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.

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Found problems: 85335

2017 Canadian Mathematical Olympiad Qualification, 2
Tags: number theory
For any positive integer n, let $\varphi(n)$ be the number of integers in the set $\{1, 2, \ldots , n\}$ whose greatest common divisor with $n$ is 1. Determine the maximum value of $\frac{n}{\varphi(n)}$ for $n$ in the set $\{2, \ldots, 1000\}$ and all values of $n$ for which this maximum is attained.

2015 Paraguayan Mathematical Olympiad, Problem 5
Tags: combinatorics
In the figure, the rectangle is formed by $4$ smaller equal rectangles. If we count the total number of rectangles in the figure we find $10$. How many rectangles in total will there be in a rectangle that is formed by $n$ smaller equal rectangles?

1998 All-Russian Olympiad Regional Round, 9.5
Tags: algebra , trinomial
The roots of the two monic square trinomials are negative integers, and one of these roots is common. Can the values of these trinomials at some positive integer point equal 19 and 98?

2018 Estonia Team Selection Test, 10
Tags: sequence , recurrence relation , algebra , inequalities
A sequence of positive real numbers $a_1, a_2, a_3, ... $ satisfies $a_n = a_{n-1} + a_{n-2}$ for all $n \ge 3$. A sequence $b_1, b_2, b_3, ...$ is defined by equations $b_1 = a_1$ , $b_n = a_n + (b_1 + b_3 + ...+ b_{n-1})$ for even $n > 1$ , $b_n = a_n + (b_2 + b_4 + ... +b_{n-1})$ for odd $n > 1$. Prove that if $n\ge 3$, then $\frac13 < \frac{b_n}{n \cdot a_n} < 1$

MOAA Team Rounds, 2018.9
Tags: geometry , team
Quadrilateral $ABCD$ with $AC = 800$ is inscribed in a circle, and $E, W, X, Y, Z$ are the midpoints of segments $BD$, $AB$, $BC$, $CD$, $DA$, respectively. If the circumcenters of $EW Z$ and $EXY$ are $O_1$ and $O_2$, respectively, determine $O_1O_2$.

1987 Mexico National Olympiad, 5
Tags: rectangle , right triangle , area , geometry
In a right triangle $ABC$, M is a point on the hypotenuse $BC$ and $P$ and $Q$ the projections of $M$ on $AB$ and $AC$ respectively. Prove that for no such point $M$ do the triangles $BPM, MQC$ and the rectangle $AQMP$ have the same area.

2005 Morocco TST, 2
Tags: combinatorics
Consider the set $A=\{1,2,...,49\}$. We partitionate $A$ into three subsets. Prove that there exist a set from these subsets containing three distincts elements $a,b,c$ such that $a+b=c$

2016 Germany National Olympiad (4th Round), 4
Tags: binomial coefficient
Find all positive integers $m,n$ with $m \leq 2n$ that solve the equation \[ m \cdot \binom{2n}{n} = \binom{m^2}{2}. \] [i](German MO 2016 - Problem 4)[/i]

2021 XVII International Zhautykov Olympiad, #2
Tags: geometry , hexagon
In a convex cyclic hexagon $ABCDEF$, $BC=EF$ and $CD=AF$. Diagonals $AC$ and $BF$ intersect at point $Q,$ and diagonals $EC$ and $DF$ intersect at point $P.$ Points $R$ and $S$ are marked on the segments $DF$ and $BF$ respectively so that $FR=PD$ and $BQ=FS.$ [b]The segments[/b] $RQ$ and $PS$ intersect at point $T.$ Prove that the line $TC$ bisects the diagonal $DB$.

2006 Turkey MO (2nd round), 3
Tags: polynomial , number theory , algebra
Find all positive integers $n$ for which all coefficients of polynomial $P(x)$ are divisible by $7,$ where \[P(x) = (x^2 + x + 1)^n - (x^2 + 1)^n - (x + 1)^n - (x^2 + x)^n + x^{2n} + x^n + 1.\]

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 $AB = AP$.