Olympiad problems

2018 German National Olympiad, 3

Given a positive integer $n$, Susann fills a square of $n \times n$ boxes. In each box she inscribes an integer, taking care that each row and each column contains distinct numbers. After this an imp appears and destroys some of the boxes. Show that Susann can choose some of the remaining boxes and colour them red, satisfying the following two conditions: 1) There are no two red boxes in the same column or in the same row. 2) For each box which is neither destroyed nor coloured, there is a red box with a larger number in the same row or a red box with a smaller number in the same column. [i]Proposed by Christian Reiher[/i]

2003 IMC, 1

Tags: linear algebra , matrix

Let $A,B \in \mathbb{R}^{n\times n}$ such that $AB+B+A=0$. Prove that $AB=BA$.

2010 Math Prize For Girls Problems, 1

Tags:

If $a$ and $b$ are nonzero real numbers such that $\left| a \right| \ne \left| b \right|$, compute the value of the expression \[ \left( \frac{b^2}{a^2} + \frac{a^2}{b^2} - 2 \right) \times \left( \frac{a + b}{b - a} + \frac{b - a}{a + b} \right) \times \left( \frac{\frac{1}{a^2} + \frac{1}{b^2}}{\frac{1}{b^2} - \frac{1}{a^2}} - \frac{\frac{1}{b^2} - \frac{1}{a^2}}{\frac{1}{a^2} + \frac{1}{b^2}} \right). \]

2007 Purple Comet Problems, 12

Tags: counting , distinguishability , factorial

If you alphabetize all of the distinguishable rearrangements of the letters in the word [b]PURPLE[/b], find the number $n$ such that the word [b]PURPLE [/b]is the $n$th item in the list.

1962 AMC 12/AHSME, 40

Tags:

The limiting sum of the infinite series, $ \frac{1}{10} \plus{} \frac{2}{10^2} \plus{} \frac{3}{10^3} \plus{} \dots$ whose $ n$th term is $ \frac{n}{10^n}$ is: $ \textbf{(A)}\ \frac19 \qquad \textbf{(B)}\ \frac{10}{81} \qquad \textbf{(C)}\ \frac18 \qquad \textbf{(D)}\ \frac{17}{72} \qquad \textbf{(E)}\ \text{larger than any finite quantity}$

2016 ASDAN Math Tournament, 13

Tags: team test

Ash writes the positive integers from $1$ to $2016$ inclusive as a single positive integer $n=1234567891011\dots2016$. What is the result obtained by successively adding and subtracting the digits of $n$? (In other words, compute $1-2+3-4+5-6+7-8+9-1+0-1+\dots$.)

2019 Moroccan TST, 2

Tags: inequalities , morocco

Let $a>1$ be a real number. Prove that for all $n\in\mathbb{N}*$ that : $\frac{a^n-1}{n}\ge \sqrt{a}^{n+1}-\sqrt{a}^{n-1}$

2024 German National Olympiad, 5

Tags: geometry , circles , circumcircle , concyclic

Let $\triangle ABC$ be a triangle and let $X$ be a point in the interior of the triangle. The second intersection points of the lines $XA,XB$ and $XC$ with the circumcircle of $\triangle ABC$ are $P,Q$ and $R$. Let $U$ be a point on the ray $XP$ (these are the points on the line $XP$ such that $P$ and $U$ lie on the same side of $X$). The line through $U$ parallel to $AB$ intersects $BQ$ in $V$ . The line through $U$ parallel to $AC$ intersects $CR$ in $W$. Prove that $Q, R, V$ , and $W$ lie on a circle.

1984 National High School Mathematics League, 4

Tags:

The number of real roots of the equation $\sin x=\lg x$ is $\text{(A)}1\qquad\text{(B)}2\qquad\text{(C)}3\qquad\text{(D)}$more than $3$

1954 Putnam, B1

Tags: number theory

Show that the equation $x^2 -y^2 =a^3$ has always integral solutions for $x$ and $y$ whenever $a$ is a positive integer.

2002 AMC 12/AHSME, 12

Tags: relatively prime , number theory

For how many integers $ n$ is $ \frac{n}{20\minus{}n}$ the square of an integer? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 10$

2002 IMC, 12

Tags: gradient , inequalities , real analysis

Let $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ be a convex function whose gradient $\nabla f$ exists at every point of $\mathbb{R}^{n}$ and satisfies the condition $$\exists L>0\; \forall x_{1},x_{2}\in \mathbb{R}^{n}:\;\; ||\nabla f(x_{1})-\nabla f(x_{2})||\leq L||x_{1}-x_{2}||.$$ Prove that $$ \forall x_{1},x_{2}\in \mathbb{R}^{n}:\;\; ||\nabla f(x_{1})-\nabla f(x_{2})||^{2}\leq L\langle\nabla f(x_{1})-\nabla f(x_{2}), x_{1}-x_{2}\rangle. $$

2021 IMO Shortlist, A2

Tags: algebra , floor function , summation , number theory

Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?

