Olympiad problems

1999 Romania National Olympiad, 2

For a finite group $G$ we denote by $n(G)$ the number of elements of the group and by $s(G)$ the number of subgroups of it. Decide whether the following statements are true or false. a) For every $a>0$ the is a finite group $G$ with $\frac{n(G)}{s(G)}<a.$ b) For every $a>0$ the is a finite group $G$ with $\frac{n(G)}{s(G)}>a.$

2008 China Team Selection Test, 3

Tags: search , arithmetic sequence , combinatorics , ramsey theory

Suppose that every positve integer has been given one of the colors red, blue,arbitrarily. Prove that there exists an infinite sequence of positive integers $ a_{1} < a_{2} < a_{3} < \cdots < a_{n} < \cdots,$ such that inifinite sequence of positive integers $ a_{1},\frac {a_{1} \plus{} a_{2}}{2},a_{2},\frac {a_{2} \plus{} a_{3}}{2},a_{3},\frac {a_{3} \plus{} a_{4}}{2},\cdots$ has the same color.

2000 India National Olympiad, 6

Tags: geometry , perimeter , inequalities , triangle inequality , combinatorics

For any natural numbers $n$, ( $n \geq 3$), let $f(n)$ denote the number of congruent integer-sided triangles with perimeter $n$. Show that (i) $f(1999) > f (1996)$; (ii) $f(2000) = f(1997)$.

1980 All Soviet Union Mathematical Olympiad, 290

Tags: combinatorics , combinatorial geometry

There are several settlements on the bank of the Big Round Lake. Some of them are connected with the regular direct ship lines. Two settlements are connected if and only if two next (counterclockwise) to each ones are not connected. Prove that you can move from the arbitrary settlement to another arbitrary settlement, having used not more than three ships.

2019 Mathematical Talent Reward Programme, SAQ: P 5

Tags: number theory

Let a fixed natural number m be given. Call a positive integer n to be an MTRP-number iff [list] [*] $n \equiv 1\ (mod\ m)$ [*] Sum of digits in decimal representation of $n^2$ is greater than equal to sum of digits in decimal representation of $n$ [/list] How many MTRP-numbers are there ?

2012 AIME Problems, 6

Tags: trigonometry , parabola , complex number , conic

The complex numbers $z$ and $w$ satisfy $z^{13} = w$, $w^{11} = z$, and the imaginary part of $z$ is $\sin\left(\frac{m\pi}n\right)$ for relatively prime positive integers $m$ and $n$ with $m < n$. Find $n$.

2000 IMO Shortlist, 4

Tags: algebra , polynomial , number theory , divisibility

Find all triplets of positive integers $ (a,m,n)$ such that $ a^m \plus{} 1 \mid (a \plus{} 1)^n$.

1955 AMC 12/AHSME, 36

Tags: geometry , rectangle

A cylindrical oil tank, lying horizontally, has an interior length of $ 10$ feet and an interior diameter of $ 6$ feet. If the rectangular surface of the oil has an area of $ 40$ square feet, the depth of the oil is: $ \textbf{(A)}\ \sqrt{5} \qquad \textbf{(B)}\ 2\sqrt{5} \qquad \textbf{(C)}\ 3\minus{}\sqrt{5} \qquad \textbf{(D)}\ 3\plus{}\sqrt{5} \\ \textbf{(E)}\ \text{either }3\minus{}\sqrt{5}\text{ or }3\plus{}\sqrt{5}$

1999 Czech and Slovak Match, 4

Tags: polynomial , algebra

Find all positive integers $k$ for which the following assertion holds: If $F(x)$ is polynomial with integer coefficients ehich satisfies $F(c) \leq k$ for all $c \in \{0,1, \cdots,k+1 \}$, then \[F(0)= F(1) = \cdots =F(k+1).\]

2003 German National Olympiad, 5

Tags: number theory

$n$ is a positive integer. Let $a(n)$ be the smallest number for which $n\mid a(n)!$ Find all solutions of:$$\frac{a(n)}{n}=\frac{2}{3}$$

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 ${d(n)}^{3} =4n$.