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

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0
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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 ...
2
votes
1answer
111 views

Find a polynomial with integral coefficients whose zeros include $\sqrt{2} + \sqrt{5}$.

Find a polynomial with integral coefficients whose zeros include $\sqrt{2} + \sqrt{5}$. I think I can use $-3= (\sqrt{2} + \sqrt{5})(\sqrt{2} - \sqrt{5})$ and a certain telescopic factorisation. The ...
30
votes
8answers
3k views

Can a pre-calculus student prove this?

a and b are rational numbers satisfying the equation $a^3 + 4a^2b = 4a^2 + b^4$ Prove $\sqrt a - 1$ is a rational square So I saw this posted online somewhere, and I kind of understand what ...
2
votes
0answers
204 views

How to prove this hard geometry

The incircle of triangle $ABC$ has center $I$ and touchs the sides $BC,CA,AB$ at the points $D,E,F$ respectively,and Let the centers of the excircles tangent to $BC,CA,AB$ be $I_{1},I_{2},I_{3}$ ...
1
vote
0answers
36 views

Prove intersection between side length and tangent to circumcircle at opposite vertex is collinear with points on perpendicular bisectors of sides

Let $ABC$ be a triangle with $AB\neq BC$. Point $E$ lies on the perpendicular bisector of $AB$ such that $BE\perp BC$. Point $F$ lies on the perpendicular bisector of $AC$ such that $CF\perp BC$. Let ...
-1
votes
1answer
42 views

Determine the symmetric sum of roots.

Please no complete solutions, ONLY HINTS REQUESTED! The complex numbers $\alpha_1$, $\alpha_2$, $\alpha_3$, and $\alpha_4$ are the four distinct roots of the equation $x^4+2x^3+2=0$. Determine the ...
0
votes
2answers
48 views

Math Team Problem Involving Powers of Powers of 3

So I am in my high school math team and I was given the following expression $$3^{3^{3^{...}}}$$ Where there are multiple powers of 3 with a total of two thousand and fifteen 3's. The question ...
1
vote
2answers
75 views

Find all values of $x$

Determine all real values of $x$ such that: $$\log_{2}(2^{x-1} + 3^{x+1}) = 2x - \log_{2}(3^x) $$ Let $u = 2^x$ and let $y = 3^x$ For ease, let $\log_{2}$ be represented by just $\log$ so: ...
3
votes
2answers
51 views

Find $s^4-18s^2-8s$

Let $a,b,c$ be the roots of $x^3-9x^2+11x-1=0$, and let $s=\sqrt{a}+\sqrt{b}+\sqrt{c}$. Find $s^4-18s^2-8s$. $s^4 - 18s^2 - 8s = (s)(s + 4)(s - 2 + \sqrt{6})(s - 2 - \sqrt{6})$ $P(x) = (x - a)(x ...
10
votes
3answers
836 views

sum of one hundred numbers

I saw this problem recently. It asks to prove that it is always possible to choose 100 numbers from 200 positive numbers such that their sum will be divisible by 100. Attempt to solve: my first step ...
2
votes
1answer
61 views

Show that there exist only $n$ solutions

Let $P(x)$ be a polynomial of degree $n>1$ with integer coefficients, and let $k$ be a positive integer. Consider the polynomial $Q(x) = P( P ( \ldots P(P(x)) \ldots ))$, where $P$ occurs $k$ ...
2
votes
1answer
102 views

Sum of powers of sine

Find $\displaystyle \sum_{n=1}^{89} \sin^6(n) = \frac{m}{n}$ Let $x = \sin(n)$ and let $y = \cos(n)$. Since $\cos(n) = \sin(90 - n)$ it follows that $= \sin^6(1) +\sin^6(1) + ... + \sin^6(45) ...
4
votes
1answer
133 views

Leningrad Mathematical Olympiad $1991$

A finite sequence $a_1, a_2, ..., a_n$ is called $p$-balanced if any sum of the form $a_k+a_{k+p} + a_{k+2p}+...$ is the same for any $k = 1, 2, 3, ..., p$. For instance the sequence $a_1 = 1$, ...
2
votes
1answer
52 views

How many sides from diagonals?

A polygon has $100$ diagonals, then it has at least: A-15, B-16, C-17, D-18 Sides? Using simple patterns, I noticed that all figures (even sides) have $\frac{n}{2}$ sides for $n$ diagonals; this ...
0
votes
3answers
65 views

Evaluate the nested square root

Evaluate: $x = \sqrt{11 - 2\sqrt{10}} - \sqrt{11 + 2\sqrt{10}}$ You may have seen my other Q/A here, but I am finding a different way, with perhaps perfect squares. If we seperate, $y = ...
4
votes
3answers
36 views

Which of the constants A,B,C,D does T depend on?

