Olympiad problems

2011 May Olympiad, 5

We consider all $14$-digit positive integers, divisible by $18$, whose digits are exclusively $ 1$ and $2$, but there are no consecutive digits $2$. How many of these numbers are there?

2003 Moldova Team Selection Test, 3

Tags: incenter , angle bisector , geometry

The sides $ [AB]$ and $ [AC]$ of the triangle $ ABC$ are tangent to the incircle with center $ I$ of the $ \triangle ABC$ at the points $ M$ and $ N$, respectively. The internal bisectors of the $ \triangle ABC$ drawn form $ B$ and $ C$ intersect the line $ MN$ at the points $ P$ and $ Q$, respectively. Suppose that $ F$ is the intersection point of the lines $ CP$ and $ BQ$. Prove that $ FI\perp BC$.

2011 Rioplatense Mathematical Olympiad, Level 3, 4

Tags: geometry , parallelogram , circles

We consider $\Gamma_1$ and $\Gamma_2$ two circles that intersect at points $P$ and $Q$ . Let $A , B$ and $C$ be points on the circle $\Gamma_1$ and $D , E$ and $F$ points on the circle $\Gamma_2$ so that the lines $A E$ and $B D$ intersect at $P$ and the lines $A F$ and $C D$ intersect at $Q$. Denote $M$ and $N$ the intersections of lines $A B$ and $D E$ and of lines $A C$ and $D F$ , respectively. Show that $A M D N$ is a parallelogram.

2016 Hanoi Open Mathematics Competitions, 1

Tags: number theory , digit

How many are there $10$-digit numbers composed from the digits $1, 2, 3$ only and in which, two neighbouring digits differ by $1$ : (A): $48$ (B): $64$ (C): $72$ (D): $128$ (E): None of the above.

2014 Czech-Polish-Slovak Junior Match, 6

Tags: minimum , maximum , min , max , algebra , inequalities

Determine the largest and smallest fractions $F = \frac{y-x}{x+4y}$ if the real numbers $x$ and $y$ satisfy the equation $x^2y^2 + xy + 1 = 3y^2$.

2013 BMT Spring, 15

Tags: geometry , equal angles

Let $ABCD$ be a convex quadrilateral with $\angle ABD = \angle BCD$, $AD = 1000$, $BD = 2000$, $BC = 2001$, and $DC = 1999$. Point $E$ is chosen on segment $DB$ such that $\angle ABD = \angle ECD$. Find $AE$.

2008 ITest, 34

Tags: probability , floor function

While entertaining his younger sister Alexis, Michael drew two different cards from an ordinary deck of playing cards. Let $a$ be the probability that the cards are of different ranks. Compute $\lfloor 1000a\rfloor$.

2018 Kazakhstan National Olympiad, 1

Tags: geometry

In an equilateral trapezoid, the point $O$ is the midpoint of the base $AD$. A circle with a center at a point $O$ and a radius $BO$ is tangent to a straight line $AB$. Let the segment $AC$ intersect this circle at point $K(K \ne C)$, and let $M$ is a point such that $ABCM$ is a parallelogram. The circumscribed circle of a triangle $CMD$ intersects the segment $AC$ at a point $L(L\ne C)$. Prove that $AK=CL$.

1981 Swedish Mathematical Competition, 4

Tags: combinatorial geometry , combinatorics

A cube side $5$ is divided into $125$ unit cubes. $N$ of the small cubes are black and the rest white. Find the smallest $N$ such that there must be a row of $5$ black cubes parallel to one of the edges of the large cube.

LMT Team Rounds 2021+, 1

Tags: number theory , algebra

Given the following system of equations: $$\begin{cases} R I +G +SP = 50 \\ R I +T + M = 63 \\ G +T +SP = 25 \\ SP + M = 13 \\ M +R I = 48 \\ N = 1 \end{cases}$$ Find the value of L that makes $LMT +SPR I NG = 2023$ true.

2016 India Regional Mathematical Olympiad, 1

Tags: geometry

Let $ABC$ be an isosceles triangle with $AB=AC.$ Let $ \Gamma $ be its circumcircle and let $O$ be the centre of $ \Gamma $ . let $CO$ meet $ \Gamma$ in $D .$ Draw a line parallel to $AC$ thrugh $D.$ Let it intersect $AB$ at $E.$ Suppose $AE : EB=2:1$ .Prove that $ABC$ is an equilateral triangle.

2010 Sharygin Geometry Olympiad, 18

Tags: parallelogram , incenter , geometric transformation , homothety , projective geometry , geometry

A point $B$ lies on a chord $AC$ of circle $\omega.$ Segments $AB$ and $BC$ are diameters of circles $\omega_1$ and $\omega_2$ centered at $O_1$ and $O_2$ respectively. These circles intersect $\omega$ for the second time in points $D$ and $E$ respectively. The rays $O_1D$ and $O_2E$ meet in a point $F,$ and the rays $AD$ and $CE$ do in a point $G.$ Prove that the line $FG$ passes through the midpoint of the segment $AC.$

2023 Kazakhstan National Olympiad, 6

Tags: geometry

Inside an equilateral triangle with side $3$ there are two rhombuses with sides $1,061$ and acute angles $60^\circ$. Prove that these two rhombuses intersect each other. (The vertices of the rhombus are strictly inside the triangle.)

