Problems from or inspired by mathematics competitions. Questions regarding mathematics competitions.

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4
votes
2answers
54 views

$k$ such that $n,2n,\dots,kn$ have odd sum of digits

is the following statement true or not? for any $k\in\mathbb{N}$ there exists $n\in\mathbb{N}$ such that all numbers $n,2n,\dots,kn$ have odd sum of digits? I have no idea... it may turn out hard ...
3
votes
1answer
48 views

An upper bound on the number of moves

Given a $ k \in \mathbb{N} $, the integers $ 1,2, \cdots ,4k-1, 4k $ are written on a blackboard. A move consists of replacing the numbers $ a, b, c, a+b+c $ with the numbers $ a+b, b+c, c+a $. Prove ...
5
votes
2answers
85 views

if $x^3-x\in\mathbb{Z}$ and $x^4-x\in\mathbb{Z}$ for some $x\in\mathbb{R}$, then $x\in\mathbb{Z}$.

Assume that $x^3-x\in\mathbb{Z}$ and $x^4-x\in\mathbb{Z}$ for some $x\in\mathbb{R}$. Prove that $x\in\mathbb{Z}$. my attempt: Let $a=x^3-x$ and consider polynomial $X^3-X-a$, then $x$ is a root of it ...
1
vote
1answer
52 views

Proving a sequence formula using induction [duplicate]

Suppose for $T_n$: $$T_n=(n+4)T_{n-1}-4nT_{n-2}+(4n-8)T_{n-3}$$ $$T_0=2,\quad T_1=3,\quad T_2=6$$ For integer, $n \ge 3$ I conjectured that: $$T_n = 2^n + n!$$ The above is actually TRUE. Using ...
0
votes
2answers
284 views

Putnam 1990 A1 Induction Help

A1. $(150,9,1,0,0,0,0,0,1,1,6,33)$ Let $$T_0=2,\quad T_1=3,\quad T_2=6,$$ and for $n\ge3$, $$T_n=(n+4)T_{n-1}-4nT_{n-2}+(4n-8)T_{n-3}.$$ The first few terms are ...
0
votes
1answer
48 views

Solving for $z^2 = x^2 -xy + y^2$

Recently, I came across the following solution to finding integer solutions for $z^2 = x^2 - xy + y^2$: $x = k(-n^2 -2mn)$ $y = k(m^2 - n^2)$ $z = k(mn + m^2 + n^2)$ I've been scratching my head ...
5
votes
1answer
70 views

A Representation Theory Problem in Putnam Competition

The following was the B6 problem of 1985 Putnam Competition: Suppose $G$ is a finite group (under matrix multiplication) of real $n\times n$ matrices $\{M_i\}, 1\leq i\leq r$. Suppose that ...
0
votes
1answer
29 views

powers, last 3 digits

For given $a\in\mathbb{N}$ define $x_1(a)=a$ and $x_{k+1}(a)=a^{x_k}$ for $k=1,2,\dots$. Find the last 3 digits of $\sum_{i=1}^{9}x_i(i)$. the obvious attempt is to work modulo 1000, but maybe ...
0
votes
1answer
147 views

Finding the largest 3-digit number $\; \overline{abc}\;$ s.t $\; \overline{abc}=100a+10b+c \equiv a+b^2+c^3$

This question comes from a maths contest (infer no calculators or other electronic calculating aids) for 14-16 year olds (infer no use of complicated theorems, but those accessible to high-school ...
6
votes
1answer
77 views

Composite $n$ such that $3^{n-1}-2^{n-1}$ is divisible by $n$

I'm stuck with the following olympiad problem (the solution to which I unfortunately do not possess): Show that there are infinitely many composite (i. e. nonprime) numbers $n$ such that ...
1
vote
4answers
93 views

How many rectangles can be found in this?

I saw there are 3 in each row, and 3 in each column hence, $$9 \cdot 9 = 81$$ But the answer is $$441$$ somehow? how do they get the answer?
2
votes
1answer
95 views

IMO problem number theory

Determine the greatest positive integer $k$ that satisfies the following property. The set of positive integers can be partitioned into $k$ subsets $A_1,A_2,A_3,\ldots,A_k$ such that for all integers ...
1
vote
2answers
72 views

Partitioning positive divisors of 100!

Is it possible to partition all positive divisors of 100! (including 1 and 100!) into 2 subsets so that each subset has the same number of integers and the product of all the divisors making up the ...
0
votes
1answer
45 views

How many possible different arrangements?

There are $1000$ numbers $1,2,3,...999,1000$ to be arranged in a line so that every number other than the rightmost differs by 1 from one or more of the numbers to its right. How many different ...
8
votes
1answer
114 views

A positive integer $n$ such that $S(n) = 1996\cdot S(3n)$

[Ireland 1996] Find a positive integer $n$ such that $S(n) = 1996\cdot S(3n)$, where $S$ stands for the sum of digits. The book "104 Number theory problems" gives the following solution: ...
0
votes
0answers
36 views

Is it possible to find out how many results were unexpected?

