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

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3
votes
3answers
66 views

Solve the following simple congruence

$$560x \equiv 1 \pmod{429}$$ I am close, I used Euclid's algorithm but the remainder is hard to go backwards. $$560 = 1(429) + 131 $$ $$429 = 3(131) + 36$$ $$131 = 3(36) + 23$$ $$36 = ...
3
votes
4answers
272 views

Integer solutions using PIE

Find the number of integer solutions to $a+b+c+d=18$ with $ 0≤a,b,c,d≤6$. With no restrictions there are: $$\binom{21}{3} = 1330$$ Ones that are invalid are: $a, b, c, d \ge 7$. But how do I ...
0
votes
1answer
53 views

Find the number of polynomials satisfying the root conditions

Let $S$ be the set of all polynomials of the form $z^3+az^2+bz+c$, where $a$, $b$, and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisfies either ...
5
votes
1answer
100 views

Jensen's Inequality Clarification

I'm working on some competition inequalities using some of the inequality theorems I just learned. I am trying to prove: $$ \frac{a}{\sqrt{a^2 + 8bc}} + \frac{b}{\sqrt{b^2 + 8ca}} + ...
2
votes
1answer
45 views

Prove that a given binary operation on a list that reduces an (arbitrary) pair at a time always results in the same single element (contest question)

I believe I have proven part (a), but would like some feedback on whether(and how) this should be made more rigorous (especially at the end), and whether there is a simpler way that doesn't involve ...
5
votes
3answers
120 views

Find $f(2015)$ given the values it attains at $k=0,1,2,\cdots,1007$

Let $f$ be a polynomial of degree $1007$ such that $f(k)=2^k$ for $k=0,1,2,\cdots, 1007$. Determine $f(2015)$. Taking $f(x)=\sum_{n=0}^{1007} a_n x^n $, (whence $a_0=1$), I tried to combine ...
4
votes
2answers
87 views

Self-avoiding rook walks on small rectangular chessboards (contest question)

I am not sure how to get a closed-form formula for $R(3,n)$ as the recursion involves a summation. Maybe the best that can be achieved is a recursion that does not involve a summation having an upper ...
9
votes
3answers
637 views

Given functional equation $f(x,y)=f(x,y+1)+f(x+1,y)+f(x-1,y)+f(x,y-1)$, show that $f(0,0)=0$ [closed]

Let $f:\mathbb{Z}\times \mathbb{Z}\to \mathbb{R}$, such that for all $x,y$ $$f(x,y)=f(x,y+1)+f(x+1,y)+f(x-1,y)+f(x,y-1),$$ and if $m,n\in \mathbb{Z},mn\neq 0$,we have $f(2m,2n)=0$. Show that ...
0
votes
2answers
34 views

Finding a point of a rotation (complex numbers)

I posted early but got a very tough response. Point $A = 2 + 0i$ and point $B = 2 + i2\sqrt{3}$ find the point $C$ $60$ degrees ($\pm$) such that Triangle $ABC$ is equilateral. Okay, so I'll begin ...
21
votes
3answers
331 views

Is it possible to cover $\{1,2,…,100\}$ with $20$ geometric progressions?

Recall that a sequence $A=(a_n)_{n\ge 1}$ of reals is said to be a geometric progression whenever $a_{n+1}/a_n$ is constant for each $n\ge 1$. Then, replacing $20$ with $12$, the following question ...
12
votes
1answer
324 views

Alternative proof of simple integral inequality

Problem. Let $f\in C^1(\mathbb R)$ such that $f(0) = 0$ and $0 < f'(x) \le 1$. Prove that for all $x\ge 0$ $$ \int_0^x f^3(t)\,dt \le \left(\int_0^x f(t)\,dt\right)^{\!\!2}. $$ Below is my ...
5
votes
1answer
50 views

Can three diagonals in a $2k+1$-gon intersect?

