Olympiad problems

2019 Junior Balkan Team Selection Tests - Moldova, 7

Tags: geometry

Point $H$ is the orthocenter of the acute triangle $\Delta ABC$ and point $K$,situated on the line $(BC)$, is the foot of the perpendicular from point $A$ .The circle $\Omega$ passes through points $A$ and $K$ ,intersecting the sides $(AB)$ and $(AC)$ in points $M$ and $N$ .The line that passes through point $A$ and is parallel with $BC$ intersects again the circumcircles of triangles $\Delta AHM$ and $\Delta AHN$ in points $X$ and $Y$.Prove that $XY =BC$.

2007 Singapore Junior Math Olympiad, 1

Tags: trapezium , geometry , trapezoid , area

Let $ABCD$ be a trapezium with $AB// DC, AB = b, AD = a ,a<b$ and $O$ the intersection point of the diagonals. Let $S$ be the area of the trapezium $ABCD$. Suppose the area of $\vartriangle DOC$ is $2S/9$. Find the value of $a/b$.

2014 Kyiv Mathematical Festival, 2

Tags: rotation

Can an $8\times8$ board be covered with 13 equal 5-celled figures? It's alowed to rotate the figures or turn them over. [size=85](Kyiv mathematical festival 2014)[/size]

2005 Estonia Team Selection Test, 4

Tags: polynomial , algebra , real root

Find all pairs $(a, b)$ of real numbers such that the roots of polynomials $6x^2 -24x -4a$ and $x^3 + ax^2 + bx - 8$ are all non-negative real numbers.

2016 ASDAN Math Tournament, 10

Tags:

Using the fact that $$\sum_{n=1}^\infty\frac{1}{n^2}=\frac{\pi^2}{6},$$ compute $$\int_0^1(\ln x)\ln(1-x)dx.$$

2011 ELMO Shortlist, 5

Tags: algorithm , combinatorics , logarithm

Prove there exists a constant $c$ (independent of $n$) such that for any graph $G$ with $n>2$ vertices, we can split $G$ into a forest and at most $cf(n)$ disjoint cycles, where a) $f(n)=n\ln{n}$; b) $f(n)=n$. [i]David Yang.[/i]

2003 Baltic Way, 10

Tags: calculus , integration , analytic geometry , combinatorics

A [i]lattice point[/i] in the plane is a point with integral coordinates. The[i] centroid[/i] of four points $(x_i,y_i )$, $i = 1, 2, 3, 4$, is the point $\left(\frac{x_1 +x_2 +x_3 +x_4}{4},\frac{y_1 +y_2 +y_3 +y_4 }{4}\right)$. Let $n$ be the largest natural number for which there are $n$ distinct lattice points in the plane such that the centroid of any four of them is not a lattice point. Prove that $n = 12$.

2014 Turkey EGMO TST, 5

Tags: geometry , circumcircle

Let $ABC$ be a triangle with circumcircle $\omega$ and let $\omega_A$ be a circle drawn outside $ABC$ and tangent to side $BC$ at $A_1$ and tangent to $\omega$ at $A_2$. Let the circles $\omega_B$ and $\omega_C$ and the points $B_1, B_2, C_1, C_2$ are defined similarly. Prove that if the lines $AA_1, BB_1, CC_1$ are concurrent, then the lines $AA_2, BB_2, CC_2$ are also concurrent.

2005 Greece Team Selection Test, 3

Tags: polynomial , algebra

Let the polynomial $P(x)=x^3+19x^2+94x+a$ where $a\in\mathbb{N}$. If $p$ a prime number, prove that no more than three numbers of the numbers $P(0), P(1),\ldots, P(p-1)$ are divisible by $p$.

2016 USAMO, 4

Tags: function , algebra

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all real numbers $x$ and $y$, $$(f(x)+xy)\cdot f(x-3y)+(f(y)+xy)\cdot f(3x-y)=(f(x+y))^2.$$

2016 Ecuador NMO (OMEC), 6

Tags: number theory , divisible

A positive integer $n$ is "[i]olympic[/i]" if there are $n$ non-negative integers $x_1, x_2, ..., x_n$ that satisfy that: $\bullet$ There is at least one positive integer $j$: $1 \le j \le n$ such that $x_j \ne 0$. $\bullet$ For any way of choosing $n$ numbers $c_1, c_2, ..., c_n$ from the set $\{-1, 0, 1\}$, where not all $c_i$ are equal to zero, it is true that the sum $c_1x_1 + c_2x_2 +... + c_nx_n$ is not divisible by $n^3$. Find the largest positive "olympic" integer.

2023 Thailand Online MO, 1

Tags: combinatorics , enumerative combinatorics

Let $n$ be a positive integer. Chef Kao has $n$ different flavors of ice cream. He wants to serve one small cup and one large cup for each flavor. He arranges the $2n$ ice cream cups into two rows of $n$ cups on a tray. He wants the tray to be colorful, so he arranges the ice cream cups with the following conditions: [list] [*]each row contains all ice cream flavors, and [*]each column has different sizes of ice cream cup. [/list]Determine the number of ways that Chef Kao can arrange cups of ice cream with the above conditions.

1998 Cono Sur Olympiad, 6

Tags: combinatorics

The mayor of a city wishes to establish a transport system with at least one bus line, in which: - each line passes exactly three stops, - every two different lines have exactly one stop in common, - for each two different bus stops there is exactly one line that passes through both. Determine the number of bus stops in the city.