2021 Kyiv City MO Round 1, 8.3

Tags: permutation , magic square , combinatorics

The $1 \times 1$ cells located around the perimeter of a $3 \times 3$ square are filled with the numbers $1, 2, \ldots, 8$ so that the sums along each of the four sides are equal. In the upper left corner cell is the number $8$, and in the upper left is the number $6$ (see the figure below). [img]https://i.ibb.co/bRmd12j/Kyiv-MO-2021-Round-1-8-2.png[/img] How many different ways to fill the remaining cells are there under these conditions? [i]Proposed by Mariia Rozhkova[/i]

2015 European Mathematical Cup, 2

Tags: inequalities , algebra

Let $a,b,c$ be positive real numbers such that $abc=1$. Prove that $$\frac{a+b+c+3}{4}\geqslant \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}.$$ [i]Dimitar Trenevski[/i]

2016 IMO Shortlist, G1

Tags: triangle , parallelogram , geometry

Triangle $BCF$ has a right angle at $B$. Let $A$ be the point on line $CF$ such that $FA=FB$ and $F$ lies between $A$ and $C$. Point $D$ is chosen so that $DA=DC$ and $AC$ is the bisector of $\angle{DAB}$. Point $E$ is chosen so that $EA=ED$ and $AD$ is the bisector of $\angle{EAC}$. Let $M$ be the midpoint of $CF$. Let $X$ be the point such that $AMXE$ is a parallelogram. Prove that $BD,FX$ and $ME$ are concurrent.

2012 IMO Shortlist, G6

Tags: geometry , circumcircle , incenter , reflection

Let $ABC$ be a triangle with circumcenter $O$ and incenter $I$. The points $D,E$ and $F$ on the sides $BC,CA$ and $AB$ respectively are such that $BD+BF=CA$ and $CD+CE=AB$. The circumcircles of the triangles $BFD$ and $CDE$ intersect at $P \neq D$. Prove that $OP=OI$.

2018 Federal Competition For Advanced Students, P1, 1

Tags: inequalities

Let $\alpha$ be an arbitrary positive real number. Determine for this number $\alpha$ the greatest real number $C$ such that the inequality$$\left(1+\frac{\alpha}{x^2}\right)\left(1+\frac{\alpha}{y^2}\right)\left(1+\frac{\alpha}{z^2}\right)\geq C\left(\frac{x}{z}+\frac{z}{x}+2\right)$$ is valid for all positive real numbers $x, y$ and $z$ satisfying $xy + yz + zx =\alpha.$ When does equality occur? [i](Proposed by Walther Janous)[/i]

2021 Brazil EGMO TST, 5

Tags: number theory , combinatorics , geometry , 3d geometry

Let $S$ be a set, such that for every positive integer $n$, we have $|S\cap T|=1$, where $T=\{n,2n,3n\}$. Prove that if $2\in S$, then $13824\notin S$.

2024 APMO, 3

Tags: algebra , inequality , induction

Let $n$ be a positive integer and let $a_1, a_2, \ldots, a_n$ be positive reals. Show that $$\sum_{i=1}^{n} \frac{1}{2^i}(\frac{2}{1+a_i})^{2^i} \geq \frac{2}{1+a_1a_2\ldots a_n}-\frac{1}{2^n}.$$

2018 AMC 12/AHSME, 5

Tags: number theory , prime number

How many subsets of $\{2,3,4,5,6,7,8,9\}$ contain at least one prime number? $\textbf{(A)} \text{ 128} \qquad \textbf{(B)} \text{ 192} \qquad \textbf{(C)} \text{ 224} \qquad \textbf{(D)} \text{ 240} \qquad \textbf{(E)} \text{ 256}$

2022 239 Open Mathematical Olympiad, 3

Tags: geometry , combinatorial geometry , point , color

Let $A$ be a countable set, some of its countable subsets are selected such that; the intersection of any two selected subsets has at most one element. Find the smallest $k$ for which one can ensure that we can color elements of $A$ with $k$ colors such that each selected subsets exactly contain one element of one of the colors and an infinite number of elements of each of the other colors.

2002 China Team Selection Test, 1

Tags: trapezoid , geometry

Circle $ O$ is inscribed in a trapzoid $ ABCD$, $ \angle{A}$ and $ \angle{B}$ are all acute angles. A line through $ O$ intersects $ AD$ at $ E$ and $ BC$ at $ F$, and satisfies the following conditions: (1) $ \angle{DEF}$ and $ \angle{CFE}$ are acute angles. (2) $ AE\plus{}BF\equal{}DE\plus{}CF$. Let $ AB\equal{}a$, $ BC\equal{}b$, $ CD\equal{}c$, then use $ a,b,c$ to express $ AE$.

1982 Austrian-Polish Competition, 5

Tags: algebra

Show that [0,1] cannot be partitioned into two disjoints sets A and B such that B=A+a for some real a.

1963 AMC 12/AHSME, 34

Tags: trigonometry , trig identities , law of cosines

In triangle ABC, side $a = \sqrt{3}$, side $b = \sqrt{3}$, and side $c > 3$. Let $x$ be the largest number such that the magnitude, in degrees, of the angle opposite side $c$ exceeds $x$. Then $x$ equals: $\textbf{(A)}\ 150 \qquad \textbf{(B)}\ 120\qquad \textbf{(C)}\ 105 \qquad \textbf{(D)}\ 90 \qquad \textbf{(E)}\ 60$