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

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AIME 2013 Solutions (divisiblity)

Problem 2 Find the number of five-digit positive integers, $n$, that satisfy the following conditions: (a) the number $n$ is divisible by $5,$ (b) the first and last digits of $n$ are equal, and (c) ...
13
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1answer
136 views

Positive integer solutions of $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+\cdots+\frac{n}{a_n}=\frac{a_1+a_2+a_3+\cdots+a_n}{2}$

Find all ordered tuples of positive integers $(a_1,a_2,a_3,\ldots,a_n)$ such that $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+\cdots+\frac{n}{a_n}=\frac{a_1+a_2+a_3+\cdots+a_n}{2}$ The only ...
2
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1answer
47 views

Prove that at least on of the numbers is positive

Prove that for $a,b,c \in \mathbb{R}$ at least of the the following numbers is non-negative: $$(a+b+c)^2 - 9ab; \quad (a+b+c)^2 - 9ac; \quad (a+b+c)^2-9bc$$ If not all of $a,b,c$ are negative or ...
0
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1answer
22 views

How do the sets have similar properties?

Where they say: "Assume that we have sets $S_k$ with the desired properties for all $k < n$ (line 4 in solution)" What properties are they talking about? They said: "Let $S_n = \{2a ...
0
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1answer
63 views

How to prove this form of $n$?

Show that every positive integer is a sum of one or more numbers of the form $2^r3^s,$ where $r$ and $s$ are nonnegative integers and no summand divides another. From: AOPS Putnam A1 Solution I ...
0
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1answer
38 views

In how many ways can the rooks be arranged? [duplicate]

In how many ways can 9 black and 9 white rooks be placed on a 6 × 6 chess board, so that no white rook can capture a black one? A rook can capture another piece if it is in the same rank (row) ...
0
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2answers
33 views

Probability using combinatorics

If Sapphira randomly chooses a 4-digit number (not beginning with zero) what is the probability that all four digits will be distinct? Let $$x = abcd$$ where they are digits. Lets see first how ...
2
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1answer
48 views

Geometrical Combinatorics About Rectangles

Part of a olympiad problem The answer is $$441 = 21^2$$ I fail to understand why. How do you solve this? I actually dont see why there are $9$ rectangles there either? Can someone give me a hand?
1
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1answer
28 views

Minimizing certain integral

Define $$F(a)=\int_0^{\pi/2}|\sin x-a\cos x|dx.$$ Find $a$ such that $F(a)$ is minimum. My attempt is to use differentiation under integral sign. Namely, we have $$F(a)=\int_0^{\arctan a}(a\cos ...
1
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3answers
72 views

If $2^n$ balls are divided into piles, they can always be brought into a single pile by a finite number of operations

$64$ balls are separated into several piles. At each step, one takes two different piles $A$ and $B$, having $p$ and $q$ balls respectively. Suppose $p\ge q$. Then one takes $q$ balls from pile $A$ ...
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1answer
51 views

Hazel chooses a die, rolls it and wins. Find the probability that she chose the biased die.

The probabilities of the scores on a biased dice are shown in the table below : ...
7
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1answer
152 views

Olympiad inequality (Cauchy/AM-GM sort)

Given $n$ positive numbers $x_1,\ldots,x_n$ ($n\ge 3$) such that the product $x_1x_2\cdots x_n=1$, show that ...
0
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3answers
78 views

Integral Solutions of $x+y=x^2-xy+y^2$

Find all integral solutions of $x+y=x^2-xy+y^2$ A modulo 2 analysis does not work here, only says cannot both be odd.
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2answers
53 views

Integer Solutions to Equation

Find the integer solutions to $x^2+xy+y^2=x^2y^2$ I have tried doing a modulo 2 analysis, which only says that $x, y$ are congruent modulo 2. But I cannot continue from here.
5
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1answer
302 views

Sum of the squares of numbers

Let $x$ and $y$ be the two numbers so that: $$x^2 + y^2 = A^2$$ $$xy = A^2 + 2A + 2$$ $$xy - x^2 - y^2 = 2A + 2$$ $$\frac{xy - x^2 - y^2}{2} -1 = A$$ So what can I do?
2
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1answer
39 views

Finding polynomial Coefficients

Let $f(x) = x^5 - x^4 + ax^3 + bx^2 + 8x + 4$ The root will make sure, $f(2) = 0$ Which shows: $$2^5 - 2^4 + a2^3 + 4b + 16 + 4 = 0$$ $$16 + 8a + 4b + 20 = 0 \implies 8a + 4b = -36 \implies 2a + ...
2
votes
1answer
49 views

How to solve this sum problem?

