Olympiad problems

1999 Mexico National Olympiad, 3

A point $P$ is given inside a triangle $ABC$. Let $D,E,F$ be the midpoints of $AP,BP,CP$, and let $L,M,N$ be the intersection points of $ BF$ and $CE, AF$ and $CD, AE$ and $BD$, respectively. (a) Prove that the area of hexagon $DNELFM$ is equal to one third of the area of triangle $ABC$. (b) Prove that $DL,EM$, and $FN$ are concurrent.

2018 Abels Math Contest (Norwegian MO) Final, 1

Tags: number theory , factorial , modulo , residue

For an odd number n, we write $n!! = n\cdot (n-2)...3 \cdot 1$. How many different residues modulo $1000$ do you get from $n!!$ for $n= 1, 3, 5, …$?

2013 India IMO Training Camp, 1

Tags: number theory proposed , number theory

For a prime $p$, a natural number $n$ and an integer $a$, we let $S_n(a,p)$ denote the exponent of $p$ in the prime factorisation of $a^{p^n} - 1$. For example, $S_1(4,3) = 2$ and $S_2(6,2) = 0$. Find all pairs $(n,p)$ such that $S_n(2013,p) = 100$.

2007 Bulgaria Team Selection Test, 3

Tags: inequalities proposed , inequalities

Let $n\geq 2$ is positive integer. Find the best constant $C(n)$ such that \[\sum_{i=1}^{n}x_{i}\geq C(n)\sum_{1\leq j<i\leq n}(2x_{i}x_{j}+\sqrt{x_{i}x_{j}})\] is true for all real numbers $x_{i}\in(0,1),i=1,...,n$ for which $(1-x_{i})(1-x_{j})\geq\frac{1}{4},1\leq j<i \leq n.$

MOAA Gunga Bowls, 2023.21

Tags: MOAA 2023

In obtuse triangle $ABC$ where $\angle B > 90^\circ$ let $H$ and $O$ be its orthocenter and circumcenter respectively. Let $D$ be the foot of the altitude from $A$ to $HC$ and $E$ be the foot of the altitude from $B$ to $AC$ such that $O,E,D$ lie on a line. If $OC=8$ and $OE=4$, find the area of triangle $HAB$. [i]Proposed by Harry Kim[/i]

2011 District Round (Round II), 2

Tags: geometry , geometric transformation , reflection , geometry unsolved

Let $ABC$ denote a triangle with area $S$. Let $U$ be any point inside the triangle whose vertices are the midpoints of the sides of triangle $ABC$. Let $A'$, $B'$, $C'$ denote the reflections of $A$, $B$, $C$, respectively, about the point $U$. Prove that hexagon $AC'BA'CB'$ has area $2S$.

2020 Iran RMM TST, 2

Tags: geometry

A circle $\omega$ is strictly inside triangle $ABC$. The tangents from $A$ to $\omega$ intersect $BC$ in $A_1,A_2$ define $B_1,B_2,C_1,C_2$ similarly. Prove that if five of six points $A_1,A_2,B_1,B_2,C_1,C_2$ lie on a circle the sixth one lie on the circle too.

1996 China Team Selection Test, 2

Tags: inequalities , algebra , Sums and Products , Real Roots , China

Let $\alpha_1, \alpha_2, \dots, \alpha_n$, and $\beta_1, \beta_2, \ldots, \beta_n$, where $n \geq 4$, be 2 sets of real numbers such that \[\sum_{i=1}^{n} \alpha_i^2 < 1 \qquad \text{and} \qquad \sum_{i=1}^{n} \beta_i^2 < 1.\] Define \begin{align*} A^2 &= 1 - \sum_{i=1}^{n} \alpha_i^2,\\ B^2 &= 1 - \sum_{i=1}^{n} \beta_i^2,\\ W &= \frac{1}{2} (1 - \sum_{i=1}^{n} \alpha_i \beta_i)^2. \end{align*} Find all real numbers $\lambda$ such that the polynomial \[x^n + \lambda (x^{n-1} + \cdots + x^3 + Wx^2 + ABx + 1) = 0,\] only has real roots.

