Navvye/Olympiad-Problems-CSV huggingface dataset:
2014 ITAMO
For any positive integer $n$, let $D_n$ denote the greatest common divisor of all numbers of the form $a^n + (a + 1)^n + (a + 2)^n$ where $a$ varies among all positive integers.



(a) Prove that for each $n$, $D_n$ is of the form $3^k$ for some integer $k \ge 0$.

(b) Prove that, for all $k\ge 0$, there exists an integer $n$ such that $D_n = 3^k$.
2014 ITAMO
Let $\omega$ be a circle with center $A$ and radius $R$. On the circumference of $\omega$ four distinct points $B, C, G, H$ are taken in that order in such a way that $G$ lies on the extended $B$-median of the triangle $ABC$, and H lies on the extension of altitude of $ABC$ from $B$. Let $X$ be the intersection of the straight lines $AC$ and $GH$. Show that the segment $AX$ has length $2R$.
2014 ITAMO
Prove that there exists a positive integer that can be written, in at least two ways, as a sum of $2014$-th powers of $2015$ distinct positive integers $x_1 <x_2 <\cdots <x_{2015}$.
2014 ITAMO
A $(2n + 1) \times (2n + 1)$ grid, with $n> 0$, is colored in such a way that each of the cell is white or black. A cell is called special if there are at least $n$ other cells of the same color in its row, and at least another $n$ cells of the same color in its column.



(a) Prove that there are at least $2n + 1$ special boxes.

(b) Provide an example where there are at most $4n$ special cells.

(c) Determine, as a function of $n$, the minimum possible number of special cells.
2013 ITAMO
A model car is tested on some closed circuit $600$ meters long, consisting of flat stretches, uphill and downhill. All uphill and downhill have the same slope. The test highlights the following facts:



(a) The velocity of the car depends only on the fact that the car is driving along a stretch of uphill, plane or downhill; calling these three velocities

$v_s, v_p$ and $v_d$ respectively, we have $v_s <v_p <v_d$;



(b) $v_s,v_p$ and $v_d$, expressed in meter per second, are integers.



(c) Whatever may be the structure of the circuit, the time taken to complete the circuit is always $50$ seconds.





Find all possible values of $v_s, v_p$ and $v_d$.
2013 ITAMO
In triangle $ABC$, suppose we have $a> b$, where $a=BC$ and $b=AC$. Let $M$ be the midpoint of $AB$, and $\alpha, \beta$ are inscircles of the triangles $ACM$ and $BCM$ respectively. Let then $A'$ and $B'$ be the points of tangency of $\alpha$ and $\beta$ on $CM$. Prove that $A'B'=\frac{a - b}{2}$.
2013 ITAMO
Each integer is colored with one of two colors, red or blue. It is known that, for every finite set $A$ of consecutive integers, the absolute value of the difference between the number of red and blue integers in the set $A$ is at most $1000$. Prove that there exists a set of $2000$ consecutive integers in which there are exactly $1000$ red numbers and $1000$ numbers blue.
2013 ITAMO
$\overline{5654}_b$ is a power of a prime number. Find $b$ if $b > 6$.
2013 ITAMO
$ABC$ is an isosceles triangle with $AB=AC$ and the angle in $A$ is less than $60^{\circ}$. Let $D$ be a point on $AC$ such that $\angle{DBC}=\angle{BAC}$. $E$ is the intersection between the perpendicular bisector of $BD$ and the line parallel to $BC$ passing through $A$. $F$ is a point on the line $AC$ such that $FA=2AC$ ($A$ is between $F$ and $C$).

Show that $EB$ and $AC$ are parallel and that the perpendicular from $F$ to $AB$, the perpendicular from $E$ to $AC$ and $BD$ are concurrent.
2013 ITAMO
Two magicians are performing the following game. Initially the first magician encloses the second magician in a cabin where he can neither see nor hear anything. To start the game, the first magician invites Daniel, from the audience, to put on each square of a chessboard $n \times n$, at his (Daniel's) discretion, a token black or white. Then the first magician asks Daniel to show him a square $C$ of his own choice. At this point, the first magician chooses a square $D$ (not necessarily different from $C$) and replaces the token that is on $D$ with other color token (white with black or black with white).

