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Questions tagged [contest-math]

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

-3
votes
0answers
77 views

If $f(x) = f(a-x)$ and $g(x) +g(a-x) = 2$ for $f, g$ continuous, then what is $\int_0^a f(x)g(x)dx$? [on hold]

If $f(x)$ and $g(x)$ are continuous functions satisfying $f(x) = f(a-x)$ and $g(x) +g(a-x) = 2$, then what is the integral of $f(x)g(x)$ from $x=0$ to $x=a$ equal to?
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votes
2answers
40 views

Proving a simple inequality involving positive reals

"Let $x,y,z> 0$ with $xyz=1$. Prove that $x+y+z\leq x^2+y^2+z^2$." There's the possibility that there is a typo in this question, but I've graphed this computationally and the inequality seems to ...
1
vote
2answers
88 views

find the $a_{111}$ if $a_{n+11}=a_{n+5}+2a_{n}$

Let sequence $\{a_{n}\}$ such $$a_{1}=a_{2}=\cdots=a_{10}=0,a_{11}=2$$,such $$a_{n+11}=a_{n+5}+2a_{n}$$ Find the $a_{111}$ I want show this sequence is Periodic series。following is some try $$a_{n+...
-1
votes
0answers
30 views

Chapter 1 Sets . Class 11th CBSE . [on hold]

In a group of 50 persons, 14 drink tea, but not coffee and 30 drinking tea. Find: How may drink tea and coffee both How many drink coffee but not tea
1
vote
3answers
41 views

What does it mean for points to be picked uniformly and independently?

I saw a which said that points were picked uniformly and independently. I have a feeling this is important for the solution but I am not sure what they mean by uniformly and independently. Any help ...
1
vote
1answer
52 views

A direct way to an inequality : Ferrari's identities

I would like to submit a recent answer that I gave (and I have deleted) where someone tolds me that I was "total wrong" this is the following : Begin Prove that $$\frac{x^2+y^2+z^2}{2}\geq (\alpha\...
-4
votes
2answers
66 views

Maximum area of Triangle

The points $A,B,C$ of a triangle are at the distances $7,15,15$ from the origin. What is the maximum area of the triangle $ABC$?
4
votes
1answer
58 views

Find all polynomials $P(x) \in \mathbb{Z}[x]$ such that if $P(s)$ and $P(t)$ are both integers, then $P(s+t)$ is also an integer

Find all polynomials $P(x)$ with integer coefficients such that for all real numbers $s$ and $t$, if $P(s)$ and $P(t)$ are both integers, then $P(s+t)$ is also an integer. This is a problem ...
0
votes
2answers
71 views

Putnam and Beyond: Problem 87 (Matrices and algebra)

Below is the problem as stated in Putnam and Beyond: Let $A$ and $B$ be tw0 $n\times n$ matrices that commute and such that for some positive integers $p$ and $q$, $A^p=I_n$ and $B^q=O_n$. Prove ...
3
votes
3answers
70 views

Ratio of angles in a triangle

came across this question in a math contest and can't quite figure out an approach to the question. Triangle $ABC$ has $AB=AC\neq BC$ and $∠BAC ≤ 90º$. $P$ lies on $AC$, and $Q$ lies on $AB$ such ...
-4
votes
1answer
59 views

Difficulty in this Olympiad problem [closed]

Peter was cutting a pipe with an outside diameter of 20cm. When the cut was just through the wall of the pipe, it was 10 cm in length. How thick was the wall of the pipe in centimeters?
1
vote
0answers
64 views

Putnam and Beyond (Methods of proof) Problem 20

Here is the problem as stated in "Putnam and Beyond": Given a sequence of integers $x_1, x_2,...,x_n$ whose sum is 1, prove that exactly one of the cyclic shifts $x_1,x_2,..., x_n ; x_2,...,...
0
votes
2answers
74 views

For $a$, $b$, $c$ distinct integers, and $P$ a polynomial with integer coefficients, $P(a)=b$, $P(b)=c$, $P(c)=a$ cannot be satisfied simultaneously

I have pasted the problem and its solution below. My question is about (1), (2), and (3). What does $P_1(x)$, $P_2(x)$, and $P_3(x)$ represent? Near the end they say it's a positive integer. So ...
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votes
0answers
36 views

