Olympiad problems

2016 Bundeswettbewerb Mathematik, 1

There are $\tfrac{n(n+1)}{2}$ distinct sums of two distinct numbers, if there are $n$ numbers. For which $n \ (n \geq 3)$ do there exist $n$ distinct integers, such that those sums are $\tfrac{n(n-1)}{2}$ consecutive numbers?

2023 Chile National Olympiad, 5

Tags: number theory

What is the smallest positive integer that is divisible by $225$ and that has ony the numbers one and zero as digits?

2019 New Zealand MO, 4

Tags: number theory , perfect square

Show that for all positive integers $k$, there exists a positive integer n such that $n2^k -7$ is a perfect square.

1973 Bulgaria National Olympiad, Problem 2

Tags: arithmetic sequence , algebra , sequence , geometric sequence

Let the numbers $a_1,a_2,a_3,a_4$ form an arithmetic progression with difference $d\ne0$. Prove that there are no exists geometric progressions $b_1,b_2,b_3,b_4$ and $c_1,c_2,c_3,c_4$ such that: $$a_1=b_1+c_1,a_2=b_2+c_2,a_3=b_3+c_3,a_4=b_4+c_4.$$

2014 Contests, 1

Tags: number theory

Find with proof all positive $3$ digit integers $\overline{abc}$ satisfying \[ b\cdot \overline{ac}=c \cdot \overline{ab} +10 \]

1987 All Soviet Union Mathematical Olympiad, 448

Tags: geometry , combinatorics , combinatorial geometry , odd , convex , broken line

Given two closed broken lines in the plane with odd numbers of edges. All the lines, containing those edges are different, and not a triple of them intersects in one point. Prove that it is possible to chose one edge from each line such, that the chosen edges will be the opposite sides of a convex quadrangle.

1974 Chisinau City MO, 74

Tags: algebra , cubic , parameter

Solve the equation: $x^3-2ax^2+(a^2-2\sqrt2 a -6)x + 2\sqrt2 a^2+ 8a + 4\sqrt2 =0$

2024 Harvard-MIT Mathematics Tournament, 15

Tags: guts

Let $a \star b=ab-2.$ Comute the remainder when $(((579\star569)\star559)\star\cdots\star19)\star9$ is divided by $100.$

2004 France Team Selection Test, 2

Tags: inequalities , geometry , incenter , trigonometry , inradius , perimeter , cyclic quadrilateral

Let $P$, $Q$, and $R$ be the points where the incircle of a triangle $ABC$ touches the sides $AB$, $BC$, and $CA$, respectively. Prove the inequality $\frac{BC} {PQ} + \frac{CA} {QR} + \frac{AB} {RP} \geq 6$.

2022 Bulgarian Spring Math Competition, Problem 12.2

Tags: ratio , pyramid , geometry , 3d geometry

Let $ABCDV$ be a regular quadrangular pyramid with $V$ as the apex. The plane $\lambda$ intersects the $VA$, $VB$, $VC$ and $VD$ at $M$, $N$, $P$, $Q$ respectively. Find $VQ : QD$, if $VM : MA = 2 : 1$, $VN : NB = 1 : 1$ and $VP : PC = 1 : 2$.

2014 Contests, 3

Tags: geometry

In triangle $ABC$, $I$ is the centre of the incircle. There is a circle tangent to $AI$ at $I$ which passes through $B$. This circle intersects $AB$ once more in $P$ and intersects $BC$ once more in $Q$. The line $QI$ intersects $AC$ in $R$. Prove that $|AR|\cdot |BQ|=|P I|^2$

2014 Contests, 3

Tags: combinatorial geometry , combinatorics

We have $4n + 5$ points on the plane, no three of them are collinear. The points are colored with two colors. Prove that from the points we can form $n$ empty triangles (they have no colored points in their interiors) with pairwise disjoint interiors, such that all points occurring as vertices of the $n$ triangles have the same color.

