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

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4
votes
2answers
66 views

Prove that all the roots of $p(x)=F_{n}x^{n}+..+F_{1}x+F_{0}$ can't be real

Last night I have created this problem. Let $p(x)=F_{n}x^{n}+..+F_{1}x+F_{0}$ where $F_{n}$ is $n$th Fibonacci number. Prove that all the roots of $p(x)$ can't be real. Edit 1: $n>1$.
3
votes
4answers
56 views

How many integer pairs (x, y) satisfy $x^2 + 4y^2 − 2xy − 2x − 4y − 8 = 0$?

How many integer pairs (x, y) satisfy $x^2 + 4y^2 − 2xy − 2x − 4y − 8 = 0$? My Attempt Let $f(x,y)=x^2 + 4y^2 − 2xy − 2x − 4y − 8$ . So $f(x,0)=x^2 − 2x − 8$ . $f(x,0)$ has two roots $x=4 , -2$ . ...
7
votes
1answer
81 views

Polynomial game problem: do we have winning strategy for this game?

I'm thinking about some game theory problem. Here it is, Problem: Consider the polynomial equation $x^3+Ax^2+Bx+C=0$. A priori, $A$,$B$ and $C$ are "undecided", yet and two players "Boy" and ...
2
votes
1answer
42 views

if $A,B,C$ are real numbers such that ,${ A }^{ 2 }+{ B }^{ 2 }+{ C }^{ 2 } = 1 $ and $A+B+C = 0 $ find the maximum value of $(ABC )^2$ [duplicate]

$$A,B,C$$ are real numbers such that ,$${ A }^{ 2 }+{ B }^{ 2 }+{ C }^{ 2 } = 1 $$ and $$A+B+C = 0 $$ find the maximum value of ${ (ABC) }^{ 2 }$ I don't know how can I start to solve this ...
2
votes
2answers
41 views

Find the maximum value of $4x − 3y − 2z$ subject to $2x^2 + 3y^2 + 4z^2 = 1.$

Find the maximum value of $4x − 3y − 2z$ subject to $2x^2 + 3y^2 + 4z^2 = 1.$ My Attempt let $S=4x − 3y − 2z$ and $ t=2x^2 + 3y^2 + 4z^2$. Then $t-s =2x^2 + 3y^2 + 4z^2 -(4x − 3y − 2z)= 2(x-1)^2 + ...
1
vote
1answer
24 views

Bingo-like Game

In one board game, each player has a unique 4 x 4 grid with squares randomly labeled with each integer from 1 to 16. As the integers 1 to 16 are randomly called, each player puts an "X" in the ...
0
votes
0answers
39 views

How many rational values of x are not integers and satisfy the following equation?

How many rational values of x are not integers and satisfy the following equation: $$x^7 - 6x^6 + 5x^5 - 4x^4 + 3x^3 - 2x^2 + 1 = 0 ?$$ Well, I got this question from one of the Mathcounts ...
7
votes
1answer
78 views

Tetrahedron packing in Cube

I'm thinking about following solid geometry problem. Q: Suppose you have a box of "cube" shape with edge length 1. Then, How many regular tetrahedrons(with edge length 1) can be in the box? So, this ...
2
votes
3answers
37 views

Find all possible values of $c^2$ in a system of equations.

Numbers $x,y,z,c\in \Bbb R$ satisfy the following system of equations: $$x(y+z)=20$$ $$y(z+x)=13$$ $$z(x+y)=c^2$$ Find all possible values of $c^2$. To try to solve this, I expanded the equations: ...
7
votes
2answers
60 views

Solve system of simulataneous equations in $3$ variables

Solve the following equation system: $$x+y+xy=19$$ $$y+z+yz=11$$ $$z+x+zx=14$$ I've tried substituting, adding, subtracting, multiplying... Nothing works. Could anyone drop me a few hints without ...
1
vote
1answer
40 views

Find closed form of $f(a,b,c)$

Let $$f(a,b,c)=\left|\dfrac{|b-a|}{|ab|}+\dfrac{b+a}{ab}-2c\right|+\dfrac{|b-a|}{|ab|}+\dfrac{b+a}{ab}+\dfrac{2}{c}.$$ Find closed form to $f$.
1
vote
0answers
40 views

