Olympiad problems

1989 All Soviet Union Mathematical Olympiad, 503

Find the smallest positive integer $n$ for which we can find an integer $m$ such that $\left[\frac{10^n}{m}\right] = 1989$.

2005 Abels Math Contest (Norwegian MO), 3a

Tags: geometry , angle , isosceles

In the isosceles triangle $\vartriangle ABC$ is $AB = AC$. Let $D$ be the midpoint of the segment $BC$. The points $P$ and $Q$ are respectively on the lines $AD$ and $AB$ (with $Q \ne B$) so that $PQ = PC$. Show that $\angle PQC =\frac12 \angle A $

2014 Sharygin Geometry Olympiad, 7

Tags: parallelogram , geometry , perpendicular bisector , ratio

A parallelogram $ABCD$ is given. The perpendicular from $C$ to $CD$ meets the perpendicular from $A$ to $BD$ at point $F$, and the perpendicular from $B$ to $AB$ meets the perpendicular bisector to $AC$ at point $E$. Find the ratio in which side $BC$ divides segment $EF$.

2023 Argentina National Olympiad, 3

Tags: geometry , tangent

Let $ABC$ be a triangle and $M$ be the middle point of $BC$. Let $\Omega$ be the circumference such as $A,B,C \in \Omega$. Let $P$ be the intersection of $\Omega$ and $AM$. $AF$ is a hight of the triangle, with $F\in BC$, and $H$ the orthocenter.Additionally the intersections of $MH$ and $PF$ with $\Omega$ are $K$ and $T$ respectibly. Demonstrate that the circumscribed circumference of the traingle $KTF$ is tangent with $BC$.

1996 National High School Mathematics League, 3

Tags: prime number , number theory

For a prime number $p$, there exists $n\in\mathbb{Z}_+$, $\sqrt{p+n}+\sqrt{n}$ is an integer, then $\text{(A)}$ there is no such $p$ $\text{(B)}$ there in only one such $p$ $\text{(C)}$ there is more than one such $p$, but finitely many $\text{(D)}$ there are infinitely many such $p$

2012 India IMO Training Camp, 3

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How many $6$-tuples $(a, b, c, d, e, f)$ of natural numbers are there for which $a>b>c>d>e>f$ and $a+f=b+e=c+d=30$ are simultaneously true?

2002 Moldova National Olympiad, 3

Tags: induction

Prove that for any $ n\in \mathbb N$ the number $ 1\plus{}\dfrac{1}{3}\plus{}\dfrac{1}{5}\plus{}\ldots\plus{}\dfrac{1}{2n\plus{}1}$ is not an integer.

1966 IMO Longlists, 37

Tags: parallelogram , geometry , cyclic quadrilateral

Show that the four perpendiculars dropped from the midpoints of the sides of a cyclic quadrilateral to the respective opposite sides are concurrent. [b]Note by Darij:[/b] A [i]cyclic quadrilateral [/i]is a quadrilateral inscribed in a circle.

2009 Argentina National Olympiad, 1

Tags: coloring , combinatorics , combinatorial geometry

$2009$ points have been marked on a circle. Lucía colors them with $7$ different colors of her choice. Then Ivan can join three points of the same color, thus forming monochrome triangles. Triangles cannot have points in common; not even vertices in common. Ivan's goal is to draw as many monochrome triangles as possible. Lucía's objective is to prevent Iván's task as much as possible through a good choice of colouring. How many monochrome triangles will Ivan get if they both do their homework to the best of their ability?

VMEO II 2005, 9

Tags: chessboard , queen , combinatorics

On a board with $64$ ($8 \times 8$) squares, find a way to arrange $9$ queens and $ 1$ king so that every queen cannot capture another queen.

2019 Saudi Arabia BMO TST, 3

Tags: circumcircle , incenter , excenter , arc midpoint , geometry

The triangle $ABC$ ($AB > BC$) is inscribed in the circle $\Omega$. On the sides $AB$ and $BC$, the points $M$ and $N$ are chosen, respectively, so that $AM = CN$, The lines $MN$ and $AC$ intersect at point $K$. Let $P$ be the center of the inscribed circle of triangle $AMK$, and $Q$ the center of the excircle of the triangle $CNK$ tangent to side $CN$. Prove that the midpoint of the arc $ABC$ of the circle $\Omega$ is equidistant from the $P$ and $Q$.