Let $f(x)=cos(5x)+Acos(4x)+Bcos(3x)+Ccos(2x)+Dcos(x)+E$ and $T=f(0)-f(\pi/5)+f(2\pi/5)-f(3\pi/5)+..-f(9\pi/5)$.Then out of A,B,C,D which does T depend on? Hints please! P.S:KVPY 2011 question
5
votes
1answer
61 views

Olympiad problem about finding minimum value with $x^2y^2+y^2z^2+z^2x^2\ge x^2y^2z^2$

Let $x,y,z$ be positive real numbers such that $x^2y^2+y^2z^2+z^2x^2\ge x^2y^2z^2$. Find the minimum value of $$\frac{x^2y^2} {z^3(x^2+y^2)}+\frac {y^2z^2} {x^3(y^2+z^2)}+\frac {z^2x^2} ...
17
votes
4answers
720 views

Why can't $p^p-(p-1)^{p-1}=n^2$ be a square?

Let $p$ be a prime number. Show that $p^p-(p-1)^{p-1}$ can't be a square. In other words, there is no $n\in\mathbb{N}^{+}$ such that $$p^p-(p-1)^{p-1}=n^2.$$
1
vote
1answer
47 views

First Three Digits of Powers of 2 and 5

Suppose you know that there exists positive integer $n\in \mathbb{N}$ such that the first three digits of $2^n$ and $5^n$ are the same, and that $\forall n$ that do so, the first three digits are ...
0
votes
1answer
45 views

Prove existence of 5 non-attacking rooks

Problem: There are $41$ rooks on a $10\times10$ chessboard. Prove that there must exist $5$ rooks, none of which attack each other. I could only observe that at least one of rows and at least one ...
1
vote
2answers
85 views

The sum of two positive integers is 29 , find the minimum value of the sum of their squares.

If the sum of two positive integers a and b is 29 , find the minimum value of the sum of their squares. Of course I dont need a brute force answer. What is a quick way to find what a and b is?
1
vote
1answer
47 views

Find $\theta$ in the given triange

Given an isosceles triangle ABC, AB=AC, AD=BC, angle ACB = 80 degree. What is $\theta$? (I'm looking for ways to find this angle without using a calculator, sorry I added this late)
2
votes
1answer
72 views

General form of $\sqrt{a - b} - \sqrt{a + b}$?

General form of $\sqrt{a - b} - \sqrt{a + b}$? What I would do is: let $x = \sqrt{a - b} - \sqrt{a + b}$ $x^2 = 2a - 2\sqrt{a^2 - b^2}$ Then since $a + b > a - b$ $x = -\sqrt{2a - ...
1
vote
1answer
50 views

Multiplying products of $p_1,p_2,\ldots,p_n$ gives a square.

Given $n+1$ ($n\ge 4$) arbitrary products of primes $p_1,p_2,\ldots, p_n$, prove multiplying some of the products gives a square. E.g., for $n=4$: $\{p_1,p_2,p_3,p_4,p_1p_3\}$ satisfies the ...
3
votes
1answer
62 views

A Diophantine Equation

Finding the number of $(a, b, c)$, where $a, b, c$ are positive integers, that $$ \frac{a^2+b^2-c^2}{ab}+\frac{c^2+b^2-a^2}{cb}+\frac{a^2+c^2-b^2}{ac}=2+\frac{15}{abc} $$ I factored it ...
-1
votes
2answers
71 views

Find the minimum value of $a^2 + b^2 + 5 - ab - 2a - 2b$

Find the minimum value (integer) of $a^2 + b^2 + 5 - ab - 2a - 2b$. I believe the answer is $1$, I got this from trial-and-error. $= a^2 + b^2 - ab - 2(a + b) + 5 = (a + b)^2 - 2(a+ b) - 3ab + ...
1
vote
1answer
41 views

Find $k$ such that the area is an integer

For some positive integers k, the parabola with equation $y = \frac{x^2}{k} - 5$ intersects the circle with equation $x^2 + y^2 = 25$ at exactly three distinct points A, B and C. Determine all ...
2
votes
0answers
64 views

How many numbers can be written as a sum?

How many integers can be written as a sum of 4 consecutive integers such that the numbers are < $2015$. EDITED QUESTION: If $N = n + (n + 1) + (n+ 2) + (n + 3)$ then how many such $N$ ...
2
votes
1answer
39 views

Sum of smallest roots

If 2 and -3 are the roots of a biquadratic equation, then the sum of the two smallest roots of this equation is: $\{-1, -3, -5, \text{cannot be determined} \}$ those are the options. The answer ...
53
votes
7answers
1k views

Let $k$ be a natural number . Then $3k+1$ , $4k+1$ and $6k+1$ cannot all be square numbers.

Let $k$ be a natural number . Then $3k+1$ , $4k+1$ and $6k+1$ cannot all be square numbers. I tried to prove this by supposing one of them is a square number and by substituting the corresponding $k$ ...
1
vote
3answers
56 views

How many participants required?

