Olympiad problems

2019 German National Olympiad, 2

Tags: geometry

Let $a$ and $b$ be two circles, intersecting in two distinct points $Y$ and $Z$. A circle $k$ touches the circles $a$ and $b$ externally in the points $A$ and $B$. Show that the angular bisectors of the angles $\angle ZAY$ and $\angle YBZ$ intersect on the line $YZ$.

2003 District Olympiad, 3

Tags: linear algebra , matrix

a)Prove that any matrix $A\in \mathcal{M}_4(\mathbb{C})$ can be written as a sum of four matrices $B_1,B_2,B_3,B_4\in \mathcal{M}_4(\mathbb{C})$ with the rank equal to $1$. b)$I_4$ can't be written as a sum of less than four matrices with the rank equal to $1$. [i]Manuela Prajea & Ion Savu[/i]

2022 Brazil Team Selection Test, 2

Tags: algebra , combinatorics

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ [i]Proposed by Shahjalal Shohag, Bangladesh[/i]

1979 IMO Longlists, 54

Tags: recurrence relation , inequality , sequence , number theory

Consider the sequences $(a_n), (b_n)$ defined by \[a_1=3, \quad b_1=100 , \quad a_{n+1}=3^{a_n} , \quad b_{n+1}=100^{b_n} \] Find the smallest integer $m$ for which $b_m > a_{100}.$

2020 Iran Team Selection Test, 5

Tags: fractional part , divisibility , algebra

For every positive integer $k>1$ prove that there exist a real number $x$ so that for every positive integer $n<1398$: $$\left\{x^n\right\}<\left\{x^{n-1}\right\} \Longleftrightarrow k\mid n.$$ [i]Proposed by Mohammad Amin Sharifi[/i]

1940 Moscow Mathematical Olympiad, 054

Tags: algebra , factoring , factor

Factor $(b - c)^3 + (c - a)^3 + (a - b)^3$.

2004 JBMO Shortlist, 2

Tags: geometry , combinatorial geometry , area of triangle

Let $E, F$ be two distinct points inside a parallelogram $ABCD$ . Determine the maximum possible number of triangles having the same area with three vertices from points $A, B, C, D, E, F$.

2019 Finnish National High School Mathematics Comp, 3

Tags: cyclic quadrilateral , midpoint , geometry , circles

Let $ABCD$ be a cyclic quadrilateral whose side $AB$ is at the same time the diameter of the circle. The lines $AC$ and $BD$ intersect at point $E$ and the extensions of lines $AD$ and $BC$ intersect at point $F$. Segment $EF$ intersects the circle at $G$ and the extension of segment $EF$ intersects $AB$ at $H$. Show that if $G$ is the midpoint of $FH$, then $E$ is the midpoint of $GH$.

2010 ELMO Shortlist, 3

Tags: functional equation , max and min

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x+y) = \max(f(x),y) + \min(f(y),x)$. [i]George Xing.[/i]

1992 All Soviet Union Mathematical Olympiad, 569

Tags: perpendicular , geometry , circles

Circles $C$ and $C'$ intersect at $O$ and $X$. A circle center $O$ meets $C$ at $Q$ and $R$ and meets $C'$ at $P$ and $S$. $PR$ and $QS$ meet at $Y$ distinct from $X$. Show that $\angle YXO = 90^o$.

2018 Canadian Open Math Challenge, C3

Tags:

Source: 2018 Canadian Open Math Challenge Part C Problem 3 ----- Consider a convex quadrilateral $ABCD$. Let rays $BA$ and $CD$ intersect at $E$, rays $DA$ and $CB$ intersect at $F$, and the diagonals $AC$ and $BD$ intersect at $G$. It is given that the triangles $DBF$ and $DBE$ have the same area. $\text{(a)}$ Prove that $EF$ and $BD$ are parallel. $\text{(b)}$ Prove that $G$ is the midpoint of $BD$. $\text{(c)}$ Given that the area of triangle $ABD$ is 4 and the area of triangle $CBD$ is 6, [color=transparent](C.)[/color]compute the area of triangle $EFG$.

2005 India IMO Training Camp, 1

Tags: geometry

Consider a $n$-sided polygon inscribed in a circle ($n \geq 4$). Partition the polygon into $n-2$ triangles using [b]non-intersecting[/b] diagnols. Prove that, irrespective of the triangulation, the sum of the in-radii of the triangles is a constant.

2022 ELMO Revenge, 4

Tags: functional equation , function , algebra

Find all ordered pairs of integers $(a,b)$ such that there exists a function $f\colon \mathbb{N} \to \mathbb{N}$ satisfying $$f^{f(n)}(n)=an+b$$ For all $n\in \mathbb{N}$.

