Olympiad problems

2023 IFYM, Sozopol, 7

Tags: combinatorics

In the country of Drilandia, which has at least three cities, there are bidirectional roads connecting some pairs of cities such that any city can be reached from any other. Two cities are called [i]close[/i] if one can reach the other by using at most two intermediary cities. The mayor, Drilago, fortified the road system by building a direct road between each pair of close cities that were not already connected. Prove that after the expansion, there exists a journey that starts and ends at the same city, where each city except the first is visited exactly once, and the first city is visited twice (once at the beginning and once at the end).

2019 India PRMO, 4

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An ant leaves the anthill for its morning exercise. It walks $4$ feet east and then makes a $160^\circ$ turn to the right and walks $4$ more feet. If the ant continues this patterns until it reaches the anthill again, what is the distance in feet it would have walked?

1997 Cono Sur Olympiad, 5

Tags: combinatorics , number theory

Let $n$ be a natural number $n>3$. Show that in the multiples of $9$ less than $10^n$, exist more numbers with the sum of your digits equal to $9(n - 2)$ than numbers with the sum of your digits equal to $9(n - 1)$.

PEN Q Problems, 12

Tags: polynomial , algebra , gauss

Prove that if the integers $a_{1}$, $a_{2}$, $\cdots$, $a_{n}$ are all distinct, then the polynomial \[(x-a_{1})^{2}(x-a_{2})^{2}\cdots (x-a_{n})^{2}+1\] cannot be expressed as the product of two nonconstant polynomials with integer coefficients.

2007 Alexandru Myller, 4

Tags: discrete math

At a math contest which has $ 5 $ problems, each candidate has solved $ 3 $ problems. Among these candidates, for any group of $ 5 $ candidates that we might choose, we see that there is a problem which all members of the group had solved it. Prove that there is a problem solved by all candidates.

1986 ITAMO, 2

Tags: recurrence relation , sequence , algebra

Determine the general term of the sequence ($a_n$) given by $a_0 =\alpha > 0$ and $a_{n+1} =\frac{a_n}{1+a_n}$ .

2000 Mediterranean Mathematics Olympiad, 2

Tags: geometry

Suppose that in the exterior of a convex quadrilateral $ABCD$ equilateral triangles $XAB,YBC,ZCD,WDA$ with centroids $S_1,S_2,S_3,S_4$ respectively are constructed. Prove that $S_1S_3\perp S_2S_4$ if and only if $AC=BD$.

2021 Estonia Team Selection Test, 3

Tags: inequalities , geometric inequality , geometry , disks

In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that \[O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.\] (A disk is assumed to contain its boundary.)

2020 LMT Fall, 27

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A list consists of all positive integers from $1$ to $2020$, inclusive, with each integer appearing exactly once. Define a move as the process of choosing four numbers from the current list and replacing them with the numbers $1,2,3,4$. If the expected number of moves before the list contains exactly two $4$'s can be expressed as $\frac{a}{b}$ for relatively prime positive integers, evaluate $a+b$. [i]Proposed by Richard Chen and Taiki Aiba[/i]

2018 International Zhautykov Olympiad, 2

Tags: incenter , geometry

Let $N,K,L$ be points on $AB,BC,CA$ such that $CN$ bisector of angle $\angle ACB$ and $AL=BK$.Let $BL\cap AK=P$.If $I,J$ be incenters of triangles $\triangle BPK$ and $\triangle ALP$ and $IJ\cap CN=Q$ prove that $IQ=JP$

1985 IMO Longlists, 94

Tags: geometry , circumcircle , angle , triangle

A circle with center $O$ passes through the vertices $A$ and $C$ of the triangle $ABC$ and intersects the segments $AB$ and $BC$ again at distinct points $K$ and $N$ respectively. Let $M$ be the point of intersection of the circumcircles of triangles $ABC$ and $KBN$ (apart from $B$). Prove that $\angle OMB=90^{\circ}$.

1997 Putnam, 4

Tags: function

Let $G$ be group with identity $e$ and $\phi :G\to G$ be a function such that : \[ \phi(g_1)\cdot \phi(g_2)\cdot \phi(g_3)=\phi(h_1)\cdot \phi(h_2)\cdot \phi(h_3) \] Whenever $g_1\cdot g_2\cdot g_3=e=h_1\cdot h_2\cdot h_3$ Show there exists $a\in G$ such that $\psi(x)=a\phi(x)$ is a homomorphism. (that is $\psi(x\cdot y)=\psi (x)\cdot \psi(y)$ for all $x,y\in G$ )

2012 Ukraine Team Selection Test, 9

Tags: geometry , collinear , incircle , centroid

The inscribed circle $\omega$ of the triangle $ABC$ touches its sides $BC, CA$ and $AB$ at points $A_1, B_1$ and $C_1$, respectively. Let $S$ be the intersection point of lines passing through points $B$ and $C$ and parallel to $A_1C_1$ and $A_1B_1$ respectively, $A_0$ be the foot of the perpendicular drawn from point $A_1$ on $B_1C_1$, $G_1$ be the centroid of triangle $A_1B_1C_1$, $P$ be the intersection point of the ray $G_1A_0$ with $\omega$. Prove that points $S, A_1$, and $P$ lie on a straight line.

