Questions tagged [contest-math]

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

Filter by
Sorted by
Tagged with
1 vote
0 answers
46 views

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 ...
user avatar
0 votes
0 answers
46 views

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. ...
user avatar
1 vote
3 answers
69 views

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 ...
user avatar
  • 1,369
0 votes
0 answers
50 views

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 ...
user avatar
-4 votes
1 answer
57 views

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 ...
user avatar
-4 votes
0 answers
34 views

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.
user avatar
0 votes
1 answer
30 views

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.
user avatar
1 vote
1 answer
67 views

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: $$x=\sqrt[3]{y^3+2y^2+1}$$ ...
user avatar
  • 392
0 votes
0 answers
31 views

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. ...
user avatar
1 vote
0 answers
71 views

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 ...
user avatar
0 votes
1 answer
213 views

$ \sum_{i \in I} a_i = \sum_{j \in J} a_j $ for distinct $ I,J \subseteq \{1,\ldots,n\} $ if $ \sum_{i=1}^n a_i < 2^n - 1 $? [closed]

Let $ a_1,a_2,\ldots,a_n $ be natural numbers with $ \sum_{i=1}^n a_i < 2^n - 1 $. Is it true that there exist distinct subsets $ I,J \subseteq \{1,\ldots,n\} $ with $ \sum_{i \in I} a_i = \sum_{j \...
user avatar
  • 6,073
0 votes
0 answers
26 views

How to know if the function is convex

How I can know if this function is a convex and why ? My understanding is the exp is convex and if we add positive number to a convex we return a convex then the log for convex sometimes return convex ...
user avatar
  • 1
-2 votes
0 answers
35 views

What is total number of ways in which Dex can distribute $10$ distinct chocolates among his $9$ wives such that each wife gets at least one chocolate? [closed]

What is total number of ways in which Dex can distribute $10$ distinct chocolates among his $9$ wives such that each wife gets at least one chocolate? Selected $9$ chocolates out of $10$ = $^{10}C_9 =...
user avatar
  • 1,369
0 votes
0 answers
34 views

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 ...
user avatar
1 vote
1 answer
33 views

If $|28 - x^2| < |3x|$ ,then what is the product of all possible integer values of $x$?

$|28 - x^2| < |3x|$ $\Rightarrow |x^2-28| < |3x| $ Case $1$ when $3x \geq 0 \Rightarrow x \geq0$ :- $|x^2-28| < 3x$ $\Rightarrow -3x < x^2-28 < 3x $ $\Rightarrow x \in (4,7) \Rightarrow ...
user avatar
  • 1,369
2 votes
0 answers
73 views

Find bounds of $\prod_{i=1}^k (x_i-x_{i+1})$ where $\sum_{i=1}^k x^2_i=1$

Let $x_1,x_2,...,x_k$ be real numbers such that $\sum_{i=1}^k x^2_i=1$. Determine the minimum and maximum (if there is) value of $$\prod_{i=1}^k (x_i-x_{i+1})$$ and determine all values of $(x_1,x_2,...
user avatar
  • 51
1 vote
1 answer
40 views

Find the number of second-degree polynomials $f(x)$ with integer coefficients and integer zeros for which $f(0)=2010.$

Find the number of second-degree polynomials $f(x)$ with integer coefficients and integer zeros for which $f(0)=2010.$ I know that if the coefficients and roots are integers then for every $r \in \...
user avatar
0 votes
2 answers
55 views

Prove that ABCD is a rectangle

From the 2018 Moroccan Mathematics Olympiad: Let $E,F, $ and $B$ three distinct points on the plane such that: $B \in [EF]$. Let the semi-circles $(C_1), (C_2)$ and $(C_3)$ with diameters, ...
user avatar
  • 95
1 vote
1 answer
80 views

Find the smallest value of the product $ab$

From the 2018 Moroccan Mathematics Olympiad: Let $(a,b) \in \mathbb{Z^2}$ such that $a+b$ is a solution of the equation $x^2+ax+b=0$. Find the smallest value of the product $ab$. ($\mathbb Z$ ...
user avatar
  • 95
1 vote
2 answers
28 views

Regarding the solution of finding the remainder of $g(x^{12})$ divided by $g(x)$

Let $g(x) = x^5 + x^4 + x^3 + x^2 + x + 1.$ What is the remainder when the polynomial $g(x^{12})$ is divided by the polynomial $g(x)$? I'm reading the solution for this and I don't understand how can ...
user avatar
0 votes
1 answer
42 views

