Olympiad problems

2005 Harvard-MIT Mathematics Tournament, 4

If $a,b,c>0$, what is the smallest possible value of $ \left\lfloor \dfrac {a+b}{c} \right\rfloor + \left\lfloor \dfrac {b+c}{a} \right\rfloor + \left\lfloor \dfrac {c+a}{b} \right\rfloor $? (Note that $ \lfloor x \rfloor $ denotes the greatest integer less than or equal to $x$.)

2006 Cuba MO, 8

Tags: power of 2 , number theory , Last digit , Digits

Prove that for any integer $k$ ($k \ge 2$) there exists a power of $2$ that among its last $k$ digits, the nines constitute no less than half. For example, for $k = 2$ and $k = 3$ we have the powers $2^{12} = ... 96$ and $2^{53} = ... 992$. [hide=original wording] Probar que para cualquier k entero existe una potencia de 2 que entre sus ultimos k dıgitos, los nueves constituyen no menos de la mitad. [/hide]

2014-2015 SDML (High School), 2

Tags: function

What is the maximum value of the function $$\frac{1}{\left|x+1\right|+\left|x+2\right|+\left|x-3\right|}?$$ $\text{(A) }\frac{1}{3}\qquad\text{(B) }\frac{1}{4}\qquad\text{(C) }\frac{1}{5}\qquad\text{(D) }\frac{1}{6}\qquad\text{(E) }\frac{1}{7}$

2002 India IMO Training Camp, 8

Tags: number theory proposed , number theory

Let $\sigma(n)=\sum_{d|n} d$, the sum of positive divisors of an integer $n>0$. [list] [b](a)[/b] Show that $\sigma(mn)=\sigma(m)\sigma(n)$ for positive integers $m$ and $n$ with $gcd(m,n)=1$ [b](b)[/b] Find all positive integers $n$ such that $\sigma(n)$ is a power of $2$.[/list]

2005 AMC 12/AHSME, 14

Tags: geometry , symmetry , power of a point , AMC

A circle having center $ (0,k)$, with $ k > 6$, is tangent to the lines $ y \equal{} x, y \equal{} \minus{} x$ and $ y \equal{} 6$. What is the radius of this circle? $ \textbf{(A)}\ 6 \sqrt 2 \minus{} 6\qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ 6 \sqrt 2\qquad \textbf{(D)}\ 12\qquad \textbf{(E)}\ 6 \plus{} 6 \sqrt 2$

2014 Harvard-MIT Mathematics Tournament, 6

Tags: geometry , 3D geometry

[5] Find all integers $n$ for which $\frac{n^3+8}{n^2-4}$ is an integer.

2018 Caucasus Mathematical Olympiad, 6

Tags: algebra , quadratic trinomial , conics , parabola

Two graphs $G_1$ and $G_2$ of quadratic polynomials intersect at points $A$ and $B$. Let $O$ be the vertex of $G_1$. Lines $OA$ and $OB$ intersect $G_2$ again at points $C$ and $D$. Prove that $CD$ is parallel to the $x$-axis.

1977 IMO Longlists, 52

Tags: calculus , geometry unsolved , geometry

Two perpendicular chords are drawn through a given interior point $P$ of a circle with radius $R.$ Determine, with proof, the maximum and the minimum of the sum of the lengths of these two chords if the distance from $P$ to the center of the circle is $kR.$

2010 IMC, 5

Tags: inequalities unsolved , inequalities

Suppose that $a,b,c$ are real numbers in the interval $[-1,1]$ such that $1 + 2abc \geq a^2+b^2+c^2$. Prove that $1+2(abc)^n \geq a^{2n} + b^{2n} + c^{2n}$ for all positive integers $n$.

2019 Paraguay Mathematical Olympiad, 2

Tags: algebra , combinatorial geometry , combinatorics

Nair has puzzle pieces shaped like an equilateral triangle. She has pieces of two sizes: large and small. [img]https://cdn.artofproblemsolving.com/attachments/a/1/aedfbfb2cb17bf816aa7daeb0d35f46a79b6e9.jpg[/img] Nair build triangular figures by following these rules: $\bullet$ Figure $1$ is made up of $4$ small pieces, Figure $2$ is made up of $2$ large pieces and $8$ small, Figure $3$ by $6$ large and $12$ small, and so on. $\bullet$ The central column must be made up exclusively of small parts. $\bullet$ Outside the central column, only large pieces can be placed. [img]https://cdn.artofproblemsolving.com/attachments/5/7/e7f6340de0e04d5b5979e72edd3f453f2ac8a5.jpg[/img] Following the pattern, how many pieces will Nair use to build Figure $20$?