What is the meaning of the following statement about subsequences? [closed]

Sum of subsequence * GCD of subsequence over all K length subsequence of a given array. array=[1,2] K=1 answer = 5 How the answer is 5?
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votes
1answer
269 views

question asked in recent hiring [closed]

Consider an n∗n matrix A consisting of all zeros. Find the number of ways to fill this matrix with exactly n+2 ones such that the permanent of the matrix is zero
1
vote
1answer
86 views

Putnam 1985 B5 Integration by Substitution

Putnam 1985 B5: Evaluate $\int_{0}^\infty t^{-1/2}e^{-1985(t+t^{-1})}dt$. You may assume that $\int_{-\infty}^\infty e^{-x^2} dx=\sqrt{\pi}$. Following the solution on https://artofproblemsolving.com/...
0
votes
1answer
59 views

From book : Putnam and beyond

Original at https://i.stack.imgur.com/Zn8RE.jpg Lucas' theorem. The zeros of the derivative $P'(z)$ of a polynomial $P(z)$ lie in the convex hull of the zeros of $P(z)$. Proof. Because any convex ...
-1
votes
1answer
52 views

Show that $x-1$ is a factor of $P(x)$ where $P(x^5)+xQ(x^5)+x^2R(x^5)=\big(x^4+x^3+x^2+x+1\big)S(x)$ [duplicate]

For the problem below, my first step was to rewrite the right side as $$\big(x^4+x^3+x^2+x+1\big)S(x)=\frac{x^5-1}{x-1}S(x)$$ From there I can isolate $P(x^5)$ which gives me $$P(x^5)=\dfrac{S(x)(x^...
2
votes
2answers
69 views

Proving a line perpendicular to another in a circle

I got a question at an exam recently, and was unable to solve it. Let $\Gamma_1$ be a circle with a chord $CD$ and a diameter $AB\perp CD$ at $N$, with $AN\geq BN$. A circle $\Gamma_2$ is drawn ...
-1
votes
0answers
60 views

Question about a inequality of Mister Rozenberg

Hello I would like to prove this : Let $a$,$b$,$c$ be real positive numbers differents of zero such that $|a|\geq |b|\geq |c|\geq 1$ then we have $$0.5+\sum_{cyc}\frac{ab}{a^2+b^2+3c^2}\geq \frac{...
0
votes
1answer
29 views

Points A, B and C on a circle of radius r are situated so that AB = AC, AB > r, and the length of minor arc BC is r.

Points A, B and C on a circle of radius r are situated so that AB = AC, AB > r, and the length of minor arc BC is r. If angles are measured in radians, then AB/BC = ? A) 1/2 csc(1/4) B) 2 cos(1/2) C) ...
2
votes
1answer
96 views

A question about IMO 1986 P3

IMO 1986 P3: To each vertex of a pentagon, we assign an integer $x_i$ with sum $s=\sum x_i>0$. If $x,y,z$ are numbers assigned to three successive vertices and if $y<0$, then we replace $(x,y,z)$...
13
votes
3answers
187 views

Putnam 2007 A5: Finite group $n$ elements order $p$, prove either $n=0$ or $p$ divides $n+1$

Putnam 2007 Question A5: "Suppose that a finite group has exactly $n$ elements of order $p$, where $p$ is a prime. Prove that either $n=0$ or $p$ divides $n+1$." I split this problem into two cases: ...
0
votes
2answers
120 views

Prove that $\frac{21n+4}{14n+3}$ is in lowest terms for any natural $n$. [closed]

Prove that the fraction $\dfrac{21n+4}{14n+3}$ is in lowest terms for any natural value of $n$.
0
votes
5answers
94 views

Solution to Putnam 2007 A-1

2007 A-1: Find the values of $\alpha$ for which the curves $y=\alpha x^2 + \alpha x + \frac{1}{24}$ and $x=\alpha y^2 + \alpha y + \frac{1}{24}$ are tangent to each other. My solution: Notice that ...
-1
votes
2answers
84 views

Integer part problem. [closed]

Find all positive integers which $\frac{n+2017}{[\sqrt{n+1}]}$ and $\frac{n+2018}{[\sqrt{n}]}$ are natural numbers.
2
votes
2answers
165 views

Making the sum of 5th power of integers, a perfect square.

