Olympiad problems

2016 Baltic Way, 14

Tags: combinatorics

A cube consists of $4^3$ unit cubes each containing an integer. At each move, you choose a unit cube and increase by $1$ all the integers in the neighbouring cubes having a face in common with the chosen cube. Is it possible to reach a position where all the $4^3$ integers are divisible by $3,$ no matter what the starting position is?

2019 PUMaC Algebra B, 2

Tags: algebra

If $x$ is a real number so $3^x=27x$, compute $\log_3 \left(\tfrac{3^{3^x}}{x^{3^3}}\right)$.

2012 Israel National Olympiad, 5

Tags: number theory , Diophantine equation

Find all integer solutions of the equation $a^3+3ab^2+7b^3=2011$.

2007 Nicolae Coculescu, 3

Tags: Gemetry , angle bisector , geometry

Let $ M,N $ be points on the segments $ AB,AC, $ respectively, of the triangle $ ABC. $ Also, let $ P,Q, $ be the midpoints of the segments $ MN,BC, $ respectively. Knowing that $ PQ $ is parallel to the bisector of $ \angle BAC , $ show that $ BM=CN. $ [i]Gheorghe Duță[/i]

1970 Putnam, B5

Tags: Putnam , function , continuity

Let $u_n$ denote the ramp function $$ u_n (x) =\begin{cases} -n \;\; \text{for} \;\; x \leq -n, \\ \; x \;\;\; \text{for} \;\; -n \leq x \leq n,\\ \;n \;\; \; \text{for} \;\; n \leq x, \end{cases}$$ and let $f$ be a real function of a real variable. Show that $f$ is continuous if and only if $u_n \circ f$ is continuous for all $n.$

2015 Mathematical Talent Reward Programme, SAQ: P 1

Tags: combinatorics

In a room there is a series of bulbs on a wall and corresponding switches on the opposite wall. If you put on the $n$ -th switch the $n$ -th bulb will light up. There is a group of men who are operating the switches according to the following rule: they go in one by one and starts flipping the switches starting from the first switch until he has to turn on a bulb; as soon as he turns a bulb on, he leaves the room. For example the first person goes in, turns the first switch on and leaves. Then the second man goes in, seeing that the first switch on turns it off and then lights the second bulb. Then the third person goes in, finds the first switch off and turns it on and leaves the room. Then the fourth person enters and switches off the first and second bulbs and switches on the third. The process continues in this way. Finally we find out that first 10 bulbs are off and the 11 -th bulb is on. Then how many people were involved in the entire process?

2020/2021 Tournament of Towns, P5

Tags: combinatorics , Process , Tournament of Towns

A hundred tourists arrive to a hotel at night. They know that in the hotel there are single rooms numbered as $1, 2, \ldots , n$, and among them $k{}$ (the tourists do not know which) are under repair, the other rooms are free. The tourists, one after another, check the rooms in any order (maybe different for different tourists), and the first room not under repair is taken by the tourist. The tourists don’t know whether a room is occupied until they check it. However it is forbidden to check an occupied room, and the tourists may coordinate their strategy beforehand to avoid this situation. For each $k{}$ find the smallest $n{}$ for which the tourists may select their rooms for sure. [i]Fyodor Ivlev[/i]

2003 JBMO Shortlist, 6

Tags: inequalities , JBMO , JBMO Shortlist

Parallels to the sides of a triangle passing through an interior point divide the inside of a triangle into $6$ parts with the marked areas as in the figure. Show that $\frac{a}{\alpha}+\frac{b}{\beta}+\frac{c}{\gamma}\ge \frac{3}{2}$ [img]https://cdn.artofproblemsolving.com/attachments/a/a/b0a85df58f2994b0975b654df0c342d8dc4d34.png[/img]

1994 Taiwan National Olympiad, 6

Tags: algebra , polynomial , algebra proposed

For $-1\leq x\leq 1$ and $n\in\mathbb N$ define $T_{n}(x)=\frac{1}{2^{n}}[(x+\sqrt{1-x^{2}})^{n}+(x-\sqrt{1-x^{2}})^{n}]$. a)Prove that $T_{n}$ is a monic polynomial of degree $n$ in $x$ and that the maximum value of $|T_{n}(x)|$ is $\frac{1}{2^{n-1}}$. b)Suppose that $p(x)=x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0}\in\mathbb{R}[x]$ is a monic polynomial of degree $n$ such that $p(x)>-\frac{1}{2^{n-1}}$ forall $x$, $-1\leq x\leq 1$. Prove that there exists $x_{0}$, $-1\leq x_{0}\leq 1$ such that $p(x_{0})\geq\frac{1}{2^{n-1}}$.

1955 Putnam, A3

Tags: Putnam

Suppose that $\sum^\infty_{i=1} x_i$ is a convergent series of positive terms which monotonically decrease (that is, $x_1 \ge x_2 \ge x_3 \ge \cdots$). Let $P$ denote the set of all numbers which are sums of some (finite or infinite) subseries of $\sum^\infty_{i= 1} x_i.$ Show that $P$ is an interval if and only if \[ x_n \le \sum^\infty_{i = n + 1} x_i\] for every integer $n.$

2012 HMNT, 4

Tags: algebra

Find the sum of all real solutions for $x$ to the equation $(x^2 + 2x + 3)^{(x^2+2x+3)^{(x^2+2x+3)}} = 2012$.

1955 Polish MO Finals, 3

Tags: geometry , Equilateral

An equilateral triangle $ ABC $ is inscribed in a circle; prove that if $ M $ is any point of the circle, then one of the distances $ MA $, $ MB $, $ MC $ is equal to the sum of the other two.

