Olympiad problems

2019 HMNT, 1

Tags: combinatorics , geometry

Dylan has a $100\times 100$ square, and wants to cut it into pieces of area at least $1$. Each cut must be a straight line (not a line segment) and must intersect the interior of the square. What is the largest number of cuts he can make?

2021 MOAA, 9

Tags:

William is biking from his home to his school and back, using the same route. When he travels to school, there is an initial $20^\circ$ incline for $0.5$ kilometers, a flat area for $2$ kilometers, and a $20^\circ$ decline for $1$ kilometer. If William travels at $8$ kilometers per hour during uphill $20^\circ$ sections, $16$ kilometers per hours during flat sections, and $20$ kilometers per hour during downhill $20^\circ$ sections, find the closest integer to the number of minutes it take William to get to school and back. [i]Proposed by William Yue[/i]

2024 Iran Team Selection Test, 4

Tags: algebra

Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that for any real numbers $x , y$ this equality holds : $$f(yf(x)+f(x)f(y))=xf(y)+f(xy)$$ [i]Proposed by Navid Safaei[/i]

1996 Cono Sur Olympiad, 2

Tags: sequence

Consider a sequence of real numbers defined by: $a_{n + 1} = a_n + \frac{1}{a_n}$ for $n = 0, 1, 2, ...$ Prove that, for any positive real number $a_0$, is true that $a_{1996}$ is greater than $63$.

2023 Romania National Olympiad, 2

Tags: algebra , sum of digits

We say that a natural number is called special if all of its digits are non-zero and any two adjacent digits in its decimal representation are consecutive (not necessarily in ascending order). a) Determine the largest special number $m$ whose sum of digits is equal to $2023$. b) Determine the smallest special number $n$ whose sum of digits is equal to $2022$.

2015 FYROM JBMO Team Selection Test, 3

Tags: inequalities , 3-variable inequality

Let $a, b$ and $c$ be positive real numbers. Prove that $\prod_{cyc}(16a^2+8b+17)\geq2^{12}\prod_{cyc}(a+1)$.

2013 Dutch Mathematical Olympiad, 5

Tags: digit , sum , number theory

The number $S$ is the result of the following sum: $1 + 10 + 19 + 28 + 37 +...+ 10^{2013}$ If one writes down the number $S$, how often does the digit `$5$' occur in the result?

1986 Bulgaria National Olympiad, Problem 6

Tags: inequalities , recurrence relation , algebra , sequence

Let $0<k<1$ be a given real number and let $(a_n)_{n\ge1}$ be an infinite sequence of real numbers which satisfies $a_{n+1}\le\left(1+\frac kn\right)a_n-1$. Prove that there is an index $t$ such that $a_t<0$.

KoMaL A Problems 2023/2024, A. 882

Tags: number theory , gcd , greatest common divisor

Let $H_1, H_2,\ldots, H_m$ be non-empty subsets of the positive integers, and let $S$ denote their union. Prove that \[\sum_{i=1}^m \sum_{(a,b)\in H_i^2}\gcd(a,b)\ge\frac1m \sum_{(a,b)\in S^2}\gcd(a,b).\] [i]Proposed by Dávid Matolcsi, Berkeley[/i]

2010 Baltic Way, 12

Tags: arithmetic sequence , geometry

Let $ABCD$ be a convex quadrilateral with precisely one pair of parallel sides. $(a)$ Show that the lengths of its sides $AB,BC,CD, DA$ (in this order) do not form an arithmetic progression. $(b)$ Show that there is such a quadrilateral for which the lengths of its sides $AB ,BC,CD,DA$ form an arithmetic progression after the order of the lengths is changed.

2020 Taiwan TST Round 1, 3

Tags: number theory

Let $N>2^{5000}$ be a positive integer. Prove that if $1\leq a_1<\cdots<a_k<100$ are distinct positive integers then the number \[\prod_{i=1}^{k}\left(N^{a_i}+a_i\right)\] has at least $k$ distinct prime factors. Note. Results with $2^{5000}$ replaced by some other constant $N_0$ will be awarded points depending on the value of $N_0$. [i]Proposed by Evan Chen[/i]

1996 Putnam, 1

Tags:

Define a $\emph{selfish}$ set to be a set which has its own cardinality as its element. And a set is a $\emph{minimal }\text{ selfish}$ set if none of its proper subsets are $\emph{selfish}$. Find with proof the number of $\text{minimal selfish}$ subsets of $\{1,2,\cdots ,n\}$.

2014 Romania National Olympiad, 4

Tags: geometry

Let $ ABCD $ be a quadrilateral inscribed in a circle of diameter $ AC. $ Fix points $ E,F $ of segments $ CD, $ respectively, $ BC $ such that $ AE $ is perpendicular to $ DF $ and $ AF $ is perpendicular to $ BE. $ Show that $ AB=AD. $

1994 AIME Problems, 4

Tags: floor function , calculus , derivative , geometric series , logarithm

Find the positive integer $n$ for which \[ \lfloor \log_2{1}\rfloor+\lfloor\log_2{2}\rfloor+\lfloor\log_2{3}\rfloor+\cdots+\lfloor\log_2{n}\rfloor=1994. \] (For real $x$, $\lfloor x\rfloor$ is the greatest integer $\le x.$)

II Soros Olympiad 1995 - 96 (Russia), 11.6

Tags: perfect cube , number theory

For what natural number $x$ will the value of the polynomial $x^3+7x^2+6x+1$ be the cube of a natural number?