Is it possibly to find three diagonals in a regular $2k+1$-gon that intersect? More particularly can they intersect inside the polygon? I am looking for an elementary solution. Can one be found ...
1
vote
1answer
34 views

How to find the These numbers?

Suppose $n \in N$. $n$ can be described as the sum of square of three number's. That is $ n = a^2 + b^2 + c^2$ . My Question is: How to find $a, b, c$?
0
votes
2answers
40 views

Finding a point using complex geometry

In the Cartesian plane let $A = (1,0)$ and $B = \left( 2, 2\sqrt{3} \right)$. Equilateral triangle $ABC$ is constructed so that $C$ lies in the first quadrant. Let $P=(x,y)$ be the center of ...
1
vote
2answers
82 views

Seating people in a circular table

It has always been an interesting question. If we have $10$ chairs and a round table, how many ways are there of seating $10$ people? I would say there are $10!$ ways to seat the people due to ...
-1
votes
2answers
42 views

Does the order in a circular arrangement matter?

I posted a question a while ago: Ten chairs are arranged in a circle. Find the number of subsets of this set of chairs that contain at least three adjacent chairs. My question here is: imagine a ...
2
votes
3answers
74 views

Why would the cubic have $5$ roots?

The polynomial $P(x)$ is cubic. What is the largest value of $k$ for which the polynomials $Q_{1}(x) = x^{2}+(k-29)x-k$ and $Q_{2}(x) = 2x^{2}+(2k-43)x+k$ are both factors of $P(x)$? $P(x) = ...
2
votes
2answers
48 views

Find the least $N$ so there is no square

Find the least positive integer $N$ such that the set of $1000$ consecutive integers beginning with $1000 \cdot N$ contains no square of an integer. Let $x^2$ appear before $1000N$ so: $(x+1)^2 ...
0
votes
2answers
66 views

How many divisors of the combination of numbers?

Find the number of positive integers that are divisors of at least one of $A=10^{10}, B=15^7, C=18^{11}$ Instead of the PIE formula, I would like to use intuition. $10^{10}$ has $121$ divisors, ...
5
votes
2answers
229 views

Problem 7 IMC 2015 - Integral and Limit

I'm trying to solve problem 7 from the IMC 2015, Blagoevgrad, Bulgaria (Day 2, July 30). Here is the problem Compute $$\large\lim_{A\to\infty}\frac{1}{A}\int_1^A A^\frac{1}{x}\,\mathrm dx$$ ...
1
vote
1answer
50 views

How to use Principle of Inclusion-Exclusion here?

A while ago I posted a question: Coloring a Grid. Online, I seem to have stumbled upon a usage of PIE AOPS Wiki Solution AIME II #9. (1) Now, I have experience with PIE, but I do not see how to ...
1
vote
0answers
80 views

IMC 2014, Problem 4 [Day 2]

We say that a subset of $\mathbb{R}^{n}$ is $k$-almost contained by a hyperplane if there are less than $k$ points in that set which do not belong to the hyperplane. We call a finite set of points ...
1
vote
2answers
90 views

Ways of coloring the $7\times1$ grid (with three colors)

Hints only please! A $7 \times 1$ board is completely covered by $m \times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the ...
1
vote
3answers
73 views

find the complex number $z^4$

Let $z = a + bi$ be the complex number with $|z| = 5$ and $b > 0$ such that the distance between $(1 + 2i)z^3$ and $z^5$ is maximized, and let $z^4 = c + di$. Find $c+d$. I got that the ...
2
votes
1answer
60 views

Partition a square

Compute the smallest positive integer $n$ such that, for any given integer $p\geq n$, we can partition a given square into $p$ number of squares (the small squares are not necessarily congruent) I ...
1
vote
1answer
88 views

