Olympiad problems

2003 Belarusian National Olympiad, 5

Tags: combinatorics

Let $m,n,k$ be positive integers, $m> n> k$. An $1 \times m$ strip of paper is divided into the $1 \times 1$ cells. A teacher asks Bill and Pit to place numbers $0$ and $1$ in the cells of the strip so that the sum of the numbers in any $n$ consecutive cells is equal to $k$. After the task was performed it turned out that the sum $S(B)$ of all numbers on the strip of Bill was different from the sum $S(P)$ of Pit. Find the largest possible value of $|S(B) - S(P) |$. (I. Voronovich)

2020 Junior Balkan Team Selection Tests - Moldova, 8

Tags: number theory

Find the pairs of real numbers $(a,b)$ such that the biggest of the numbers $x=b^2-\frac{a-1}{2}$ and $y=a^2+\frac{b+1}{2}$ is less than or equal to $\frac{7}{16}$

1988 Bundeswettbewerb Mathematik, 2

Tags: inradius , geometry

Let $h_a$, $h_b$ and $h_c$ be the heights and $r$ the inradius of a triangle. Prove that the triangle is equilateral if and only if $h_a + h_b + h_c = 9r$.

1983 Kurschak Competition, 1

Tags: algebra , rational

Let $x, y$ and $z$ be rational numbers satisfying $$x^3 + 3y^3 + 9z^3 - 9xyz = 0.$$ Prove that $x = y = z = 0$.

2009 Jozsef Wildt International Math Competition, W. 26

Tags: inequalities

If $a_i >0$ ($i=1, 2, \cdots , n$) and $\sum \limits_{i=1}^n a_i^k=1$, where $1\leq k\leq n+1$, then $$\sum \limits_{i=1}^n a_i + \frac{1}{\prod \limits_{i=1}^n a_i} \geq n^{1-\frac{1}{k}}+n^{\frac{n}{k}}$$

2009 National Olympiad First Round, 5

Tags: geometry , perimeter

What is the perimeter of the right triangle whose exradius of the hypotenuse is $ 30$ ? $\textbf{(A)}\ 40 \qquad\textbf{(B)}\ 45 \qquad\textbf{(C)}\ 50 \qquad\textbf{(D)}\ 60 \qquad\textbf{(E)}\ 75$

2006 Czech-Polish-Slovak Match, 6

Tags: geometry

Find out if there is a convex pentagon $A_1A_2A_3A_4A_5$ such that, for each $i = 1, \dots , 5$, the lines $A_iA_{i+3}$ and $A_{i+1}A_{i+2}$ intersect at a point $B_i$ and the points $B_1,B_2,B_3,B_4,B_5$ are collinear. (Here $A_{i+5} = A_i$.)

2013 Harvard-MIT Mathematics Tournament, 10

Tags: geometry

Triangle $ABC$ is inscribed in a circle $\omega$. Let the bisector of angle $A$ meet $\omega$ at $D$ and $BC$ at $E$. Let the reflections of $A$ across $D$ and $C$ be $D'$ and $C'$, respectively. Suppose that $\angle A = 60^o$, $AB = 3$, and $AE = 4$. If the tangent to $\omega$ at $A$ meets line $BC$ at $P$, and the circumcircle of APD' meets line $BC$ at $F$ (other than $P$), compute $FC'$.

2002 Croatia Team Selection Test, 3

Tags: number theory , prime

Prove that if $n$ is a natural number such that $1 + 2^n + 4^n$ is prime then $n = 3^k$ for some $k \in N_0$.

2007 Iran Team Selection Test, 2

Tags: combinatorics

Suppose $n$ lines in plane are such that no two are parallel and no three are concurrent. For each two lines their angle is a real number in $[0,\frac{\pi}2]$. Find the largest value of the sum of the $\binom n2$ angles between line. [i]By Aliakbar Daemi[/i]

2014 Contests, 1

Tags: function , algebra , functional equation , fe , real

Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that \[ xf(y) + f(xf(y)) - xf(f(y)) - f(xy) = 2x + f(y) - f(x+y)\] holds for all $x,y \in \mathbb{R}$.

2010 Contests, 4

Tags: prime number , number theory

Determine the smallest positive integer $m$ with the property that $m^3-3m^2+2m$ is divisible by both $79$ and $83$.

2009 Postal Coaching, 2

Tags: diophantine , number theory , diophantine equation

Find all non-negative integers $a, b, c, d$ such that $7^a = 4^b + 5^c + 6^d$

2006 ISI B.Stat Entrance Exam, 3

Tags: induction , modular arithmetic , number theory , prime number

Prove that $n^4 + 4^{n}$ is composite for all values of $n$ greater than $1$.