2019 Singapore MO Open, 5

Tags: combinatorics

In a $m\times n$ chessboard ($m,n\ge 2$), some dominoes are placed (without overlap) with each domino covering exactly two adjacent cells. Show that if no more dominoes can be added to the grid, then at least $2/3$ of the chessboard is covered by dominoes. [i]Proposed by DVDthe1st, mzy and jjax[/i]

1955 Moscow Mathematical Olympiad, 304

Tags: geometry , acute , excenter

The centers $O_1, O_2$ and $O_3$ of circles exscribed about $\vartriangle ABC$ are connected. Prove that $O_1O_2O_3$ is an acute-angled one.

2014 ELMO Shortlist, 1

Tags: inequalities , trigonometry , calculus , topology , three variable inequality

In a non-obtuse triangle $ABC$, prove that \[ \frac{\sin A \sin B}{\sin C} + \frac{\sin B \sin C}{\sin A} + \frac{\sin C \sin A}{ \sin B} \ge \frac 52. \][i]Proposed by Ryan Alweiss[/i]

2000 Turkey MO (2nd round), 1

Tags: trigonometry , geometry , perpendicular bisector , geometry proposed

A circle with center $O$ and a point $A$ in this circle are given. Let $P_{B}$ is the intersection point of $[AB]$ and the internal bisector of $\angle AOB$ where $B$ is a point on the circle such that $B$ doesn't lie on the line $OA$, Find the locus of $P_{B}$ as $B$ varies.

2016 NIMO Problems, 1

Tags: arithmetic sequence

Find the value of $\lfloor 1 \rfloor + \lfloor 1.7 \rfloor +\lfloor 2.4 \rfloor +\lfloor 3.1 \rfloor +\cdots+\lfloor 99 \rfloor$. [i]Proposed by Jack Cornish[/i]

2012 AIME Problems, 2

Tags: ratio , AMC

Two geometric sequences $ a_1,a_2,a_3,\ldots$ and $b_1,b_2,b_3\ldots $have the same common ratio, with $a_1=27$,$b_1=99$, and $a_{15}=b_{11}$. Find $a_9.$

1984 Czech And Slovak Olympiad IIIA, 1

Tags: geometry , 3D geometry , cube

A cube $A_1A_2A_3A_4A_5A_6A_7A_8$ is given in space. We will mark its center with the letter $S$ (intersection of solid diagonals). Find all natural numbers $k$ for which there exists a plane not containing the point $S$ and intersecting just $k$ of the rays $SA_1, SA_2, .. SA_8$

2022 IFYM, Sozopol, 2

Tags: algebra

We say that a rectangle and a triangle are [i]similar[/i], if they have the same area and the same perimeter. Let $P$ be a rectangle for which the ratio of the longer to the shorter side is at least $\lambda -1+\sqrt{\lambda (\lambda -2)}$ where $\lambda =\frac{3\sqrt{3}}{2}$. Prove that there exists a tringle that is [i]similar[/i] to $P$.

1991 Bundeswettbewerb Mathematik, 4

Tags: combinatorics , combinatorial geometry , geometry , rectangle

A strip of width $1$ is to be divided by rectangular panels of common width $1$ and denominations long $a_1$, $a_2$, $a_3$, $. . .$ be paved without gaps ($a_1 \ne 1$). From the second panel on, each panel is similar but not congruent to the already paved part of the strip. When the first $n$ slabs are laid, the length of the paved part of the strip is $sn$. Given $a_1$, is there a number that is not surpassed by any $s_n$? The accuracy answer has to be proven.

2022 HMNT, 31

Tags:

Given positive integers $a_1, a_2, \ldots, a_{2023}$ such that $$a_k = \sum_{i=1}^{2023} |a_k - a_i|$$ for all $1 \le k \le 2023,$ find the minimum possible value of $a_1+a_2+\ldots+a_{2023}.$

1960 IMO, 5

Tags: geometry , 3D geometry , cube , Locus , Locus problems , IMO , IMO 1960

Consider the cube $ABCDA'B'C'D'$ (with face $ABCD$ directly above face $A'B'C'D'$). a) Find the locus of the midpoints of the segments $XY$, where $X$ is any point of $AC$ and $Y$ is any piont of $B'D'$; b) Find the locus of points $Z$ which lie on the segment $XY$ of part a) with $ZY=2XZ$.

2014 NIMO Problems, 4

Tags: trigonometry , geometry , trapezoid , ratio , trig identities , Law of Sines , power of a point

Points $A$, $B$, $C$, and $D$ lie on a circle such that chords $\overline{AC}$ and $\overline{BD}$ intersect at a point $E$ inside the circle. Suppose that $\angle ADE =\angle CBE = 75^\circ$, $BE=4$, and $DE=8$. The value of $AB^2$ can be written in the form $a+b\sqrt{c}$ for positive integers $a$, $b$, and $c$ such that $c$ is not divisible by the square of any prime. Find $a+b+c$. [i]Proposed by Tony Kim[/i]

2010 ELMO Shortlist, 6

Tags: geometry , circumcircle , geometry proposed

Let $ABC$ be a triangle with circumcircle $\Omega$. $X$ and $Y$ are points on $\Omega$ such that $XY$ meets $AB$ and $AC$ at $D$ and $E$, respectively. Show that the midpoints of $XY$, $BE$, $CD$, and $DE$ are concyclic. [i]Carl Lian.[/i]