This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 85335

2016 Regional Competition For Advanced Students, 3
Tags: game theory
On the occasion of the 47th Mathematical Olympiad 2016 the numbers 47 and 2016 are written on the blackboard. Alice and Bob play the following game: Alice begins and in turns they choose two numbers $a$ and $b$ with $a > b$ written on the blackboard, whose difference $a-b$ is not yet written on the blackboard and write this difference additionally on the board. The game ends when no further move is possible. The winner is the player who made the last move. Prove that Bob wins, no matter how they play. (Richard Henner)

1954 Putnam, B2
Tags: series , harmonic series , limit
Let $s$ denote the sum of the alternating harmonic series. Rearrange this series as follows $$1 + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} +\frac{1}{7} - \frac{1}{4} + \frac{1}{9} + \frac{1}{11} - \ldots$$ Assume as known that this series converges as well and denote its sum by $S$. Denote by $s_k, S_k$ respectively the $k$-th partial sums of both series. Prove that $$ \!\!\!\! \text{i})\; S_{3n} = s_{4n} +\frac{1}{2} s_{2n}.$$ $$ \text{ii}) \; S\ne s.$$

2015 Balkan MO Shortlist, C2
Tags: combinatorics , game , game strategy
Isaak and Jeremy play the following game. Isaak says to Jeremy that he thinks a few $2^n$ integers $k_1,..,k_{2^n}$. Jeremy asks questions of the form: ''Is it true that $k_i<k_j$ ?'' in which Isaak answers by always telling the truth. After $n2^{n-1}$ questions, Jeramy must decide whether numbers of Isaak are all distinct each other or not. Prove that Jeremy has bo way to be ''sure'' for his final decision. (UK)

2015 NZMOC Camp Selection Problems, 4
Tags: number theory , diophantine equation , diophantine
For which positive integers $m$ does the equation: $$(ab)^{2015} = (a^2 + b^2)^m$$ have positive integer solutions?

1981 IMO Shortlist, 16
Tags: function , limit , algebra , sequence , recurrence relation
A sequence of real numbers $u_1, u_2, u_3, \dots$ is determined by $u_1$ and the following recurrence relation for $n \geq 1$: \[4u_{n+1} = \sqrt[3]{ 64u_n + 15.}\] Describe, with proof, the behavior of $u_n$ as $n \to \infty.$

2017 Romania Team Selection Test, P2
Tags: algebra
Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that \[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]

1972 Spain Mathematical Olympiad, 7
Tags: number theory , multiple , divide , divisible
Prove that for every positive integer $n$, the number $$A_n = 5^n + 2 \cdot 3^{n-1} + 1$$ is a multiple of $8$.

2009 Grand Duchy of Lithuania, 1
Tags: number theory , multiple , sum of digits
The natural number $N$ is a multiple of $2009$ and the sum of its (decimal) digits equals $2009$. (a) Find one such number. (b) Find the smallest such number.

1949 Moscow Mathematical Olympiad, 159
Tags: disc , geometry , perimeter , combinatorial geometry
Consider a closed broken line of perimeter $1$ on a plane. Prove that a disc of radius $\frac14$ can cover this line.

2021 Caucasus Mathematical Olympiad, 5
Tags: number theory
Let $a, b, c$ be positive integers such that the product $$\gcd(a,b) \cdot \gcd(b,c) \cdot \gcd(c,a) $$ is a perfect square. Prove that the product $$\operatorname{lcm}(a,b) \cdot \operatorname{lcm}(b,c) \cdot \operatorname{lcm}(c,a) $$ is also a perfect square.

2020 Ukrainian Geometry Olympiad - December, 4
Tags: geometry , isosceles , equal segments , angle
In an isosceles triangle $ABC$ with an angle $\angle A= 20^o$ and base $BC=12$ point $E$ on the side $AC$ is chosen such that $\angle ABE= 30^o$ , and point $F$ on the side $AB$ such that $EF = FC$ . Find the length of $FC$.

2015 Indonesia MO, 5
Tags: divisibility , number theory
Given positive integers $a,b,c,d$ such that $a\mid c^d$ and $b\mid d^c$. Prove that \[ ab\mid (cd)^{max(a,b)} \]

Ukrainian From Tasks to Tasks - geometry, 2015.10
Tags: geometry , perimeter
Can the sum of the lengths of the median, angle bisector and altitude of a triangle be equal to its perimeter if a) these segments are drawn from three different vertices? b) these segments are drawn from one vertex?