2023 ELMO Shortlist, C1

Tags: Elmo , combinatorics

Elmo has 2023 cookie jars, all initially empty. Every day, he chooses two distinct jars and places a cookie in each. Every night, Cookie Monster finds a jar with the most cookies and eats all of them. If this process continues indefinitely, what is the maximum possible number of cookies that the Cookie Monster could eat in one night? [i]Proposed by Espen Slettnes[/i]

2024 AIME, 4

Tags: AMC , AIME , algebra , system of equations , number theory , relatively prime , AIME II

Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations: $$\log_2\left({x \over yz}\right) = {1 \over 2}$$ $$\log_2\left({y \over xz}\right) = {1 \over 3}$$ $$\log_2\left({z \over xy}\right) = {1 \over 4}$$ Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is ${m \over n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$

2007 Harvard-MIT Mathematics Tournament, 7

Tags:

An infinite sequence of positive real numbers is defined by $a_0=1$ and $a_{n+2}=6a_n-a_{n+1}$ for $n=0,1,2,\cdots$. Find the possible value(s) of $a_{2007}$.

2017 239 Open Mathematical Olympiad, 4

Tags: combinatorics

An invisible tank is on a $100 \times 100 $ table. A cannon can fire at any $60$ cells of the board after that the tank will move to one of the adjacent cells (by side). Then the progress is repeated. Can the cannon grantee to shoot the tank?

2006 Iran MO (2nd round), 2

Tags: algebra , polynomial , algebra proposed

Determine all polynomials $P(x,y)$ with real coefficients such that \[P(x+y,x-y)=2P(x,y) \qquad \forall x,y\in\mathbb{R}.\]

2016 ASMT, 8

Tags: geometry , rectangle , angles , asmt

In rectangle $ABCD$, point $E$ is chosen on $AB$ and $F$ is the foot of $E$ onto side $CD$ such that the circumcircle of $\vartriangle ABF$ intersects line segments $AD$ and $BC$ at points $G$ and $H$ respectively. Let $S$ be the intersection of $EF$ and $GH$, and $T$ the intersection of lines $EC$ and $DS$. If $\angle SF T = 15^o$ , compute the measure of $\angle CSD$.

2014 Turkey Junior National Olympiad, 4

Tags: geometry , circumcircle , trapezoid , geometry proposed

$ABC$ is an acute triangle with orthocenter $H$. Points $D$ and $E$ lie on segment $BC$. Circumcircle of $\triangle BHC$ instersects with segments $AD$,$AE$ at $P$ and $Q$, respectively. Prove that if $BD^2+CD^2=2DP\cdot DA$ and $BE^2+CE^2=2EQ\cdot EA$, then $BP=CQ$.

2022 Princeton University Math Competition, A3 / B5

Tags: combinatorics

Randy has a deck of $29$ distinct cards. He chooses one of the $29!$ permutations of the deck and then repeatedly rearranges the deck using that permutation until the deck returns to its original order for the first time. What is the maximum number of times Randy may need to rearrange the deck?

1969 Miklós Schweitzer, 7

Tags: calculus , derivative , advanced fields , advanced fields unsolved

Prove that if a sequence of Mikusinski operators of the form $ \mu e^{\minus{}\lambda s}$ ( $ \lambda$ and $ \mu$ nonnegative real numbers, $ s$ the differentiation operator) is convergent in the sense of Mikusinski, then its limit is also of this form. [i]E. Geaztelyi[/i]

1957 Kurschak Competition, 3

Tags: algebra , inequalities , permutation , combinatorics

What is the largest possible value of $|a_1 - 1| + |a_2-2|+...+ |a_n- n|$ where $a_1, a_2,..., a_n$ is a permutation of $1,2,..., n$?

2005 Czech-Polish-Slovak Match, 4

Tags: floor function , algebra , polynomial , combinatorics unsolved , combinatorics

We distribute $n\ge1$ labelled balls among nine persons $A,B,C, \dots , I$. How many ways are there to do this so that $A$ gets the same number of balls as $B,C,D$ and $E$ together?

2019 BMT Spring, Tie1

Tags: geometry

We inscribe a circle $\omega$ in equilateral triangle $ABC$ with radius $1$. What is the area of the region inside the triangle but outside the circle?

2023 JBMO TST - Turkey, 4

Tags: number theory , Divisibility

For a prime number $p$. Can the number of n positive integers that make the expression \[\dfrac{n^3+np+1}{n+p+1}\] an integer be $777$?

2022 South Africa National Olympiad, 3

Tags: number theory , GCD , number theory unsolved

Let a, b, and c be nonzero integers. Show that there exists an integer k such that $$gcd\left(a+kb, c\right) = gcd\left(a, b, c\right)$$

2001 Saint Petersburg Mathematical Olympiad, 9.2

Tags: quadratics , algebra , quadratic trinomial , roots , St. Petersburg MO

Define a quadratic trinomial to be "good", if it has two distinct real roots and all of its coefficients are distinct. Do there exist 10 positive integers such that there exist 500 good quadratic trinomials coefficients of which are among these numbers? [I]Proposed by F. Petrov[/i]