Olympiad problems

2018 China Team Selection Test, 6

Tags: function , Divisibility , Iteration , algebra , functional equation

Let $M,a,b,r$ be non-negative integers with $a,r\ge 2$, and suppose there exists a function $f:\mathbb{Z}\rightarrow\mathbb{Z}$ satisfying the following conditions: (1) For all $n\in \mathbb{Z}$, $f^{(r)}(n)=an+b$ where $f^{(r)}$ denotes the composition of $r$ copies of $f$ (2) For all $n\ge M$, $f(n)\ge 0$ (3) For all $n>m>M$, $n-m|f(n)-f(m)$ Show that $a$ is a perfect $r$-th power.

2021 AMC 10 Spring, 7

Tags: geometry

In a plane, four circles with radii $1,3,5,$ and $7$ are tangent to line $l$ at the same point $A,$ but they may be on either side of $l$. Region $S$ consists of all the points that lie inside exactly one of the four circles. What is the maximum possible area of region $S$? $\textbf{(A) }24\pi \qquad \textbf{(B) }32\pi \qquad \textbf{(C) }64\pi \qquad \textbf{(D) }65\pi \qquad \textbf{(E) }84\pi$

Brazil L2 Finals (OBM) - geometry, 2001.3

Tags: geometry , number theory

Given a positive integer $h$, show that there are a finite number of triangles with integer sides $a, b, c$ and altitude relative to side $c$ equal to $h$ .

2017 Harvard-MIT Mathematics Tournament, 4

Tags: number theory , Diophantine equation

Find the number of ordered triples of nonnegative integers $(a, b, c)$ that satisfy \[(ab + 1)(bc + 1)(ca + 1) = 84.\]

1993 ITAMO, 5

Tags: inequalities

Prove the following inequality for any positive real numbers a,b,c not exceeding 1 $a^2b+b^2c+c^2a+1\ge a^2+b^2+c^2$

2018 Pan-African Shortlist, N2

Tags: number theory , Divisibility , Digits

A positive integer is called special if its digits can be arranged to form an integer divisible by $4$. How many of the integers from $1$ to $2018$ are special?

1958 Poland - Second Round, 3

Tags: algebra , polynomial

Prove that if the polynomial $ f(x) = ax^3 + bx^2 + cx + d $ with integer coefficients takes odd values for $ x = 0 $ and $ x = 1 $, then the equation $ f(x) = 0 $ has no integer roots.

2015 Postal Coaching, Problem 1

Tags: geometry

A circle, its chord $AB$ and the midpoint $W$ of the minor arc $AB$ are given. Take an arbitrary point $C$ on the major arc $AB$. The tangent to the circle at $C$ meets the tangents at $A$ and $B$ at points $X$ and $Y$ respectively. Lines $WX$ and $WY$ meet $AB$ at points $N$ and $M$. Prove that the length of segment $NM$ doesn’t depend on point $C$.

2016 International Zhautykov Olympiad, 3

Tags: graph theory , combinatorics , Probabilistic Method

There are $60$ towns in $Graphland$ every two countries of which are connected by only a directed way. Prove that we can color four towns to red and four towns to green such that every way between green and red towns are directed from red to green

2009 Brazil Team Selection Test, 1

Tags: right triangle , orthocenter , geometry , Cyclic , pentagon

Let $A, B, C, D, E$ points in circle of radius r, in that order, such that $AC = BD = CE = r$. The points $H_1, H_2, H_3$ are the orthocenters of the triangles $ACD$, $BCD$ and $BCE$, respectively. Prove that $H_1H_2H_3$ is a right triangle .

2004 National Olympiad First Round, 28

Tags:

What is the largest possible value of $8x^2+9xy+18y^2+2x+3y$ such that $4x^2 + 9y^2 = 8$ where $x,y$ are real numbers? $ \textbf{(A)}\ 23 \qquad\textbf{(B)}\ 26 \qquad\textbf{(C)}\ 29 \qquad\textbf{(D)}\ 31 \qquad\textbf{(E)}\ 35 $

2005 Belarusian National Olympiad, 5

Tags: algebra , trigonometry

For $0<a,b,c,d<\frac{\pi}{2}$ is true that $$\cos 2a+\cos 2b+ \cos 2c+ \cos 2d= 4 (\sin a \sin b \sin c \sin d -\cos a \cos b \cos c \cos d)$$ Find all possible values of $a+b+c+d$

2021 Junior Balkan Team Selection Tests - Romania, P2

Tags: number theory , Subsets , romania , Romanian TST

For any non-empty subset $X$ of $M=\{1,2,3,...,2021\}$, let $a_X$ be the sum of the greatest and smallest elements of $X$. Determine the arithmetic mean of all the values of $a_X$, as $X$ covers all the non-empty subsets of $M$.

