Navvye/Olympiad-Problems-CSV huggingface dataset:
2003 ITAMO
A museum has the shape of a $n \times n$ square divided into $n^2$ rooms of the shape of a unit square $(n>1)$. Between every two adjacent rooms (i.e. sharing a wall) there is a door. A night guardian wants to organize an inspection journey through the museum according to the following rules. He starts from some room and, whenever he enters a room, he stays there for exactly one minute and then proceeds to another room. He is allowed to enter a room more than once, but at the end of his journey he must have spent exactly $k$ minutes in every room. Find all $n$ and $k$ for which it is possible to organize such a journey.
2003 ITAMO
Let a semicircle is given with diameter $AB$ and centre $O$ and let $C$ be a arbitrary point on the segment $OB$. Point $D$ on the semicircle is such that $CD$ is perpendicular to $AB$. A circle with centre $P$ is tangent to the arc $BD$ at $F$ and to the segment $CD$ and $AB$ at $E$ and $G$ respectively. Prove that the triangle $ADG$ is isosceles.
2003 ITAMO
There are two sorts of people on an island: knights, who always talk truth, and scoundrels, who always lie. One day, the people establish a council consisting of $2003$ members. They sit around a round table, and during the council each member said: "Both my neighbors are scoundrels". In a later day, the council meets again, but one member could not come due to illness, so only $2002$ members were present. They sit around the round table, and everybody said: "Both my neighbors belong to the sort different from mine". Is the absent member a knight or a scoundrel?
2003 ITAMO
In each lattice-point of an $m \times n$ grid and in the centre of each of the formed unit squares a pawn is placed.



a) Find all such grids with exactly $500$ pawns.

b) Prove that there are infinitely many positive integers $k$ for which therer is no grid containing exactly $k$ pawns.
2003 ITAMO
Every of $n$ guests invited to a dinner has got an invitation denoted by a number from $1$ to $n$. The guests will be sitting around a round table with $n$ seats. The waiter has decided to derve them according to the following rule. At first, he selects one guest and serves him/her at any place. Thereafter, he selects the guests one by one: having chosen a guest, he goes around the table for the number of seats equal to the preceeding guest's invitation number (starting from the seat of the preceeding guest), and serves the guest there.

Find all $n$ for which he can select the guests in such an order to serve all the guests.
2002 ITAMO
Find all $3$-digit positive integers that are $34$ times the sum of their digits.
2002 ITAMO
The plan of a house has the shape of a capital $L$, obtained by suitably placing side-by-side four squares whose sides are $10$ metres long. The external walls of the house are $10$ metres high. The roof of the house has six faces, starting at the top of the six external walls, and each face forms an angle of $30^\circ$ with respect to a horizontal plane.

Determine the volume of the house (that is, of the solid delimited by the six external walls, the six faces of the roof, and the base of the house).
2002 ITAMO
Let $A$ and $B$ are two points on a plane, and let $M$ be the midpoint of $AB$. Let $r$ be a line and let $R$ and $S$ be the projections of $A$ and $B$ onto $r$. Assuming that $A$, $M$, and $R$ are not collinear, prove that the circumcircle of triangle $AMR$ has the same radius as the circumcircle of $BSM$.
2002 ITAMO
Find all values of $n$ for which all solutions of the equation $x^3-3x+n=0$ are integers.
2002 ITAMO
Prove that if $m=5^n+3^n+1$ is a prime, then $12$ divides $n$.
2002 ITAMO
We are given a chessboard with 100 rows and 100 columns. Two squares of the board are said to be adjacent if they have a common side. Initially all squares are white.



a) Is it possible to colour an odd number of squares in such a way that each coloured square has an odd number of adjacent coloured squares?



b) Is it possible to colour some squares in such a way that an odd number of them have exactly $4$ adjacent coloured squares and all the remaining coloured squares have exactly $2$ adjacent coloured squares?



c) Is it possible to colour some squares in such a way that an odd number of them have exactly $2$ adjacent coloured squares and all the remaining coloured squares have exactly $4$ adjacent coloured squares?
2001 ITAMO
A hexagon has all its angles equal, and the lengths of four consecutive sides are $5$, $3$, $6$ and $7$, respectively. Find the lengths of the remaining two edges.
2001 ITAMO
In a basketball tournament every two teams play two matches. As usual, the winner of a match gets $2$ points, the loser gets $0$, and there are no draws. A single team wins the tournament with $26$ points and exactly two teams share the last position with $20$ points. How many teams participated in the tournament?
2001 ITAMO
Consider the equation

$$ x^{2001}=y^x .$$



Find all pairs $(x,y)$ of solutions where $x$ is a prime number and $y$ is a positive integer.