During a school year Andrew was given 40 mathematical problems as part of his assessment, one problem per week. As a result of marking he could receive 2,3,4 or 5 marks for each problem. Andrew called ...
6
votes
1answer
80 views

What is the maximum area two non intersecting circles can cover if they are contained in a unit square?

What is the maximum area two non intersecting circles can cover if they are contained in a unit square? I think that they cover the most area when in the following position: However I haven't been ...
0
votes
1answer
42 views

Perimeter problem involving different sized sticks?

Could you please help me find the answer to this question. I think it has something to do with grouping or pairing some numbers.I would appreciate easy-to-understand solutions. Thank you. There are ...
0
votes
1answer
25 views

Intersection point is in the triangle

On $X={\bf R}^2$ or $S^2(1)$, we have a triangle $\triangle ABC$ whose perimeter is small. On $D\in \overline{BC}$, let $$ r_1:=|BD|,\ r_2:=|CD| $$ Consider spheres $S(B,r_1),\ S(C,r_2),\ S(A,r)$. ...
0
votes
3answers
52 views

Order of Natural Numbers in Algorithms

Could anyone describe, why this is a True statements? if $f_i$ be a function of natural numbers to natural numbers and $f_i(n)=O(n)$ then $\Sigma_{i=1}^{n} f_i(n)=O(n^2) $
0
votes
2answers
71 views

Assuming $a_k + b_k = 1$ (Putnam 2003) [duplicate]

I do not understand as I wrote in a previous question: solution: I see that we can scale: $u_k$ but I do not understand why it is legal to say $a_k + b_k = 1$ what is $a_1 = 20$ and $b_1 = 1$ ...
0
votes
0answers
13 views

Proof for Scaling homogeneous inequalities. [duplicate]

Apparently, there exists a theorem, which says if a inequalities is homoegenous the terms can be multiplied by a scale $u_k$ like: $$(a_1 a_2 ... a_n)^{1/n} + (b_1 b_2 ... b_n)^{1/n} \le [(a_1 + ...
4
votes
2answers
101 views

prove that $f$ is periodic

A function $f\colon\mathbb{R}\to(0,\infty)$ satisfies equation $f(x)=f(x+64)+f(x+1999)-f(x+2063)$. Prove that $f$ is periodic. I'm quite sure that 1999 and 64 are random numbers (probably 1999 = year ...
8
votes
4answers
155 views

$k$ with an even sum of digits for all multiples of $k$?

Is there a number $k\in\mathbb{N}$ such that $k\cdot n$ has an even sum of digits for all $n\in\mathbb{N}$? I would be grateful for any ideas of how to attack this problem...
1
vote
1answer
20 views

Possible combinations

There are 1 to 24 numbers (1,2,3...24). How many possible combinations of 12 combine numbers will result with a sum of 146 when you add those 12 numbers? So meaning you have to combine 12 numbers ...
3
votes
4answers
236 views

How should you prove product rules by induction?

For example: $$\prod_{i=2}^n\left(1-\frac{1}{i^2}\right)=\frac{n+1}{2n}$$ For every $n$ greater than or equal to $2$ my approach for this was that I need to prove that: $$ ...
1
vote
0answers
47 views

What is the smallest number written as a sum of cubes?

What is the smallest number of the kind $\overline{999a}$, which can be presented as a sum of two natural cubes? ($a$ is a digit). I do NOT multiply below (when I write $999a$) $$999a = x^3 + y^3$$ ...
0
votes
2answers
75 views

Putnam and Beyond AM-GM help

From Putnam and Beyond: The Solution is: The only part I do NOT understand is how: $a_k + b_k = 1$ for every $k$? The problem just specifies nonnegative numbers?
4
votes
3answers
111 views

If numbers $a$ and $b$ are rational.

If $a$ and $b$ are rational numbers such that $\sqrt{4 -2 \sqrt{3}} = a + b\sqrt{3}$ Then what is the value of $a$? The answer is $-1$. $$\sqrt{4 - 2\sqrt{3}} = a + b\sqrt{3}$$ $$4 - 2\sqrt{3} = ...
6
votes
0answers
160 views

when $F_n^2+F_m^2$ is a square for fibonacci numbers

This is a curiosity question I'm trying to solve a Diophantine equation and I need some results about fibonnacci numbers, I encountered this problem: For which couple of integers $(n,m)$ the ...
2
votes
3answers
51 views

Complex numbers modulus inequality

Let $a, b \in \mathbb{C}$. Prove that $|az + b\bar{z}| \leq 1$, for any $z \in \mathbb{C}$, $|z| = 1$, if and only if $|a| + |b| \leq 1$. I want to know if my demonstration is correct. Here is how I ...
1
vote
0answers
40 views

four different integers exist, what is the least product value? [duplicate]

Four different positive integers $a, b, c, d$ are such that $a^2 + b^2 = c^2 + d^2$. What is the smallest possible value of $abcd$? $$a^2 - c^2 = d^2 - b^2$$ $$(a-c)(a+c) = (d-b)(d+b)$$ ...
2
votes
1answer
64 views

Generalization of Euler's totient theorem (aka Fermat–Euler theorem)

I am solving some math competition questions, and I realized that I do not know of a rigorous solution for this problem: What is the units digit of $2^{2015}$? We can easily see that the units ...
6
votes
3answers
138 views

The number $90$ is a polite number, what is its politeness?