Yesterday this question was posed in a contest. It contains pretty easy questions like asking range of $ab+bc+ca$ when $a^2+b^2+c^2=1$, etc. But this question is something else. I haven't been able ...
0
votes
2answers
62 views

Prove $\sqrt{x^2+1} + \frac{1}{\sqrt{x^2 +1}}\geq 2$ [duplicate]

I want to show why the last inequality in the problem below $\sqrt{x^2+1} + \frac{1}{\sqrt{x^2 +1}}\geq 2$ holds. It's clear that $x^2\geq 0$ and that equality holds when $x=0$ but how can I clearly ...
0
votes
1answer
53 views

AIME I 2000 Problem 8: Calculating the height of the liquid given Fraction of the Volume

As the title says, I'm looking at problem number 8 from AIME I 2000. https://artofproblemsolving.com/wiki/index.php?title=2000_AIME_I_Problems/Problem_8 I'm currently looking at solution 3 and I ...
-3
votes
2answers
55 views

Find an integer $a$ such that $(x-a)(x-10)+1=(x-b)(x-c)$ for some integers $b$ and $c$ [closed]

Can someone help with this Olympiad question? Find an integer $a$ such that $$(x-a) (x-10) +1$$ can be factored as $$(x-b) (x-c)$$ with $b$ and $c$ integer.
19
votes
0answers
412 views

Proving $10240…02401$ composite

I got this question recently, and have been unable to solve it. Prove that $1024\underbrace{00 \ldots\ldots 00}_{2014 \text{ times}}2401$ is composite. I have two different ways in mind. First is ...
1
vote
1answer
92 views

Prove that $\sqrt{1-x^2} + \sqrt{1-y^2} + \sqrt{1-z^2}\geq 4\sqrt{\frac{3-(x^2+y^2+z^2)}{5+x^2+y^2+z^2}}$.

If $x,y,z \in[0,1/2]$, with $x+y+z=1$, then prove that: $$\sqrt{1-x^2} + \sqrt{1-y^2} + \sqrt{1-z^2}\geq 4\sqrt{\frac{3-(x^2+y^2+z^2)}{5+x^2+y^2+z^2}}$$ OK so... I've tried to square the expression ...
1
vote
1answer
71 views

How to find the least positive $K$ such that $N^K \equiv 1 \pmod{P}$ where $P$ is prime and $P$ doesn't divide $N$?

I noticed that this $K$ is one of the divisors of $P-1$. So my solutions is looping on the divisors of $P-1$ in ascending order, till I find the first divisor $d$ where $N^d \equiv 1 \pmod{P}$. ...
0
votes
1answer
42 views

$2010$ $G1$, proving quadrilateral concyclic

I was solving the following question, from $2010\text{ IMO}$ shortlist. $G1.$ Let $ABC$ be an acute triangle with $D, E, F$ the feet of the altitudes lying on $BC, CA, AB$ respectively. One of the ...
8
votes
2answers
197 views

If $ab \mid c(c^2-c+1)$ and $c^2+1 \mid a+b$ then prove that $\{a, b\}=\{c, c^2-c+1 \}$

If $ab \mid c(c^2-c+1)$ and $c^2+1 \mid a+b$ then prove that $\{a, b\}=\{c, c^2-c+1 \}$ (equal sets), where $a$, $b$, and $c$ are positive integers. This is math contest problem (I don't know the ...
1
vote
1answer
34 views

Effectively reconstructing all original 5-tuples from a subset of their respective 4-tuples

Let's take integer numbers from [1..36]. We can generate 376992 different (order is not important) five-number-combinations like (1,3,5,7,12), etc. Such five-number-combinations always have five ...
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votes
1answer
38 views

Fundamental Solution for this Problem Solving Question

This question appeared in my math/ problem solving competition. It goes like this : Six competitors in a chess tournament played against each other once only. (That totals fifteen games). A win is ...
7
votes
2answers
543 views