PEN D Problems, 19

Tags: modular arithmetic , congruence

Let $a_{1}$, $\cdots$, $a_{k}$ and $m_{1}$, $\cdots$, $m_{k}$ be integers with $2 \le m_{1}$ and $2m_{i}\le m_{i+1}$ for $1 \le i \le k-1$. Show that there are infinitely many integers $x$ which do not satisfy any of congruences \[x \equiv a_{1}\; \pmod{m_{1}}, x \equiv a_{2}\; \pmod{m_{2}}, \cdots, x \equiv a_{k}\; \pmod{m_{k}}.\]

2016 Korea Junior Math Olympiad, 5

Tags: polynomial , number theory , algebra

$n \in \mathbb {N^+}$ Prove that the following equation can be expressed as a polynomial about $n$. $$\left[2\sqrt {1}\right]+\left[2\sqrt {2}\right]+\left[2\sqrt {3}\right]+ . . . +\left[2\sqrt {n^2}\right]$$

2010 Stanford Mathematics Tournament, 15

Tags: analytic geometry , graphing lines , slope , continued fraction

Find the best approximation of $\sqrt{3}$ by a rational number with denominator less than or equal to $15$

2012 Balkan MO, 3

Tags: induction , inequalities , floor function , combinatorics , logarithm

Let $n$ be a positive integer. Let $P_n=\{2^n,2^{n-1}\cdot 3, 2^{n-2}\cdot 3^2, \dots, 3^n \}.$ For each subset $X$ of $P_n$, we write $S_X$ for the sum of all elements of $X$, with the convention that $S_{\emptyset}=0$ where $\emptyset$ is the empty set. Suppose that $y$ is a real number with $0 \leq y \leq 3^{n+1}-2^{n+1}.$ Prove that there is a subset $Y$ of $P_n$ such that $0 \leq y-S_Y < 2^n$

2010 Romania Team Selection Test, 3

Tags: geometry

Let $\mathcal{L}$ be a finite collection of lines in the plane in general position (no two lines in $\mathcal{L}$ are parallel and no three are concurrent). Consider the open circular discs inscribed in the triangles enclosed by each triple of lines in $\mathcal{L}$. Determine the number of such discs intersected by no line in $\mathcal{L}$, in terms of $|\mathcal{L}|$. [i]B. Aronov et al.[/i]

1969 IMO Shortlist, 47

Tags: geometry , circumcircle , reflection , tangent

$C$ is a point on the semicircle diameter $AB$, between $A$ and $B$. $D$ is the foot of the perpendicular from $C$ to $AB$. The circle $K_1$ is the incircle of $ABC$, the circle $K_2$ touches $CD,DA$ and the semicircle, the circle $K_3$ touches $CD,DB$ and the semicircle. Prove that $K_1,K_2$ and $K_3$ have another common tangent apart from $AB$.

2019 Iranian Geometry Olympiad, 2

Tags: geometry

As shown in the figure, there are two rectangles $ABCD$ and $PQRD$ with the same area, and with parallel corresponding edges. Let points $N,$ $M$ and $T$ be the midpoints of segments $QR,$ $PC$ and $AB$, respectively. Prove that points $N,M$ and $T$ lie on the same line. [img]http://s4.picofile.com/file/8372959484/E02.png[/img] [i]Proposed by Morteza Saghafian[/i]

2023 India Regional Mathematical Olympiad, 2

Tags: geometry

Let $\omega$ be a semicircle with $A B$ as the bounding diameter and let $C D$ be a variable chord of the semicircle of constant length such that $C, D$ lie in the interior of the arc $A B$. Let $E$ be a point on the diameter $A B$ such that $C E$ and $D E$ are equally inclined to the line $A B$. Prove that (a) the measure of $\angle C E D$ is a constant; (b) the circumcircle of triangle $C E D$ passes through a fixed point.

OIFMAT III 2013, 4

Tags: perfect square , perfect power , number theory

Show that there exists a set of infinite positive integers such that the sum of an arbitrary finite subset of these is never a perfect square. What happens if we change the condition from not being a perfect square to not being a perfect power?

2003 Czech And Slovak Olympiad III A, 1

Tags: system of equations , algebra

Solve the following system in the set of real numbers: $x^2 -xy+y^2 = 7$, $x^2y+xy^2 = -2$.

1956 Moscow Mathematical Olympiad, 323

Tags: number theory , divisor , divide

a) Find all integers that can divide both the numerator and denominator of the ratio $\frac{5m + 6}{8m + 7}$ for an integer $m$. b) Let $a, b, c, d, m$ be integers. Prove that if the numerator and denominator of the ratio $\frac{am + b}{cm+ d}$ are both divisible by $k$, then so is $ad - bc$.

1999 Mongolian Mathematical Olympiad, Problem 1

Tags: fe , functional equation , algebra

Suppose that a function $f:\mathbb R\to\mathbb R$ is such that for any real $h$ there exist at most $19990509$ different values of $x$ for which $f(x)\ne f(x+h)$. Prove that there is a set of at most $9995256$ real numbers such that $f$ is constant outside of this set.