A test consisting of 20 problems is given at a math competition. Each correct answer to each problem gains 4 points; each wrong answer takes away 1 point, and each problem left without an answer ...
1
vote
0answers
39 views

puzzle-coloring problem-olmpiad

A $23\times23$ square is completely tiled by $1\times1, 2\times2$ and $3\times3$ tiles. What is the smallest number of $1\times1$ tiles needed? This is the solution If we color the rows of the ...
2
votes
0answers
101 views

Multivariable Factor Theorem

By my previous questions here and here I have been inspired to ask about the factor theorem; the multivariable case of it. So take $f(a, b, c) = (a-b)^3 + (b-c)^3 + (c-a)^3$ $f(a, a, c) = f(b, b, c) ...
5
votes
3answers
122 views

Olympiad inequality problem with $a+b+c+abc=4$

If $a,b,c \in \mathbb R_{> 0}$ and $a+b+c+abc=4$, prove that $$({a\over {\sqrt {b+c}}}+{b\over {\sqrt {c+a}}}+{c\over {\sqrt {a+b}}})^2(ab+bc+ca) \ge {\frac 12}(4-abc)^3$$ This can be solved by ...
2
votes
2answers
47 views

Factoring a polynomial (multivariable)

Factor $ (a - b)^3 + (b - c)^3 + (c-a)^3$ by SYMMETRY. Okay, this is the problem. Let $f(a) = (a - b)^3 + (b-c)^3 + (c-a)^3$ obviously, if you let $a = b$ then, $f(b) = 0$, thus $(a - b)$ is a ...
0
votes
1answer
31 views

Giving this formula in DNF and CNF propositional logic

The formula I am trying to turn into conjunctive normal form and disjunctive normal form is: $P \rightarrow (Q \land R)$ could anyone please help me give two answers, CNF and DNF? I have managed to ...
1
vote
3answers
49 views

Find the sum from the system of equations

If $x,y, z$ satisfy: $$x + y = z^2 + 1, y + z = x^2 + 1, x + z = y^2 + 1 $$ Find the value of $2x +3y + 4z$. This gives us (by getting $x + y + z$ that) $z^2 + z + 1 = x^2 + x + 1 = y^2 + y + ...
4
votes
2answers
145 views

Prove every integer is obtained from functions

We are given the following operations: $$f(n)=10n, g(n)=10n+4, h(n)=\frac{n}{2}$$, where $n$ is a positive integer (n must be even for $h(n)$. Show that, beginning with $n=4$, every positive ...
1
vote
2answers
41 views

show this diophantine equation has at least is $3n+3\lfloor \frac{n+1}{3}\rfloor+1$ postive integer solution

For any postive integer $n\ge 4$, let $s(n)$ denote the number of ordered pairs $(x,y,z)$ of positive integers for which $$\color{red}{xy+yz+xz=n(x+y+z)}$$ show that $$s(n)\ge 3n+3\lfloor ...
6
votes
3answers
119 views

Board game on a $m\times n$ board - winning strategy

Two friends, $A$ and $B$, play a game with one single game piece on a rectangular board with $m$ rows and $n$ columns. $A$ begins the game by moving the game piece from its starting point $(1, 1)$ to ...
3
votes
2answers
72 views

Finding an angle in a circle

In circle $O$, $PA\perp AO,AE\perp PO,\angle BCO=30^{\circ},\angle BFO=20^{\circ}$,find $\angle DAF$. It is obvious that $\angle EAD=\angle PAD=\frac{1}{2}\angle AOP$, but I can't get more ...
1
vote
1answer
62 views

Confused by a step in a solution to the problem

I'm pretty confused by the step $$ \prod_{n=1}^{45}\sin(2n^\circ)=\sum_{n=1}^{45}\frac{\omega^n-1}{2i\omega^{n/2}} $$ in the official solution of this problem from 2010 PUMaC Algebra A7: The ...
6
votes
2answers
316 views

How many solutions to the rational equation?

If $a$ and $b$ and $c$ are parameters, how many solutions for: $$\frac{(x-b)(x-c)}{(a-b)(a-c)} + \frac{(x-a)(x-c)}{(b-a)(b-c)} + \frac{(x-a)(x-b)}{(c-a)(c-b)} = 1$$ I would say $3 \implies ...
1
vote
2answers
86 views

How many numbers less than $1000$ with digit sum to $11$ and divisible by $11$

How many positive (integers) numbers less than $1000$ with digit sum to $11$ and divisible by $11$? There are $\lfloor 1000/11 \rfloor = 90$ numbers less than $1000$ divisible by $11$. $N = 100a + ...
0
votes
2answers
43 views

For how many integers $a$ does this equation have three solutions?

For how many integers $a$ does the equation $(x^2-a^2 ) \sqrt{(5-x)}=0$ have three different solutions? The options were: $10, 9, 8, $other. I say other. No matter what, $\sqrt{5-x} = 0$ ...
0
votes
1answer
34 views

Determine the maximum GCD

The sum of $10$ natural numbers is $2014$. Determine the greatest possible value of the GCD of these numbers. Is this a trial and error type of problem? $a_1 + a_2 + ... + a_{10} = 2014$. ...