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

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9
votes
2answers
132 views

What's the minimal $k$ satisfying these conditions? Graph theory problem.

I'm thinking following problem. There are five pairs of couples (So, ten people total) and $k$ clubs satisfying following three conditions. Let $A,B$ are arbitrary people among those 10, ...
1
vote
1answer
91 views

How many perfect squares exist? [closed]

Consider a set of $1985$ positive integers not necessarily distinct. Every number in set can be written in the form $p_1^{{\alpha _1}}p_2^{{\alpha _2}} \cdots p_9^{{\alpha _9}}$ where ...
3
votes
1answer
26 views

Counting the set with restrictions

This is an interesting problem from a high school programming contest in China. Define a set $S=\{1,2,\ldots , n\}$ with $n$ elements. We need to choose some subset $A_{i,j}(A_{i,j}\subseteq ...
3
votes
2answers
54 views

1987 AIME Problem #4

The Question: Find the area of the region enclosed by the graph of $|x-60|+|y|=|\frac x4|$. Answer: What I know: Because of all the absolute values I only need to find one side of the graph of ...
10
votes
2answers
293 views

Russia (2000) contest:Prove the existence of a pair of rows and columns with intersections differently coloured

We have a $100\times100$ board divided into $10^4$ unit squares. These squares are coloured with four colours so that every row and every column has $25$ squares of each colour. Prove that there ...
2
votes
2answers
56 views

Combinatorics - Number of ways to fill a 3x3 grid with 0's and 1's such that there is at least one zero in each column and row

There seems to be a simple answer for this problem, but I just can't figure it out. I know there must be at least 3-9 zeros for a valid arrangement, and that there are $3!$ (6) possible combinations ...
1
vote
4answers
84 views

Find $N$ with $N^2=10^4M+N$

I'm a high school student in France and I participated in a math olympiad, and there was a question which I found impossible to solve. Maybe they are here some people who can help me: We take a ...
0
votes
2answers
19 views

Computing first k digits and last k digits of a large number using logarithm

How do we compute the first $k$ digits and last $k$ digits of a large number say $2^{N-1}$ for bigger values of $N$ using logarithms? An example for the algorithm will be greatly appreciated. I got ...
0
votes
0answers
54 views

Prove $(a-b)^2+(b-c)^2+(c-a)^2 \geq a^2+b^2+c^2-3\sqrt[3]{a^2b^2c^2}$ [duplicate]

For nonnegative real numbers $a,b$ and $c$, prove $$(a-b)^2+(b-c)^2+(c-a)^2 \geq a^2+b^2+c^2-3\sqrt[3]{a^2b^2c^2}$$ It is clear that the inequality is equivalent to $a^2+b^2+c^2 + ...
1
vote
2answers
75 views

Geometry - Tangent circles

Let chords AC and BD of a circle ω intersect at P. A smaller circle ω1 is tangent to ω at T and to segments AP and DP at E and F respectively. (a) Prove that ray T E bisects arc ABC of ω. (b) Let ...
0
votes
1answer
25 views

Geometry - Angle chasing

An interior point $P$ is chosen in the rectangle ABCD such that $∠AP D + ∠BP C = 180◦$ . Find $∠DAP + ∠BCP$ Since also $\angle APB + \angle CPD = 180◦$ and by symmetry (you can swap $B$ and $D$) ...
6
votes
2answers
133 views

Show that $x=y+z$ for all $x \in S$

We are given a set $S$ as a subset of the rational numbers defined by: $0 \notin S$ If $s_1 , s_2 \in S$, then $\frac {s_1}{s_2} \in S$ There exists a nonzero rational number $q \notin S$ such ...
9
votes
1answer
535 views

2016 Spain Math Olympiad final stage, problem 2

Given a prime $p$. Prove that there exist $\alpha$ such that $p|\alpha(\alpha-1)+3$, if and only if there exist $\beta$ such that $p|\beta(\beta-1)+25$. My solution: Using quadratic residuu we ...
3
votes
1answer
57 views

