Olympiad problems

2015 JBMO Shortlist, 1

Tags: geometry , JBMO

Around the triangle $ABC$ the circle is circumscribed, and at the vertex ${C}$ tangent ${t}$ to this circle is drawn. The line ${p}$, which is parallel to this tangent intersects the lines ${BC}$ and ${AC}$ at the points ${D}$ and ${E}$, respectively. Prove that the points $A,B,D,E$ belong to the same circle. (Montenegro)

1954 AMC 12/AHSME, 4

Tags: algorithm , number theory , Euclidean algorithm

If the Highest Common Divisor of $ 6432$ and $ 132$ is diminished by $ 8$, it will equal: $ \textbf{(A)}\ \minus{}6 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ \minus{}2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

2019 Junior Balkan Team Selection Tests - Romania, 2

Tags: geometry , circumcircle , homothety , Triangle , IMO Shortlist , Hi

Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$. The line $AO$ meets the side $BC$ at $D$. The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$, respectively. Extend the sides $BA$ and $CA$ beyond $A$, and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$. [i]Proposed by Hojoo Lee, Korea[/i]

2005 VJIMC, Problem 3

Tags: calculus , integration , limits

Let $f:[0,1]\times[0,1]\to\mathbb R$ be a continuous function. Find the limit $$\lim_{n\to\infty}\left(\frac{(2n+1)!}{(n!)^2}\right)^2\int^1_0\int^1_0(xy(1-x)(1-y))^nf(x,y)\text dx\text dy.$$

1969 IMO, 2

Tags: function , trigonometry , Trigonometric Equations , IMO , IMO 1969

Let $f(x)=\cos(a_1+x)+{1\over2}\cos(a_2+x)+{1\over4}\cos(a_3+x)+\ldots+{1\over2^{n-1}}\cos(a_n+x)$, where $a_i$ are real constants and $x$ is a real variable. If $f(x_1)=f(x_2)=0$, prove that $x_1-x_2$ is a multiple of $\pi$.

2009 Today's Calculation Of Integral, 500

Tags: calculus , integration , calculus computations

Let $ a,\ b,\ c$ be positive real numbers. Prove the following inequality. \[ \int_1^e \frac {x^{a \plus{} b \plus{} c \minus{} 1}[2(a \plus{} b \plus{} c) \plus{} (c \plus{} 2a)x^{a \minus{} b} \plus{} (a \plus{} 2b)x^{b \minus{} c} \plus{} (b \plus{} 2c)x^{c \minus{} a} \plus{}(2a \plus{} b)x^{a \minus{} c} \plus{} (2b \plus{} c)x^{b \minus{} a} \plus{} (2c \plus{} a)x^{c \minus{} b}]}{(x^a \plus{} x^b)(x^b \plus{} x^c)(x^c \plus{} x^a)}\geq a \plus{} b \plus{} c.\] I have just posted 500 th post. [color=blue]Thank you for your cooperations, mathLinkers and AOPS users.[/color] I will keep posting afterwards. Japanese Communities Modeartor kunny

2024 Romania National Olympiad, 1

Tags: function , real analysis

Let $I \subset \mathbb{R}$ be an open interval and $f:I \to \mathbb{R}$ a twice differentiable function such that $f(x)f''(x)=0,$ for any $x \in I.$ Prove that $f''(x)=0,$ for any $x \in I.$

2020 CMIMC Geometry, 7

Tags: geometry , 2020

In triangle $ABC$, points $D$, $E$, and $F$ are on sides $BC$, $CA$, and $AB$ respectively, such that $BF = BD = CD = CE = 5$ and $AE - AF = 3$. Let $I$ be the incenter of $ABC$. The circumcircles of $BFI$ and $CEI$ intersect at $X \neq I$. Find the length of $DX$.

2010 ELMO Shortlist, 2

Tags: geometry , parallelogram , trapezoid , geometry proposed

Given a triangle $ABC$, a point $P$ is chosen on side $BC$. Points $M$ and $N$ lie on sides $AB$ and $AC$, respectively, such that $MP \parallel AC$ and $NP \parallel AB$. Point $P$ is reflected across $MN$ to point $Q$. Show that triangle $QMB$ is similar to triangle $CNQ$. [i]Brian Hamrick.[/i]

1995 Czech and Slovak Match, 5

Tags: convex quadrilateral , diagonals , reflection , Concyclic , orthogonal , geometry

The diagonals of a convex quadrilateral $ABCD$ are orthogonal and intersect at point $E$. Prove that the reflections of $E$ in the sides of quadrilateral $ABCD$ lie on a circle.

2008 All-Russian Olympiad, 1

Tags: algebra , polynomial , algebra proposed

Numbers $ a,b,c$ are such that the equation $ x^3 \plus{} ax^2 \plus{} bx \plus{} c$ has three real roots.Prove that if $ \minus{} 2\leq a \plus{} b \plus{} c\leq 0$,then at least one of these roots belongs to the segment $ [0,2]$

VMEO IV 2015, 10.3

Tags: number theory , divides , divisible

Given a positive integer $k$. Find the condition of positive integer $m$ over $k$ such that there exists only one positive integer $n$ satisfying $$n^m | 5^{n^k} + 1,$$

Novosibirsk Oral Geo Oly VIII, 2019.5

Tags: geometry , geometric inequality

Two turtles, the leader and the slave, are crawling along the plane from point $A$ to point $B$. They crawl in turn: first the leader crawls some distance, then the slave crawls some distance in a straight line towards the leading one. Then the leader crawls somewhere again, after which the slave crawls towards the leader, etc. Finally, they both crawl to $B$. Prove that the slave turtle crawled no more than the leading one.