2016 LMT, 21

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Let $S$ be the set of positive integers $n$ such that \[3\cdot \varphi (n)=n,\] where $\varphi (n)$ is the number of positive integers $k\leq n$ such that $\gcd (k, n)=1$. Find \[\sum_{n\in S} \, \frac{1}{n}.\] [i]Proposed by Nathan Ramesh

2010 Regional Competition For Advanced Students, 4

Tags: number theory

Let $(b_n)_{n \ge 0}=\sum_{k=0}^{n} (a_0+kd)$ for positive integers $a_0$ and $d$. We consider all such sequences containing an element $b_i$ which equals $2010$. Determine the greatest possible value of $i$ and for this value the integers $a_0$ and $d$. [i](41th Austrian Mathematical Olympiad, regional competition, problem 4)[/i]

2024 Yasinsky Geometry Olympiad, 5

Tags: parallelogram , geometry

On side $ AC $ of triangle $ ABC $, a point $ P $ is chosen such that $ AP = \frac{1}{3} AC $, and on segment $ BP $, a point $ S $ is chosen such that $ CS \perp BP $. A point $ T $ is such that $ BCST $ is a parallelogram. Prove that $ AB = AT $. [i]Proposed by Bohdan Zheliabovskyi[/i]

1995 Balkan MO, 4

Tags: induction , analytic geometry , combinatorics

Let $n$ be a positive integer and $\mathcal S$ be the set of points $(x, y)$ with $x, y \in \{1, 2, \ldots , n\}$. Let $\mathcal T$ be the set of all squares with vertices in the set $\mathcal S$. We denote by $a_k$ ($k \geq 0$) the number of (unordered) pairs of points for which there are exactly $k$ squares in $\mathcal T$ having these two points as vertices. Prove that $a_0 = a_2 + 2a_3$. [i]Yugoslavia[/i]

PEN A Problems, 46

Tags: divisibility theory

Let $a$ and $b$ be integers. Show that $a$ and $b$ have the same parity if and only if there exist integers $c$ and $d$ such that $a^2 +b^2 +c^2 +1 = d^2$.

2014 NIMO Problems, 2

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In the figure below, how many ways are there to select two squares which do not share an edge? [asy] size(3cm); for (int t = -2; t <= 2; t=t+1) { draw( shift((t,0))*unitsquare ) ; if (t!=0) draw( shift((0,t))*unitsquare ); } [/asy] [i]Proposed by Evan Chen[/i]

2015 Taiwan TST Round 3, 1

Tags: function , algebra , induction , number theory

Let $\mathbb{Q}^+$ be the set of all positive rational numbers. Find all functions $f:\mathbb{Q}^+\rightarrow \mathbb{Q}^+$ satisfying $f(1)=1$ and \[ f(x+n)=f(x)+nf(\frac{1}{x}) \forall n\in\mathbb{N},x\in\mathbb{Q}^+\]

1993 Chile National Olympiad, 1

Tags: geometry , path , distance , square

There are four houses, located on the vertices of a square. You want to draw a road network, so that you can go from any house to any other. Prove that the network formed by the diagonals is not the shortest. Find a shorter network.

1947 Moscow Mathematical Olympiad, 124

Tags: combinatorics , consecutive , relatively prime , number theory

a) Prove that of $5$ consecutive positive integers one that is relatively prime with the other $4$ can always be selected. b) Prove that of $10$ consecutive positive integers one that is relatively prime with the other $9$ can always be selected.

1999 Ukraine Team Selection Test, 9

Tags: algebra , function , functional equation

Find all functions $u : R \to R$ for which there is a strictly increasing function $f : R \to R$ such that $f(x+y) = f(x)u(y)+ f(y)$ for all $x,y \in R$.

2019 Miklós Schweitzer, 8

Tags: real analysis

Let $f: \mathbb{R} \to \mathbb{R}$ be a measurable function such that $f(x+t) - f(x)$ is locally integrable for every $t$ as a function of $x$. Prove that $f$ is locally integrable.

2018 AMC 12/AHSME, 5

Tags: polynomial , algebra

What is the sum of all possible values of $k$ for which the polynomials $x^2 - 3x + 2$ and $x^2 - 5x + k$ have a root in common? $ \textbf{(A) }3 \qquad \textbf{(B) }4 \qquad \textbf{(C) }5 \qquad \textbf{(D) }6 \qquad \textbf{(E) }10 \qquad $

1990 IberoAmerican, 4

Tags: parabola , radical axis , conic , geometry , power of a point , rectangle

Let $\Gamma_{1}$ be a circle. $AB$ is a diameter, $\ell$ is the tangent at $B$, and $M$ is a point on $\Gamma_{1}$ other than $A$. $\Gamma_{2}$ is a circle tangent to $\ell$, and also to $\Gamma_{1}$ at $M$. a) Determine the point of tangency $P$ of $\ell$ and $\Gamma_{2}$ and find the locus of the center of $\Gamma_{2}$ as $M$ varies. b) Show that there exists a circle that is always orthogonal to $\Gamma_{2}$, regardless of the position of $M$.

2022 CMIMC, 1.7

Tags: number theory , algebra

Let $f(n)$ count the number of values $0\le k\le n^2$ such that $43\nmid\binom{n^2}{k}$. Find the least positive value of $n$ such that $$43^{43}\mid f\left(\frac{43^{n}-1}{42}\right)$$ [i]Proposed by Adam Bertelli[/i]