2003 China Girls Math Olympiad, 1

Tags: inequalities , cauchy inequality , geometry

Let $ ABC$ be a triangle. Points $ D$ and $ E$ are on sides $ AB$ and $ AC,$ respectively, and point $ F$ is on line segment $ DE.$ Let $ \frac {AD}{AB} \equal{} x,$ $ \frac {AE}{AC} \equal{} y,$ $ \frac {DF}{DE} \equal{} z.$ Prove that (1) $ S_{\triangle BDF} \equal{} (1 \minus{} x)y S_{\triangle ABC}$ and $ S_{\triangle CEF} \equal{} x(1 \minus{} y) (1 \minus{} z)S_{\triangle ABC};$ (2) $ \sqrt [3]{S_{\triangle BDF}} \plus{} \sqrt [3]{S_{\triangle CEF}} \leq \sqrt [3]{S_{\triangle ABC}}.$

2007 China Northern MO, 1

Tags: geometry

Let $ ABC$ be acute triangle. The circle with diameter $ AB$ intersects $ CA,\, CB$ at $ M,\, N,$ respectively. Draw $ CT\perp AB$ and intersects above circle at $ T$, where $ C$ and $ T$ lie on the same side of $ AB$. $ S$ is a point on $ AN$ such that $ BT = BS$. Prove that $ BS\perp SC$.

2005 Portugal MO, 2

Tags: geometry , right triangle , geometric inequality

Consider the triangles $[ABC]$ and $[EDC]$, right at $A$ and $D$, respectively. Show that, if $E$ is the midpoint of $[AC]$, then $AB <BD$. [img]https://cdn.artofproblemsolving.com/attachments/c/3/75bc1bda1a22bcf00d4fe7680c80a81a9aaa4c.png[/img]

1996 IMC, 9

Tags: group theory , abstract algebra , linear algebra

Let $G$ be the subgroup of $GL_{2}(\mathbb{R})$ generated by $A$ and $B$, where $$A=\begin{pmatrix} 2 &0\\ 0&1 \end{pmatrix},\; B=\begin{pmatrix} 1 &1\\ 0&1 \end{pmatrix}$$. Let $H$ consist of the matrices $\begin{pmatrix} a_{11} &a_{12}\\ a_{21}& a_{22} \end{pmatrix}$ in $G$ for which $a_{11}=a_{22}=1$. a) Show that $H$ is an abelian subgroup of $G$. b) Show that $H$ is not finitely generated.

2020 AMC 10, 20

Tags: geometry , 3d geometry , prism

Let $B$ be a right rectangular prism (box) with edges lengths $1,$ $3,$ and $4$, together with its interior. For real $r\geq0$, let $S(r)$ be the set of points in $3$-dimensional space that lie within a distance $r$ of some point $B$. The volume of $S(r)$ can be expressed as $ar^{3} + br^{2} + cr +d$, where $a,$ $b,$ $c,$ and $d$ are positive real numbers. What is $\frac{bc}{ad}?$ $\textbf{(A) } 6 \qquad\textbf{(B) } 19 \qquad\textbf{(C) } 24 \qquad\textbf{(D) } 26 \qquad\textbf{(E) } 38$

2019 India PRMO, 24

Tags: combinatorics , geometry , rectangle

A $1 \times n$ rectangle ($n \geq 1 $) is divided into $n$ unit ($1 \times 1$) squares. Each square of this rectangle is colored red, blue or green. Let $f(n)$ be the number of colourings of the rectangle in which there are an even number of red squares. What is the largest prime factor of $f(9)/f(3)$? (The number of red squares can be zero.)

2011 China Team Selection Test, 2

Tags: modular arithmetic , number theory

Let $n>1$ be an integer, and let $k$ be the number of distinct prime divisors of $n$. Prove that there exists an integer $a$, $1<a<\frac{n}{k}+1$, such that $n \mid a^2-a$.

1970 Putnam, A3

Tags: number theory

Note that $12^2=144$ ends in two $4$s and $38^2=1444$ end in three $4$s. Determine the length of the longest string of equal nonzero digits in which the square of an integer can end.

2010 Bundeswettbewerb Mathematik, 1

Tags: inequalities , algebra , sidelengths , geometric inequality

Let $a, b, c$ be the side lengths of an non-degenerate triangle with $a \le b \le c$. With $t (a, b, c)$ denote the minimum of the quotients $\frac{b}{a}$ and $\frac{c}{b}$ . Find all values that $t (a, b, c)$ can take.

2005 Serbia Team Selection Test, 2

Tags: geometry

A convex angle $xOy$ and a point $M$ inside it are given in the plane. Prove that there is a unique point $P$ in the plane with the following property: - For any line $l$ through $M$, meeting the rays $x$ and $y$ (or their extensions) at $X$ and $Y$, the angle $XPY$ is not obtuse.

1956 AMC 12/AHSME, 19

Tags:

Two candles of the same height are lighted at the same time. The first is consumed in $ 4$ hours and the second in $ 3$ hours. Assuming that each candle burns at a constant rate, in how many hours after being lighted was the first candle twice the height of the second? $ \textbf{(A)}\ \frac {3}{4} \qquad\textbf{(B)}\ 1\frac {1}{2} \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 2\frac {2}{5} \qquad\textbf{(E)}\ 2\frac {1}{2}$