2011 All-Russian Olympiad, 1

Tags: inequalities , number theory , relatively prime , number theory unsolved

Two natural numbers $d$ and $d'$, where $d'>d$, are both divisors of $n$. Prove that $d'>d+\frac{d^2}{n}$.

2009 IMC, 3

Tags: linear algebra , matrix , algebra , polynomial , IMC , college contests

Let $A,B\in \mathcal{M}_n(\mathbb{C})$ be two $n \times n$ matrices such that \[ A^2B+BA^2=2ABA \] Prove there exists $k\in \mathbb{N}$ such that \[ (AB-BA)^k=\mathbf{0}_n\] Here $\mathbf{0}_n$ is the null matrix of order $n$.

2021 AMC 12/AHSME Spring, 1

Tags: AMC , AMC 12 , AMC 12 A

What is the value of $$2^{1+2+3}-(2^1+2^2+2^3)?$$ $\textbf{(A) }0 \qquad \textbf{(B) }50 \qquad \textbf{(C) }52 \qquad \textbf{(D) }54 \qquad \textbf{(E) }57$ Proposed by [b]djmathman[/b]

2006 Moldova Team Selection Test, 2

Tags: combinatorics proposed , combinatorics

Let $n\in N$ $n\geq2$ and the set $X$ with $n+1$ elements. The ordered sequences $(a_{1}, a_{2},\ldots,a_{n})$ and $(b_{1},b_{2},\ldots b_{n})$ of distinct elements of $X$ are said to be $\textit{separated}$ if there exists $i\neq j$ such that $a_{i}=b_{j}$. Determine the maximal number of ordered sequences of $n$ elements from $X$ such that any two of them are $\textit{separated}$. Note: ordered means that, for example $(1,2,3)\neq(2,3,1)$.

May Olympiad L2 - geometry, 2015.5

Tags: geometry , combinatorics

If you have $65$ points in a plane, we will make the lines that passes by any two points in this plane and we obtain exactly $2015$ distinct lines, prove that least $4$ points are collinears!!

III Soros Olympiad 1996 - 97 (Russia), 11.7

Tags: geometry , tangent circles

On the plane there are two circles $a$ and $b$ and a line $\ell$ perpendicular to the line passing through the centers of these circles. It is known that there are $4$ unequal circles, each of which touches $a$, $b$ and $\ell$. Find the radius of the smallest of these four circles if the radii of the other three are $2$, $3$ and $6$. Also find the ratio of the radii of the circles $a$ and $b$.

2004 Putnam, B2

Tags: Putnam , inequalities , probability , LaTeX , function , combinatorics , logarithms

Let $m$ and $n$ be positive integers. Show that $\frac{(m+n)!}{(m+n)^{m+n}} < \frac{m!}{m^m}\cdot\frac{n!}{n^n}$

1998 Polish MO Finals, 3

Tags: analytic geometry , invariant , symmetry , combinatorics unsolved , combinatorics

$S$ is a board containing all unit squares in the $xy$ plane whose vertices have integer coordinates and which lie entirely inside the circle $x^2 + y^2 = 1998^2$. In each square of $S$ is written $+1$. An allowed move is to change the sign of every square in $S$ in a given row, column or diagonal. Can we end up with exactly one $-1$ and $+1$ on the rest squares by a sequence of allowed moves?

2019 Latvia Baltic Way TST, 15

Tags: number theory

Determine all tuples of integers $(a,b,c)$ such that: $$(a-b)^3(a+b)^2 = c^2 + 2(a-b) + 1$$

2024 Malaysian APMO Camp Selection Test, 5

Tags: geometry

Let $ABC$ be a scalene triangle and $D$ be the feet of altitude from $A$ to $BC$. Let $I_1$, $I_2$ be incenters of triangles $ABD$ and $ACD$ respectively, and let $H_1$, $H_2$ be orthocenters of triangles $ABI_1$ and $ACI_2$ respectively. The circles $(AI_1H_1)$ and $(AI_2H_2)$ meet again at $X$. The lines $AH_1$ and $XI_1$ meet at $Y$, and the lines $AH_2$ and $XI_2$ meet at $Z$. Suppose the external common tangents of circles $(BI_1H_1)$ and $(CI_2H_2)$ meet at $U$. Prove that $UY=UZ$. [i]Proposed by Ivan Chan Kai Chin[/i]

2005 AMC 10, 3

Tags:

A gallon of paint is used to paint a room. One third of the paint is used on the first day. On the second day, one third of the remaining paint is used. What fraction of the original amount of paint is available to use on the third day? $ \textbf{(A)}\ \frac{1}{10}\qquad \textbf{(B)}\ \frac{1}{9}\qquad \textbf{(C)}\ \frac{1}{3}\qquad \textbf{(D)}\ \frac{4}{9}\qquad \textbf{(E)}\ \frac{5}{9}$

2010 LMT, 21

Tags:

Jae and Yoon are playing SunCraft. The probability that Jae wins the $n$-th game is $\frac{1}{n+2}.$ What is the probability that Yoon wins the first six games, assuming there are no ties?