2016 ASDAN Math Tournament, 9

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Compute $$\int_0^\infty\frac{\ln\left(\frac{1+x^{11}}{1+x^3}\right)}{(1+x^2)\ln x}dx.$$

2021 Purple Comet Problems, 3

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The diagram shows a semicircle with diameter $20$ and the circle with greatest diameter that fits inside the semicircle. The area of the shaded region is $N\pi$, where $N$ is a positive integer. Find $N$.

2023 Brazil Undergrad MO, 5

Tags: modular arithmetic , generating functions , recurrence relation , algebra

A drunken horse moves on an infinite board whose squares are numbered in pairs $(a, b) \in \mathbb{Z}^2$. In each movement, the 8 possibilities $$(a, b) \rightarrow (a \pm 1, b \pm 2),$$ $$(a, b) \rightarrow (a \pm 2, b \pm 1)$$ are equally likely. Knowing that the knight starts at $(0, 0)$, calculate the probability that, after $2023$ moves, it is in a square $(a, b)$ with $a \equiv 4 \pmod 8$ and $b \equiv 5 \pmod 8$.

2014 Contests, 2

Tags: quick nt

2. What’s the closest number to $169$ that’s divisible by $9$?

2010 Canada National Olympiad, 2

Tags:

Let $A,B,P$ be three points on a circle. Prove that if $a,b$ are the distances from $P$ to the tangents at $A,B$ respectively, and $c$ is the distance from $P$ to the chord $AB$, then $c^2 =ab$.

2018 IFYM, Sozopol, 6

Tags: number theory

Find all sets $(a, b, c)$ of different positive integers $a$, $b$, $c$, for which: [b]*[/b] $2a - 1$ is a multiple of $b$; [b]*[/b] $2b - 1$ is a multiple of $c$; [b]*[/b] $2c - 1$ is a multiple of $a$.

1978 Polish MO Finals, 2

Tags: combinatorics , combinatorial geometry

In a coordinate plane, consider the set of points with integer cooedinates at least one of which is not divisible by $4$. Prove that these points cannot be partitioned into pairs such that the distance between points in each pair equals $1$. In other words, an infinite chessboard, whose cells with both coordinates divisible by $4$ are cut out, cannot be tiled by dominoes.

2002 Dutch Mathematical Olympiad, 3

Tags: analytic geometry , geometry , perimeter

$A, B$ and $C$ are points in the plane with integer coordinates. The lengths of the sides of triangle $ABC$ are integer numbers. Prove that the perimeter of the triangle is an even number.

VII Soros Olympiad 2000 - 01, 11.2

Tags: parameterization , algebra , trinomial

For all valid values of $a, b$, and $c$, solve the equation $$\frac{a (x-b) (x-c) }{(a-b) (a-c)} + \frac{b (x-c) (x-a)}{(b-c) (b-a)} +\frac{c (x-a) (x-b) }{(c-a ) (c-b)} = x^2$$

1988 Brazil National Olympiad, 3

Tags: number theory , function

Find all functions $f:\mathbb{N}^* \rightarrow \mathbb{N}$ such that [list] [*] $f(x \cdot y) = f(x) + f(y)$ [*] $f(30) = 0$ [*] $f(x)=0$ always when the units digit of $x$ is $7$ [/list]

1995 IMC, 10

Tags: function , approximation , real analysis

a) Prove that for every $\epsilon>0$ there is a positive integer $n$ and real numbers $\lambda_{1},\dots,\lambda_{n}$ such that $$\max_{x\in [-1,1]}|x-\sum_{k=1}^{n}\lambda_{k}x^{2k+1}|<\epsilon.$$ b) Prove that for every odd continuous function $f$ on $[-1,1]$ and for every $\epsilon>0$ there is a positive integer $n$ and real numbers $\mu_{1},\dots,\mu_{n}$ such that $$\max_{x\in [-1,1]}|f(x)-\sum_{k=1}^{n}\mu_{k}x^{2k+1}|<\epsilon.$$

2021 New Zealand MO, 5

Tags: diophantine equation , number theory , diophantine

Find all pairs of integers $x, y$ such that $y^5 + 2xy = x^2 + 2y^4.$ .