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

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0
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2answers
67 views

Find all integer solutions to $x^2+y^2+z^2=2xyz$ [closed]

I am working on some of these types of problems for fun, just want to see a couple solved as examples.
2
votes
0answers
66 views

Which positive integers satisfies $a^{b^2} = b^a$

How one can find all integers satisfying $a\geq 1,b\geq 1,a^{b^2} = b^a$? I think that the solutions are $ (a,b)=(1, 1), (16, 2),(27, 3)$.
0
votes
1answer
51 views

$a,b,c$ are distinct real number

$a,b,c$ are distinct real number such that $a^3=3(b^2+c^2)-25$, $b^3=3(c^2+a^2)-25$, $c^3=3(a^2+b^2)-25$. Find the numerical value of $abc$
3
votes
1answer
62 views

Show that $\frac{a_1}{b_1}+\frac{a_2}{b_2}+…+\frac{a_n}{b_n} \geq n$

Let $a_1$, $a_2$,..., $a_n$ be the sequence of positive numbers, and let $b_1$, $b_2$,..., $b_n$ be any permutation of the first sequence. Show that ...
2
votes
3answers
68 views

How can this sum be maximized?

Suppose that $a_1, a_2, a_3, a_4, a_5, a_6, a_7$ are distinct integers from $1$ to $7$. What is, then, the maximum value of the sum $$|a_1-a_2|+|a_2-a_3|+|a_3-a_4|+|a_4-a_5|+|a_5-a_6|+|a_6-a_7|+a_7$$? ...
1
vote
1answer
44 views

Find all real polynomials $P(x)$ which satisfy the equation$ P(x)P(-x)=P(x^2-1)$

Find all real polynomials $P(x)$ having only real zeros and which satisfy the equation $$P(x)P(-x)=P(x^2-1)$$ Please explain me the process and refer some books to learn polynomials. Thanks ...
0
votes
1answer
54 views

Two polynomials that differ by polynomial expansion of $e$

Let $h(n)=\sum_{k=1}^{n}\frac{1}{k!}$. Does there exist real polynomials $f(x)$ and $g(x)$ such that $f(n)=h(n)g(n)$ for every positive integer $n$? So far, I got that $f(x)$ and $g(x)$ needs to be ...
3
votes
1answer
74 views

How many numbers have unit digit $1$?

Let $f(n)$ be the number of positive integers that have exactly $n$ digits and whose digits have a sum of $5$. Determine, with proof, how many of the $2014$ integers $f(1), f(2), . . . , ...
0
votes
1answer
35 views

Arithmetic sequence to geometric sequence.

The numbers $a_1, a_2, a_3, . . .$ form an arithmetic sequence with $a_1 \ne a_2$. The three numbers $a_1, a_2, a_6$ form a geometric sequence in that order. Determine all possible positive ...
3
votes
0answers
36 views

Finding the maximum cardinality of a set

Let $B$ be a subset of $A$ such that for any two elements $b_1$ and $b_2$ in $B$, we always have $2b_1\not \equiv{0}\pmod{b_2}$ if $2b_1\ge b_2$. If $A=\{1,2,...,n\}$ then find the maximum possible ...
1
vote
2answers
39 views

Maximum possible number of elements in a subset given a condition

Let $B$ be a subset of $A$ such that no element in $B$ is twice the other. Find the maximum number of elements possible in $B$ if $A=\{1,2,...,n\}$.
2
votes
1answer
25 views

Let $a_1, a_2, a_3…$ be the sequence of all positive integers relatively prime to 75. Find the value of $a_{2008}$.

Let $a_1, a_2, a_3...$ be the sequence of all positive integers relatively prime to 75, where $a_1<a_2<a_3...$ with $a_1=1, a_2=2, a_3=4, a_4=7$. Find the value of $a_{2008}$. What I have done: ...
0
votes
1answer
28 views

Inequality with logs

Let $n>1$ be a integer, show that there exists a constant $t$ such that $$\displaystyle \sum_{j=0}^{\lfloor \log_4(n)\rfloor}\lfloor \frac{n+2^{2j}}{2^{2j+1}}\rfloor-\frac{4n}{9}\le t\log_{10} n ...
2
votes
1answer
18 views

Find the expected area of a randomly chosen triangle.

The set of numbers $(x,y)$ are positive natural numbers such that $x+y=n$. 2 points are chosen from this set. What is the expected area of the triangle formed by the origin and the two points?
1
vote
2answers
78 views

Counting the number of words made of $2n$ letters

Compute the number of words made of $2n$ letters taken from the alphabet $\{a_1, a_2,\ldots,a_n\}$ such that each letters occurs exactly twice and no two consecutive letters are equal. I started ...
3
votes
5answers
84 views

What is the best way to solve an equation of the form $(f(x))^2-a(f(x))+b=x$?

