Olympiad problems

2011 Gheorghe Vranceanu, 1

Tags: circumcircle , geometry

2017 Hanoi Open Mathematics Competitions, 5

Tags: number theory , maximum , digit

Let $a, b, c$ be two-digit, three-digit, and four-digit numbers, respectively. Assume that the sum of all digits of number $a+b$, and the sum of all digits of $b + c$ are all equal to $2$. The largest value of $a + b + c$ is (A): $1099$ (B): $2099$ (C): $1199$ (D): $2199$ (E): None of the above.

1979 VTRMC, 4

Tags: real analysis

Let $f(x)$ be continuously differentiable on $(0,\infty)$ and suppose $ \lim _ { x \rightarrow \infty } f ^ { \prime } ( x ) = 0 $. Prove that $ \lim _ { x \rightarrow \infty } f ( x ) / x = 0 $.

2019 Iran MO (2nd Round), 6

Tags: graph theory , combinatorics

Consider lattice points of a $6*7$ grid.We start with two points $A,B$.We say two points $X,Y$ connected if one can reflect several times WRT points $A,B$ and reach from $X$ to $Y$.Over all choices of $A,B$ what is the minimum number of connected components?

1976 IMO Shortlist, 10

Tags: number theory , product , maximization

Determine the greatest number, who is the product of some positive integers, and the sum of these numbers is $1976.$

2001 Baltic Way, 6

Tags: parallelogram , geometry

The points $A, B, C, D, E$ lie on the circle $c$ in this order and satisfy $AB\parallel EC$ and $AC\parallel ED$. The line tangent to the circle $c$ at $E$ meets the line $AB$ at $P$. The lines $BD$ and $EC$ meet at $Q$. Prove that $|AC|=|PQ|$.

2016 Ukraine Team Selection Test, 11

Tags: geometry

Let $ABC$ be a triangle with $\angle{C} = 90^{\circ}$, and let $H$ be the foot of the altitude from $C$. A point $D$ is chosen inside the triangle $CBH$ so that $CH$ bisects $AD$. Let $P$ be the intersection point of the lines $BD$ and $CH$. Let $\omega$ be the semicircle with diameter $BD$ that meets the segment $CB$ at an interior point. A line through $P$ is tangent to $\omega$ at $Q$. Prove that the lines $CQ$ and $AD$ meet on $\omega$.

2005 CentroAmerican, 2

Tags: number theory

Show that the equation $a^{2}b^{2}+b^{2}c^{2}+3b^{2}-c^{2}-a^{2}=2005$ has no integer solutions. [i]Arnoldo Aguilar, El Salvador[/i]

2011 National Olympiad First Round, 26

Tags: modular arithmetic

The integers $0 \leq a < 2^{2008}$ and $0 \leq b < 8$ satisfy the equivalence $7(a+2^{2008}b) \equiv 1 \pmod{2^{2011}}$. Then $b$ is $\textbf{(A)}\ 3 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ \text{None}$

2000 May Olympiad, 2

Tags: geometry

Given a parallelogram with area $1$ and we will construct lines where this lines connect a vertex with a midpoint of the side no adjacent to this vertex; with the $8$ lines formed we have a octagon inside of the parallelogram. Determine the area of this octagon

2019 Sharygin Geometry Olympiad, 20

Tags: geometry

Let $O$ be the circumcenter of triangle ABC, $H$ be its orthocenter, and $M$ be the midpoint of $AB$. The line $MH$ meets the line passing through $O$ and parallel to $AB$ at point $K$ lying on the circumcircle of $ABC$. Let $P$ be the projection of $K$ onto $AC$. Prove that $PH \parallel BC$.

1993 All-Russian Olympiad Regional Round, 10.7

Tags: parallelogram , ratio , geometry

Points $ M,N$ are taken on sides $ BC,CD$ respectively of parallelogram $ ABCD$. Let $ E\equal{}BD\cap AM, F\equal{}BD\cap AN$. Diagonal $ BD$ cuts triangle $ AMN$ into two parts. Prove that these two parts have equal area if and only if the point $ K$ given by $ EK\parallel{}AD, FK\parallel{}AB$ lies on segment $ MN$.