2021 JHMT HS, 2

Tags: calculus , integration , algebra , floor function , function

Compute the smallest positive integer $n$ such that $\int_{0}^{n} \lfloor x\rfloor\,dx$ is at least $2021.$

1999 Mongolian Mathematical Olympiad, Problem 2

Tags: inequality , inequalities

Let $a,b,c$ be the real numbers with $a\ge\frac85b>0$ and $a\ge c>0$. Prove the inequality $$\frac45\left(\frac1a+\frac1b\right)+\frac2c\ge\frac{27}2\cdot\frac1{a+b+c}.$$

2007 Indonesia MO, 7

Tags: circumcircle , geometry

Points $ A,B,C,D$ are on circle $ S$, such that $ AB$ is the diameter of $ S$, but $ CD$ is not the diameter. Given also that $ C$ and $ D$ are on different sides of $ AB$. The tangents of $ S$ at $ C$ and $ D$ intersect at $ P$. Points $ Q$ and $ R$ are the intersections of line $ AC$ with line $ BD$ and line $ AD$ with line $ BC$, respectively. (a) Prove that $ P$, $ Q$, and $ R$ are collinear. (b) Prove that $ QR$ is perpendicular to line $ AB$.

2002 IMO Shortlist, 4

Tags: geometry , circles , circumcenter

Circles $S_1$ and $S_2$ intersect at points $P$ and $Q$. Distinct points $A_1$ and $B_1$ (not at $P$ or $Q$) are selected on $S_1$. The lines $A_1P$ and $B_1P$ meet $S_2$ again at $A_2$ and $B_2$ respectively, and the lines $A_1B_1$ and $A_2B_2$ meet at $C$. Prove that, as $A_1$ and $B_1$ vary, the circumcentres of triangles $A_1A_2C$ all lie on one fixed circle.

2021 Kyiv City MO Round 1, 11.2

Tags: combinatorics , chessboard

Chess piece called [i]skew knight[/i], if placed on the black square, attacks all the gray squares. [img]https://i.ibb.co/HdTDNjN/Kyiv-MO-2021-Round-1-11-2.png[/img] What is the largest number of such knights that can be placed on the $8\times 8$ chessboard without them attacking each other? [i]Proposed by Arsenii Nikolaiev[/i]

2004 Germany Team Selection Test, 2

Tags: modular arithmetic , induction , number theory

Find all pairs of positive integers $\left(n;\;k\right)$ such that $n!=\left( n+1\right)^{k}-1$.

2010 Contests, 3

Tags: induction , geometry , perimeter , combinatorics

All sides and diagonals of a convex $n$-gon, $n\ge 3$, are coloured one of two colours. Show that there exist $\left[\frac{n+1}{3}\right]$ pairwise disjoint monochromatic segments. [i](Two segments are disjoint if they do not share an endpoint or an interior point).[/i]

2007 Korea - Final Round, 5

Tags: geometry , trigonometry , function , angle bisector

For the vertex $ A$ of a triangle $ ABC$, let $ l_a$ be the distance between the projections on $ AB$ and $ AC$ of the intersection of the angle bisector of ∠$ A$ with side $ BC$. Define $ l_b$ and $ l_c$ analogously. If $ l$ is the perimeter of triangle $ ABC$, prove that $ \frac{l_a l_b l_c}{l^3}\le\frac{1}{64}$.

1983 Bulgaria National Olympiad, Problem 5

Tags: polynomial , solve equation , complex number , algebra

Can the polynomials $x^{5}-x-1$ and $x^{2}+ax+b$ , where $a,b\in Q$, have common complex roots?

1955 AMC 12/AHSME, 17

Tags: logarithm

If $ \log x\minus{}5 \log 3\equal{}\minus{}2$, then $ x$ equals: $ \textbf{(A)}\ 1.25 \qquad \textbf{(B)}\ 0.81 \qquad \textbf{(C)}\ 2.43 \qquad \textbf{(D)}\ 0.8 \qquad \textbf{(E)}\ \text{either 0.8 or 1.25}$

2024 AMC 12/AHSME, 24

Tags: 3d geometry

A $\textit{disphenoid}$ is a tetrahedron whose triangular faces are congruent to one another. What is the least total surface area of a disphenoid whose faces are scalene triangles with integer side lengths? $\textbf{(A) }\sqrt{3}\qquad\textbf{(B) }3\sqrt{15}\qquad\textbf{(C) }15\qquad\textbf{(D) }15\sqrt{7}\qquad\textbf{(E) }24\sqrt{6}$

May Olympiad L1 - geometry, 2022.5

Tags: geometry , isosceles

Vero had an isosceles triangle made of paper. Using scissors, he divided it into three smaller triangles and painted them blue, red and green. Having done so, he observed that: $\bullet$ with the blue triangle and the red triangle an isosceles triangle can be formed, $\bullet$ with the blue triangle and the green triangle an isosceles triangle can be formed, $\bullet$ with the red triangle and the green triangle an isosceles triangle can be formed. Show what Vero's triangle looked like and how he might have made the cuts to make this situation be possible.