2021 Polish MO Finals, 3

Tags: circumcircle , geometry

Let $\omega$ be the circumcircle of a triangle $ABC$. Let $P$ be any point on $\omega$ different than the verticies of the triangle. Line $AP$ intersects $BC$ at $D$, $BP$ intersects $AC$ at $E$ and $CP$ intersects $AB$ at $F$. Let $X$ be the projection of $D$ onto line passing through midpoints of $AP$ and $BC$, $Y$ be the projection of $E$ onto line passing through $BP$ and $AC$ and let $Z$ be the projection of $F$ onto line passing through midpoints of $CP$ and $AB$. Let $Q$ be the circumcenter of triangle $XYZ$. Prove that all possible points $Q$, corresponding to different positions of $P$ lie on one circle.

1991 Arnold's Trivium, 79

Tags: trigonometry

How many solutions has the boundary-value problem \[u_{xx}+\lambda u=\sin x,\;u(0)=u(\pi)=0\]

2016 ASDAN Math Tournament, 7

Tags:

What is $$\sum_{n=1996}^{2016}\lfloor\sqrt{n}\rfloor?$$

2021 Yasinsky Geometry Olympiad, 2

Tags: isosceles , geometry

In the triangle $ABC$, it is known that $AB = BC = 20$ cm, and $AC = 24$ cm. The point $M$ lies on the side $BC$ and is equidistant from sides $AB$ and $AC$. Find this distance. (Alexander Shkolny)

1971 Miklós Schweitzer, 9

Tags: function , real analysis

Given a positive, monotone function $ F(x)$ on $ (0, \infty)$ such that $ F(x)/x$ is monotone nondecreasing and $ F(x)/x^{1+d}$ is monotone nonincreasing for some positive $ d$, let $ \lambda_n >0$ and $ a_n \geq 0 , \;n \geq 1$. Prove that if \[ \sum_{n=1}^{\infty} \lambda_n F \left( a_n \sum _{k=1}^n \frac{\lambda_k}{\lambda_n} \right) < \infty,\] or \[ \sum_{n=1}^{\infty} \lambda_n F \left( \sum _{k=1}^n a_k \frac{\lambda_k}{\lambda_n} \right) < \infty,\] then $ \sum_{n=1}^ {\infty} a_n$ is convergent. [i]L. Leindler[/i]

2020 Tournament Of Towns, 2

Tags:

$ What~ is~ the~ maximum~ number~ of~ distinct~ integers~ in~ a~ row~ such~ that~ the~sum~ of~ any~ 11~ consequent~ integers~ is~ either~ 100~ or~ 101~?$ I'm posting this problem for people to discuss

2011 ELMO Shortlist, 2

Tags: function , induction , functional equation , algebra

Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that whenever $a>b>c>d>0$ and $ad=bc$, \[f(a+d)+f(b-c)=f(a-d)+f(b+c).\] [i]Calvin Deng.[/i]

2018 Serbia National Math Olympiad, 2

Tags: quadratic residue , sequence , number theory

Let $n>1$ be an integer. Call a number beautiful if its square leaves an odd remainder upon divison by $n$. Prove that the number of consecutive beautiful numbers is less or equal to $1+\lfloor \sqrt{3n} \rfloor$.

1989 IMO Longlists, 89

Tags: combinatorics , graph theory , extremal graph theory , extremal combinatorics

155 birds $ P_1, \ldots, P_{155}$ are sitting down on the boundary of a circle $ C.$ Two birds $ P_i, P_j$ are mutually visible if the angle at centre $ m(\cdot)$ of their positions $ m(P_iP_j) \leq 10^{\circ}.$ Find the smallest number of mutually visible pairs of birds, i.e. minimal set of pairs $ \{x,y\}$ of mutually visible pairs of birds with $ x,y \in \{P_1, \ldots, P_{155}\}.$ One assumes that a position (point) on $ C$ can be occupied simultaneously by several birds, e.g. all possible birds.

1992 AMC 12/AHSME, 7

Tags: ratio

The ratio of $w$ to $x$ is $4:3$, of $y$ to $z$ is $3:2$ and of $z$ to $x$ is $1:6$. What is the ratio of $w$ to $y$? $ \textbf{(A)}\ 1:3\qquad\textbf{(B)}\ 16:3\qquad\textbf{(C)}\ 20:3\qquad\textbf{(D)}\ 27:4\qquad\textbf{(E)}\ 12:1 $

2004 Unirea, 2

Tags: linear algebra , matrix

Let be two matrices $ A,N\in\mathcal{M}_2(\mathbb{R}) $ that commute and such that $ N $ is nilpotent. Show that: [b]a)[/b] $ \det (A+N)=\det (A) $ [b]b)[/b] if $ A $ is general linear, then the matrix $ A+N $ is invertible and $ (A+N)^{-1}=(A-N)A^{-2} . $

1986 IMO Longlists, 50

Tags: incenter , perimeter , geometry

Let $D$ be the point on the side $BC$ of the triangle $ABC$ such that $AD$ is the bisector of $\angle CAB$. Let $I$ be the incenter of$ ABC.$ [i](a)[/i] Construct the points $P$ and $Q$ on the sides $AB$ and $AC$, respectively, such that $PQ$ is parallel to $BC$ and the perimeter of the triangle $APQ$ is equal to $k \cdot BC$, where $k$ is a given rational number. [i](b) [/i]Let $R$ be the intersection point of $PQ$ and $AD$. For what value of $k$ does the equality $AR = RI$ hold? [i](c)[/i] In which case do the equalities $AR = RI = ID$ hold?