Olympiad Problem on Modular Arithmetic

Suppose $a,b,c,d$ are integers such that $$(3a+5b)(7b+11c)(13c+17d)(19d+23a)=2001^{2001},$$ prove that $a$ is even. We have $2001=3\cdot 23\cdot 29$, hence we have $3a+5b=3^{e_1}23^{e_2}29^{e_3}$ ...
2
votes
0answers
57 views

Cycle triplets: A beats B beats C beats A. Minimum and maximum number of triplets for round-robin tournament of $2n+1$ teams? (contest question)

From the 2006 Canada National Olympiad: Consider a round-robin tournament with $2n + 1$ teams, where each team plays each other team exactly once. We say that three teams $X, Y\text{ and }Z$, ...
1
vote
2answers
138 views

Probability distribution of number of waiting customers in front of a counter [closed]

The number of customers arriving at a bank counter is in accordance with a Poisson distribution with mean rate of 5 customers in 3 minutes. Service time at the counter follows exponential distribution ...
10
votes
2answers
347 views

IMC 2015 - Problem 10 - Inequality between polynomials and exponential

This is problem 10 from the International Mathematical Competition for University Students of 2015, from day 2, in Bulgaria. I think it is an interesting problem! Let $n$ be a positive integer, and ...
1
vote
0answers
75 views

Find the number of “p-safe numbers”

For a positive integer $p$, define the positive integer $n$ to be $p$-safe if $n$ differs in absolute value by more than $2$ from all multiples of $p$. For example, the set of $10$-safe numbers is ...
0
votes
0answers
34 views

Let $T$ be the set of all positive integer divisors of $2004^{100}$. Size of largest subset $S$ of $T$ such that no element in $S$ divides another?

I am getting an answer slightly over $100^2$. Is this right (working below), or is there a better way of selecting elements of S? The following question appeared on the 2004 Canada National Olympiad: ...
3
votes
1answer
65 views

Let $S$ be a set of $n$ points in the plane with min spacing of 1. Prove $S$ has a subset of $\ge n/7$ points with min spacing of $\sqrt{3}$.

I believe I have proven the case $n=8,|T|=2$, but welcome feedback. I need help proving the case for general $|T|>2$. From the 2003 Canada National Olympiad: Let $S$ be a set of $n$ points in ...
6
votes
5answers
168 views

Show that $(1+p/n)^n$ is a Cauchy sequence for arbitrary $p$

It is a generalization of this question. I am looking for a similar derivation as in here. Can we prove that $(1+p/n)^n$ is a Cauchy sequence for any $p \in [a, b]$ by showing that $$ \Bigg| \left( ...
1
vote
2answers
37 views

How to show that this interesting difference of products is $O \left( \frac{1}{n^2} \right) $

Let $k \leq n$. Consider the following difference of products: $$ \prod_{i=1}^{k-1} \left( 1 - \frac{i}{n+1} \right) - \prod_{i=1}^{k-1} \left( 1 - \frac{i}{n} \right)$$ For $n=1,2,3$, this is ...
1
vote
2answers
51 views

Given 4 points with 2 on different radius. Obtain the center of the circle.

I'm struggle on a math question that states the following: Black holes have an overwhelming gravity, such that the nearest stars begin spinning around them (Example). Every affected star keeps ...
0
votes
0answers
49 views

Can you verify the combinatoric recurrence?

There are $2^{10} = 1024$ possible 10-letter strings in which each letter is either an A or a B. Find the number of such strings that do not have more than 3 adjacent letters that are identical. ...
1
vote
1answer
103 views

AMT - Three whole numbers add up to 149 and multiply to give 987. What is the largest of the three number

So about this question I'm not too sure... Can't find out what I should start off with. If anyone can help me I'll be very greatly appreciated. The question is: Three whole numbers add up to 149 ...
0
votes
0answers
35 views

Additional explanation needed for the solution to a spesific sequence

Some of my attempts include trying to use the the term formula for geometric sequences and some other manipulations in hope of getting a more clearer, workable expression, though without success. ...
1
vote
3answers
65 views

Right answer, wrong explanation, probability of grids?