2021-IMOC, A9

Tags: sum , algebra

For a given positive integer $n,$ find $$\sum_{k=0}^{n} \left(\frac{\binom{n}{k} \cdot (-1)^k}{(n+1-k)^2} - \frac{(-1)^n}{(k+1)(n+1)}\right).$$

2016 Purple Comet Problems, 20

Tags:

The 24 unshaded squares in the 5 × 5 grid below can be tiled with twelve 1 × 2 tiles. One such tiling is shown. Find the number of ways the grid can be tiled. [center][img]https://snag.gy/KMoPrF.jpg[/img][/center]

2015 İberoAmerican, 1

Tags: number theory

The number $125$ can be written as a sum of some pairwise coprime integers larger than $1$. Determine the largest number of terms that the sum may have.

1985 Austrian-Polish Competition, 1

Tags: inequalities , algebra

Show that if $a+b+c=0$ then $(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b})(\frac{b-c}{a}+\frac{c-a}{b}+\frac{a-b}{c})=9$.

2002 Croatia Team Selection Test, 1

Tags: combinatorics , max

In a certain language there are $n$ letters. A sequence of letters is a word, if there are no two equal letters between two other equal letters. Find the number of words of the maximum length.

2013 Princeton University Math Competition, 1

Tags: inequalities , function , calculus , integration , derivative

Prove that \[ \frac{1}{a^2+2} + \frac{1}{b^2+2} + \frac{1}{c^2+2} \le \frac{1}{6ab+c^2} + \frac{1}{6bc+a^2} + \frac{1}{6ca+b^2} \] for all positive real numbers $a$, $b$ and $c$ satisfying $a^2+b^2+c^2=1$.

2002 Belarusian National Olympiad, 7

Tags: combinatorics

Several clocks lie on the table. It is known that at some moment the sum of distances between a point $X$ of the table and the ends of their minute hands is not equal to the sum of distances between $X$ and the ends of their hour hands. Prove that there is a moment when the sum of distances between $X$ and the ends of their minute hands is greater than the sum of distances between $X$ and the ends of their hour hands. (E. Barabanov, I. Voronovich)

2022 Lusophon Mathematical Olympiad, 5

Tags: geometry

Tow circumferences of radius $R_1$ and $R_2$ are tangent externally between each other. Besides that, they are both tangent to a semicircle with radius of 1, as shown in the figure. (Diagram is in the attachment) a) If $A_1$ and $A_2$ are the tangency points of the two circumferences with the diameter of the semicircle, find the length of $\overline{A_1 A_2}$. b) Prove that $R_{1}+R_{2}=2\sqrt{R_{1}R_{2}}(\sqrt{2}-\sqrt{R_{1}R_{2}})$.

2023 Iran MO (3rd Round), 2

Tags: function , algebra

find all $f : \mathbb{C} \to \mathbb{C}$ st: $$f(f(x)+yf(y))=x+|y|^2$$ for all $x,y \in \mathbb{C}$

2020 Jozsef Wildt International Math Competition, W33

Tags: calculus , integration , function

Let $p\in\mathbb N,f:[0,1]\to(0,\infty)$ be a continuous function and $$a_n=\int^1_0x^p\sqrt[n]{f(x)}dx,n\in\mathbb N,n\ge2.$$ Demonstrate that: a) $\lim_{n\to\infty}a_n=\frac1{p+1}$ b) $\lim_{n\to\infty}((p+1)a_n)^n=\exp\left((p+1)\int^1_0x^p\ln f(x)dx\right)$ [i]Proposed by Nicolae Papacu[/i]

2017 Harvard-MIT Mathematics Tournament, 4

Tags: number theory

Find all pairs $(a,b)$ of positive integers such that $a^{2017}+b$ is a multiple of $ab$.

2012 BAMO, 5

Tags: relative prime , polynomial , integer polynomial

Find all nonzero polynomials $P(x)$ with integers coefficients that satisfy the following property: whenever $a$ and $b$ are relatively prime integers, then $P(a)$ and $P(b)$ are relatively prime as well. Prove that your answer is correct. (Two integers are [b]relatively prime[/b] if they have no common prime factors. For example, $-70$ and $99$ are relatively prime, while $-70$ and $15$ are not relatively prime.)

1973 All Soviet Union Mathematical Olympiad, 177

Tags: geometry , equal segments

Given an angle with the vertex $O$ and a circle touching its sides in the points $A$ and $B$. A ray is drawn from the point $A$ parallel to $[OB)$. It intersects with the circumference in the point $C$. The segment $[OC]$ intersects the circumference in the point $E$. The straight lines $(AE)$ and $(OB)$ intersect in the point $K$. Prove that $|OK| = |KB|$.