A theorem about binomial coefficient module prime

For any integer $r$ and prime $p$, there is a integer $n$ which $\binom{2n}{n}\equiv r \pmod{p}$. I tried Lucas's theorem, but I was stuck. Suppose $r\neq 0$, otherwise we can let $n=p$. Let ...
1
vote
3answers
80 views

The total amount Edgar paid for a slice of pizza and a tip of exactly $24\%$ was between $\$2.50$ and $\$3.00$. What was the price of the pizza slice?

The total amount Edgar paid for a slice of pizza and a tip of exactly 24% was between $\$2.50$ and $\$3.00$. What was the price of the pizza slice? Well, I did the trial and error method and ...
1
vote
4answers
90 views

Problem about simple probability

I guess that this will be really simple for you guys, but i have no foundation in probability. Please, help me to find not only the answer but also what i need to learn in order to be able to solve ...
1
vote
2answers
96 views

Problem from Olympiads of mathematics about elementary number theory

Can you please help me with this problem from the Italian selection of the Olympiads of mathematics? Let $p(x)$ be a polynomial with integer coefficients and let $p(0)=6$. Exactly $40$ $p(n)$ with ...
1
vote
1answer
42 views

Quadratic recurrence relation (from a math-contest)

It's given the following quadratic relation: $$a_n = \frac{a_{n-1}^2+61}{a_{n-2}}$$ Find $a_{10}$. Note that I can't use a calculator or a computer, instead I was wondering if there's a trick to find ...
2
votes
1answer
34 views

Finding a path on a coordinate plane

On a coordinate plane, a path consists of a series of moves in the positive $x$- or $y$- direction. If the first move is 1 unit in length, the second move is 2 units, the third move is 3 units, and ...
2
votes
2answers
40 views

“Stairstep Numbers”

I've been preparing for Mathcounts competition, but this one question confused me a bit. If a stairstep number is defined as a number whose digits are strictly increasing in value from left to ...
-1
votes
1answer
55 views

What is the largest six-digit number with an odd number of positive factors?

What is the largest six-digit number with an odd number of positive factors? So I know the number must be a perfect square, but how do I know six-digit number perfect squares? I'm pretty sure there's ...
-1
votes
2answers
55 views

How many cubes must be randomly selected to ensure that at least one pair of each color has been removed from the bag?

There are $15$ red, $11$ blue and $13$ green cubes in a bag. All cubes are identical, except for color. How many cubes must be randomly selected to ensure that at least one pair of each color has been ...
5
votes
4answers
1k views

Factorial question: number of trailing zeroes in 125! [duplicate]

How many zeros are after the last nonzero digit of 125! ? The answer is 31, but how do you solve it?
4
votes
1answer
35 views

Example of $2$ nonisomorphic simple graphs satisfying conditions.

What is an example of $2$ simple graphs which: have the same degree sequence; for any $n \ge 2$ have the same number of copies of $K_n$; for any given $k \ge 3$ have the same number of induced ...
3
votes
3answers
105 views

Proving that $1^n+2^n+3^n+4^n$ $(n\in \Bbb N)$ is divisible by 10 when $n$ is not divisible by 4

I was solving some math problems to prepare for math contests and came across this one: Prove that $1^n+2^n+3^n+4^n$ $(n\in N)$ is divisible by 10 if and only if $n$ is not divisible by 4. So, from ...
1
vote
2answers
52 views

2015 AIME #3: Where did I go wrong?

This is a question conerning 2015 AIME #3. The problem goes as follows: There is a prime number $p$ such that $\displaystyle 16p+1$ is the cube of a positive integer. Find $p$. Here is my ...
2
votes
3answers
67 views

Do the lengths of all three segments necessarily have the same distribution?

Let $A$ and $B$ be independent $U(0, 1)$ random variables. Divide $(0, 1)$ into three line segments, where $A$ and $B$ are the dividing points. Do the lengths of all three segments necessarily have ...
1
vote
4answers
86 views

Among the following, which is closest to $\sqrt{0.016}$?