KoMaL A Problems 2021/2022, A. 816
Tags: combinatorics
Peter has $2022$ pieces of magnetic railroad cars, which are of two types: some have the front with north and the rear with south magnetic polarity, and some have the rear with north and the rear with south magnetic polarity (on these railroad cars the front and the rear can be distinguished). Peter wants to decide whether there is the same number of both types of cars. He can try to fit together two cars in one try. What is the least number of tries needed? [i]Proposed by Dömötör Pálvölgyi, Budapest[/i]

1995 ITAMO, 4
Tags: geometry , circumcircle , equal segments , angle bisector
An acute-angled triangle $ABC$ is inscribed in a circle with center $O$. The bisector of $\angle A$ meets $BC$ at $D$, and the perpendicular to $AO$ through $D$ meets the segment $AC$ in a point $P$. Show that $AB = AP$.

1956 AMC 12/AHSME, 17
Tags:
The fraction $ \frac {5x \minus{} 11}{2x^2 \plus{} x \minus{} 6}$ was obtained by adding the two fractions $ \frac {A}{x \plus{} 2}$ and $ \frac {B}{2x \minus{} 3}$. The values of $ A$ and $ B$ must be, respectively: $ \textbf{(A)}\ 5x, \minus{} 11 \qquad\textbf{(B)}\ \minus{} 11,5x \qquad\textbf{(C)}\ \minus{} 1,3 \qquad\textbf{(D)}\ 3, \minus{} 1 \qquad\textbf{(E)}\ 5, \minus{} 11$

2011 ELMO Shortlist, 2
Tags: modular arithmetic , number theory
Let $p\ge5$ be a prime. Show that \[\sum_{k=0}^{(p-1)/2}\binom{p}{k}3^k\equiv 2^p - 1\pmod{p^2}.\] [i]Victor Wang.[/i]

2024 Mathematical Talent Reward Programme, 1
Tags: algebra
Hari the milkman delivers milk to his customers everyday by travelling on his cycle. Each litre of milk costs him Rs. $20$, and he sells it at Rs. $24$. One day while riding his cycle with $20$L, Hari trips and loses $5$L of it, and he decides to mix some water with the rest of the milk. His customers can detect if the milk is more than $10$% impure ($1$L water in $10$L misture). Given that he doesn't wish to make his customers angry, what is his maximum profit for the day? $(A)$ Rs $12$ profit $(B)$ Rs $24$ profit $(C)$ No profit $(D)$ Rs $12$ loss

2003 Gheorghe Vranceanu, 1
Tags: floor function , algebra , equation
Solve in $ \mathbb{R}^2 $ the equation $ \lfloor x/y-y/x \rfloor =x^2/y+y/x^2. $

2017 CMIMC Number Theory, 4
Tags: number theory
Let $a_1, a_2, a_3, a_4, a_5$ be positive integers such that $a_1, a_2, a_3$ and $a_3, a_4, a_5$ are both geometric sequences and $a_1, a_3, a_5$ is an arithmetic sequence. If $a_3 = 1575$, find all possible values of $\vert a_4 - a_2 \vert$.

2009 Baltic Way, 20
Tags: analytic geometry , graphing lines , slope , combinatorics
In the future city Baltic Way there are sixteen hospitals. Every night exactly four of them must be on duty for emergencies. Is it possible to arrange the schedule in such a way that after twenty nights every pair of hospitals have been on common duty exactly once?

1999 Akdeniz University MO, 3
Tags: algebra , inequalities
Let $a$,$b$,$c$ and $d$ positive reals. Prove that $$\frac{1}{a+b+c+d} \leq \frac{1}{64}(\frac{1}{a}+\frac{1}{b}+\frac{4}{c}+\frac{16}{d})$$

1977 Chisinau City MO, 138
Tags: geometry , isosceles , angle
In an isosceles triangle $BAC$ ($| AC | = | AB |$) , point $D$ is marked on the side $AC$. Determine the angles of the triangle $BDC$ if $\angle A = 40^o$ and $|BC|: |AD|= \sqrt3$.

2000 IMC, 6
Tags: linear algebra , matrix , algebra , polynomial
Let $A$ be a real $n\times n$ Matrix and define $e^{A}=\sum_{k=0}^{\infty} \frac{A^{k}}{k!}$ Prove or disprove that for any real polynomial $P(x)$ and any real matrices $A,B$, $P(e^{AB})$ is nilpotent if and only if $P(e^{BA})$ is nilpotent.

2018 BMT Spring, 6
Tags:
Let $x,y,z \in \mathbb{R}$ and $7x^2 + 7y^2 + 7z^2 + 9xyz = 12$. The minimum value of $x^2 + y^2 + z^2$ can be expressed as $\dfrac{a}{b}$ where $a,b \in \mathbb{Z}, \gcd(a,b) = 1$. What is $a + b$?