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

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6
votes
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 $79$ ...
0
votes
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 ...
1
vote
2answers
72 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
59 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
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 ...
2
votes
2answers
33 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 ...
0
votes
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 ...
2
votes
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
votes
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
vote
1answer
27 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
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\...
4
votes
1answer
62 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 $a\...
1
vote
2answers
54 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 it'...
5
votes
1answer
64 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
40 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
28 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
47 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 nonnegative ...
2
votes
1answer
69 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, & \text{if}~-1\...
2
votes
1answer
37 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
50 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
46 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 a ...
2
votes
1answer
120 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 ...
31
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 the ...
2
votes
0answers
213 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
44 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
49 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
76 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: Then, $\...
3
votes
2answers
53 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
849 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
69 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 $k$ times....
4
votes
1answer
129 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
160 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 $a_1 = 1$, $a_2 ...
2
votes
1answer
56 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
76 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 = \sqrt{11 - ...
4
votes
3answers
37 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
67 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} {y^3(z^2+x^2)}...
17
votes
4answers
749 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 $$p^p-(p-1)^{p-1}=n^2.$$
3
votes
2answers
80 views

What is the smallest possible value of $\lfloor (a+b+c)/d\rfloor+\lfloor (a+b+d)/c\rfloor+\lfloor (a+d+c)/b\rfloor+\lfloor (d+b+c)/a\rfloor$?

What is the smallest possible value of $$\left\lfloor\frac{a+b+c}{d}\right\rfloor+\left\lfloor\frac{a+b+d}{c}\right\rfloor+\left\lfloor\frac{a+d+c}{b}\right\rfloor+\left\lfloor\frac{d+b+c}{a}\right\...
1
vote
1answer
56 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
48 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
169 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
50 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)
3
votes
1answer
82 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 - 2\sqrt{a^2 ...
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
68 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 in ...