Find all pairs $(x,y)$ of solutions where $x$ and $y$ are positive integers.





(Remember that $2001=3 \cdot 23 \cdot 29$.)
2001 ITAMO
A positive integer is called monotone if has at least two digits and all its digits are nonzero and appear in a strictly increasing or strictly decreasing order.



(a) Compute the sum of all monotone five-digit numbers.

(b) Find the number of final zeros in the least common multiple of all monotone numbers (with any number of digits).
2001 ITAMO
Let $ABC$ be a triangle and $\gamma$ the circle inscribed in $ABC$. The circle $\gamma$ is tangent to side $AB$ at the point $T$. Let $D$ be the point of $\gamma$ diametrically opposite to $T$, and $S$ the intersection point of the line through $C$ and $D$ with side $AB$.

Prove that $AT=SB$.
2001 ITAMO
A panel contains $100$ light bulbs, arranged in a $10$ by $10$ square array. Some of them are on, the others are off.

The electrical system is such that when the switch corresponding to a light bulb is pressed, all the light bulbs that are on the same row or column of it (including the bulb linked to the pressed switch) change their state (that is they are turned on or off).



From which starting configurations, pressing the right sequence of switches, is it possible to achieve that all bulbs are on at the same time?



What is the answer to the previous question if the bulbs are $81$, arranged in a $9$ by $9$ panel?
2000 ITAMO
A possitive integer is called special if all its decimal digits are equal and it can be represented as the sum of squares of three consecutive odd integers.

(a) Find all $4$-digit special numbers

(b) Are there $2000$-digit special numbers?
2000 ITAMO
Let $ABCD$ be a convex quadrilateral, and write $\alpha=\angle DAB$; $\beta=\angle ADB$; $\gamma=\angle ACB$; $\delta= \angle DBC$; and $\epsilon=\angle DBA$. Assuming that $\alpha<\pi/2$, $\beta+\gamma=\pi /2$, and $\delta+2\epsilon=\pi$, prove that

$$(DB+BC)^2=AD^2+AC^2$$



[Moderator edit: Also discussed at  .]
2000 ITAMO
A pyramid with the base $ABCD$ and the top $V$ is inscribed in a sphere. Let $AD = 2BC$ and let the rays $AB$ and $DC$ intersect in point $E$. Compute the ratio of the volume of the pyramid $VAED$ to the volume of the pyramid $VABCD$.
2000 ITAMO
Let $n > 1$ be a fixed integer. Alberto and Barbara play the following game:

(i) Alberto chooses a positive integer,

(ii) Barbara chooses an integer greater than $1$ which is a multiple or submultiple of the number Alberto chose (including itself),

(iii) Alberto increases or decreases the Barbara’s number by $1$.

Steps (ii) and (iii) are alternatively repeated. Barbara wins if she succeeds to reach the number $n$ in at most $50$ moves. For which values of $n$ can she win, no matter how Alberto plays?
2000 ITAMO
A man disposes of sufficiently many metal bars of length $2$ and wants to construct a grill of the shape of an $n \times n$ unit net. He is allowed to fold up two bars at an endpoint or to cut a bar into two equal pieces, but two bars may not overlap or intersect. What is the minimum number of pieces he must use?
2000 ITAMO
Let $p(x)$ be a polynomial with integer coefficients such that $p(0) = 0$ and $0 \le  p(1) \le  10^7$. Suppose that there exist positive integers $a,b$ such that $p(a) = 1999$ and $p(b) = 2001$. Determine all possible values of $p(1)$.