For the first radical section. $$\sqrt{1\times 2\times 3\times 4 + 1} - 1 = 1 + 3 + 1 - 1= 5 - 1 = 4$$ The second radical section. $$(\sqrt{2\times 3\times 4\times 5 + 1}) = 4 = 4 + 6 + 1 - 4 = ...
0
votes
1answer
23 views

Modulus after X concatenation

Given a number $N$ we can form another number $Y$ by concatenating $N$, $X$ times towards right. How to compute $Y \mod \space M$ efficiently? For example: if $N = 456$ and $X = 3$ then $Y = ...
1
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2answers
81 views

Placing $9$ black and $9$ white rooks on $ 6\times6$ board without any attacks between different colours

In how many ways can $9$ black and $9$ white rooks be placed on a $6 × 6$ chess board, so that no white rook can capture a black one? A rook can capture another piece if it is in the same rank (row) ...
2
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3answers
57 views

Give an example of four different subsets A, B, C and D of {1, 2, 3, 4} such that all intersections of two subsets are different.

My work, Suppose E={1,2,3,4} then power set of E is P(E)={ {}, {1}, {2}, {3}, {4} {1,2}, {2,3}, {3,4}, {1,3}, {1,4}, {2,4}, {1,2,3},{2,3,4}, {1,2,4}, {1,3,4}, {1,2,3,4} } Shows the possible subsets ...
5
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1answer
54 views

$\lim_{x\to 0}\frac{\sin(x)-x+\frac{x^3}{3!}-\frac{x^5}{5!}}{m x^n}=\frac{8}{7!}$

If $$\lim_{x\to 0}\dfrac{\sin(x)-x+\dfrac{x^3}{3!}-\dfrac{x^5}{5!}}{m x^n}=\dfrac{8}{7!}$$ then find $m+n$: My attempts: note that $$\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - ...
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2answers
61 views

Compare $4^x+1$ and $2^x+3^x$ for non-negative real $x$

Is it possible to find which on is bigger without calculus?I've thought that since $4^x=2^{2x}$ there could be a quadratic but it doesn't seem right,other then that I tried dividing by $2^x$,$3^x$ but ...
1
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0answers
53 views

Biggest number divisible by $99$

John has $1012$ stickers on which the numbers $1000,1001,\cdots,2010,2011$ are written.He wants to put them(not necessarily all) in a row so that he gets the biggest number which is divisible by ...
4
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1answer
43 views

Getting a specific formula for a sequence.

If: $$a_0 = \frac{5}{2}, a_k = a_{k-1}^{2} - 2$$ for $k \ge 1$. How do I get a general formula for $a_k$? With induction proof. I even tried calculating $a_1, a_2 ...$: $$a_0 = \frac{5}{2}$$ ...
4
votes
2answers
54 views

$k$ such that $n,2n,\dots,kn$ have odd sum of digits

is the following statement true or not? for any $k\in\mathbb{N}$ there exists $n\in\mathbb{N}$ such that all numbers $n,2n,\dots,kn$ have odd sum of digits? I have no idea... it may turn out hard ...
3
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1answer
48 views

An upper bound on the number of moves

Given a $ k \in \mathbb{N} $, the integers $ 1,2, \cdots ,4k-1, 4k $ are written on a blackboard. A move consists of replacing the numbers $ a, b, c, a+b+c $ with the numbers $ a+b, b+c, c+a $. Prove ...
5
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2answers
84 views

if $x^3-x\in\mathbb{Z}$ and $x^4-x\in\mathbb{Z}$ for some $x\in\mathbb{R}$, then $x\in\mathbb{Z}$.

Assume that $x^3-x\in\mathbb{Z}$ and $x^4-x\in\mathbb{Z}$ for some $x\in\mathbb{R}$. Prove that $x\in\mathbb{Z}$. my attempt: Let $a=x^3-x$ and consider polynomial $X^3-X-a$, then $x$ is a root of it ...
1
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1answer
51 views

Proving a sequence formula using induction [duplicate]

Suppose for $T_n$: $$T_n=(n+4)T_{n-1}-4nT_{n-2}+(4n-8)T_{n-3}$$ $$T_0=2,\quad T_1=3,\quad T_2=6$$ For integer, $n \ge 3$ I conjectured that: $$T_n = 2^n + n!$$ The above is actually TRUE. Using ...
0
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2answers
279 views

Putnam 1990 A1 Induction Help

A1. $(150,9,1,0,0,0,0,0,1,1,6,33)$ Let $$T_0=2,\quad T_1=3,\quad T_2=6,$$ and for $n\ge3$, $$T_n=(n+4)T_{n-1}-4nT_{n-2}+(4n-8)T_{n-3}.$$ The first few terms are ...
0
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1answer
48 views

Solving for $z^2 = x^2 -xy + y^2$

Recently, I came across the following solution to finding integer solutions for $z^2 = x^2 - xy + y^2$: $x = k(-n^2 -2mn)$ $y = k(m^2 - n^2)$ $z = k(mn + m^2 + n^2)$ I've been scratching my head ...
5
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1answer
68 views

A Representation Theory Problem in Putnam Competition

The following was the B6 problem of 1985 Putnam Competition: Suppose $G$ is a finite group (under matrix multiplication) of real $n\times n$ matrices $\{M_i\}, 1\leq i\leq r$. Suppose that ...
0
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1answer
29 views

powers, last 3 digits

For given $a\in\mathbb{N}$ define $x_1(a)=a$ and $x_{k+1}(a)=a^{x_k}$ for $k=1,2,\dots$. Find the last 3 digits of $\sum_{i=1}^{9}x_i(i)$. the obvious attempt is to work modulo 1000, but maybe ...
0
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1answer
138 views

Finding the largest 3-digit number $\; \overline{abc}\;$ s.t $\; \overline{abc}=100a+10b+c \equiv a+b^2+c^3$

This question comes from a maths contest (infer no calculators or other electronic calculating aids) for 14-16 year olds (infer no use of complicated theorems, but those accessible to high-school ...
6
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1answer
76 views

Composite $n$ such that $3^{n-1}-2^{n-1}$ is divisible by $n$

I'm stuck with the following olympiad problem (the solution to which I unfortunately do not possess): Show that there are infinitely many composite (i. e. nonprime) numbers $n$ such that ...
1
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4answers
82 views

How many rectangles can be found in this?

