is made up of unit moves upward or rightward. It is balanced if the sum of the x-coordinates of its $ 2n+1$ vertices equals the sum of their y-coordinates. Show that a balanced path divides the square with vertices $ (0,0)$, $ (n,0)$, $ (n,n)$, $ (0,n)$ into two parts with equal area.
2001 CentroAmerican
Two players $ A$, $ B$ and another 2001 people form a circle, such that $ A$ and $ B$ are not in consecutive  positions. $ A$ and $ B$ play in alternating turns, starting with $ A$. A play consists of touching one of the people neighboring you, which such person once touched leaves the circle. The winner is the last man standing.



Show that one of the two players has a winning strategy, and give such strategy.



Note: A player has a winning strategy if he/she is able to win no matter what the opponent does.
2001 CentroAmerican
Let $ AB$ be the diameter of a circle with a center $ O$ and radius $ 1$. Let $ C$ and $ D$ be two points on the circle such that $ AC$ and $ BD$ intersect at a point $ Q$ situated inside of the circle, and $ \angle AQB= 2 \angle COD$. Let $ P$ be a point that intersects the tangents to the circle that pass through the points $ C$ and $ D$.



Determine the length of segment $ OP$.
2001 CentroAmerican
Find all the real numbers $ N$ that satisfy these requirements:



1. Only two of the digits of $ N$ are distinct from $ 0$, and one of them is $ 3$.



2. $ N$ is a perfect square.
2001 CentroAmerican
Determine the smallest positive integer $ n$ such that there exists positive integers $ a_1,a_2,\cdots,a_n$, that smaller than or equal to $ 15$ and are not necessarily distinct, such that the last four digits of the sum,



$$ a_1!+a_2!+\cdots+a_n!$$



Is $ 2001$.
2001 CentroAmerican
Let $ a,b$ and $ c$ real numbers such that the equation $ ax^2+bx+c=0$ has two distinct real solutions $ p_1,p_2$ and the equation $ cx^2+bx+a=0$ has two distinct real solutions $ q_1,q_2$. We know that the numbers $ p_1,q_1,p_2,q_2$ in that order, form an arithmetic progression. Show that $ a+c=0$.
2001 CentroAmerican
In a circumference of a circle, $ 10000$ points are marked, and they are numbered from $ 1$ to $ 10000$ in a clockwise manner. $ 5000$ segments are drawn in such a way so that the following conditions are met:



1. Each segment joins two marked points.

2. Each marked point belongs to one and only one segment.

3. Each segment intersects exactly one of the remaining segments.

4. A number is assigned to each segment that is the product of the number assigned to each end point of the segment.



Let $ S$ be the sum of the products assigned to all the segments.



Show that $ S$ is a multiple of $ 4$.
2000 CentroAmerican
Find all three-digit numbers $ abc$ (with $ a \neq 0$) such that $ a^{2}+b^{2}+c^{2}$ is a divisor of 26.