On a math contest I was told to solve the equation $$(x^2-3x+1)^2-3(x^2-3x+1)+1=x$$ For this particular problem I simplified by letting $$a\equiv x^2-3x+1$$ Then I continued with ...
2
votes
1answer
63 views

question on right angle triangle

Let ABC and DBC be two equilateral triangle on the same base BC,a point P is taken on the circle with centre D,radius BD. Show that PA,PB,PC are the sides of a right triangle.
5
votes
1answer
71 views

Why are there no recreational math events for adults?

Possibly off topic because it's about mathematical community rather than mathematics directly. Assuming that the answer to Mathematical competitions for adults. is still up to date... The ...
2
votes
1answer
50 views

How to prove this bound of $L^\infty$ norm.

A differentiable function $ f:\mathbb R\to \mathbb R$ satisfies such conditions, $ $\begin{cases} \lim_{x\to\infty} f(x)=\lim_{x\to-\infty} f(x)=0, &\\ ...
0
votes
4answers
48 views

Function to repeat number N times

I am not a math person, but is it possible to repeat one number N times without programming langs or programs? If yes, which type of function can I use to do it? For example: number ...
0
votes
1answer
32 views

Existence of positive integers $a_1,a_2,…,a_k$

Let $x$ and $y$ be positive integers such that $\arctan(\frac1x)+\arctan(\frac1y)<\frac{\pi}2$. Show that there exists positive integers $a_1,a_2,...,a_k$ none of which equals $x$ or $y$ such that ...
5
votes
4answers
100 views

If $x$ is a positive integer such that $x(x+1)(x+2)(x+3)+1=379^2$, find $x$

If $x$ is a positive integer such that $x(x+1)(x+2)(x+3)+1=379^2$, find $x$ This is a 1989 ARML problem. One, ugly way to solve this is: What's a nicer way? Hint
0
votes
0answers
83 views

Combinatorics : # of ways to invite the guests

At the moment we are doing combinatorics and probability at school and it is a branch of mathematics that interests me probably more than anything else. Upon doing some of my own research I´ve come ...
2
votes
3answers
65 views

Find the least $n$ such that the expression is divisible by $700$.

What is the sum of the digits of the smallest positive integer $n^4 + 6n^3 + 11n + 6$ is divisible by $700$. Hints please. I got that $P(n) = n(n+1)(n+2)(n+3) \equiv 0 \pmod{700}$ I cannot ...
6
votes
5answers
397 views

Prove that in every sequence of 79 consecutive positive numbers written in decimal system there is a number whose sum of the digits is divisible by 13

Prove that in every sequence of $79$ consecutive positive numbers written in decimal notation there is a number the sum of whose digits is divisible by $13$. I tried to take one by one sets of ...
0
votes
1answer
23 views

Cyclic hexagon with every other side equal

Let $ABCDEF$ be a cyclic hexagon with $AB=CD=EF$. Let $AC\cap BD=P, CE\cap DF=Q, EA\cap FB=R$. Prove that $\triangle PQR\sim\triangle BDF$. This problem seems simple, but I'm having trouble figuring ...
0
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0answers
40 views

Breaking a stick to form a triangle

A stick is randomly broken into $n$ pieces. What is the minimum value of $n$ such that there always exists three pieces that can form a non-degenerate triangle? Preferably without calculus. I know ...
2
votes
0answers
48 views

Polynomial can be written as a sum of two monic polynomials

Hints only Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree $n$ with real coefficients is the average of two monic polynomials of degree $n$ with $n$ real roots. ...
1
vote
2answers
58 views

How to prove a combinatoric statement?

From Number 10B with PICTURE. Suppose there are n plates equally spaced around a circular table. Ross wishes to place an identical gift on each of k plates, so that no two neighbouring plates ...
2
votes
2answers
29 views

Divisibility of numbers without a digit

How many of the integers from $0,1, 2, ... ,999$ are neither divisible by $9$ nor contain the digit $9$. Let $N$ be an integer, so, $N \equiv 1, 2, 3, 4, 5, 6, 7, 8 \pmod{9}$. That is $8$ ...
0
votes
0answers
26 views

What is the fastest way to perform below operation?

Assuming an array A of integers of size m and n to be some random number. What is the fastest way to calculate the following, A[i]%n + A[i+1]%n + ----A[m]%n One ...
2
votes
3answers
75 views

How to find the value of this expression?

I just saw this question in one exam. Please help me solve it. I am not able to find any clue on where to begin. (ignore that tick it might be wrong)
2
votes
1answer
32 views

Number of ordered positive rationals (x,y,z) satisfying following conditions.

How many ordered triples $(x,y,z)$ of positive rational numbers satisfy the conditions: $x+y+z$, $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$, and $xyz$ are all integers.
3
votes
1answer
49 views

Integer coefficients polynomial. Find largest number of roots.

The polynomial $p(x)$ has integer coefficients, and $p(100)=100$. Let $r_1, r_2, …, r_k$ be distinct integers that satisfy the equation $p(x)=x^3$. What is the largest possible value of $k$?
1
vote
1answer
19 views

Minimal rank of special matrix

Let $n\geq 2$ be an integer and $A=(a_{ij})$ an $n\times n$ matrix whose elements are $1,2,\dots,n^2$. I am supposed to find the minimal and maximal possible rank of $A$. (In this question, I'm not at ...
1
vote
2answers
41 views

Find all possible values of $\lambda$ which satisfy the given equation.

