# Questions tagged [contest-math]

For questions about mathematics competitions or the questions that typically appear in math competitions. Provide enough information about the source to confirm the question doesn't come from a live contest.

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I am training for math olympiad and I do exercise from old contest and I have read some book. I read for example Modern Olympiad Number Theory Aditya Khurmi, Evan Chen geometry book, Olympiad ...
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### A problem from Titu

For a positive integer $A$ with decimal representation $$A=\overline{a_na_{n-1}\dots a_0},$$ we set $$F(A)=a_n+2a_{n-1}+\dots+2^na_0$$ and consider the sequence $A_0=A,A_1=F(A_0),A_2=F(A_1),\dots$. ...
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### Can $\ln$ be written as a ratio of polynomials?

Is it possible that $\ln(x)=\frac{p(x)}{q(x)}$ for all $x>0,$ where $p$ and $q$ are polynomials with real coefficients? I think the answer is no. Suppose two such polynomials did exist. Take the ...
21 views

### Parabola Equations [closed]

Plot five parabolas. Each parabola should pass through the origin and a pair of color-coordinated points. One is done for you. In desmos. Parabola Equation Red y = x^2 Blue Green Orange ...
43 views

### A problem on counting total number of ways of arranging numbers in a grid.

Let $N$ be the number of ways to place the integers $1$ through $12$ in the $12$ cells of a $2×6$ grid so that for any two cells sharing a side, the difference between the numbers in those cells is ...
171 views

### Does there exist an $f: \mathbb{R} \to \mathbb{R}$ such that $f(f(x)) = x^3 + 1$?

This (question 6) was from the November $2022$ NZ maths olympiad workshop (I couldn't find the online one, even on the Wayback Machine so I'm emailing them). All I know is that $f$ is bijective (...
110 views

### define $a@b=\frac{b^2+3a}{a+33b},$ calculate (1@2@...@100)×3303

Find the following question in a middle school math competition: define $a@b=\frac{b^2+3a}{a+33b}$, then what is $(1@2@3@\cdots@100)\times3303$? If we assume that $1@2@3=(1@2)@3$, the code below is ...
1 vote
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### A geometry question. Source: RMO 2019 P5

In an acute angled triangle ABC, let H be the orthocenter, and let D, E, F be the feet of altitudes from A, B, C to the opposite sides, respectively. Let L, M, N be midpoints of segments AH, EF, BC, ...
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### Some help on a combinatoric problem

Problem Solution I don't understand the solution. I haven't studied much combinatorics so I do not no what they mean by "5-cycle" and "3 and 2-cycle", how there is 4! 5-cycles ...
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### If $1! \cdot 2! \cdot 3! \cdot\cdots\cdot 12! = m! \, n^2$, then what is $m$?

Here's the problem: The super factorial number $1! \cdot 2! \cdot 3! \cdot\cdots\cdot 12!$ can be written as a factorial times a perfect square, that is, in the form $m! \cdot n^2$. What is the value ...
37 views

### Help understanding the solution to this problem

Here is the problem:﻿ There are sixteen different ways of writing four-digit strings using 1s and Os. Three of these strings are 1010, 0100 and 1001. These three can be found as substrings of 101001. ...
1 vote
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### $a_{m+n}+a_{m-n}=\frac{1}{2}(a_{2m}+a_{2n}), a_1=1$, find $a_{1995}$ (craft)

I understand the solution of $a_{m+n}+a_{m-n}=\frac{1}{2}(a_{2m}+a_{2n}), a_1=1$, find $a_{1995}$ and I was able to derive it myself, however, in my first attempts I conjectured something very ...
1 vote
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1 vote
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### how to solve $\int_{\frac{-\pi}{4}}^{\frac{\pi}{4}}\frac{x^2}{\sin(x)} \ln\left(2^{\sin^3(x)}+ 5^{\cos^3(x)} \right)dx$?

Edit the question had a typo that made it impossible to solve the question was $\int_{\frac{-\pi}{4}}^{\frac{\pi}{4}}\frac{x^2}{\sin(x)} \ln\left(2^{\sin^3(x)}+ 5^{\sin^3(x)} \right)dx$ (to solve it ...
52 views

### Olympiad number theory problem with primes [duplicate]

Find all pairs of $x, y \in \mathbb Z$ that satisfy the following equation: $$1 + 1996x+1998y=xy$$ (Irish Mathematical Olympiad 1997) I am stuck and I have not been able to make even a start to a ...
1 vote
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### Closed form of $\prod_{n=0}^{\infty}\frac1{1+x^{2^n}}$

In the qualifying exam for the MIT integration bee 2023, the following question was asked: $$\int_0^1\prod_{n=0}^{\infty}\frac1{1+x^{2^n}}dx$$I graphed the integrand (denoted as $f(x)$) and from what ...
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### A product of Cosines [duplicate]

Question came from my math olympiad exercise book which doesn't have any solutions on the back of it, here is the problem Prove that $\prod_{k=1}^{n}\cos\frac{2^{k}\pi}{2^{n}-1}=\frac{1}{2^{n}}$ Now ...
42 views

### Why does the following condition holds in the geometric optimisation problem?

Given seven points on the plane, the distance between them is expressed by numbers $a_1,a_2,...,a_{21}$. What is the maximal number of times that we may have the same number among those $21$ distances?...
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### BMO2 2011/12 question on cyclic quadrilateral and showing two circumcircles have same radius

This problem is from the 2011/12 BMO2. The diagonals AC and BD of a cyclic quadrilateral meet at E. The midpoints of the sides AB, BC, CD and DA are P, Q, R and S respectively. Prove that the circles ...
1 vote
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### Verification of the maximum number of pairwise non-disjoint subsets of $\{1,2,\dots,100\}$ [duplicate]

Let $A=\{1,2,\cdots ,100\}$. Let $S$ be some set of subsets of $A$ such that any two elements of $S$ have a nonempty intersection. Then what is the maximum possible cardinality of $S$? My answer- My ...
1 vote
34 views

### Sum of the roots of the equality. [closed]

Sum of the roots in the range $\left(-\frac{\pi}{2},\frac{\pi}{2} \right)$ of the equation $\sin x\tan x=x^2$ is $\frac{\pi}{2}$ $0$ $1$ None of these This is a contest problem so I do not wish to ...
Middle School Math Club Question : The stable, $6$ yards by $6$ yards with concrete walls, is divided by internal wooden partitions into stalls $1$ yard by $2$ yards. What could be the total length of ...
### Given that $a,b,c>0$ and $abc=1$, prove that $a+b+c+\frac{3}{ab+bc+ca} \geq 4$
I was given some exercises from Math olympiads, and I am stuck with the one below, which seems soluble, yet I can't come up with something that works. Given that $a,b,c>0$ and $abc=1$, prove that \$...