Then he opens the cabin in which the second magician was held. Looking at the chessboard, the second magician guesses what is the square $C$. For what value of $n$, the two magicians have a strategy such that the second magician makes a successful guess.
2012 ITAMO
On the sides of a triangle $ABC$ right angled at $A$ three points $D, E$ and $F$ (respectively $BC, AC$

and $AB$) are chosen so that the quadrilateral $AFDE$ is a square. If $x$ is the length of the side of the square, show that

$$\frac{1}{x}=\frac{1}{AB}+\frac{1}{AC}$$
2012 ITAMO
Determine all positive integers that are equal to $300$ times the sum of their digits.
2012 ITAMO
Let $n$ be an integer greater than or equal to $2$. There are $n$ people in one line, each of which is either a scoundrel (who always lie) or a knight (who always tells the truth). Every person, except the first, indicates a person in front of him/her and says "This person is a scoundrel" or "This person is a knight." Knowing that there are strictly more scoundrel than knights, seeing the statements show that it is possible to determine each person whether he/she is a scoundrel or a knight.
2012 ITAMO
Let $x_1,x_2,x_3, \cdots$ be a sequence defined by the following recurrence relation:

$$ \begin{cases}x_{1}&= 4\\ x_{n+1}&= x_{1}x_{2}x_{3}\cdots x_{n}+5\text{ for }n\ge 1\end{cases} $$

The first few terms of the sequence are $x_1=4,x_2=9,x_3=41 \cdots$



Find all pairs of positive integers $\{a,b\}$ such that $x_a x_b$ is a perfect square.
2012 ITAMO
$ABCD$ is a square. Describe the locus of points $P$, different from $A, B, C, D$, on that plane for which

$$\widehat{APB}+\widehat{CPD}=180^\circ$$
2012 ITAMO
Determine all pairs $\{a, b\}$ of positive integers with the property that, in whatever manner you color the positive integers with two colors $A$ and $B$, there always exist two positive integers of color $A$ having their difference equal to $a$ or of color $B$ having their difference equal to $b$.
2011 ITAMO
A trapezium is given with parallel bases having lengths $1$ and $4$. Split it into two trapeziums by a cut, parallel to the bases, of length $3$. We now want to divide the two new trapeziums, always by means of cuts parallel to the bases, in $m$ and $n$ trapeziums, respectively, so that all the $m + n$ trapezoids obtained have the same area. Determine the minimum possible value for $m + n$ and the lengths of the cuts to be made to achieve this minimum value.
2011 ITAMO
A sequence of positive integers $a_1, a_2,\ldots, a_n$ is called ladder of length $n$ if it consists of $n$ consecutive integers in ascending order.

(a) Prove that for every positive integer $n$ there exist two ladders of length $n$, with no elements in common,

$a_1, a_2,\ldots, a_n$ and $b_1, b_2,\ldots, b_n$, such that for all $i$ between $1$ and $n$, the greatest common divisor of $a_i$ and $b_i$ is equal to $1$.

(b) Prove that for every positive integer $n$ there exist two ladders of length $n$, with no elements in common,

$a_1, a_2,\ldots, a_n$ and $b_1, b_2,\ldots, b_n$, such that for all $i$ between $1$ and $n$, the greatest common divisor of  $a_i$ and $b_i$ is greater than $1$.
2011 ITAMO
Integers between $1$ and $7$ are written on a blackboard. It is possible that not all the numbers from 1 to 7 are present, and it is also possible that one, some or all of the numbers are repeated, one or more times.

A move consists of choosing one or more numbers on the blackboard, where all distinct, delete them and write different numbers in their place, such that the written numbers together with those deleted form the whole set $\{1, 2, 3, 4, 5, 6 , 7\}$



For example, moves allowed are:

• delete a $4$ and a $5$, and write in their place the numbers $1, 2, 3, 6$ and $7$;

• deleting a $1$, a $2$, a $3$, a $4$, a $5$, a $6$ and a $7$ and write nothing in their place.



