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

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1answer
58 views

Using Lagrange Multipliers to find the maximum of a asymmetric value

This problem is from Korean Mathematical Olympiad 2015 P3. The problem asks to find, with proof, the maximum value of $$(ax+by)^2+(bx+cy)^2$$ with the constraint of $$a^2+b^2+c^2+x^2+y^2=1$$ Now, I ...
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1answer
46 views

Length of segment $PA$ in rectangle $ABCD$

In rectangle $ABCD$ ,$AB=10$ and $BC=15$. A point $P$ inside the rectangle such that $PB=12$ and $PC=9$.What is the length of $PA$ ? I've calculated that $PA=10$ by using the law of cosines applied ...
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0answers
56 views

An Elementary Solution to a Polynomial Problem?

The following problem is from Larson's problem solving through problems: If $a,b$ and $c$ are the roots of the equation $x^3-x^2-x-1=0$, show that $$ \frac{a^{1000}-b^{1000}}{a-b}+ \frac{b^{...
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1answer
83 views

Different ways to arrange a set of numbers, so X can be seen from the left and Y can be seen from the right

Given an set of unique integers of length N. What are number of different ways you can rearrange the array so that, you can only see X numbers of integers from the left and Y numbers of integers from ...
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5answers
39 views

How to determine the weight of a coin from each of the $4$ bags.

We are given $4$ bags of coins such that (a) all coins in a given bag weigh the same, and (b) the coins of a given bag weigh either $1,2, $ or $3$ ounces. Take $1$ coin from bag $1,3$ coins from ...
7
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3answers
526 views

How does aptitude at solving Olympiad problems relate to success at further mathematical studies? [duplicate]

I spent last 6 years mostly practising my problem solving skills so I do well in my national Math Olympiad. Out of curiosity I did some reading on basics of what undergraduate students are taught - ...
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1answer
39 views

Using Affine Transformation to prove Concurrency

Let $ABCDE$ be a convex pentagon with $F=BC\cap DE, G=CD\cap EA, H=DE\cap AB, I=EA\cap BC, J=AB\cap CD$, Suppose that the areas of $\triangle AHI, \triangle BIJ, \triangle CJF, \triangle DFG, \...
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1answer
39 views

Lineal functions problem; interpreting $\;{g}^{-1}(x)=g(x)$

I'm preparing for a local math competition/olympiad, so I've been researching for past exams, and I've found this problem: Consider the lineal functions $f$ and $g$ such that $f(2)=1$,$\;g(2x)=-2f(...
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2answers
69 views

Number of Polynomials with Integer Coefficients that are bounded by $x^2$ and $x^4+1$

What is the number of polynomials $p(x)$ with integer coefficients, such that $x^2≤p(x)≤x^4+1$ for all real numbers $x$?
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3answers
140 views

Does there exist $n$ such that all numbers $n,2n,\dots,2000n$ have the same digits?

Does there exist a number $n$ such that all numbers $n, 2n, 3n, 4n, \dots, 2000n$ have the same multi-sets of digits except zeroes? (Having the same multi-sets of digits excepts zeroes means having ...
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4answers
197 views

Prove that $\pi>3$ using geometry

I was asked this question today in an interview. Question: Prove that $\pi>3$ using geometry. They gave me hints about drawing a unit circle and then inscribing an equilateral triangle and then ...
1
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1answer
31 views

Finding number of combinations

So I'm trying to figure out how many 3-number combinations can be made in a specific range, but the combinations can ONLY be in increasing numeric value. So for example: If I get the range 1-50, a ...
3
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1answer
67 views

GCD and LCM trouble

Question: Let $a_1$, $b_1$, $c_1$ be natural numbers. We define: $$a_2 = (b_1, c_1), b_2 = (c_1, a_1), c_2 = (a_1, b_1)$$and $$a_3 = [b_2, c_2], b_3 = [c_2, a_2], c_3 = [a_2, b_2]$$ Show ...
7
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2answers
119 views

Can I show $\sum_{i=1}^n 1/{i^2} \le 2$ without calculus?

I'm solving a number theory problem and it suffices to show that $$\sum_{i=1}^n \frac {1}{i^2} \le 2$$ In fact, I only need $$\sum_{d|n} \frac{1}{d^2} \le 2$$ Trying wolfram alpha suggests they are ...
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2answers
69 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.
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0answers
73 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)$.
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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
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1answer
64 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 $$\frac{a_1}{b_1}+\frac{a_2}{b_2}+...+\frac{a_n}...
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3answers
72 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 $$|a_1-a_2|+|a_2-a_3|+|a_3-a_4|+|a_4-a_5|+|a_5-a_6|+|a_6-a_7|+a_7$$?
2
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1answer
50 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 ...
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1answer
55 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
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1answer
96 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 $f(1), f(2), . . . , f(2014)$ ...
0
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1answer
43 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 ...
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0answers
41 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 ...
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2answers
49 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 $A=\{1,2,...,n\}$.
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1answer
36 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
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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
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1answer
22 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?
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2answers
100 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
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5answers
88 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 $$a^2-3a+1-x=0$$...
2
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1answer
72 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.
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1answer
85 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 motivation:...
2
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1answer
53 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, &\\ \int_{-\infty}^{\infty}|{f(x)}|^{2}dx<\...
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4answers
60 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
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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
112 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
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0answers
96 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 ...
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3answers
66 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 seem ...
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5answers
410 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 $79$ ...
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1answer
26 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 ...
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2answers
85 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
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0answers
61 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. ...
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2answers
59 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 have ...
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2answers
35 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 $8$ numbers ...
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0answers
27 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 ...
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3answers
77 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
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1answer
33 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
51 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
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1answer
29 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
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2answers
48 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 $\pi/2$ $${\sin\lambda\...