2007 Moldova National Olympiad, 9.4

Tags: Diophantine equation

Find all rational terms of sequence defined by formula $ a_n=\sqrt{\frac{9n-2}{n+1}}, n \in N $

2005 ISI B.Math Entrance Exam, 4

Tags: induction , function , number theory proposed , number theory

For a set $S$ we denote its cardinality by $|S|$. Let $e_1,e_2,\ldots,e_k$ be non-negative integers. Let $A_k$ (respectively $B_k$) be the set of all $k$-tuples $(f_1,f_2,\ldots,f_k)$ of integers such that $0\leq f_i\leq e_i$ for all $i$ and $\sum_{i=1}^k f_i$ is even (respectively odd). Show that $|A_k|-|B_k|=0 \textrm{ or } 1$.

1998 AMC 8, 3

Tags: AMC

$ \cfrac{\cfrac{3}{8}+\cfrac{7}{8}}{\cfrac{4}{5}}= $ $ \text{(A)}\ 1\qquad\text{(B)}\frac{25}{16}\qquad\text{(C)}\ 2\qquad\text{(D)}\ \frac{43}{20}\qquad\text{(E)}\ \frac{47}{16} $

2017 F = ma, 25

Tags: orbit

25) A planet orbits around a star S, as shown in the figure. The semi-major axis of the orbit is a. The perigee, namely the shortest distance between the planet and the star is 0.5a. When the planet passes point $P$ (on the line through the star and perpendicular to the major axis), its speed is $v_1$. What is its speed $v_2$ when it passes the perigee? A) $v_2 = \frac{3}{\sqrt{5}}v_1$ B) $v_2 = \frac{3}{\sqrt{7}}v_1$ C) $v_2 = \frac{2}{\sqrt{3}}v_1$ D) $v_2 = \frac{\sqrt{7}}{\sqrt{3}}v_1$ E) $v_2 = 4v_1$

2009 Vietnam Team Selection Test, 2

Tags: algebra , polynomial , trigonometry , algebra proposed

Let a polynomial $ P(x) \equal{} rx^3 \plus{} qx^2 \plus{} px \plus{} 1$ $ (r > 0)$ such that the equation $ P(x) \equal{} 0$ has only one real root. A sequence $ (a_n)$ is defined by $ a_0 \equal{} 1, a_1 \equal{} \minus{} p, a_2 \equal{} p^2 \minus{} q, a_{n \plus{} 3} \equal{} \minus{} pa_{n \plus{} 2} \minus{} qa_{n \plus{} 1} \minus{} ra_n$. Prove that $ (a_n)$ contains an infinite number of nagetive real numbers.

1999 China National Olympiad, 1

Tags: modular arithmetic , number theory unsolved , number theory

Let $m$ be a positive integer. Prove that there are integers $a, b, k$, such that both $a$ and $b$ are odd, $k\geq0$ and \[2m=a^{19}+b^{99}+k\cdot2^{1999}\]

1989 Putnam, B6

Tags: probability , expected value , calculus , integration

Let $(x_1,x_2,\ldots,x_n)$ be a point chosen at random in the $n$-dimensional region defined by $0<x_1<x_2<\ldots<x_n<1$, denoting $x_0=0$ and $x_{n+1}=1$. Let $f$ be a continuous function on $[0,1]$ with $f(1)=0$. Show that the expected value of the sum $$\sum_{i=0}^n(x_{i+1}-x_i)f(x_{i+1})$$is $\int^1_0f(t)P(t)dt$., where $P$ is a polynomial of degree $n$, independent of $f$, with $0\le P(t)\le1$ for $0\le t\le1$.

PEN J Problems, 7

Tags: Divisor Functions

Show that if the equation $\phi(x)=n$ has one solution, it always has a second solution, $n$ being given and $x$ being the unknown.

2015 Purple Comet Problems, 5

Tags: geometry , rectangle

The diagram below shows a rectangle with one side divided into seven equal segments and the opposite side divided in half. The rectangle has area 350. Find the area of the shaded region. For Diagram go to purplecomet.org/welcome/practice, the $2015$ middle school contest, and #5.

2021 MMATHS, 6

Tags: Yale , MMATHS

Ben flips a coin 10 times and then records the absolute difference between the total number of heads and tails he's flipped. He then flips the coin one more time and records the absolute difference between the total number of heads and tails he's flipped. If the probability that the second number he records is greater than the first can be expressed as $\frac{a}{b}$ for positive integers $a,b$ with $\gcd (a,b) = 1$, then find $a + b$. [i]Proposed by Vismay Sharan[/i]

2023 Math Prize for Girls Problems, 7

Tags:

An arithmetic expression is created by inserting either a plus sign or a multiplication sign in each of the 11 spaces between consecutive $\sqrt{3}$’s in a row of twelve $\sqrt{3}$’s. The signs are chosen uniformly and independently at random. What is the probability that the resulting expression evaluates to $12\sqrt{3}$?

2023 India IMO Training Camp, 1

Tags: number theory

Let $\mathbb{N}$ be the set of all positive integers. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that $f(x) + y$ and $f(y) + x$ have the same number of $1$'s in their binary representations, for any $x,y \in \mathbb{N}$.