(Note: $1999$ is a prime number.)
1999 ITAMO
A rectangular sheet with sides $a$ and $b$ is fold along a diagonal. Compute the area of the overlapping triangle.
1999 ITAMO
An integer is balance if the number of digit in its decimal representation is equal to the number of its distinct prime factors (For example, 15 is balanced, but not 49).

Prove that there are finite balanced number.
1999 ITAMO
Let $r_1,r_2,r$, with $r_1 < r_2 < r$, be the radii of three circles $\Gamma_1,\Gamma_2,\Gamma$, respectively. The circles $\Gamma_1,\Gamma_2$ are internally tangent to $\Gamma$ at two distinct points $A,B$ and intersect in two distinct points. Prove that the segment $AB$ contains an intersection point of $\Gamma_1$ and $\Gamma_2$ if and only if $r_1 +r_2 = r$.
1999 ITAMO
Albert and Barbara play the following game. On a table there are $1999$ sticks, and each player in turn removes some of them: at least one stick, but at most half of the currently remaining sticks. The player who leaves just one stick on the table loses the game. Barbara moves first. Decide which player has a winning strategy and describe that strategy.
1999 ITAMO
There is a village of pile-built dwellings on a lake, set on the gridpoints of an $m \times n$ rectangular grid. Each dwelling is connected by exactly $p$ bridges to some of the neighboring dwellings (diagonal connections are not allowed, two dwellings can be connected by more than one bridge). Determine for which values $m,n, p$ it is possible to place the bridges so that from any dwelling one can reach any other dwelling.
1999 ITAMO
(a) Find all pairs $(x,k)$ of positive integers such that $3^k -1 = x^3$ .

(b) Prove that if $n > 1$ is an integer, $n \ne 3$, then there are no pairs $(x,k)$ of positive integers such that $3^k -1 = x^n$.
1998 ITAMO
Calculate the sum  $\sum_{n=1}^{1.000.000}[ \sqrt{n} ]$ .

[You may use the formula$\sum_{i=1}^{k} i^2=\frac{k(k +1)(2k +1)}{6}$ without a proof.]
1998 ITAMO
Prove that in each polyhedron there exist two faces with the same number of edges.
1998 ITAMO
Alberto wants to organize a poker game with his friends this evening. Bruno and Barbara together go to gym once in three evenings, whereas Carla, Corrado, Dario and Davide are busy once in two evenings (not necessarily the same day). Moreover, Dario is not willing to play with Davide, since they have a quarrel over a girl. A poker game requires at least four persons (including Alberto). What is the probability that the game will be played?
1998 ITAMO
Let $ABCD$ be a trapezoid with the longer base $AB$ such that its diagonals $AC$ and $BD$ are perpendicular. Let O be the circumcenter of the triangle $ABC$ and $E$ be the intersection of the lines $OB$ and $CD$. Prove that $BC^2 = CD \cdot CE$.
1998 ITAMO
Suppose $a_1,a_2,a_3,a_4$ are distinct integers and $P(x)$ is a polynomial with integer coefficients satisfying $P(a_1) = P(a_2) = P(a_3) = P(a_4) = 1$.

(a) Prove that there is no integer $n$ such that $P(n) = 12$.

(b) Do there exist such a polynomial and $a_n$ integer $n$ such that $P(n) = 1998$?
1998 ITAMO
We say that a function $f : N \to N$ is increasing if $f(n) < f(m)$ whenever $n < m$, multiplicative if $f(nm) = f(n)f(m)$ whenever $n$ and $m$ are coprime, and completely multiplicative if $f(nm) = f(n)f(m)$ for all $n,m$.

(a) Prove that if $f$ is increasing then $f(n) \ge n$ for each $n$.

(b) Prove that if $f$ is increasing and completely multiplicative and $f(2) = 2$, then $f(n) = n$ for all $n$.