Number theory question with floor function

Define $[a]$ as the largest integer not greater than $a$. For example, $\left[\frac{11}3\right]=3$. Given the function $$f(x)=\left[\frac x7\right]\left[\frac{37}x\right],$$ where $x$ is an ...
2
votes
0answers
34 views

If every point of the three dimensional space is coloured red, green or blue prove that one colour attains all distances

The problem as stated in the title is taken from "Putnam and beyond". Below is the problem as stated in the book: Every point of the three-dimensional space is coloured red, green, or blue. Prove ...
3
votes
2answers
92 views

There are $n$ different 3-element subsets $A_1,A_2,…,A_n$ of the set $\{1,2,…,n\}$, with $|A_i \cap A_j| \not= 1$ for all $i \not= j$.

Determine all possible values of positive integer $n$, such that there are $n$ different 3-element subsets $A_1,A_2,...,A_n$ of the set $\{1,2,...,n\}$, with $|A_i \cap A_j| \not= 1$ for all $i \not= ...
1
vote
1answer
903 views

A rope is hanging between two poles find this

Two poles of 50 m height are connected with 80m rope hanging between them. The rope is hanging 20m above the ground. What is the distance between the two poles? My attempt: I tried finding the ...
7
votes
2answers
179 views

Hour, second and minute hands

I've never found any such problem here on MathSE, so here is the problem: Given that hour, minute and second hands move continuously, how much time in the period from 0:00 to 12:00 (12 hours total) ...
4
votes
2answers
54 views

Is it true that $\sqrt{(a_1+b_1+c_1)(a_2+b_2+c_2)}\geq \sqrt{a_1a_2}+\sqrt{b_1b_2}+\sqrt{c_1c_2}$? [closed]

Let $a_1,b_1,c_1,a_2,b_2,c_2$ be positive real numbers. Is it true that $\sqrt{(a_1+b_1+c_1)(a_2+b_2+c_2)}\geq \sqrt{a_1a_2}+\sqrt{b_1b_2}+\sqrt{c_1c_2}$ If this is true, then I will have proved ...
2
votes
3answers
88 views

Angles on a point inside a triangle

Let $ABC$ be an isosceles triangle with $AB=AC$ and $∠BAC = 100$. A point $P$ inside the triangle $ABC$ satisfies that $∠CBP=35$ and $∠PCB= 30$. Find the measure, in degrees, of angle $∠BAP$. ...
0
votes
2answers
56 views

Show $\measuredangle ATB=\measuredangle CTA$ in excircle configuration

I recently came across the following problem in olympiad training material: A circle $k$ is internally tangent to sides $AB, AC$ of $\Delta ABC$ and its circumcircle in points $X, Y$ and $Z$, ...
-1
votes
2answers
59 views

Factorize $ (kx-y+z)(x+ky-z)(x-y-kz)-(kx+y-z)(x-ky-z)(x-y+kz) $ [closed]

I tried opening the brackets but it keeps getting complex and complex. Does anyone have any easier way of doing it. Please tell how did you observe.
0
votes
1answer
63 views

$3×7×11×\dots×2003$. Find last three digits of product [closed]

What are the last three digits of $3×7×11×\dots×2003$? $$\prod_{k=0}^{500}(4k+3)\bmod1000$$ I have tried a lot and spent a lot of time to but don't seem to be getting anywhere. I tried creating ...
1
vote
4answers
115 views

Line from incenter bisects side

I got a problem recently, and have been unable to solve it. Let $\Delta ABC$ with incenter $I$ and the incircle tangent to $BC$ at $D$. Let $M$ be the midpoint of $AD$. Prove that $MI$ bisects $BC$....
-4
votes
0answers
89 views

Geometry Problem From IMO 2018(Problem 1) [duplicate]

Let Γ be the circumcircle of an acute-angled triangle ABC. Points D and E lie on segments AB and AC, respectively, such that AD=AE. The perpendicular bisectors of BD and CE intersect the minor arcs AB ...
7
votes
2answers
867 views

Octagon inside a circle

An octagon which has side lengths 3, 3, 11, 11, 15, 15, 15 and 15 is inscribed in a circle. What is the area of the octagon? I tried using the cosine law on the triangles made when connected with ...