Olympiad problems

2010 Romanian Master of Mathematics, 6

Tags: algebra , polynomial , induction , pigeonhole principle , number theory , relatively prime

Given a polynomial $f(x)$ with rational coefficients, of degree $d \ge 2$, we define the sequence of sets $f^0(\mathbb{Q}), f^1(\mathbb{Q}), \ldots$ as $f^0(\mathbb{Q})=\mathbb{Q}$, $f^{n+1}(\mathbb{Q})=f(f^{n}(\mathbb{Q}))$ for $n\ge 0$. (Given a set $S$, we write $f(S)$ for the set $\{f(x)\mid x\in S\})$. Let $f^{\omega}(\mathbb{Q})=\bigcap_{n=0}^{\infty} f^n(\mathbb{Q})$ be the set of numbers that are in all of the sets $f^n(\mathbb{Q})$, $n\geq 0$. Prove that $f^{\omega}(\mathbb{Q})$ is a finite set. [i]Dan Schwarz, Romania[/i]

2022 MMATHS, 11

Tags: probability , combinatorics

Every time Josh and Ron tap their screens, one of three emojis appears, each with equal probability: barbecue, bacon, or burger. Josh taps his screen until he gets a sequence of barbecue, bacon, and burger consecutively (in that specific order.) Ron taps his screen until he gets a sequence of three bacons in a row. Let $M$ and $N$ be the expected number of times Josh and Ron tap their screens, respectively. What is $|M-N|$?

2005 Romania National Olympiad, 2

Tags: function , algebra

Find all functions $f:\mathbb{R}\to\mathbb{R}$ for which \[ x(f(x+1)-f(x)) = f(x), \] for all $x\in\mathbb{R}$ and \[ | f(x) - f(y) | \leq |x-y| , \] for all $x,y\in\mathbb{R}$. [i]Mihai Piticari[/i]

1996 Tournament Of Towns, (489) 2

Tags: geometry , common tangent

An exterior common tangent to two non-intersecting circles with centers and $O_2$ touches them at the points $A_1$ and $A_2$ respectively. The segment $O_1O_2$ intersects the circles at the points $B_1$ and $B_2$ respectively. $C$ is the point where the straight lines $A_1B_1$ and $A_2B_2$ meet. $D$ is the point on the line $A_1A_2$ such that $CD$ is perpendicular to $B_1B_2$. Prove that $A_1D = DA_2$.

2011 Swedish Mathematical Competition, 4

Tags: combinatorics

Towns $A$, $B$ and $C$ are connected with a telecommunications cable. If you for example want to send a message from $A$ to $B$ is assigned to either a direct line between $A$ and $B$, or if necessary, a line via $C$. There are $43$ lines between $A$ and $B$, including those who go through $C$, and $29$ lines between $B$ and $C$, including those who go via $A$. How many lines, are there between $A$ and $C$ (including those who go via $B$)?

the 6th XMO, 5

Tags: geometry , tangent circle

As shown in the figure, $\odot O$ is the circumcircle of $\vartriangle ABC$, $\odot J$ is inscribed in $\odot O$ and is tangent to $AB$, $AC$ at points $D$ and E respectively, line segment $FG$ and $\odot O$ are tangent to point $A$, and $AF =AG=AD$, the circumscribed circle of $\vartriangle AFB$ intersects $\odot J$ at point $S$. Prove that the circumscribed circle of $\vartriangle ASG$ is tangent to $\odot J$. [img]https://cdn.artofproblemsolving.com/attachments/a/a/62d44e071ea9903ebdd68b43943ba1d93b4138.png[/img]

1994 All-Russian Olympiad, 5

Tags: recursive , sequence , power of 2 , number theory

Let $a_1$ be a natural number not divisible by $5$. The sequence $a_1,a_2,a_3, . . .$ is defined by $a_{n+1} =a_n+b_n$, where $b_n$ is the last digit of $a_n$. Prove that the sequence contains infinitely many powers of two. (N. Agakhanov)

1959 AMC 12/AHSME, 11

Tags: logarithm

The logarithm of $.0625$ to the base $2$ is: $ \textbf{(A)}\ .025 \qquad\textbf{(B)}\ .25\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ -4\qquad\textbf{(E)}\ -2 $

Ukrainian TYM Qualifying - geometry, 2010.6

Tags: geometry , max , geometric inequality

Find inside the triangle $ABC$, points $G$ and $H$ for which, respectively, the geometric mean and the harmonic mean of the distances to the sides of the triangle acquire maximum values. In which cases is the segment $GH$ parallel to one of the sides of the triangle? Find the length of such a segment $GH$.

2003 China Team Selection Test, 1

Tags: trigonometry , calculus , geometry

Let $ ABCD$ be a quadrilateral which has an incircle centered at $ O$. Prove that \[ OA\cdot OC\plus{}OB\cdot OD\equal{}\sqrt{AB\cdot BC\cdot CD\cdot DA}\]

CNCM Online Round 3, 7

Tags:

A subset of the positive integers $S$ is said to be a \emph{configuration} if 200 $\notin S$ and for all nonnegative integers $x$, $x \in S$ if and only if both 2$x\in S$ and $\left \lfloor{\frac{x}{2}}\right \rfloor\in S$. Let the number of subsets of $\{1, 2, 3, \dots, 130\}$ that are equal to the intersection of $\{1, 2, 3, \dots, 130\}$ with some configuration $S$ equal $k$. Compute the remainder when $k$ is divided by 1810. [i]Proposed Hari Desikan (HariDesikan)[/i]

2019 Latvia Baltic Way TST, 3

Tags: algebra

All integers are written on an axis in an increasing order. A grasshopper starts its journey at $x=0$. During each jump, the grasshopper can jump either to the right or the left, and additionally the length of its $n$-th jump is exactly $n^2$ units long. Prove that the grasshopper can reach any integer from its initial position.