(c) Does (b) remain true if the word ”completely” is omitted?
1997 ITAMO
An infinite rectangular stripe of width $3$ cm is folded along a line. What is the minimum possible area of the region of overlapping?
1997 ITAMO
Let a real function $f$ defined on the real numbers satisfy the following conditions:

(i) $f(10+x) = f(10- x)$

(ii) $f(20+x) = - f(20- x)$

for all $x$. Prove that f is odd and periodic.
1997 ITAMO
The positive quadrant of a coordinate plane is divided into unit squares by lattice lines. Is it possible to color the squares in black and white so that:

(i) In every square of side $n$ ($n \in N$) with a vertex at the origin and sides are parallel to the axes, there are more black than white squares;

(ii) Every diagonal parallel to the line $y = x$ intersects only finitely many black squares?
1997 ITAMO
Let $ABCD$ be a tetrahedron. Let $a$ be the length of $AB$ and let $S$ be the area of the projection of the tetrahedron onto a plane perpendicular to $AB$. Determine the volume of the tetrahedron in terms of $a$ and $S$.
1997 ITAMO
Let $X$ be the set of natural numbers whose all digits in the decimal representation are different. For $n \in N$, denote by $A_n$ the set of numbers whose digits are a permutation of the digits of $n$, and $d_n$ be the greatest common divisor of the numbers in $A_n$. (For example, $A_{1120} =\{112,121,...,2101,2110\}$, so $d_{1120}  = 1$.)

Find the maximum possible value of $d_n$.
1997 ITAMO
A tourist wants to visit each of the ten cities shown on the picture. The continuous segments on the picture denote railway lines, whereas the dashed segments denote air lines. A railway line costs $150000$ lires, and an air line costs $250000$ lires. What is the minimum possible price of a desired route?
1996 ITAMO
Among all the triangles which have a fixed side $l$ and a fixed area $S$, determine for which triangles the product of the altitudes is maximum.
1996 ITAMO
Show that the equation $a^2 + b^2 = c^2 + 3$ has infinetely many triples of integers $a, b, c$ that are solutions.
1996 ITAMO
Given a cube of unit side. Let $A$ and $B$ be two opposite vertex. Determine the radius of the sphere, with center inside the cube, tangent to the three faces of the cube with common point $A$ and tangent to the three sides with common point $B$.
1996 ITAMO
There is a list of $n$ football matches. Determine how many possible columns, with an even number of draws, there are.
1996 ITAMO
Given a circle $C$ and an exterior point $A$. For every point $P$ on the circle construct the square $APQR$ (in counterclock order). Determine the locus of the point $Q$ when $P$ moves on the circle $C$.
1996 ITAMO
What is the minimum number of squares that is necessary to draw on a white sheet to obtain a square grid of side $n$?
1995 ITAMO
Determine for which values of $n$ it is possible to tile a square of side $n$ with figures of the type shown in the picture
1995 ITAMO
No two of $20$ students in a class have the same scores on both written and oral examinations in mathematics. We say that student $A$ is better than $B$ if his two scores are greater than or equal to the corresponding scores of $B$. The scores are integers between $1$ and $10$.

(a) Show that there exist three students $A,B,C$ such that $A$ is better than $B$ and $B$ is better than $C$.

(b) Would the same be true for a class of $19$ students?
1995 ITAMO
In a town there are four pubs, $A,B,C,D$, and any two of them are connected to each other except $A$ and $D$. A drunkard wanders about the pubs starting with $A$ and, after having a drink, goes to any of the pubs directly connected, with equal probability.

(a) What is the probability that the drunkard is at $C$ at its fifth drink?

(b) Where is the drunkard most likely to be after $n$ drinks ($n > 5$)?
1995 ITAMO
An acute-angled triangle $ABC$ is inscribed in a circle with center $O$. The bisector of $\angle A$ meets $BC$ at $D$, and the perpendicular to $AO$ through $D$ meets the segment $AC$ in a point $P$. Show that $AB = AP$.
1995 ITAMO
Two non-coplanar circles in space are tangent at a point and have the same tangents at this point. Show that both circles lie on some sphere.
1995 ITAMO
Find all pairs of positive integers $x,y$ such that $x^2 +615 = 2^y$
1994 ITAMO
Show that there exists an integer $N$ such that for all $n \ge N$ a square can be partitioned into $n$ smaller squares.
1994 ITAMO
solve this diophantine equation y^2 = x^3 - 16
1994 ITAMO
A journalist wants to report on the island of scoundrels and knights, where all inhabitants are either scoundrels (and they always lie) or knights (and they always tell the truth). The journalist interviews each inhabitant exactly once and

gets the following answers:

$A_1$: On this island there is at least one scoundrel,

$A_2$: On this island there are at least two scoundrels,

$...$

$A_{n-1}$: On this island there are at least $n-1$ scoundrels,

$A_n$: On this island everybody is a scoundrel.