2013 Today's Calculation Of Integral, 883

Tags: calculus , integration , trigonometry , logarithms , calculus computations , Integral inequality

Prove that for each positive integer $n$ \[\frac{4n^2+1}{4n^2-1}\int_0^{\pi} (e^{x}-e^{-x})\cos 2nx\ dx>\frac{e^{\pi}-e^{-\pi}-2}{4}\ln \frac{(2n+1)^2}{(2n-1)(n+3)}.\]

2006 China Team Selection Test, 1

Tags: geometry , circumcircle , geometry unsolved

$H$ is the orthocentre of $\triangle{ABC}$. $D$, $E$, $F$ are on the circumcircle of $\triangle{ABC}$ such that $AD \parallel BE \parallel CF$. $S$, $T$, $U$ are the semetrical points of $D$, $E$, $F$ with respect to $BC$, $CA$, $AB$. Show that $S, T, U, H$ lie on the same circle.

2018 BMT Spring, 1

Tags: number theory

How many multiples of $20$ are also divisors of $17!$?

1989 IMO Longlists, 21

Tags: geometry , 3D geometry , tetrahedron , induction , combinatorics unsolved , combinatorics

Let $ ABC$ be an equilateral triangle with side length equal to $ N \in \mathbb{N}.$ Consider the set $ S$ of all points $ M$ inside the triangle $ ABC$ satisfying \[ \overrightarrow{AM} \equal{} \frac{1}{N} \cdot \left(n \cdot \overrightarrow{AB} \plus{} m \cdot \overrightarrow{AC} \right)\] with $ m, n$ integers, $ 0 \leq n \leq N,$ $ 0 \leq m \leq N$ and $ n \plus{} m \leq N.$ Every point of S is colored in one of the three colors blue, white, red such that [b](i) [/b]no point of $ S \cap [AB]$ is coloured blue [b](ii)[/b] no point of $ S \cap [AC]$ is coloured white [b](iii)[/b] no point of $ S \cap [BC]$ is coloured red Prove that there exists an equilateral triangle the following properties: [b](1)[/b] the three vertices of the triangle are points of $ S$ and coloured blue, white and red, respectively. [b](2)[/b] the length of the sides of the triangle is equal to 1. [i]Variant:[/i] Same problem but with a regular tetrahedron and four different colors used.

2018 Azerbaijan Senior NMO, 5

Tags: inequalities , Tuymaada , three variable inequality

Prove that if $x$, $y$, $z$ are positive real numbers and $xyz = 1$ then \[\frac{x^3}{x^2+y}+\frac{y^3}{y^2+z}+\frac{z^3}{z^2+x}\geq \dfrac {3} {2}.\] [i]A. Golovanov[/i]

2009 Today's Calculation Of Integral, 459

Tags: calculus , integration , limit , logarithms , calculus computations

Find $ \lim_{x\to\infty} \int_{e^{\minus{}x}}^1 \left(\ln \frac{1}{t}\right)^ n\ dt\ (x\geq 0,\ n\equal{}1,\ 2,\ \cdots)$.

2008 Kazakhstan National Olympiad, 1

Tags: geometry , rectangle , perimeter , ceiling function , floor function , combinatorics proposed , combinatorics

Let $ F_n$ be a set of all possible connected figures, that consist of $ n$ unit cells. For each element $ f_n$ of this set, let $ S(f_n)$ be the area of that minimal rectangle that covers $ f_n$ and each side of the rectangle is parallel to the corresponding side of the cell. Find $ max(S(f_n))$,where $ f_n\in F_n$? Remark: Two cells are called connected if they have a common edge.

2004 Vietnam Team Selection Test, 1

Tags: algebra unsolved , algebra

Let $ \left\{x_n\right\}$, with $ n \equal{} 1, 2, 3, \ldots$, be a sequence defined by $ x_1 \equal{} 603$, $ x_2 \equal{} 102$ and $ x_{n \plus{} 2} \equal{} x_{n \plus{} 1} \plus{} x_n \plus{} 2\sqrt {x_{n \plus{} 1} \cdot x_n \minus{} 2}$ $ \forall n \geq 1$. Show that: [b](1)[/b] The number $ x_n$ is a positive integer for every $ n \geq 1$. [b](2)[/b] There are infinitely many positive integers $ n$ for which the decimal representation of $ x_n$ ends with 2003. [b](3)[/b] There exists no positive integer $ n$ for which the decimal representation of $ x_n$ ends with 2004.