Functional Equation - Rational

Fing all functions $g: R \to R$ such that, $g(x+y) + g(x)g(y) = g(xy) + g(x) + g(y)$ I have shown that $g(x) = 0$ for all $x$ and $g(x) = 2$ for all $x$ are solutions. I have also show that $g(x) = ...
0
votes
1answer
50 views

Inequalities - AM-GM

Let $H_n = 1 + 1/2 + 1/3 + ... + 1/n$ Prove that; $H_n + n$ $\geq$ n$(n+1)^\frac{1}{n}$ for $n$ $\leq$ $2$ I have tried writing $H_n + n = 1/2 + 1/3 +...+ 1/n + (n+1)$ but am left with an $n!$ in ...
4
votes
1answer
80 views

Spanish Math Olympiad

In the circumscircle of a triangle $ABC$, let $A_1$ be the point diametrically opposed to the vertex $A$. Let $A'$ the intersection point of $AA'$ and $BC$. The perpendicular to the line $AA'$ from ...
0
votes
1answer
56 views

complex integrals springing from IMO question?

I was looking at problem $A2$ here: https://www.imo-official.org/problems/IMO2006SL.pdf The comment following $A2$ suggests that complex contours lead to a nice expression, but I don’t see the ...
11
votes
3answers
134 views

Doubt regarding divisibility of the expression: $1^{101}+2^{101} \cdot \cdot \cdot +2016^{101}$

In an interesting contest question I recently encountered, I chanced upon a question I couldn't solve. $$\sum^{2016}_{i=1}i^{101}$$ is divisible by: (a)2014 (b)2015 (c)2016 (d)2017 How would I ...
0
votes
0answers
16 views

Two polylines could form a convex quadrilateral

A close polyline $\Delta$ with length $n$ here means a sequence of segments $A_1A_2,\ldots, A_{n-1}A_n$ and $A_nA_1$ so that there are no two segments $A_iA_{i+1}$ and $A_jA_{j+1}$, with $1\leq ...
0
votes
0answers
20 views

Projective Geometry - Pole/Polar

A circle is inscribed in quadrilateral $ABCD$ so that it touches sides $AB, BC, CD, DA$ at $E, F, G, H$ respectively. (a) Show that lines $AC, EF, GH$ are concurrent. In fact, they concur at ...
0
votes
2answers
33 views

Alfonzo is a kangaroo. Each second, he takes a 2-yard hop forward with probability 60% or a 1-yard hop backward with probability 40%.

In 15 seconds, I believe that, in expectation, Alfonzo has hopped 15[(2*0.6) + (-1 * 0.4)] = 15*0.8 = 12 yards forward. How long, in seconds, before Alfonzo takes a hop backwards?
0
votes
1answer
19 views

Reflection of median concurs with intersection of diagonals of a cyclic quadrilateral

In the non-isosceles triangle $ABC$ an altitude from $A$ meets side $BC$ in $D$ . Let $M$ be the midpoint of $BC$ and let $N$ be the reflection of $M$ in $D$ . The circumcircle of triangle $AMN$ ...
4
votes
0answers
70 views

Sally and I EACH flip a fair coin.

We then each guess what the other person got: I guess what side Sally's coin landed on, and Sally guesses what side my coin landed on. We win as long as at least one of us is correct. I understand ...
3
votes
1answer
42 views

Let $x_1,x_2,\dots ,x_{50}$ be $50$ integers such that the sum of any $6$ of them is 24, then:

Let $x_1,x_2,\dots,x_{50}$ be $50$ integers such that the sum of any $6$ of them is $24$, then which option is true the largest of $x_i$ equals $6$. the smallest of $x_i$ equals $3$. ...
2
votes
0answers
95 views

Prove that $a_{n+1}a_{n-2}-a_{n}a_{n-1}=1$ is always an integer [duplicate]

We are given the sequence $a_1, ... , a_n$ defined by $a_1=a_2=a_3=1$, and $$a_{n+1}a_{n-2}-a_{n}a_{n-1}=1.$$ Prove that $a_k$ is an integer for all positive integers $k$. The most obvious idea to me ...
2
votes
2answers
39 views