1965 AMC 12/AHSME, 16

Tags: geometry , ratio , analytic geometry , pythagorean theorem , area of a triangle , heron's formula

Let line $ AC$ be perpendicular to line $ CE$. Connect $ A$ to $ D$, the midpoint of $ CE$, and connect $ E$ to $ B$, the midpoint of $ AC$. If $ AD$ and $ EB$ intersect in point $ F$, and $ \overline{BC} \equal{} \overline{CD} \equal{} 15$ inches, then the area of triangle $ DFE$, in square inches, is: $ \textbf{(A)}\ 50 \qquad \textbf{(B)}\ 50\sqrt {2} \qquad \textbf{(C)}\ 75 \qquad \textbf{(D)}\ \frac {15}{2}\sqrt {105} \qquad \textbf{(E)}\ 100$

2014 Iran Team Selection Test, 4

Tags: combinatorics

Find the maximum number of Permutation of set {$1,2,3,...,2014$} such that for every 2 different number $a$ and $b$ in this set at last in one of the permutation $b$ comes exactly after $a$

PEN J Problems, 14

Tags: divisor function

Find all positive integers $n$ such that ${d(n)}^{3} =4n$.

2017 Iran Team Selection Test, 4

Tags: number theory , prime number , modular arithmetic

We arranged all the prime numbers in the ascending order: $p_1=2<p_2<p_3<\cdots$. Also assume that $n_1<n_2<\cdots$ is a sequence of positive integers that for all $i=1,2,3,\cdots$ the equation $x^{n_i} \equiv 2 \pmod {p_i}$ has a solution for $x$. Is there always a number $x$ that satisfies all the equations? [i]Proposed by Mahyar Sefidgaran , Yahya Motevasel[/i]

1970 Regional Competition For Advanced Students, 4

Tags: number theory , greatest common divisor , polynomial , algorithm , algebra

Find all real solutions of the following set of equations: \[72x^3+4xy^2=11y^3\] \[27x^5-45x^4y-10x^2y^3=\frac{-143}{32}y^5\]

2019 Saudi Arabia JBMO TST, 1

Tags: number theory

2016 digits are written on a circle. Reading these digits counterclockwise, starting from a certain number, you get a number divisible by 81. Prove that by reading these digits clockwise, we obtain a number divisible by 81 for every starting number.

1976 Polish MO Finals, 6

Tags: algebra , logarithm , functional

An increasing function $f : N \to R$ satisfies $$f(kl) = f(k)+ f(l)\,\,\, for \,\,\, all \,\,\, k,l \in N.$$ Show that there is a real number $p > 1$ such that $f(n) =\ log_pn$ for all $n$.

2013 Mexico National Olympiad, 4

Tags: geometry , 3d geometry , analytic geometry , prism , combinatorics

A $n \times n \times n$ cube is constructed using $1 \times 1 \times 1$ cubes, some of them black and others white, such that in each $n \times 1 \times 1$, $1 \times n \times 1$, and $1 \times 1 \times n$ subprism there are exactly two black cubes, and they are separated by an even number of white cubes (possibly 0). Show it is possible to replace half of the black cubes with white cubes such that each $n \times 1 \times 1$, $1 \times n \times 1$ and $1 \times 1 \times n$ subprism contains exactly one black cube.

2005 USA Team Selection Test, 1

Tags: geometry , rectangle , combinatorics

Let $n$ be an integer greater than $1$. For a positive integer $m$, let $S_{m}= \{ 1,2,\ldots, mn\}$. Suppose that there exists a $2n$-element set $T$ such that (a) each element of $T$ is an $m$-element subset of $S_{m}$; (b) each pair of elements of $T$ shares at most one common element; and (c) each element of $S_{m}$ is contained in exactly two elements of $T$. Determine the maximum possible value of $m$ in terms of $n$.

1993 Turkey Team Selection Test, 1

Tags: modular arithmetic , arithmetic sequence , number theory

Show that there exists an infinite arithmetic progression of natural numbers such that the first term is $16$ and the number of positive divisors of each term is divisible by $5$. Of all such sequences, find the one with the smallest possible common difference.

2008 National Chemistry Olympiad, 5

Tags:

Which element is the major component in solar cells? $ \textbf{(A)}\hspace{.05in}\ce{As}\qquad\textbf{(B)}\hspace{.05in}\ce{Ge} \qquad\textbf{(C)}\hspace{.05in}\ce{P}\qquad\textbf{(D)}\hspace{.05in}\ce{Si} \qquad $

MathLinks Contest 7th, 7.3

Tags:

Let $ n$ be a positive integer, and let $ M \equal{} \{1,2,\ldots, 2n\}$. Find the minimal positive integer $ m$, such that no matter how we choose the subsets $ A_i \subset M$, $ 1\leq i\leq m$, with the properties: (1) $ |A_i\minus{}A_j|\geq 1$, for all $ i\neq j$, (2) $ \bigcup_{i\equal{}1}^m A_i \equal{} M$, we can always find two subsets $ A_k$ and $ A_l$ such that $ A_k \cup A_l \equal{} M$ (here $ |X|$ represents the number of elements in the set $ X$.)

LMT Team Rounds 2010-20, B8

Tags: geometry

In rectangle $ABCD$, $AB = 3$ and $BC = 4$. If the feet of the perpendiculars from $B$ and $D$ to $AC$ are $X$ and $Y$ , the length of $X Y$ can be expressed in the form m/n , where m and n are relatively prime positive integers. Find $m +n$.