Tagged Questions

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

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5
votes
1answer
50 views

likely open number theory problem: finite sum of $\zeta(2)$ equal to a square of rationals

Which $n$ can let $S=1+\frac14+\frac19+\cdots+\frac1{n^2}$ be a square of a rational number? Obviously, $1$ and $3$ work, but how to prove they are the only ones? I think this problem is really hard. ...
3
votes
0answers
17 views

Bounded area for any triangle formed by polygons

Let $P_1,P_2,P_3$ be closed polygons on the plane. Suppose that for any points $A\in P_1$ (meaning $A$ can be inside or on the boudary of $P_1$), $B\in P_2,C\in P_3$, we have $[ABC]\leq 1$. Is it ...
1
vote
1answer
32 views

How should I approach solving this floor function?

Prove that for all $n \in \Bbb Z, \lfloor\sqrt {(n)}+ \sqrt {(n+1)} \rfloor = \lfloor \sqrt{4n+2}\rfloor$. There must be some algebraic substitution?
6
votes
2answers
59 views

Every $3\times 3$ square has even number of painted cells

Given a $1000\times 1000$ board. We paint some cells (at least one) so that in every $3\times 3$ square, an even number of cells are painted. What is the minimum number of painted cells? One way to ...
3
votes
0answers
26 views

Condition $|x_1x_2+1|<x_1+x_2$ in quadratic polynomial

Let $x^2-ax+b$ be a polynomial with real coefficients having two nonzero roots. Given that $|b+1|<a$, and one of the roots have modulus $<1$, show that the other root has modulus $>1$. We ...
3
votes
1answer
27 views

GCD of adjacent pairs take on all possible values

Given a fixed positive integer $n$. Consider the numbers $1,2,\ldots,2n$. The GCD of any pair is one of $1,2,\ldots,n$. Suppose that all $2n$ numbers are placed around a circle. Is it possible that ...
1
vote
3answers
40 views

Finding the range of equation. Any tricks?

I m working on the following problem For real numbers $a,b$, if $a+ab+b=3$, then find the range of $m=a-ab+b$. Is there any inequalities here to use?
4
votes
2answers
58 views

Least possible number of squares with odd side length

An $n\times(n+3)$ rectangular grid ($n>10$) is cut into some squares, with all cuts being along the grid lines. What is the least possible number of squares with odd side length? [Source: Russian ...
5
votes
1answer
32 views

Quadratic polynomial with alternate negative value

Let $f(x)=-x^2+ax+b$, where $a,b\in\mathbb{R}$. Suppose there exist distinct integers $m,n$ such that $f(m)=-n^2$ and $f(n)=-m^2$. Prove that there are infinitely many pairs of integers $x,y$ such ...
1
vote
0answers
16 views

How prove $\left(\frac{D}{\sqrt{3}}+\frac{d}{2}\right)^{2}\geq n\cdot\frac{d^{2}}{4}$ for $D=\max{A_iA_j}, d=\min{A_iA_j} (1\leq i<j\leq n)$

Let be n points in the plane $A_1,A_2,A_3,...,A_n(n\geq 3)$. Let $D=\max{A_iA_j}, d=\min{A_iA_j} (1\leq i<j\leq n)$. How prove $\left(\frac{D}{\sqrt{3}}+\frac{d}{2}\right)^{2}\geq ...
1
vote
0answers
31 views

Maximum number of acute triangles

Given $n$ points on the plane, no three of which are collinear, what is the maximum number of acute triangles formed by them? [Source: Based on Hungarian competition problem]
1
vote
0answers
42 views

How solve the equation $x^{4}-y^{4}= 80z^{4}$ for x and y odd integers, and z integer

Let x and y be odd integers, and let z be an integer. How solve the equation $x^{4}-y^{4}= 80z^{4}$?
3
votes
0answers
75 views

Integer solutions of $a^3+2a+1=2^b$

What are the solutions in integers of $a^3+2a+1=2^b$? [Source: Serbian competition problem]
-1
votes
1answer
40 views

Using the difference equation to find the problem. [on hold]

let $a_1 = 2\sqrt 2$ and for any $n>1$ define $a_n = 2^{(n+1)/2}\sqrt{2^n - \sqrt{4^n - (a_{n-1})^2}}.$ Find $a_n$ in closed form and evaluate $\displaystyle\lim_{n\to\infty} a_n.$
0
votes
0answers
24 views

Game placing numbers in increasing order

Let $k\leq m\leq 100$ be positive integers. Aaron and Britney play a game on a $1\times m$ board, using $100$ paper pieces numbered from $1$ to $100$. The game has $k$ turns. In each turn, Aaron ...
7
votes
0answers
36 views

Ratio of product from one point and minimum distance

Given points $A_0,A_1,\ldots,A_n$ in the plane, let $m$ denote the minimum distance among any two points. What is the minimum value of $$\dfrac{|A_0A_1|\cdot|A_0A_2|\cdot\ldots\cdot|A_0A_n|}{m^n}?$$ ...
6
votes
1answer
86 views