1979 IMO Shortlist, 10

Tags: vector , trigonometry , geometry , geometric inequality , euclidean geometry , IMO Shortlist , IMO Longlist

Show that for any vectors $a, b$ in Euclidean space, \[|a \times b|^3 \leq \frac{3 \sqrt 3}{8} |a|^2 |b|^2 |a-b|^2\] Remark. Here $\times$ denotes the vector product.

1996 Miklós Schweitzer, 8

Tags: topology , differential topology

Prove that a simply connected, closed manifold (i.e., compact, no boundary) cannot contain a closed, smooth submanifold of codimension 1, with odd Euler characteristic.

2009 Sharygin Geometry Olympiad, 7

Tags: geometry , construction , circumcircle

Let $s$ be the circumcircle of triangle $ABC, L$ and $W$ be common points of angle's $A$ bisector with side $BC$ and $s$ respectively, $O$ be the circumcenter of triangle $ACL$. Restore triangle $ABC$, if circle $s$ and points $W$ and $O$ are given. (D.Prokopenko)

1972 AMC 12/AHSME, 18

Tags: geometry , trapezoid , similar triangles , AMC

Let $ABCD$ be a trapezoid with the measure of base $AB$ twice that of base $DC$, and let $E$ be the point of intersection of the diagonals. If the measure of diagonal $AC$ is $11$, then that of segment $EC$ is equal to $\textbf{(A) }3\textstyle\frac{2}{3}\qquad\textbf{(B) }3\frac{3}{4}\qquad\textbf{(C) }4\qquad\textbf{(D) }3\frac{1}{2}\qquad \textbf{(E) }3$

1980 Austrian-Polish Competition, 5

Tags: geometry , concurrency , IMO Shortlist

Let $A_1A_2A_3$ be a triangle and, for $1 \leq i \leq 3$, let $B_i$ be an interior point of edge opposite $A_i$. Prove that the perpendicular bisectors of $A_iB_i$ for $1 \leq i \leq 3$ are not concurrent.

2021 Dutch IMO TST, 2

Tags: algebra , system of equations , System

Find all quadruplets $(x_1, x_2, x_3, x_4)$ of real numbers such that the next six equalities apply: $$\begin{cases} x_1 + x_2 = x^2_3 + x^2_4 + 6x_3x_4\\ x_1 + x_3 = x^2_2 + x^2_4 + 6x_2x_4\\ x_1 + x_4 = x^2_2 + x^2_3 + 6x_2x_3\\ x_2 + x_3 = x^2_1 + x^2_4 + 6x_1x_4\\ x_2 + x_4 = x^2_1 + x^2_3 + 6x_1x_3 \\ x_3 + x_4 = x^2_1 + x^2_2 + 6x_1x_2 \end{cases}$$

2024 Nordic, 4

Tags: Nordic , combinatorics , set

Alice and Bob are playing a game. First, Alice chooses a partition $\mathcal{C}$ of the positive integers into a (not necessarily finite) set of sets, such that each positive integer is in exactly one of the sets in $\mathcal{C}$. Then Bob does the following operation a finite number of times. Choose a set $S \in \mathcal{C}$ not previously chosen, and let $D$ be the set of all positive integers dividing at least one element in $S$. Then add the set $D \setminus S$ (possibly the empty set) to $\mathcal{C}$. Bob wins if there are two equal sets in $\mathcal{C}$ after he has done all his moves, otherwise, Alice wins. Determine which player has a winning strategy.

1995 All-Russian Olympiad Regional Round, 11.2

Tags: 3D geometry , cube , geometry , parallelepiped

A planar section of a parallelepiped is a regular hexagon. Show that this parallelepiped is a cube.

2020-2021 OMMC, 5

Tags: ommc

How many nonempty subsets of ${1, 2, \dots, 15}$ are there such that the sum of the squares of each subset is a multiple of $5$?