# Questions tagged [contest-math]

For questions about mathematics competitions or the questions that typically appear in math competitions.

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### Intergration Bee Intergral [duplicate]

In the Columbia 2022 intergration bee, one of the problems was $$\int _0 ^{\frac{\pi}{4}} \dfrac{\cos^{2022} (x) }{\sin^{2022} (x) + \cos^{2022} (x) } \, dx.$$ I've tried solving it, but have gotten ...
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### Is contest math essential?

Context: I performed well in contests in younger years, then I abandoned math for a while. Its only after high school that I started doing math again and enrolled at math college, where I did well. ...
1 vote
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### What is the minimum value of the function $f(x)= \frac{x^2+3x-6}{x^2+3x+6}$?

I was trying to use the differentiation method to find the minimum value of the person but it did not give any result, I mean when I differentiated this function and equated to zero for finding the ...
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### Possible ways of solving an Olympiad question

"A piece of land of a square shape with dimensions 10m x 10m is divided into 100 square parcels with dimensions 1m x 1m. Initially, 9 of the parcels are overgrown by weed. If a parcel is ...
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### Functional equation $(x-y)(f(x)+f(y))=(x+y)f(x-y)$ [closed]

I have proved that the functional equation, $(x-y)(f(x)+f(y))=(x+y)f(x-y)$, has the following results: $f(0)=0$, $f$ is an odd function. It is clear that $f(x)=cx$ for $c \in \mathbb R$ is a ...
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### I have a set of five positive integers whose mean is 100. When I remove the median, the mean increases by 5 and the median decreases by 5. [closed]

What is the maximum value of the largest number I have? (Purple Comet 2016). I am struggling to find an explanation. Thank you.
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### For $\frac{16x-3}{x^3+x} = \frac{bx+c}{x^2+1}+\frac{a}{x}$, what is a+b+c?

I only got to getting rid of the denominator and turning the equation into 16x-3 = ax^2+a+bx^2+cx, but from that on I don't know what to do.
1 vote
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### Is My Solution Valid?

Question from the 1999 Bulgarian Math Olympiad: Find all pairs $(x,y)\in\mathbb{Z}$ satisfying $$x^3=y^3+2y^2+1$$ My first approach was to take the cube root of both sides: $$x=\sqrt[3]{y^3+2y^2+1}$$ ...
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### How to calculate the orthogonality error between sine and cosine wave?

As the picture below(assume the magnitude is the same),the zero-crossing points of the SIN and COS signals do not occur at the precise distance of 90°.So I want to figure out the φ which is φx-φy. ...
1 vote
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### Find sum of 5-digit numbers that can be formed using 0, 0, 1, 2, 3, 4.

Find sum of 5-digit numbers that can be formed using 0, 0, 1, 2, 3, 4. I tried solving it by cases but I don't understand how to deal with Identical digits. Hints would be more appreciated as I want ...
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### A positive integer gets reduced by nine times when one of its digits is deleted and the resultant number is divisible by 9.

The question says A positive integer gets reduced by nine times when one of its digits is deleted and the resultant number is divisible by 9. Prove that to divide the resultant number by 9, it is ...
1 vote
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### Prove the zeroes of a polynomial are all real and distinct

For a polynomial $P(x) = (x-x_1)(x-x_2)\cdots (x-x_n)$ with distinct real zeroes, $x_1 < x_2<\cdots < x_n$, prove or disprove that all zeroes of $f(x) := P'(x) - kP(x)$ are real and that for ...
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### What is the value of $a_1a_2\cdots a_{2019}$?

Let $a_1=\frac 34$ and for any $n\geq2$ $4a_n=4a_{n-1}+\frac {2n+1}{1^3+2^3+\cdots n^3 }$. What is the value of $a_1a_2\cdots a_{2019}$? I tried $1^3+2^3+\cdots +n^3=\frac {n^2(n+1)^2}{4}$ and I ...
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1 vote
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### A alone takes $a$ more days than A and B, and B alone takes $b$ more days than A and B. Find how long for A and B.

What's said in the box is not clear. The author meant to say, If A working alone takes a days more than A and B together, and B working alone takes b days more than A and B together, then the number ...
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1 vote
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### Help in proving $\sqrt{a-1}+\sqrt{b-1}+\sqrt{c-1}\leq \sqrt{a(bc+1)}$

I was practising inequality problems from an online handout. I stumbled across this inequality problem which seems easy to solve but I wasn't able to move anywhere while trying to solve it; Prove ...
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### Prove that points $E, H,$ and $F$ are collinear

Let $\triangle ABC$ be a triangle. Let $M$ be the midpoint of side $[BC]$. $H,$ and $I$ are respectively the orthocenter and incenter of $\triangle ABC$. Let $D = (MH)\cap(AI)$. $E$ and $F$ are the ...
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### Number of the Sides of a polygon can be seen from the point E.

Question: Consider a regular $2n + 1$-gon $P$ in the plane, where n is a positive integer. We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$, if ...
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### APMO 2020 Geometry Problem | Proving lines to be concurrent

PROBLEM Let $\Gamma$ be the circumcircle of $∆ABC$. Let $D$ be a point on the side $BC$. The tangent to $\Gamma$ at $A$ intersects the parallel line to $BA$ through $D$ at point $E$. The segment $CE$ ...
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### Evaluating $\arctan(u)+\arctan(v)+\arctan(w)$, where $u$, $v$, $w$ are the zeros of $P(x) = x^3 - 10x+11$

I'm trying to understand the solution to the following question, shown below: The zeros of the polynomial $P(x) = x^3 - 10x+11$ are $u,v,w$. Determine the value of $\arctan(u)+\arctan(v) +\arctan(w)$....
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### Prove that ⌊2x⌋+⌊2y⌋≥⌊x⌋+⌊y⌋+⌊x+y⌋ for all real x,y. [duplicate]

Prove that ⌊2x⌋+⌊2y⌋≥⌊x⌋+⌊y⌋+⌊x+y⌋ for all real x,y. Consider the cases that (i) {x},{y}<1/2, (ii) 1/2≤{x} and {y}<1/2 or {x}<1/2 and 1/2≤{y}, and (iii) 1/2≤{x},{y}.
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### A permutation problem regarding number of ways of a given permutation

Given a permutation of N length. Lets say the permutation is : p1,p2,....,pn. How many tuples [a,b,c,d] such that: pa < pc and pb > pd. Example: 5 3 6 1 4 2 for this permutation of 6 length, ...
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### Let $x_1,x_2,..,x_{2022}\in \Bbb{R-R^-}$ with $x_k+x_{k+1}+x_{k+2} \leq2\text{ for }k=1,2,....,2020$. Prove $\sum_{k=1}^{2020} x_kx_{k+2} \leq1010$ [closed]

QUESTION Let $x_1,x_2,...,x_{2022}$ be non-negative real numbers such that $$x_k + x_{k+1}+x_{k+2} \leq 2 \text{ for } k=1,2,....,2020$$ Prove that $$\sum_{k=1}^{2020} x_k\cdot x_{k+2} \leq 1010$$ MY ...
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### General advice on tackling Olympiad level maths [closed]

Firstly, a bit of context: I am studying for a college entrance exam, which is mainly maths at the olympiad level. I am completely new to this kind of maths. Due to circumstances outside of my control,...
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Let $n$ be an odd integer greater than $1$. Find the number of permutations $\sigma$ of the set $\{1,\cdots, n\}$ for which \$|\sigma(1) - 1| + |\sigma(2) - 2|+\cdots + |\sigma(n) - n| = \frac{n^2 - 1}...