Dividing tournament into “equal” groups

In a tournament of $n$ teams, each team plays all other teams exactly once, with no draw. For which $n$ is it always possible to divide all teams into several groups so that each group of teams won ...
-1
votes
0answers
41 views

Square root Question from GRE [closed]

A question in GRE states: What is the smallest number which when subtracted from 1.00060219 gives a perfect number? Any easy method & time saving please Update: This was what presented in a ...
5
votes
0answers
30 views

Selecting cells so that every $2\times 2$ square is odd, then even

Jacob selects some cells from a $12\times9$ table, so that every $2\times 2$ subsquare contains an odd number of selected cells. He then selects some more cells, so that every $2\times 2$ subsquare ...
6
votes
0answers
87 views
+50

Floor function inequality $\lfloor a\sqrt{2}\rfloor\lfloor b\sqrt{7}\rfloor <\lfloor ab\sqrt{14}\rfloor$

Let $a,b$ be positive integers. Show that $\lfloor a\sqrt{2}\rfloor\lfloor b\sqrt{7}\rfloor <\lfloor ab\sqrt{14}\rfloor$. [Source: Russian competition problem]
0
votes
0answers
18 views

Newton-Raphson with Exponentials

I'm having trouble getting initial values for x and y to be thrown into the Newton Raphson formulae, aka Xv1 and Yv1 respectively. Question; Show that the equation: 10e^-2x = 2x + 3x^2 has a root ...
14
votes
3answers
1k views
+400

Integral Contest

Before you answer this OP, please read all the terms and conditions below. Thank you... Today I hold an unofficial little contest on brilliant.org. Now, I will hold it here on Math S.E. It's just for ...
0
votes
2answers
27 views

Percentage Question from GRE

A question in GRE states: In a survey of a town,it was found that 65% of the people surveyed watched the news on television,40% read newspaper, and 25% read a newspaper and watched the news on ...
1
vote
1answer
38 views

Calculus Proof involving exponents.

Prove that $2015^{2013}<2014^{2014}<2013^{2015}$ without the use of a calculator. I don't know where to begin here. Any help or guidance on where to begin would be greatly appreciated.
-1
votes
1answer
39 views

Analytical Question for GRE

In a book prep. MCQ's in analytical portion a question says: "The chairs in the school hall can be set out in 35 equal rows or in 45 equal rows or in 105 equal rows are:" I'm unable to sort out ...
4
votes
1answer
331 views

Need help with GRE question

I encountered a question while preparing for GRE and am stuck. In an examination paper of 5 Questions, 5 percent of the candidates answered all of them and 5 percent none. Of the rest, 25% ...
2
votes
0answers
14 views

Reflection to get within convex polygon

Let $P$ be a convex polygon, and let $A_1$ be a point on the same plane as $P$. Prove that we can find an integer $n$, and points $A_2,A_3,\ldots,A_n$, such that $A_{i+1}$ is a reflection of $A_i$ ...
8
votes
1answer
74 views

Game replacing two numbers by mean

Alicia and Bart plays a game. Alicia first writes $100$ real numbers on the board. After that they move alternately; Bart goes first. In every move, the player chooses two numbers, erases them, and ...
0
votes
0answers
45 views

Finding examples before solving

So I've been solving some contest problems,and most of them require a solution in order to be solved. For example $$S_n=\left\{{n\choose n},{2n\choose n},{3n\choose n},\ldots,{n^2\choose n} \right\}$$ ...
5
votes
2answers
93 views

Integral involving inverse of $x^x$

My brother gave me the following problem: Let $f:[1;\infty)\to[1;\infty)$ be such that for $x≥1$ we have $f(x)=y$ where $y$ is the unique solution of $y^y=x$. Then calculate: $$ \int_0^e f(e^x)dx $$ ...
11
votes
2answers
164 views

Integer solutions of $x^3-x+9=5y^2$

What are the solutions in integers of $x^3-x+9=5y^2$? [Source: Hungarian competition problem]
6
votes
2answers
92 views

How to Solve : $ A =\frac{1}{6}\left((\log_2(3))^3-(\log_2(6))^3-(\log_2(12))^3+(\log_2(24))^3\right) $

$ A =\frac{1}{6}\left((\log_2(3))^3-(\log_2(6))^3-(\log_2(12))^3+(\log_2(24))^3\right).$ Solve for $2^A.$ (no calculators or graphs are permitted) The way I went about solving this problem was using ...
3
votes
0answers
24 views

Bound on number of breakable sets

Let $\mathcal{S}$ be a finite family of finite sets. A finite set $A$ is called breakable if for every $B\subseteq A$, there exists $S\in \mathcal{S}$ such that $A\cap S=B$. Show that at least ...
1
vote
2answers
23 views

Forming Random Team and Choosing Pair of Friends

n participants of the competition were split into m teams in some manner so that each team has at least one participant. After the competition each pair of participants from the same team became ...
0
votes
1answer
25 views

A problem regarding table decorations

My one friend Alex has r red, g green and b blue balloons. To decorate a single table for the banquet he needs exactly three balloons. Three balloons attached to some table shouldn't have the same ...
2
votes
4answers
57 views

Closed Form for Factorial Sum

I came across this question in some extracurricular problem sets my professor gave me: what is the closed form notation for the following sum: $$S_n = 1\cdot1!+2\cdot2!+ ...+n \cdot n!$$ I tried ...
6
votes
4answers
102 views

If $\sum_{n=1}^\infty \frac{1}{a_n}$ converges, must $\sum_{n=1}^\infty \frac{n}{a_1 + \dots + a_n}$ converge?