Prove that, if it is possible to find a sequence of moves, starting from the initial situation, leading to have on board a single number (written once), then this number does not depend on the sequence of moves used.
2011 ITAMO
$ABCD$ is a convex quadrilateral. $P$ is the intersection of external bisectors of $\angle DAC$ and $\angle DBC$. Prove that $\angle APD = \angle BPC$ if and only if $AD+AC=BC+BD$
2011 ITAMO
Determine all solutions $(p,n)$ of the equation

$$n^3=p^2-p-1$$

where $p$ is a prime number and $n$ is an integer
2011 ITAMO
Let $X = \{1, 2, 3, 4, 5, 6, 7, 8\}$. We want to color, using $k$ colors, all subsets of $3$ elements of $X$ in such a way that, two disjoint subsets have distinct colors.

Prove that:

(a) $4$ colors are sufficient;

(b) $3$ colors are not sufficient.
2010 ITAMO
In a mathematics test number of participants is $N < 40$. The passmark is fixed at $65$. The test results are

the following:

The average of all participants is $66$, that of the promoted $71$ and that of the repeaters $56$.

However, due to an error in the wording of a question, all scores are increased by $5$. At this point

the average of the promoted participants becomes $75$ and that of the non-promoted $59$.

(a) Find all possible values ​​of $N$.

(b) Find all possible values ​​of $N$ in the case where, after the increase, the average of the promoted had become $79$ and that of non-promoted $47$.
2010 ITAMO
Every non-negative integer is coloured white or red, so that:

• there are at least a white number and a red number;

• the sum of a white number and a red number is white;

• the product of a white number and a red number is red.

Prove that the product of two red numbers is always a red number, and the sum of two red numbers is always a red number.
2010 ITAMO
Let $ABCD$ be a convex quadrilateral. such that $\angle CAB = \angle CDA$ and $\angle BCA = \angle ACD$. If $M$ be the midpoint of $AB$, prove that $\angle BCM = \angle DBA$.
2010 ITAMO
In a trapezium $ABCD$, the sides $AB$ and $CD$ are parallel and the angles $\angle ABC$ and $\angle BAD$ are acute. Show that it is possible to divide the triangle $ABC$ into 4 disjoint triangle $X_1. . . , X_4$ and the triangle $ABD$ into 4 disjoint triangles $Y_1,. . . , Y_4$ such that the triangles $X_i$ and $Y_i$ are congruent for all $i$.
2010 ITAMO
In the land of Cockaigne, people play the following solitaire. It starts from a finite string of zeros and ones, and are granted the following moves:

(i) cancel each two consecutive ones;

(ii) delete three consecutive zeros;

(iii) if the substring within the string is $01$, one may replace this by substring $100$.

The moves (i), (ii) and (iii) must be made one at a time.  You win if you can reduce the string to a string formed by two digits or less.

(For example, starting from $0101$, one can win using move (iii) first in the last two digits, resulting in $01100$, then playing the move (i) on two 'ones', and finally the move (ii) on the three zeros, one will get the empty string.)

Among all the $1024$ possible strings of ten-digit binary numbers, how many are there from which it is not possible to win the solitary?
2010 ITAMO
Prove that there are infinitely many prime numbers that divide at least one integer of the form $2^{n^3+1}-3^{n^2+1}+5^{n+1}$ where $n$ is a positive integer.
2009 ITAMO
Let $a < b < c < d < e$ be real numbers. We calculate all possible sums in pairs of these 5 numbers. Of these 10 sums, the three smaller ones are 32, 36, 37, while the two larger ones are 48 and 51. Determine all possible values ​​that $e$ can take.
2009 ITAMO
$ABCD$ is a square with centre $O$. Two congruent isosceles triangle $BCJ$ and $CDK$ with base $BC$ and $CD$ respectively are constructed outside the square. let $M$ be the midpoint of $CJ$. Show that $OM$ and $BK$ are perpendicular to each other.
2009 ITAMO
A natural number $n$ is called nice if it enjoys the following properties:

• The expression is made ​​up of $4$ decimal digits;

• the first and third digits of $n$ are equal;

• the second and fourth digits of $n$ are equal;

• the product of the digits of $n$ divides $n^2$.

