Olympiad problems

2013 Germany Team Selection Test, 1

Tags: geometry

Two concentric circles $\omega, \Omega$ with radii $8,13$ are given. $AB$ is a diameter of $\Omega$ and the tangent from $B$ to $\omega$ touches $\omega$ at $D$. What is the length of $AD$.

2015 Nordic, 4

Tags: combinatorics

An encyclopedia consists of ${2000}$ numbered volumes. The volumes are stacked in order with number ${1}$ on top and ${2000}$ in the bottom. One may perform two operations with the stack: (i) For ${n}$ even, one may take the top ${n}$ volumes and put them in the bottom of the stack without changing the order. (ii) For ${n}$ odd, one may take the top ${n}$ volumes, turn the order around and put them on top of the stack again. How many different permutations of the volumes can be obtained by using these two operations repeatedly?

2014 Junior Regional Olympiad - FBH, 4

Tags: number theory , remainder

Positive integer $n$ when divided with number $3$ gives remainder $a$, when divided with $5$ has remainder $b$ and when divided with $7$ gives remainder $c$. Find remainder when dividing number $n$ with $105$ if $4a+3b+2c=30$

2007 Peru MO (ONEM), 2

Tags: geometry , ratio , coloring , point

Assuming that each point of a straight line is painted red or blue, arbitrarily, show that it is always possible to choose three points $A, B$ and $C$ in such a way straight, that are painted the same color and that: $$\frac{AB}{1}=\frac{BC}{2}=\frac{AC}{3}.$$

1986 IMO, 2

Tags: function , algebra , functional equation

Find all functions $f$ defined on the non-negative reals and taking non-negative real values such that: $f(2)=0,f(x)\ne0$ for $0\le x<2$, and $f(xf(y))f(y)=f(x+y)$ for all $x,y$.

2014 AMC 10, 21

Tags: trapezoid , trigonometry , geometry

Trapezoid $ABCD$ has parallel sides $\overline{AB}$ or length $33$ and $\overline{CD}$ of length $21$. The other two sides are of lengths $10$ and $14$. The angles at $A$ and $B$ are acute. What is the length of the shorter diagonal of $ABCD$? $ \textbf {(A) } 10\sqrt{6} \qquad \textbf {(B) } 25 \qquad \textbf {(C) } 8\sqrt{10} \qquad \textbf {(D) } 18\sqrt{2} \qquad \textbf {(E) } 26 $

2016 Brazil National Olympiad, 1

Tags: geometry , incenter , parallel lines

Let $ABC$ be a triangle. $r$ and $s$ are the angle bisectors of $\angle ABC$ and $\angle BCA$, respectively. The points $E$ in $r$ and $D$ in $s$ are such that $AD \| BE$ and $AE \| CD$. The lines $BD$ and $CE$ cut each other at $F$. $I$ is the incenter of $ABC$. Show that if $A,F,I$ are collinear, then $AB=AC$.

2002 Poland - Second Round, 1

Tags: modular arithmetic , number theory

Find all numbers $p\le q\le r$ such that all the numbers \[pq+r,pq+r^2,qr+p,qr+p^2,rp+q,rp+q^2 \] are prime.

2005 Morocco TST, 3

Tags: algebra , polynomial , inequalities

The real numbers $a_1,a_2,...,a_{100}$ satisfy the relationship : $a_1^2+ a_2^2 + \cdots +a_{100}^2 + ( a_1+a_2 + \cdots + a_{100})^2 = 101$ Prove that $|a_k| \leq 10$ for all $k \in \{1,2,...,100\}$

2014 AMC 10, 7

Tags: percent

Suppose $A>B>0$ and A is $x\%$ greater than $B$. What is $x$? $ \textbf {(A) } 100\left(\frac{A-B}{B}\right) \qquad \textbf {(B) } 100\left(\frac{A+B}{B}\right) \qquad \textbf {(C) } 100\left(\frac{A+B}{A}\right)\qquad \textbf {(D) } 100\left(\frac{A-B}{A}\right) \qquad \textbf {(E) } 100\left(\frac{A}{B}\right)$

2014 ELMO Shortlist, 2

Tags: geometry , induction , combinatorics

A $2^{2014} + 1$ by $2^{2014} + 1$ grid has some black squares filled. The filled black squares form one or more snakes on the plane, each of whose heads splits at some points but never comes back together. In other words, for every positive integer $n$ greater than $2$, there do not exist pairwise distinct black squares $s_1$, $s_2$, \dots, $s_n$ such that $s_i$ and $s_{i+1}$ share an edge for $i=1,2, \dots, n$ (here $s_{n+1}=s_1$). What is the maximum possible number of filled black squares? [i]Proposed by David Yang[/i]

2016 USAMTS Problems, 5:

Tags: geometry , perimeter

Let $ABCD$ be a convex quadrilateral with perimeter $\tfrac{5}{2}$ and $AC=BD=1$. Determine the maximum possible area of $ABCD$.