1972 Bundeswettbewerb Mathematik, 2

Tags: number theory

Prove: out of $ 79$ consecutive positive integers, one can find at least one whose sum of digits is divisible by $ 13$. Show that this isn't true for $ 78$ consecutive integers.

MathLinks Contest 3rd, 2

Tags: inequalities , algebra

Prove that for all positive reals $a, b, c$ the following double inequality holds: $$\frac{a+b+c}{3}\ge \sqrt[3]{\frac{(a+b)(b+c)(c+a)}{8}}\ge \frac{\sqrt{ab}+\sqrt{bc}\sqrt{ca}}{3}$$

1985 IMO Longlists, 78

Tags: function , recurrence relation , fixed point , calculus , limit

The sequence $f_1, f_2, \cdots, f_n, \cdots $ of functions is defined for $x > 0$ recursively by \[f_1(x)=x , \quad f_{n+1}(x) = f_n(x) \left(f_n(x) + \frac 1n \right)\] Prove that there exists one and only one positive number $a$ such that $0 < f_n(a) < f_{n+1}(a) < 1$ for all integers $n \geq 1.$

2014 ELMO Shortlist, 8

Tags: function , inequalities

Let $a, b, c$ be positive reals with $a^{2014}+b^{2014}+c^{2014}+abc=4$. Prove that \[ \frac{a^{2013}+b^{2013}-c}{c^{2013}} + \frac{b^{2013}+c^{2013}-a}{a^{2013}} + \frac{c^{2013}+a^{2013}-b}{b^{2013}} \ge a^{2012}+b^{2012}+c^{2012}. \][i]Proposed by David Stoner[/i]

2012 Online Math Open Problems, 37

Tags: geometry , angle bisector

In triangle $ABC$, $AB = 1$ and $AC = 2$. Suppose there exists a point $P$ in the interior of triangle $ABC$ such that $\angle PBC = 70^{\circ}$, and that there are points $E$ and $D$ on segments $AB$ and $AC$, such that $\angle BPE = \angle EPA = 75^{\circ}$ and $\angle APD = \angle DPC = 60^{\circ}$. Let $BD$ meet $CE$ at $Q,$ and let $AQ$ meet $BC$ at $F.$ If $M$ is the midpoint of $BC$, compute the degree measure of $\angle MPF.$ [i]Authors: Alex Zhu and Ray Li[/i]

2007 Pre-Preparation Course Examination, 13

Tags: number theory

Let $\{a_i\}_{i=1}^{\infty}$ be a sequence of positive integers such that $a_1<a_2<a_3\cdots$ and all of primes are members of this sequence. Prove that for every $n<m$ \[\dfrac{1}{a_n} + \dfrac{1}{a_{n+1}} + \cdots + \dfrac{1}{a_m} \not \in \mathbb N\]

2024 Harvard-MIT Mathematics Tournament, 7

Tags:

Let $ABCDEF$ be a regular hexagon with $P$ as a point in its interior. Prove that of the three values $\tan \angle APD$, $\tan \angle BPE$ and $\tan \angle CPF$, two of them sum to the third one.

2013 Kurschak Competition, 2

Tags: geometry

Consider the closed polygonal discs $P_1$, $P_2$, $P_3$ with the property that for any three points $A\in P_1$, $B\in P_2$, $C\in P_3$, we have $[\triangle ABC]\le 1$. (Here $[X]$ denotes the area of polygon $X$.) (a) Prove that $\min\{[P_1],[P_2],[P_3]\}<4$. (b) Give an example of polygons $P_1,P_2,P_3$ with the above property such that $[P_1]>4$ and $[P_2]>4$.

2009 Sharygin Geometry Olympiad, 21

Tags: parallelogram , geometry

The opposite sidelines of quadrilateral $ ABCD$ intersect at points $ P$ and $ Q$. Two lines passing through these points meet the side of $ ABCD$ in four points which are the vertices of a parallelogram. Prove that the center of this parallelogram lies on the line passing through the midpoints of diagonals of $ ABCD$.

Kvant 2024, M2817

Tags: geometry , circumcircle

We are given fixed circles $\Omega$ and $\omega$ such that there exists a hexagon $ABCDEF$ inscribed in $\Omega$ and circumscribed around $\omega$. (Note that then, by virtue of Poncelet's theorem, there is an infinite family of such hexagons.) Prove that the value of $\dfrac{S_{ABCDEF}}{AD+BE+CF}$ it does not depend on the choice of the hexagon $ABCDEF$. [i]A. Zaslavsky and Tran Quang Hung[/i]