Can the journalist decide whether there are more scoundrels or more knights?
1994 ITAMO
Let $ABC$ be a triangle contained in one of the halfplanes determined by a line $r$. Points $A',B',C'$ are the reflections of $A,B,C$ in $r,$ respectively. Consider the line through $A'$ parallel to $BC$, the line through $B'$ parallel to $AC$ and the line through $C'$ parallel to $AB$. Show that these three lines have a common point.
1994 ITAMO
Let $OP$ be a diagonal of a unit cube. Find the minimum and the maximum value of the area of the intersection of the cube with a plane through $OP$.
1994 ITAMO
The squares of a $10 \times 10$ chessboard are labelled with $1,2,...,100 $ in the usual way: the $i$−th row contains the numbers $10i - 9,10i - 8,...,10i$ in increasing order. The signs of fifty numbers are changed so that each row and each column contains exactly five negative numbers. Show that after this change the sum of all numbers on the chessboard is zero.
1993 ITAMO
Let be given points $A,B,C$ on a line, with $C$ between $A$ and $B$. Three semicircles with diameters $AC,BC,AB$ are drawn on the same side of line $ABC$. The perpendicular to $AB$ at $C$ meets the circle with diameter $AB$ at $H$. Given that $CH =\sqrt2$, compute the area of the region bounded by the three semicircles.
1993 ITAMO
Find all pairs $(p,q)$ of positive primes such that the equation $3x^2 - px + q = 0$ has two distinct rational roots.
1993 ITAMO
Consider an infinite chessboard whose rows and columns are indexed by positive integers. At most one coin can be put on any cell of the chessboard. Let be given two arbitrary sequences ($a_n$) and ($b_n$) of positive integers ($n \in N$). Assuming that infinitely many coins are available, prove that they can be arranged on the chessboard so that there are $a_n$ coins in the $n$-th row and $b_n$ coins in the $n$-th column for all $n$.
1993 ITAMO
Let $P$ be a point in the plane of a triangle $ABC$, different from its circumcenter. Prove that the triangle whose vertices are the projections of $P$ on the perpendicular bisectors of the sides of $ABC$, is similar to $ABC$.
1993 ITAMO
Prove the following inequality for any positive real numbers a,b,c not exceeding 1

$a^2b+b^2c+c^2a+1\ge a^2+b^2+c^2$
1993 ITAMO
A unit cube $C$ is rotated around one of its diagonals for the angle $\pi /3$ to form a cube $C'$. Find the volume of the intersection of $C$ and $C'$.
1992 ITAMO
A cube is divided into $27$ equal smaller cubes. A plane intersects the cube. Find the maximum possible number of smaller cubes the plane can intersect.
1992 ITAMO
A convex quadrilateral of area $1$ is given. Prove that there exist four points in the interior or on the sides of the quadrilateral such that each triangle with the vertices in three of these four points has an area greater than or equal to $1/4$.
1992 ITAMO
Prove that for each $n \ge  3$ there exist $n$ distinct positive divisors $d_1,d_2, ...,d_n$ of $n!$ such that $n! = d_1 +d_2 +...+d_n$.
1992 ITAMO
A jury of $9$ persons should decide whether a verdict is guilty or not. Each juror votes independently with the probability $1/2$ for each of the two possibilities, and noone is allowed to be abstinent. Find the probability that a fixed juror will be a part of the majority. In the case of a jury of $n$ persons, find the values of n for which the probability of being a part of the majority is greater than, equal to, and smaller than $1/2$, respectively. (For $n = 2k$, $k +1$ votes are needed for a majority.)
1992 ITAMO
$a$, $b$, $c$ are real numbers. Show that

$\min((a-b)^2,(b-c)^2,(c-a)^2)\leq \frac{a^2+b^2+c^2}{2}$
1992 ITAMO
Let $a$ and $b$ be integers. Prove that if $\sqrt[3]{a}+\sqrt[3]{b}$ is a rational number, then both $a$ and $b$ are perfect cubes.
1991 ITAMO
For every triangle $ABC$ inscribed in a circle  $\Gamma$ , let $A',B',C'$ be the intersections of the bisectors of the angles at $A,B,C$ with  $\Gamma$ . Consider the triangle $A'B'C'$ .