1997 Romania National Olympiad, 1

Tags: superior algebra , rings , units , gaussian integer

Let $\alpha \in \mathbb{C} \setminus \mathbb{Q}$ be such that the set $A= \{ a+b \alpha : a,b \in \mathbb{Z} \}$ is a ring with respect to the usual operations of $\mathbb{C}.$ If the ring $A$ has exactly four invertible elements, prove that $A= \mathbb{Z}[i].$

PEN K Problems, 14

Tags: function , inequalities , functional equation

Find all functions $f:\mathbb{Z} \to \mathbb{Z}$ such that for all $m\in\mathbb{Z}$: [list][*] $f(m+8) \le f(m)+8$, [*] $f(m+11) \ge f(m)+11$.[/list]

2016 India Regional Mathematical Olympiad, 1

Tags: inequalities , number theory , construction

Find distinct positive integers $n_1<n_2<\dots<n_7$ with the least possible sum, such that their product $n_1 \times n_2 \times \dots \times n_7$ is divisible by $2016$.

PEN G Problems, 15

Tags: inequalities , irrational number , logarithm

Prove that for any $ p, q\in\mathbb{N}$ with $ q > 1$ the following inequality holds: \[ \left\vert\pi\minus{}\frac{p}{q}\right\vert\ge q^{\minus{}42}.\]

2022 IFYM, Sozopol, 8

Tags: number theory , modulo

Determine the number of ordered quadruples of integers $(a,b,c,d)$ for which $0\leq a,b,c,d\leq 36$ and $37|a^2+b^2-c^3-d^3$.

2017 EGMO, 2

Tags: function , algebra , coloring , functional equation

Find the smallest positive integer $k$ for which there exists a colouring of the positive integers $\mathbb{Z}_{>0}$ with $k$ colours and a function $f:\mathbb{Z}_{>0}\to \mathbb{Z}_{>0}$ with the following two properties: $(i)$ For all positive integers $m,n$ of the same colour, $f(m+n)=f(m)+f(n).$ $(ii)$ There are positive integers $m,n$ such that $f(m+n)\ne f(m)+f(n).$ [i]In a colouring of $\mathbb{Z}_{>0}$ with $k$ colours, every integer is coloured in exactly one of the $k$ colours. In both $(i)$ and $(ii)$ the positive integers $m,n$ are not necessarily distinct.[/i]

2010 AMC 10, 24

Tags: modular arithmetic , ratio , geometric sequence

A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than $ 100$ points. What was the total number of points scored by the two teams in the first half? $ \textbf{(A)}\ 30 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 33 \qquad \textbf{(E)}\ 34$

V Soros Olympiad 1998 - 99 (Russia), 11.8

Tags: geometry , angle

Inside triangle $ABC$, point $P$ is taken so that angles $\angle ARB= \angle BPC = \angle CPA= 120^o$. Lines $BP$ and $CP$ intersect lines $AC$ and $AB$ at points $M$ and $K$. It is known that the quadrilateral $AMPK$ has same areq with the triangle $BCP$. What is the angle $\angle BAC$?

2013 Estonia Team Selection Test, 3

Tags: inequalities , sum , algebra

Let $x_1,..., x_n$ be non-negative real numbers, not all of which are zeros. (i) Prove that $$1 \le \frac{\left(x_1+\frac{x_2}{2}+\frac{x_3}{3}+...+\frac{x_n}{n}\right)(x_1+2x_2+3x_3+...+nx_n)}{(x_1+x_2+x_3+...+x_n)^2} \le \frac{(n+1)^2}{4n}$$ (ii) Show that, for each $n > 1$, both inequalities can hold as equalities.

2012 Grand Duchy of Lithuania, 2

Tags: geometry , trapezoid , right angle

The base $AB$ of a trapezium $ABCD$ is longer than the base $CD$, and $\angle ADC$ is a right angle. The diagonals $AC$ and $BD$ are perpendicular. Let $E$ be the foot of the altitude from $D$ to the line $BC$. Prove that $$\frac{AE}{BE} =\frac{ AC \cdot CD}{AC^2 - CD^2}$$ .