The question is to find the values of a real number $\lambda$ for which the following equation is satisfied for all real values of $\alpha$ which are not integral multiples of $\pi/2$ ...
4
votes
1answer
47 views

Prove that the integer $a$ represented in base $b$ has at least $n$ non-zero digits

Let $a,b,n$ be integers greater than $1$. Suppose $(b^n-1)|a$. Prove that the integer $a$ represented in base $b$ has at least $n$ non-zero digits. I observe that $b^n\equiv 1 \pmod{b^n-1}$ and ...
0
votes
2answers
47 views

Quadratic Equation to prove $ax^2+bx+c=0$

"Prove that there is one and only quadratic equation for which the sum of the roots is $3$ and the cubed of the roots is $63$" I'm practicing for the Maths Olympiad. I'm a high school student and ...
5
votes
1answer
58 views

Prove $S$ is composite

HINTS ONLY Let $a, b, c, d, e, f$ be positive integers. Suppose that $S = a + b + c + d + e + f$ divides both $abc + def $ and $ab + bc + ca − de − ef − fd$. Prove that $S$ is composite. Must ...
1
vote
3answers
36 views

Finding maximum $b$ in $x^5-20x^4+bx^3+cx^2+dx+e=0$

Let $b, c, d, e$ be real numbers such that the following equation $$x^5-20x^4+bx^3+cx^2+dx+e=0$$ has real roots only. Find the largest possibe value of $b$. What I have done is: Let $x_1, x_2, x_3, ...
0
votes
1answer
27 views

Maximum and minimum of a function.

Given a function $f(x) = C(x, 2) + C(N-x,2)$, where N is a constant and C(N, K) is the binomial coefficient choose K from N, we need to find minimum and maximum value. Also, $x > 0$. So, f(x) = ...
3
votes
1answer
46 views

Proof of existing degree $n$ binomial

Let $P(x)$ be a polynomial with real coefficients such that $P(x) > 0$ for all $x \ge 0$. Prove that there exists a positive integer $n$ such that $(x + 1)^n P(x)$ is a polynomial with ...
0
votes
0answers
17 views

Why is this proof for Enestrom-Kakeya theorem invalid?

Let $a_n \ge a_{n - 1} \ge \dots \ge a_1 \ge a_0 \ge 0$ with $a_n > 0$. Prove that every complex root of the polynomial $f(x) = a_n x^n + a_{n - 1} x^{n - 1} + \dots + a_1 x + a_0$ satisfies ...
2
votes
1answer
66 views

Find polynomials $f(x), g(x)$, and $h(x)$

Find polynomials $f(x), g(x)$, and $h(x)$, if they exist, such that for all $x$, $$\mid f(x)\mid-\mid g(x) \mid+h(x)= \begin{cases} -1, & \text{if}~x<-1 \\ 3x+2, & ...
2
votes
1answer
35 views

Prove that $\frac{x_1 + x_2 + x_3 +x_4}{4}$ is independent of the line, and compute its value.

Consider the lines that meet the graph $y = 2x^4 + 7x^3 + 3x − 5$ in four distinct points $P_i=[x_i, y_i]$, $i = 1, 2, 3, 4$. Prove that $\frac{x_1 + x_2 + x_3 +x_4}{4}$ is independent of the line, ...
3
votes
1answer
45 views

How many solutions exist for a non-linear system

How many solutions exist to the following system: $$ \begin{eqnarray} xy+xz &=& 54+x^2 \\ yx+yz &=& 64+y^2 \\ xz+yz &=& 70+z^2 \end{eqnarray} $$ I have guessed that the ...
0
votes
1answer
23 views

Comparing absolute values

If $|i - (a + bi)| < 1$ does $|i - (a - bi)| < 1$ also? I would say yes, because the absolute value shouldn't differ by more than $1$? Where $i = \sqrt{-1}$
2
votes
1answer
39 views

find an invariant

I've been reading about the use of invariants in contest math. I saw the following problem (in my own words): There are $N = 2n$ numbers placed on a circle. Then we increase two any consecutive ...
1
vote
0answers
50 views

Prove that there is no integer $k$ with $P(k)=8$

Let $P(x)= x^n + a_{n-1}x^{n-1}+...+a_1x+a_0$be a polynomial with integral coefficients. Suppose that there exists four distinct integers $a$, $b$, $c$, $d$ with $P(a)=P(b)=P(c)=P(d)=5$. Prove ...
0
votes
1answer
41 views

Can I get three roots $a'$, $b'$ and $c'$ such that $P(x)=(x-a')(x-b')(x-c')$?

If I have $(x-a)(x-b)(x-c)=1$ ($a,b,c \in \mathbb{Z}$) for the polynomial $P(x)=(x-a)(x-b)(x-c)-1$, can I get three roots $a'$, $b'$ and $c'$ such that $P(x)=(x-a')(x-b')(x-c')$? This is only ...