Prove that the area of their union is greater than $\frac{2}{9}S$

A finite set of unit circles is given in a plane such that the area of their union $U$ is $S$. Prove that there exists a subset of mutually disjoint circles such that the area of their union is ...
6
votes
5answers
774 views

Do there exist several positive real numbers such that their sum is $1$ and sum of their squares is less than $0.01$

Do there exist several positive real numbers such that their sum is $1$ and sum of their squares is less than $0.01$? My Attempt: Let there are $n$ real numbers and we call them ...
2
votes
1answer
101 views

Divisibility of a summation

Let $n , l, k, p$ be positive integers, and $1\leq p\leq n$. Let $B(n, l, k, p)$ be the cardinality of the following set \begin{eqnarray} \{(a_1, a_2, \cdots, a_n)\in\mathbb{Z}^{\oplus n}|\ \ 0\leq ...
1
vote
1answer
82 views

Find the largest integer $n$ such that $n^2$ is the difference of two consecutive cubes and $2n +79$ is a perfect square.

Find the largest integer $n$ such that $n^2$ is the difference of two consecutive cubes and $2n +79$ is a perfect square. This is an AIME problem. I have been trying and have been going round in ...
4
votes
1answer
104 views

Problem PUTNAM of the day - Harvard Mathematics department [closed]

Let $f$ be a twice-differentiable real-valued function satisfying $f(x)+f''(x)= -xg(x)f'(x)$, where $g(x) \geq 0$ for all real $x$. Prove that $|f(x)|$ is bound. Honnestly I worked on this problem ...
1
vote
3answers
121 views

What's the ratio between these two lengths? plane geometry problem

I'm thinking following plane geometry problem. Question: There is a parallelogram $ABCD$ such that $\overline{AC}:\overline{BD}=2:1$ and $\overline{AB}\neq\overline{BC}$. Draw a line which is ...
4
votes
1answer
67 views

Math Problem: Forty-nine points

49 points are marked on a sheet of paper in a square. Adjacent points horizontally or vertically are separated by exactly 1 centimetre. How many straight lines of length 5 centimetres can be ...
2
votes
1answer
52 views

How can I get better at algorithmic thinking?

I have been practising for a an upcoming algorithmic thinking competition but have always found that when doing the past papers, I have never had enough time left to finish. I can do basically all of ...
1
vote
2answers
85 views

If $\det(A+B)=\det(A+2B)=\det(A+3B)=1$ and $AB=BA$ then $B^2=0$

Prove that if $\det(A+B)=\det(A+2B)=\det(A+3B)=1$ and $AB=BA$ then $B^2=0$. A problem from a math competition. $A$, $B$ are 2 by 2 complex matrices. I've tried using Cayley Hamilton theorem, on ...
6
votes
1answer
78 views

$A^2$ $B=A^2-B$ then $AB=BA$

If for $2$ real $n$ by $n$ matrices we have $A^2B=A^2-B$ then prove that the two matrices commute. This is a problem from a competition. I've tried several manipulations but none of them work. ...
9
votes
0answers
167 views

solve in positive integers sum of squares of sines equation

Find all positive integer triples $(l,m,n)$ such that $\sin^2\frac{\pi}{n}+\sin^2\frac{\pi}{m}=\sin^2\frac{\pi}{l}$. I have found the solutions $(m,m,1)$ for any $m\in\mathbb{Z}^+$, and also ...
4
votes
0answers
37 views

Functional division $\max(f(x+y),f(x-y))\mid \min(xf(y)-yf(x), xy)$

As the title suggests, the problem here is: Find all functions $f:\mathbb{Z}\to\mathbb{N}$ such that, for every $x,y\in\mathbb{Z}$, we have $$\max(f(x+y),f(x-y))\mid \min(xf(y)-yf(x), xy)$$ I ...
2
votes
2answers
284 views

Indistinguishable pairs, distinguishable triples of metal circles in key-ring jumble.