Among the following, which is closest in value to $\sqrt{0.016}$? A. $0.4$ B. $0.04$ C. $0.2$ D. $0.02$ E. $0.13$ My Approach: $(\frac{16}{1000})^\frac{1}{2} = (\frac{4}{250})^\frac{1}{2} = ...
3
votes
0answers
50 views

No eight digit perfect fourth powers with distinct digits and not containing 3.

Problem: Prove that there are no perfect fourth powers that have eight distinct digits in their base 10 representation and also don't contain 3 as a digit. My attempt: Since the problem is about ...
0
votes
3answers
38 views

Modular Arithmetic Related Question

I've been trying to solve this problem but I couldn't and I don't get the solutions either (I don't think I get how to use modular arithmetic to solve problems in general. And though I tried to ...
3
votes
1answer
114 views

$15$ integers $m_1 \ldots m_{15}$such that $ \sum _{k=1} ^{15} m_k \arctan {k} = \arctan 16$

Determine whether or not there exist $15$ integers $m_1 \ldots m_{15}$ such that $ \sum _{k=1} ^{15} m_k \cdot \arctan (k) = \arctan (16)$. This is a question from IMC 2015 Day 1 Problem. Here ...
1
vote
2answers
31 views

What is the probability that the two-seed makes the finals?

I can solve the following problem using brute-force combinatorics, but I'm looking for an elegant way to think about it, since there is a rather elegant answer. Suppose there is a tournament of ...
2
votes
3answers
82 views

Find the value of $\frac{1}{20} + \frac{1}{30} + \frac{1}{42} + \frac{1}{56} + \frac{1}{72} + \frac{1}{90}$

Find the value of $p+q$, where $p$ and $q$ are two positive integers such that $p$ and $q$ have no common factor larger than $1$ and $$\frac{1}{20} + \frac{1}{30} + \frac{1}{42} + \frac{1}{56} + ...
0
votes
2answers
39 views

Proving angles are supplementary in isosceles triangle

Let $ABC$ be a triangle with $AC=BC$, and let $P$ be a point inside $\triangle ABC$, satisfying $\angle PAB=\angle PBC$. If $M$ is the midpoint of $AB$, show that $\angle APM+\angle BPC=180^{\circ}$. ...
3
votes
2answers
53 views

how we can calculate $ \frac {\sqrt {x^2} + \sqrt {y^2} }{2 \sqrt {xyz}}$? [closed]

I teach math for Schools. How can Help me in the following past Olympiad question? If $y,z$ be two negative distinct number and $x$ and $y$ be negate of each other, how we can calculate $ ...
-1
votes
1answer
229 views

Set of all $n$; $n={d^2_1 + d^2_2 + d^2_3 +d^2_4}$

$A$ is the set of all $n$ numbers where $n={d^2_1 + d^2_2 + d^2_3 +d^2_4}$. Here $1=d_1<d_2<d_3<d_4$ where $d_1,d_2,d_3,d_4$ are the $4$ smallest divisors of $n$. As an example ...
6
votes
1answer
254 views

Game theory, olympiad question

I've seen the following question in the brazilian olympiad for university students, and I couldn't solve it. Thor and Loki play the game: Thor chooses an integer $n_1 \ge 1$ , Loki chooses $n_2 \gt ...
2
votes
2answers
58 views

Does probability depend on knowledge?

There is at least $2/3$ probability that this question is rather silly, but being an almost absolute beginner in Probability, I will ask it anyway. Consider the following problem, proposed at AIME ...
-1
votes
2answers
36 views

How many possible $4$-digit integer $x$ are there if $y-x=3177$?