1957 AMC 12/AHSME, 6

Tags: geometry , 3d geometry , prism

An open box is constructed by starting with a rectangular sheet of metal $ 10$ in. by $ 14$ in. and cutting a square of side $ x$ inches from each corner. The resulting projections are folded up and the seams welded. The volume of the resulting box is: $ \textbf{(A)}\ 140x \minus{} 48x^2 \plus{} 4x^3 \qquad \textbf{(B)}\ 140x \plus{} 48x^2 \plus{} 4x^3\qquad \\\textbf{(C)}\ 140x \plus{} 24x^2 \plus{} x^3\qquad \textbf{(D)}\ 140x \minus{} 24x^2 \plus{} x^3\qquad \textbf{(E)}\ \text{none of these}$

2020 AIME Problems, 7

Tags: counting , committees , vandermondeaaaaaaaaaa

A club consisting of $11$ men and $12$ women needs to choose a committee from among its members so that the number of women on the committee is one more than the number of men on the committee. The committee could have as few as $1$ member or as many as $23$ members. Let $N$ be the number of such committees that can be formed. Find the sum of the prime numbers that divide $N$.

2017 India IMO Training Camp, 1

Tags: geometry , incenter

Let $ABC$ be an acute angled triangle with incenter $I$. Line perpendicular to $BI$ at $I$ meets $BA$ and $BC$ at points $P$ and $Q$ respectively. Let $D, E$ be the incenters of $\triangle BIA$ and $\triangle BIC$ respectively. Suppose $D,P,Q,E$ lie on a circle. Prove that $AB=BC$.

1989 Iran MO (2nd round), 2

Tags: polynomial , algebra

Let $n$ be a positive integer. Prove that the polynomial \[P(x)= \frac{x^n}{n!}+\frac{x^{n-1}}{(n-1)!}+...+x+1 \] Does not have any rational root.

2016 NIMO Problems, 7

Tags:

Given two positive integers $m$ and $n$, we say that $m\mid\mid n$ if $m\mid n$ and $\gcd(m,\, n/m)=1$. Compute the smallest integer greater than \[\sum_{d\mid 2016}\sum_{m\mid\mid d}\frac{1}{m}.\] [i]Proposed by Michael Ren[/i]

2015 ASDAN Math Tournament, 3

Tags:

You have a circular necklace with $10$ beads on it, all of which are initially unpainted. You randomly select $5$ of these beads. For each selected bead, you paint that selected bead and the two beads immediately next to it (this means we may paint a bead multiple times). Once you have finished painting, what is the probability that every bead is painted?

2006 Hanoi Open Mathematics Competitions, 9

Tags: algebra , inequalities

Let $x,y,z$ be real numbers such that $x^2+y^2+z^2=1$.Find the largest posible value of $$|x^3+y^3+z^3-xyz|$$

1997 Bundeswettbewerb Mathematik, 2

Tags: number theory , prime number

Find a prime number $p$ such that $\frac{p+1}{2}$ and $\frac{p^2+1}{2}$ are perfect square

2016 Brazil Undergrad MO, 1

Tags: real analysis

Let $(a_n)_{n \geq 1}$ s sequence of reals such that $\sum_{n \geq 1}{\frac{a_n}{n}}$ converges. Show that $\lim_{n \rightarrow \infty}{\frac{1}{n} \cdot \sum_{k=1}^{n}{a_k}} = 0$

1998 Cono Sur Olympiad, 2

Tags: circumcircle , geometry

Let $H$ be the orthocenter of the triangle $ABC$, $M$ is the midpoint of the segment $BC$. Let $X$ be the point of the intersection of the line $HM$ with arc $BC$(without $A$) of the circumcircle of $ABC$, let $Y$ be the point of intersection of the line $BH$ with the circle, show that $XY = BC$.

2005 India IMO Training Camp, 2

Tags: function , number theory , algebra , divisibility , functional equation

Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying \[ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}\] for any two positive integers $ m$ and $ n$. [i]Remark.[/i] The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers: $ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$. By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$). [i]Proposed by Mohsen Jamali, Iran[/i]

2018 Belarusian National Olympiad, 10.3

Tags: number theory

For a fixed integer $n\geqslant2$ consider the sequence $a_k=\text{lcm}(k,k+1,\ldots,k+(n-1))$. Find all $n$ for which the sequence $a_k$ increases starting from some number.

2015 Taiwan TST Round 3, 1

Tags: number theory

For any positive integer $n$, let $a_n=\sum_{k=1}^{\infty}[\frac{n+2^{k-1}}{2^k}]$, where $[x]$ is the largest integer that is equal or less than $x$. Determine the value of $a_{2015}$.