I saw there are 3 in each row, and 3 in each column hence, $$9 \cdot 9 = 81$$ But the answer is $$441$$ somehow? how do they get the answer?
2
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1answer
93 views

IMO problem number theory

Determine the greatest positive integer $k$ that satisfies the following property. The set of positive integers can be partitioned into $k$ subsets $A_1,A_2,A_3,\ldots,A_k$ such that for all integers ...
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2answers
67 views

Partitioning positive divisors of 100!

Is it possible to partition all positive divisors of 100! (including 1 and 100!) into 2 subsets so that each subset has the same number of integers and the product of all the divisors making up the ...
0
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1answer
43 views

How many possible different arrangements?

There are $1000$ numbers $1,2,3,...999,1000$ to be arranged in a line so that every number other than the rightmost differs by 1 from one or more of the numbers to its right. How many different ...
8
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1answer
113 views

A positive integer $n$ such that $S(n) = 1996\cdot S(3n)$

[Ireland 1996] Find a positive integer $n$ such that $S(n) = 1996\cdot S(3n)$, where $S$ stands for the sum of digits. The book "104 Number theory problems" gives the following solution: ...
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0answers
34 views

Is it possible to find out how many results were unexpected?

During a school year Andrew was given 40 mathematical problems as part of his assessment, one problem per week. As a result of marking he could receive 2,3,4 or 5 marks for each problem. Andrew called ...
6
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1answer
77 views

What is the maximum area two non intersecting circles can cover if they are contained in a unit square?

What is the maximum area two non intersecting circles can cover if they are contained in a unit square? I think that they cover the most area when in the following position: However I haven't been ...
0
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1answer
42 views

Perimeter problem involving different sized sticks?

Could you please help me find the answer to this question. I think it has something to do with grouping or pairing some numbers.I would appreciate easy-to-understand solutions. Thank you. There are ...
0
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1answer
25 views

Intersection point is in the triangle

On $X={\bf R}^2$ or $S^2(1)$, we have a triangle $\triangle ABC$ whose perimeter is small. On $D\in \overline{BC}$, let $$ r_1:=|BD|,\ r_2:=|CD| $$ Consider spheres $S(B,r_1),\ S(C,r_2),\ S(A,r)$. ...
0
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3answers
50 views

Order of Natural Numbers in Algorithms

Could anyone describe, why this is a True statements? if $f_i$ be a function of natural numbers to natural numbers and $f_i(n)=O(n)$ then $\Sigma_{i=1}^{n} f_i(n)=O(n^2) $
2
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2answers
42 views

Proof by Contradiction Minimum Value Proof $f(x)$

Focusing on $x=a$ first. My Proof: Assume $f'(a) < 0$ $f(x) \le f(x_1)$ for all $x$, this follows from the extreme value theorem. $$f'(x_1) = 0$$ Because it is a maximum. $$\exists x_4 ...
0
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2answers
71 views

Assuming $a_k + b_k = 1$ (Putnam 2003) [duplicate]

I do not understand as I wrote in a previous question: solution: I see that we can scale: $u_k$ but I do not understand why it is legal to say $a_k + b_k = 1$ what is $a_1 = 20$ and $b_1 = 1$ ...
0
votes
0answers
13 views

Proof for Scaling homogeneous inequalities. [duplicate]

Apparently, there exists a theorem, which says if a inequalities is homoegenous the terms can be multiplied by a scale $u_k$ like: $$(a_1 a_2 ... a_n)^{1/n} + (b_1 b_2 ... b_n)^{1/n} \le [(a_1 + ...
4
votes
2answers
100 views

prove that $f$ is periodic

A function $f\colon\mathbb{R}\to(0,\infty)$ satisfies equation $f(x)=f(x+64)+f(x+1999)-f(x+2063)$. Prove that $f$ is periodic. I'm quite sure that 1999 and 64 are random numbers (probably 1999 = year ...
8
votes
4answers
148 views

$k$ with an even sum of digits for all multiples of $k$?

Is there a number $k\in\mathbb{N}$ such that $k\cdot n$ has an even sum of digits for all $n\in\mathbb{N}$? I would be grateful for any ideas of how to attack this problem...
1
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1answer
20 views

Possible combinations

There are 1 to 24 numbers (1,2,3...24). How many possible combinations of 12 combine numbers will result with a sum of 146 when you add those 12 numbers? So meaning you have to combine 12 numbers ...