Two unit squares are selected at random without replacement from an $n\times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected squares are ...
6
votes
2answers
235 views

Finding all real numbers x such that $x \lceil x \lceil x \lceil x \rceil \rceil \rceil = 88$

Question: Find all real numbers x such that $x \lceil x \lceil x \lceil x \rceil \rceil \rceil = 88$. The notation $\lceil x \rceil$ means: The least integer which is not less than $x$. My ...
4
votes
2answers
114 views

Find the maximum value of the fraction

Let $a$ and $b$ be positive integers satisfying $\frac{ab+1}{a+b}<\frac{3}{2}$. The maximum possible value of $\frac{a^3b^3+1}{a^3+b^3}$ is $\frac{p}{q}$, where $p$ and $q$ are relatively prime ...
0
votes
2answers
31 views

ind $\tan \alpha$ in the square

let say the square has sides of 2 units, $DM = DN = AN = AP = 1$, $NP = \sqrt 2$, $NQ = QP = \frac{\sqrt 2}{2}$, and $AR \ne AP$ (?) we have know that $\tan \alpha = \frac 2{RP}$, but what's the ...
2
votes
2answers
83 views

Find a recursion (combinatorial)

Consider sequences that consist entirely of $ A$'s and $ B$'s and that have the property that every run of consecutive $ A$'s has even length, and every run of consecutive $ B$'s has odd length. ...
0
votes
2answers
54 views

Difficult nonlinear system based on max value

Let $ (a,b,c)$ be the real solution of the system of equations $ x^3 - xyz = 2$, $ y^3 - xyz = 6$, $ z^3 - xyz = 20$. The greatest possible value of $ a^3 + b^3 + c^3$ can be written in the form $ ...
4
votes
6answers
99 views

find x in $\sqrt[3]{6+\sqrt x} + \sqrt[3]{6-\sqrt x} = \sqrt[3] {3}$

Which one satisfies the equation $\sqrt[3]{6+\sqrt x} + \sqrt[3]{6-\sqrt x} = \sqrt[3] {3}$ (A)$27$ (B)$32$ (C)$45$ (D)$52$ (E)$63$ let $a = 6+\sqrt x , b=6-\sqrt x$ cube each side ...
2
votes
4answers
80 views

Find $x$ in the triangle

the triangle without point F is drawn on scale, while I made the point F is explained below So, I have used $\sin, \cos, \tan$ to calculate it Let $\angle ACB = \theta$, $\angle DFC = \angle ...
3
votes
3answers
89 views

Values of the sums $\sum\limits_{k=1}^{n}\cos^4(πk/(2n+1))$

I have been given a question which asks you to prove that $$ \sum_{k=1}^{n}\cos^4\left(\frac{πk}{2n+1}\right)=\frac{6n-5}{16} $$ The main problem I have with solving this is that since the summands ...
1
vote
4answers
65 views

remainder of $a^2+3a+4$ divided by 7

If the remainder of $a$ is divided by $7$ is $6$, find the remainder when $a^2+3a+4$ is divided by 7 (A)$2$ (B)$3$ (C)$4$ (D)$5$ (E)$6$ if $a = 6$, then $6^2 + 3(6) + 4 = 58$, and $a^2+3a+4 ...
4
votes
3answers
153 views

Determine all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $xf(y)+yf(x)=(x+y)f(x^2+y^2)$ for all $x,y\in\mathbb{N}$ (contest question)

The question below is from the 2002 Canada National Olympiad. I have found one family of functions but need help in finding (or proving the non-existence) of others. Suggestions on how to improve the ...
0
votes
1answer
27 views

Probability the range is disjoint

Let $A=\{1,2,3,4\}$, and $f$ and $g$ be randomly chosen (not necessarily distinct) functions from $A$ to $A$. The probability that the range of $f$ and the range of $g$ are disjoint is ...