Determine all nice numbers.
2009 ITAMO
A flea is initially at the point $(0, 0)$ in the Cartesian plane. Then it makes $n$ jumps. The direction of the jump is taken in a choice of the four cardinal directions. The first step is of length $1$, the second of length $2$, the third of length $4$, and so on. The $n^{th}$-jump is of length $2^{n-1}$. Prove that, if you know the final position flea, then it is possible to uniquely determine its position after each of the $n$ jumps.
2009 ITAMO
Let $ABC$ be an acute-angled scalene triangle and $\Gamma$ be its circumcircle. $K$ is the foot of the internal bisector of $\angle BAC$ on $BC$. Let $M$ be the midpoint of the arc $BC$ containing $A$. $MK$ intersect $\Gamma$ again at $A'$. $T$ is the intersection of the tangents at $A$ and $A'$. $R$ is the intersection of the perpendicular to $AK$ at $A$ and perpendicular to $A'K$ at $A'$. Show that $T, R$ and $K$ are collinear.
2009 ITAMO
A natural number $k$ is said $n$-squared if by colouring the squares of a $2n \times k$ chessboard, in any manner, with $n$ different colours, we can find $4$ separate unit squares of the same colour, the centers of which are vertices of a rectangle having sides parallel to the sides of the board. Determine, in function of $n$, the smallest natural $k$ that is $n$-squared.
2008 ITAMO
Let $ ABCDEFGHILMN$ be a regular dodecagon, let $ P$ be the intersection point of the diagonals $ AF$ and $ DH$. Let $ S$ be the circle which passes through $ A$ and $ H$, and which has the same radius of the circumcircle of the dodecagon, but is different from the circumcircle of the dodecagon. Prove that:

1. $ P$ lies on $ S$

2. the center of $ S$ lies on the diagonal $ HN$

3. the length of $ PE$ equals the length of the side of the dodecagon
2008 ITAMO
A square $ (n - 1) \times (n - 1)$ is divided into $ (n - 1)^2$ unit squares in the usual manner. Each of the $ n^2$ vertices of these squares is to be coloured red or blue. Find the number of different colourings such that each unit square has exactly two red vertices. (Two colouring schemse are regarded as different if at least one vertex is coloured differently in the two schemes.)
2008 ITAMO
Find all functions $ f: Z \rightarrow R$ that verify the folowing two conditions:

(i) for each pair of integers $ (m,n)$ with $ m<n$ one has $ f(m)<f(n)$;

(ii) for each pair of integers $ (m,n)$ there exists an integer $ k$ such that $ f(m)-f(n)=f(k)$.
2008 ITAMO
Find all triples $ (a,b,c)$ of positive integers such that $ a^2+2^{b+1}=3^c$.
2008 ITAMO
Let $ ABC$ be a triangle, all of whose angles are greater than $ 45^{\circ}$ and smaller than $ 90^{\circ}$.

(a) Prove that one can fit three squares inside $ ABC$ in such a way that: (i) the three squares are equal (ii) the three squares have common vertex $ K$ inside the triangle (iii) any two squares have no common point but $ K$ (iv) each square has two opposite vertices onthe boundary of $ ABC$, while all the other points of the square are inside $ ABC$.

(b) Let $ P$ be the center of the square which has $ AB$ as a side and is outside $ ABC$. Let $ r_{C}$ be the line symmetric to $ CK$ with respect to the bisector of $ \angle BCA$. Prove that $ P$ lies on $ r_{C}$.
2008 ITAMO
Francesca and Giorgia play the following game. On a table there are initially coins piled up in some stacks, possibly in different numbers in each stack, but with at least one coin. In turn, each player chooses exactly one move between the following:

(i) she chooses a stack that has an even non-zero number of coins $ 2k$ and breaks it into two identical stacks of coins, i.e. each stack contains $ k$ coins;

(ii) she removes from the table the stacks with coins in an odd number, i.e. all such in odd number, not just those with a specific odd number.

At each turn, a player necessarily moves: if one choice is not available, the she must take the other.

Francesca moves first. The one who removes the last coin from the table wins.

1. If initially there is only one stack of coins on the table, and this stack contains $ 2008^{2008}$ coins, which of the players has a winning strategy?