The number $90$ is a polite number, what is its politeness? A. $12$ B. $9$ C. $6$ D. $14$ E. $3$ How did you get that answer? I tried Wikipedia to figure out what a polite number was ...
2
votes
1answer
105 views

Learning Olympiad Level Combinatorics

Combinatorics has always been my weakest point, I want to improve it. There are such problems like: "Five friends should give each other gifts. They have made a gift each, as they should give away ...
2
votes
2answers
103 views

Expected number of times to roll die before getting higher number

Consider the following cute problem: I roll a fiar 10-sided die (with sides labeled 1-10) until I get a number greater than or equal to my previous roll. If the epxexted value for the number of rolls ...
1
vote
1answer
40 views

how can I solve this equation

I know it is an easy equation but I do not remember how we solve it I have the $Ev$ and the $N$ and I want to find the t the thing I did is that I take the exp for the both sides of the equation ...
1
vote
1answer
59 views

positive integers on the circle

we have $101$ positive integers with sum $300$ arranged on the circle. prove that there exists an arc such that all numbers on it have sum $200$. probably Dirichlet's Pigeonhole principle should be ...
0
votes
0answers
36 views

Help needed in understanding a contradiction [duplicate]

Will someone please help me? I have posted this question before once, but I got no avail, I am trying one more time. How to solve this derivative of f proof The answer is interesting. "A ...
3
votes
0answers
77 views

A Tricky Quetiones on Creative Algorithm in Graph

an agent is works between n producer and m consumers. i'th producer, generate $s_i$ candy and j'th consumer, consumes $b_j$ candy, in this year. for each candy that sales, agent get 1 dollar payoff. ...
2
votes
0answers
63 views

Graph Algorithm and Cycle Detection

In $O(|V|+|E|)$, we can detect whether a Directed Graph has a cycle or not. ---> True In depth-first seach on DAG, there is no Back Edge. ---> True With known Number of Edges, in $O(|V|)$ and not ...
0
votes
1answer
53 views

Give an example of a function that is not strictly increasing. Draw its graph and prove that the function is not strictly increasing

I picked x^4 to be a function which is not strictly increasing for all real numbers. Since to be not strictly increasing means that for the function y=f(x) x1 < x2 then f(x1)< f(x2) but ...
0
votes
1answer
22 views

Computation Operation in one Recurrence Relation

We want to calculate $T(n)$ from recurrence relation $ T(n)= \Sigma_{i=1}^{n-1} T(i) \times T(i-1)$` and we know $T(0)=T(1)=2$. How many computation operation, an Efficient Algorithm needs for ...
0
votes
0answers
61 views

A room contains n people. Everybody wants to shake everyone else’s hand (but not their own).

(a) Suppose that n people require hn handshakes. If an (n + 1)th person enters the room, how many additional handshakes are required? Obtain a recurrence relation for hn+1 in terms of hn. (b) ...
1
vote
1answer
27 views

How does the sequence have $n$ possibilities?

From: Solution 2003 A1 and problem: Problem 2003 A1 The first part of the solution is fine: The problem is here: The issue is this: (1) They say "once $a_1$ is fixed..." there are $k$ different ...
7
votes
3answers
175 views

When is $2^x+3^y$ a perfect square?

If $x$ and $y$ are positive integers, then when is $2^x+3^y$ a perfect square? I tried this question a lot but failed. I tried dividing cases into when $x,y$ are even/odd, but still have no idea ...
11
votes
1answer
222 views

if $n$ is not divisible by any prime less than $2014$, then $n+c$ divides $a^n+b^n+n$

Find all triples $(a,b,c)$ of positive integers such that if $n$ is not divisible by any prime less than $2014$, then $\color{red}n+\color{red}c$ divides ...
2
votes
0answers
19 views

Can you recommend me a book about integration? [duplicate]

I'm new to this site. I'm an university student in korea and I major in engineering. Recently, I've been quite interested in calculating integration. So nowadays as my hobby, I seek many interesting ...
3
votes
3answers
831 views

Number of six-digit integers with increasing digits [duplicate]

How many six-digit positive integers can you write, if each number must have strictly increasing digits from left to right? How is it allowed to use: $$ \binom{9}{3}$$ Because this says out of $9$ ...
1
vote
3answers
94 views

What does $\binom{a}{b}$ represent?

Problem: How many six-digit positive integers can you write, if each number must have strictly increasing digits from left to right From the other link, How do I know if I use $\binom{a}{b}$ or ...