The following problem was part of a $\pi$-day contest sponsored by Pizza Hut and written by John H. Conway: My key-rings are metal circles of diameter about two inches. They are all linked ...
0
votes
2answers
155 views

Guess the number. Maximizing expected winnings? [closed]

A man in a trench coat approaches you and pulls an envelope from his pocket. He tells you that it contains a sum of money in bills, anywhere from 1 dollar up to 1,000 dollars. He says that if you can ...
2
votes
4answers
101 views

Prove $x^2+y^4=1994$

Let $x$ and $y$ positive integers with $y>3$, and $$x^2+y^4=2(x-6)^2+2(y+1)^2$$ Prove that $x^2+y^4=1994$. I've tried finding an upper bound on the value of $x$ or $y$, but without sucess. Can ...
2
votes
1answer
42 views

The Functional Equation $f(mn)=f(m)f(n)$ where $f:\mathbb{N}\rightarrow \mathbb{R}$, $f(2)=2$, and $f(m) > fn)$ if $m>n$.

The following is exercise 3.3 from Terence Tao's "Solving Mathematical Problems." Emphasis added. Suppose $f$ is a function on the positive integers which takes real values with the following ...
3
votes
1answer
34 views

An infimum of a double integral on the unit disk

The following question comes from Arnold's Trivium of $1991$ and it is problem $68$. I do not have a solution neither can I come up with something. Find $$\inf \iint \limits_{x^2+y^2 \leq 1} \left[ ...
-3
votes
1answer
31 views

Get rate in Excel without using “RATE” command [closed]

I want your help to calculate rate of interest in excel without using RATE command. (so that I can try to convert the formula in php) This is the code we use in Excel B5= 60 B4= 16070 B3= 750000 B6= ...
0
votes
7answers
142 views

How many real roots does the equation $e^x-x^2=0$ have?

How many real roots does the equation $e^x-x^2=0$ have? I can see from wolfram that the curve cuts X-axis only once. How do you go about solving it?
0
votes
0answers
19 views

What are the valid deductions of a congruence equation?

So I was just sitting here, doing math, and I came over this: $9+16a\equiv 12 \space(mod 5) $ Obviously, through some simple manipulations: $9+16a-15a-9\equiv 12-9 \space(mod 5) $ $a\equiv 3 $ ...
0
votes
1answer
32 views

Write the set of all positive integers in triangular array as

1 3 6 10 15 . . 2 5 9 14 . . . 4 8 13 . . . . 7 12 . . . . . 11 Find the row number and column number where 20096 occurs. For example 8 appears in the third row and second column.
0
votes
8answers
164 views

The number of real roots of $x^5 + 2x^3 + x^2 + 2 = 0 $ is

The number of real roots of $x^5 + 2x^3 + x^2 + 2 = 0 $ is A. 0; B. 3; C. 5; D. 1. I don't know how to solve this.
3
votes
2answers
56 views

Number of common roots of $x^3 + 2 x^2 +2x +1 = 0$ and $x^{200} + x^{130} + 1 = 0 $

The equations $x^3 + 2 x^2 +2x +1 = 0$ and $x^{200} + x^{130} + 1 = 0 $ have exactly one common root; no common root; exactly three common roots; exactly two common roots. I factored the first ...
-2
votes
1answer
45 views

Let $a_1 = 2$ and for all natural number n, define $a_{n+1}= a_{n}(a_{n}+1)$. Then as $n\rightarrow \infty$, the number of prime factors of $a_{n}$ [closed]

Let $a_1 = 2$ and for all natural number n, define $a_{n+1}= a_{n}(a_{n}+1)$. Then as $n\rightarrow \infty$, the number of prime factors of $a_{n}$: goes to infinity. goes to a finite limit. ...
3
votes
0answers
114 views

(2016 China team selection Test) with a complex inequality

Let $z_{1},z_{2},z_{3}$ be complex numbers, such that: $z_{1}+z_{2}+z_{3}=0,|z_{i}|<1,i=1,2,3$. Find the minimum of the positive $A$ such that: ...