2. For which initial configurations of stacks of coins does Francesca have a winning strategy?
2007 ITAMO
It is given a regular hexagon in the plane. Let P be a point of the plane. Define s(P) as the sum of the distances from P to each side of the hexagon, and v(P) as the sum of the distances from P to each vertex.

a) Find the locus of points P that minimize s(P)

b) Find the locus of points P that minimize v(P)
2007 ITAMO
We define two polynomials with integer coefficients P,Q to be similar if the coefficients of P are a permutation of the coefficients of Q.

a) if P,Q are similar, then $P(2007)-Q(2007)$ is even

b) does there exist an integer $k > 2$ such that $k \mid P(2007)-Q(2007)$ for all similar polynomials P,Q?
2007 ITAMO
Let ABC be a triangle, G its centroid, M the midpoint of AB, D the point on the line $AG$ such that $AG = GD, A \neq D$, E the point on the line $BG$ such that $BG = GE, B \neq E$. Show that the quadrilateral BDCM is cyclic if and only if $AD = BE$.
2007 ITAMO
Today is Barbara's birthday, and Alberto wants to give her a gift playing the following game. The numbers 0,1,2,...,1024 are written on a blackboard. First Barbara erases $2^{9}$ numbers, then Alberto erases $2^{8}$ numbers, then Barbara $2^{7}$ and so on, until there are only two numbers a,b left. Now Barbara earns $|a-b|$ euro.

Find the maximum number of euro that Barbara can always win, independently of Alberto's strategy.
2007 ITAMO
The sequence of integers $(a_{n})_{n \ge 1}$ is defined by $a_{1}= 2$, $a_{n+1}= 2a_{n}^{2}-1$.

Prove that for each positive integer n, $n$ and $a_{n}$ are coprime.
2007 ITAMO
a) For each $n \ge 2$, find the maximum constant $c_{n}$ such that

$\frac 1{a_{1}+1}+\frac 1{a_{2}+1}+\ldots+\frac 1{a_{n}+1}\ge c_{n}$

for all positive reals $a_{1},a_{2},\ldots,a_{n}$ such that $a_{1}a_{2}\cdots a_{n}= 1$.



b) For each $n \ge 2$, find the maximum constant $d_{n}$ such that

$\frac 1{2a_{1}+1}+\frac 1{2a_{2}+1}+\ldots+\frac 1{2a_{n}+1}\ge d_{n}$

for all positive reals $a_{1},a_{2},\ldots,a_{n}$ such that $a_{1}a_{2}\cdots a_{n}= 1$.
2006 ITAMO
Two people play the following game: there are $40$ cards numbered from $1$ to $10$ with $4$ different signs. At the beginning they are given $20$ cards each. Each turn one player either puts a card on the table or removes some cards from the table, whose sum is $15$. At the end of the game, one player has a $5$ and a $3$ in his hand, on the table there's a $9$, the other player has a card in his hand. What is it's value?
2006 ITAMO
Solve $p^n+144=m^2$ where $m,n\in \mathbb{N}$ and $p$ is a prime number.
2006 ITAMO
Let $A$ and $B$ be two distinct points on the circle $\Gamma$, not diametrically opposite. The point $P$, distinct from $A$ and $B$, varies on $\Gamma$. Find the locus of the orthocentre of triangle $ABP$.
2006 ITAMO
The squares of an infinite chessboard are numbered $1,2,\ldots $ along a spiral, as shown in the picture. A rightline  is the sequence of the numbers in the squares obtained by starting at one square at going to the right.

a) Prove that exists a rightline without multiples of $3$.

b) Prove that there are infinitely many pairwise disjoint rightlines not containing multiples of $3$.
2006 ITAMO
Consider the inequality

$$(a_1+a_2+\dots+a_n)^2\ge 4(a_1a_2+a_2a_3+\cdots+a_na_1).$$

a) Find all $n\ge 3$ such that the inequality is true for positive reals.

b) Find all $n\ge 3$ such that the inequality is true for reals.
2006 ITAMO
Alberto and Barbara play the following game. Initially, there are some piles of coins on a table. Each player in turn, starting with Albert, performs one of the two following ways:

1) take a coin from an arbitrary pile;

2) select a pile and divide it into two non-empty piles.