Given any $4$-digit positive integer $x$ not ending in '$0$', we can reverse the digits to obtain another integer $y$. How many possible $4$-digit integer $x$ are there if $y-x=3177$? Denote ...
1
vote
4answers
69 views

Compute the sum $\sum_{k=1}^{10}{\dfrac{k}{2^k}}$ [duplicate]

Compute the sum $$\sum_{k=1}^{10}{\dfrac{k}{2^k}}$$ This question is taken from SMO junior (I can't remember which year it is). I have no idea how to start. Can anyone give some hint? By writing ...
1
vote
1answer
45 views

Plane geometry problem, Suppose ABP,BCP,CAP have same area&perimeter…

I'm trying to solve following geometry question, but it is quite challenging.(at least for me!) Thanks for your help in advance. On plane, there is some triangle ABC. Also, there is a point P ...
1
vote
2answers
37 views

Let $S$ be the smallest positive multiple of $15$, that comprises exactly $3k$ digits with $k$ $0$'s, $k$ $3$'s and $k$ $8$'s.

The following is taken from Singapore Mathematical Olympiad $2013$ Junior Round $1$. Let $S$ be the smallest positive multiple of $15$, that comprises exactly $3k$ digits with $k$ $0$'s, $k$ $3$'s ...
3
votes
2answers
70 views

Variance of the random variable $|X \cup Y|$? [closed]

Let $X$ and $Y$ be random subsets of $\{1, 2, \dots, k-1, k\}$ picked uniformly at random from all $2^k$ subsets, independent of each other. What is the variance of the random variable $|X \cup Y|$?
16
votes
3answers
828 views

How many sewings are there on a soccer ball?

A soccer ball is obtained by sewing $20$ hexagonal pieces of leather and $12$ pieces of leather of pentagonal shape. A sewing joins together the sides of two adjacent pieces. How many sewings ...
1
vote
2answers
225 views

Weight of watermelons after percentage of water is evaporated.

A stock of watermelons of the initial weight of $500 \space\text{kg}$ has been put in a store for a week. Initially the percentage of water in the watermelons makes up $99 \% $ of the ...
1
vote
2answers
54 views

Arrangement of points in a circle

From the 2015 Moscow Mathematical Olympiad: The numbers $1$ to $1000$ are arranged on a circle such that each number divides the sum of its two neighbors. Suppose that the number $k$ has two odd ...
0
votes
0answers
27 views

$f\in \mathbb{Z}[x], f(x) = y^2, f(y) = z^2, f(z)=x^2 \implies x=y=z$?

Given that $f\in \mathbb{Z}[x], f(x) = y^2, f(y) = z^2, f(z)=x^2$ for some real numbers $(x,y,z)$, does it follow that $x=y=z$? It is well known that if $f\in \mathbb{Z}[x]$ and $f(x)=y, f(y)=z, ...
8
votes
4answers
140 views

Is there $n$ such that $n,n^2,n^3$ start with the same digit ($\neq 1)$

From the 2015 Moscow Mathematical Olympiad: Is there some $n>2$ such that $n,n^2$ and $n^3$ start with the same digit (this digit being different from $1$) Using a computer I found that $99$ ...
1
vote
1answer
62 views

Find minimum number of coins with Largest value coins?

There is a greedy algorithm for coin change problem : using most valuable coin as possible. How We can find a quick method to see which of following sets of coin values this algoithms cannot find ...
3
votes
1answer
38 views

Probability: Finding the Number of Pears Given Two Scenarios

You have a bag containing 20 apples, 10 oranges, and an unknown number of pears. If the probability that you select 2 apples and 2 oranges is equal to the probability that you select 1 apple, 1 ...
1
vote
1answer
72 views

High-school group-theory problem(given in a contest)

Let $G$ be a finite group and let $ H \le G $ be a subgroup of $G$. Suppose there is some $ \emptyset \neq S \subset G$ such that for any $x\in S$ we have $x^2 \notin H$. Prove that there is $T ...
4
votes
1answer
32 views

How do you find $∠XPC$ + $∠XPB$ such that $PB+PC$ is maximum where $P$ is a point on $f(x) = (x-1)(x-3)(x-5)$?

Problem: $f(x) = (x-1)(x-3)(x-5)$ intersects the x axis at $A(1,0)$, $B(3,0)$ and $C(5,0)$. A point $P(t,f(t))$ is selected on the curve such that $PB+PC$ is maximum and $t \in (3,5).$ Let $PX$ be ...