(a) Do triangles $A'B'C'$ go over all possible triangles inscribed in $\Gamma$ as $\vartriangle ABC$ varies? If not, what are the constraints?

(b) Prove that the angle bisectors of $\vartriangle ABC$ are the altitudes of $\vartriangle A',B',C'$ .
1991 ITAMO
Prove that no number of the form $a^3+3a^2+a$, for a positive integer $a$, is a perfect square.
1991 ITAMO
We consider the sums of the form $\pm 1 \pm 4 \pm 9\pm ... \pm n^2$. Show that every integer can be represented in this form for some $n$. (For example, $3 = -1 + 4$ and $8 = 1-4-9+16+25-36-49+64$.)
1991 ITAMO
The squares of an $8 \times 8$ board are colored black and white in such a way that every row and every column contains exactly four black squares. Prove that the number of pairs of neighboring white squares is the same as the number of pairs of neighboring black squares. (Two squares are neighboring if they have a side in common.)
1991 ITAMO
For which values of $n$ does there exist a convex polyhedron with $n$ edges?
1991 ITAMO
We say that each positive number $x$ has two sons: $x+1$ and $\frac{x}{x+1}$. Characterize all the descendants of number $1$.
1990 ITAMO
A cube of edge length $3$ consists of $27$ unit cubes. Find the number of lines passing through exactly three centers of these $27$ cubes, as well as the number of those passing through exactly two such centers.
1990 ITAMO
In a triangle $ABC$, the bisectors of the angles at $B$ and $A$ meet the opposite sides at $P$ and $Q$, respectively. Suppose that the circumcircle of triangle $PQC$ passes through the incenter $R $of $\vartriangle ABC$. Given that $PQ = l$, find all sides of triangle $PQR$.
1990 ITAMO
Let $a,b,c$ be distinct real numbers and $P(x)$ a polynomial with real coefficients. Suppose that the remainders of $P(x)$ upon division by $(x-a), (x-b)$ and $(x-c)$ are $a,b$ and $c$, respectively. Find the polynomial that is obtained as the remainder of $P(x)$ upon division by $(x-a)(x-b)(x-c)$.
1990 ITAMO
Let $a,b,c$ be side lengths of a triangle with $a+b+c = 1$. Prove that $a^2 +b^2 +c^2 +4abc \le \frac12$ .
1990 ITAMO
Prove that, for any integer $x$, $x^2 +5x+16$ is not divisible by $169$.
1990 ITAMO
Some marbles are distributed over $2n + 1$ bags. Suppose that, whichever bag is removed, it is possible to divide the remaining bags into two groups of $n$ bags such that the number of marbles in each group is the same. Prove that all the bags contain the same number of marbles.
1989 ITAMO
Determine whether the equation $x^2 +xy+y^2 = 2$ has a solution $(x,y)$ in rational numbers.
1989 ITAMO
There are $30$ men with their $30$ wives sitting at a round table. Show that there always exist two men who are on the same distance from their wives. (The seats are arranged at vertices of a regular polygon.)
1989 ITAMO
Prove that, for every tetrahedron $ABCD$, there exists a unique point $P$ in the interior of the tetrahedron such that the tetrahedra $PABC,PABD,PACD,PBCD$ have equal volumes.
1989 ITAMO
Points $A,M,B,C,D$ are given on a circle in this order such that $A$ and $B$ are equidistant from $M$. Lines $MD$ and $AC$ intersect at $E$ and lines $MC$ and $BD$ intersect at $F$. Prove that the quadrilateral $CDEF$ is inscridable in a circle.
1989 ITAMO
A fair coin is repeatedly tossed. We receive one marker for every ”head” and two markers for every ”tail”. We win the game if, at some moment, we possess exactly $100$ markers. Is the probability of winning the game greater than, equal to, or less than $2/3$?