The winner is the player who removes the last coin on the table. Determine which player has a winning strategy with respect to the initial state.
2005 ITAMO
Let $ABC$ be a right angled triangle with hypotenuse $AC$, and let $H$ be the foot of the altitude from $B$ to $AC$. Knowing that there is a right-angled triangle with side-lengths $AB, BC, BH$, determine all the possible values ​​of $\frac{AH}{CH}$
2005 ITAMO
Prove that among any $18$ consecutive positive integers not exceeding $2005$ there is at least one divisible by the sum of its digits.
2005 ITAMO
In each cell of a $4 \times 4$ table a digit $1$ or $2$ is written. Suppose that the sum of the digits in each of the four $3 \times 3$ sub-tables is divisible by $4$, but the sum of the digits in the entire table is not divisible by $4$. Find the greatest and the smallest possible value of the sum of the $16$ digits.
2005 ITAMO
Determine all $n \geq 3$ for which there are $n$ positive integers $a_1, \cdots , a_n$ any two of which have a common divisor greater than $1$, but any three of which are coprime. Assuming that, moreover, the numbers $a_i$ are less than $5000$, find the greatest possible $n$.
2005 ITAMO
Let $h$ be a positive integer. The sequence $a_n$ is defined by $a_0 = 1$ and



$$a_{n+1} = \{\begin{array}{c} \frac{a_n}{2} \text{  if  } a_n \text{  is  even  }\\\\a_n+h \text{  otherwise  }.\end{array}$$



For example, $h = 27$ yields $a_1=28, a_2 = 14, a_3 = 7, a_4 = 34$ etc. For which $h$ is there an $n > 0$ with $a_n = 1$?
2005 ITAMO
Two circles $\gamma_1, \gamma_2$ in a plane, with centers $A$ and $B$ respectively, intersect at $C$ and $D$. Suppose that the circumcircle of $ABC$ intersects $\gamma_1$ in $E$ and $\gamma_2$ in $F$, where the arc $EF$ not containing $C$ lies outside $\gamma_1$ and $\gamma_2$. Prove that this arc $EF$ is bisected by the line $CD$.
2004 ITAMO
Observing the temperatures recorded in Cesenatico during the December and January, Stefano noticed an interesting coincidence: in each day of this period, the low temperature is equal to the sum of the low temperatures the preceeding day and the succeeding day.

Given that the low temperatures in December $3$ and January $31$ were $5^\circ \text C$ and $2^\circ \text C$ respectively, find the low temperature in December $25$.
2004 ITAMO
Two parallel lines $r,s$ and two points $P \in r$ and $Q \in s$ are given in a plane. Consider all pairs of circles $(C_P, C_Q)$ in that plane such that $C_P$ touches $r$ at $P$ and $C_Q$ touches $s$ at $Q$ and which touch each other externally at some point $T$. Find the locus of $T$.
2004 ITAMO
(a) Is $2005^{2004}$ the sum of two perfect squares?

(b) Is $2004^{2005}$ the sum of two perfect squares?
2004 ITAMO
Antonio and Bernardo play the following game. They are given two piles of chips, one with $m$ and the other with $n$ chips. Antonio starts, and thereafter they make in turn one of the following moves:

(i) take a chip from one pile;

(ii) take a chip from each of the piles;

(ii) remove a chip from one of the piles and put it onto the other.

Who cannot make any more moves, loses. Decide, as a function of $m$ and $n$ if one of the players has a winning strategy, and in the case of the affirmative answer describe that strategy.
2004 ITAMO
Decide if the following statement is true or false:

For every sequence $\{x_n\}_{n\in \mathbb{N}}$ of non-negative real numbers, there exist sequences $\{a_n\}_{n\in\mathbb{N}}$ and $\{b_n\}_{n\in\mathbb{N}}$ of non-negative real numbers such that:

(a) $x_n = a_n + b_n$ for all $n$;

(b) $a_1 + \cdots + a_n \le n$ for infinitely many values of $n$;

(c) $b_1 + \cdots + b_n \le n$ for infinitely many values of $n$.
2004 ITAMO
Let $P$ be a point inside a triangle $ABC$. Lines $AP,BP,CP$ meet the opposite sides of the triangle at points $A',B',C'$ respectively. Denote $x =\frac{AP}{PA'}, y = \frac{BP}{PB'}$ and $z = \frac{CP}{PC'}$. Prove that $xyz = x+y+z+2$.
2003 ITAMO
Find all three digit numbers $n$ which are equal to the number formed by three last digit of $n^2$.