# Tagged Questions

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

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### 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 $3$. $x_{16}=x_{34}$....
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### 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 ...
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### 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 ...
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### 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 $x_{1},x_{2},..,x_{n}$...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $A+B$...
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### $A^2$ $B=A^2-B$ 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. Can'...
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### Squares, midpoints and heights

Let $ABC$ be a traingle, we draw squares on the sides $AB$ and $AC$, now we draw a segment from the vertexes of the square which are closer and then it forms a triangle, so prove that the line throw A ...
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### Filling a grid square with 0,1,2 [duplicate]

Each of the 25 cells in a five-by-five grid square is filled with a 0, 1, or 2 in such a way that the numbers written in neighboring cells differ from the number in that cell by 1. Two cells are ...
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### Absolute difference and probability [closed]

Fifty tickets numbered with consecutive integers are in a jar. Two are drawn at random and without replacement. What is the probability that the absolute difference between the two numbers is 10 or ...
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### Pentagon Problem

In a regular pentagon ABCDE, point M is the midpoint of side AE, and segments AC and BM intersect at point Z. If ZA = 3, what is the value of AB? (The answer is supposed to be in simplest radical form....
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### Shortest distance between two circles

What is the shortest distance, in units, between the circles $(x - 9)^2 + (y - 5)^2 = 6.25$ and $(x + 6)^2 + (y + 3)^2 = 49$? Express your answer as a decimal to the nearest tenth. So I know that ...
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### heart rate problem [closed]

The average heart rate of a shrew is 800 beats per minute, while an elephant has a heart rate of 25 beats per minute. If 1 billion heartbeats is a natural life span for each animal, on average, how ...
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### How many different paths from top to bottom spell ALGEBRA?

Starting with the A on top and only moving one letter at a time down to the left or down to the right, how many different paths from top to bottom spell ALGEBRA? ...
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### How many tokens would person A have under these conditions?

Persons A and B each have a positive integer number of tokens, and the number of tokens B has is a square number less than 100. B says to A, "If you give me all of your tokens, my total number of ...
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### Three-digit numbers whose digits and digit sum are all prime

How many 3-digit numbers are there such that each of the digits is prime, and the sum of the digits is prime? Shouldn't it be $0$, because the only one digit primes are $2,3,5,7$, and so the ...
### Find the minimum value of $\frac{4}{4-x^2} + \frac{9}{9-y^2}$
Let $x, y ∈ (−2, 2)$ and $xy = −1$. Find the minimum value of $\frac{4}{4-x^2} + \frac{9}{9-y^2}$ ? My Attempt let $t=\frac{4}{4-x^2} + \frac{9}{9-y^2}$ , replacing $y$ by $- \frac{1}{x}$ we get \$t=\...