Suppose $\sum_{n=1}^\infty \frac{1}{a_n} = A$ is summable, with $a_n > 0,$ $n = 1,2,3,\cdots.$ How can we prove that $\sum_{n=1}^\infty \frac{n}{a_1 + \dots + a_n}$ is also summable? This question ...
4
votes
1answer
41 views

Some Strange Minimum and proof

I read following on Norm chapter in one book. $$\begin{align}\left|\left\|x-y\right\|-\left\|w-z\right\|\right| \leq & \min \{\|x-w\| + \|y-z\| , \|x-z\| + \|y-w\|\}\\\text{ or, }&\min ...
2
votes
1answer
74 views

On the equality $\sqrt[n]{a_1}+\sqrt[n]{a_2}+\cdots+\sqrt[n]{a_k}= \sqrt[n]{b_1}+\sqrt[n]{b_2}+\cdots+\sqrt[n]{b_m}$

Let $k,m\in \mathbb{N}$. Let $a_1,a_2,\cdots,a_k\ >0$ and $b_1,b_2,\cdots,b_m \ >0$ such that $$\sqrt[n]{a_1}+\sqrt[n]{a_2}+\cdots+\sqrt[n]{a_k}= ...
6
votes
1answer
57 views

Functional Equation $f(mn)=f(m)f(n)$.

If $f: \mathbb N \mapsto \mathbb N$ is one-to-one and $f(mn) = f(m)f(n)$, what is the smallest possible value of $f(999)$? Easily $f(1)=1$, and I think $f(n)=n$ must be the only map, but not able to ...
6
votes
2answers
83 views

Eating chocolate game on grid

Given is a chocolate of size $m\times n$. Anne and Birgitte plays a game, with Anne starting. In each turn, the player has to divide the chocolate into two rectangular parts along the lines, and eat ...
1
vote
1answer
77 views

For which values of $a$ is the solution for $x^2 - y^2 = a^3$ unique?

For which values of $a$ is the solution unique? $$x^2-y^2=a^3$$ I'm not sure how to do this, so I've been looking at this guy's solution. $x^2 - y^2 = a^3$ is factored into $(x-y)(x+y) = a^3$. ...
-1
votes
0answers
44 views

Count edges that can be removed

Given are N nodes and M edges, each edge connects two nodes. The edges are bidirectional , i.e., substance can flow in either direction through the edge. We start from node 1 and end up at node N. ...
-1
votes
1answer
74 views

Check if $N$ is of form $6A + 8B$

Given a number $N$ we need to check if its of form $6A + 8B$ .If its of this form then we need to check if $B$ can be greater than equal to $1$ or not. Like $24$ is of form $6A + 8B$. Also $B$ can ...
4
votes
0answers
134 views

Can I use any theorem I know at an IMO? [closed]

What if I happen to know a (fairly well-known) theorem that trivializes a given problem set at a math contest? Could my answer be rejected (unless I provide proof)? For example, see this question on ...
0
votes
1answer
47 views

Expected value for Head/Tails

There are $N$ coins placed in a line. A coin may be facing head/tail direction with $0.5$ probability. Now I need to find number of pairs of coins $(i,j)$ such that $i<j$ and on index $i$ , I ...
1
vote
2answers
49 views

Prove that if $abc\ne0$ and $ab+bc+ac=0$ then $a+b+c\ne0$

I tried to do proof by contradiction, but problem is how to get from $ab+bc+ac$ to $a+b+c$ Assuming $a+b+c=0$ my approachs: Adding $ab+ac+bc=0$ and $a+b+c=0$ and try to factor Deriving ...
-1
votes
2answers
92 views

Check if we can turn a string into a palindrome by reversing a substring

Given a string consisting of lower-case characters from English alphabets, we want to reverse a substring from the string such that the string becomes a palindrome. Note : A Palindrome is a string ...
-4
votes
3answers
85 views

Expected number of good pair of coins [closed]

N coins are being put in a line, each of them is either facing Heads or Tail with equal probability.A pair of indices (i,j) is called good coin pair if coin at index i is facing Heads, and coin at ...
2
votes
1answer
78 views

Find different sequences of game to find winner

Alice and Bob are having a racing competition to see who is the best runner. They don't want to decide this in a single race, so they choose a number N which is the minimum number of points one of ...