Problems in understanding 2008 AMC 12B Problem 19

A function $f$ is defined by $f(z) = (4 + i) z^2 + \alpha z + \gamma$ for all complex numbers $z$, where $\alpha$ and $\gamma$ are complex numbers and $i^2 = - 1$. Suppose that $f(1)$ and $f(i)$ are ...
user avatar
0 votes
0 answers
57 views

Columbia Integration Bee 2022 Finals [duplicate]

I want to find the definite integral shown below, but I'm not quite sure where to start. The fastest solution apparently involved some sort of change of variables, but I can't quite find a ...
user avatar
1 vote
1 answer
99 views

Prove that $\frac{2022}{n} + 4n$ is a perfect square iff $\frac{2022}{n} - 8n$ is a perfect square

Prove that $\frac{2022}{n} + 4n$ is a perfect square iff $\frac{2022}{n} - 8n$ is a perfect square My solution was to substitute all the positive divisors of $2022$ into the $2$ expressions and ...
user avatar
  • 310
0 votes
1 answer
58 views

How many factors of $2400$ are not factors of $3600$? [closed]

I solved this question by writing all the factors and then just selecting the factors as per the question requirement. But I want to know is there any other way to solve this? Please help !!! Thanks ...
user avatar
  • 1,369
0 votes
0 answers
14 views

prove that the sets $\{A_k\}$ are pairwise disjoint and that all numbers in $A_k$ are $\equiv (\frac{1}2 3^{k-1}, 3^{k-1}] \mod 3^{k}$

For a number n, write it in base three as $a_t\cdots a_2a_1$. Let $B$ be a positive integer and let $[B] = \{1,\cdots, B\}$. Construct the sets $A_1,A_2,\cdots$ inductively as follows. $A_1$ consists ...
user avatar
4 votes
1 answer
82 views

Prove that there is a circle containing exactly $2018$ points

Problem Given a set $\mathtt{E}$ containing $2017^{2019}$ points on the plane. Prove that there is a circle containing exactly $2018$ points from the set $\mathtt{E}$ (these points are on the open ...
user avatar
  • 95
1 vote
0 answers
58 views

Let $f(x)$ be a polynomial with real coefficients such that $f(0) = 1$, $f(2) + f(3) = 125,$ and for all $x$, $f(x)f(2x^2) = f(2x^3 +x)$. Find $f(5)$ [duplicate]

Let $f(x)$ be a polynomial with real coefficients such that $f(0) = 1$, $f(2) + f(3) = 125,$ and for all $x$, $f(x)f(2x^2) = f(2x^3 +x)$. Find $f(5)$. If $r$ is a root of $f$, then $f(r)f(2r^2)=f(2r^...
user avatar
1 vote
1 answer
47 views

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 ...
user avatar
2 votes
1 answer
58 views

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 ...
user avatar
  • 460
1 vote
1 answer
33 views

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 ...
user avatar
  • 313
1 vote
2 answers
47 views

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 ...
user avatar
7 votes
3 answers
179 views

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 ...
user avatar
  • 95
2 votes
0 answers
29 views

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 ...
user avatar
  • 51
5 votes
2 answers
86 views

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 $CE$ ...
user avatar
  • 107
0 votes
3 answers
60 views

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 $\arctan(u)+\arctan(v) +\arctan(w)$....
user avatar
  • 883
1 vote
1 answer
37 views

Maximum number of students who have passed exactly in one subject : $4$ set Venn diagram problem

$50$ students attempted the midterm tests of class $5th$, the midterm consists of $4$ subjects English, Hindi, Maths and Science. The number of students passed in these subjects is $35$,$45$,$25$ and $...
user avatar
  • 1,369
4 votes
1 answer
27 views

find the probability both planes can park at a gate

Two airplanes are supposed to park at the same gate of a concourse. The arrival times of the airplanes are independent and randomly distributed throughout the 24 hours of the day. What is the ...
user avatar
  • 883
1 vote
2 answers
39 views

Solve for reals $[2022+x(2+\sqrt{x})][111-x(2+\sqrt{x})]=-52^3$

Solve in real numbers: $$\sqrt[3]{2022+x(2+\sqrt{x})}+\sqrt[3]{111-x(2+\sqrt{x})}=9$$ Cubing both sides, we get$$ 2022+111+3\sqrt[3]{2022+x(2+\sqrt{x})}\sqrt[3]{111-x(2+\sqrt{x})}\Big(\sqrt[3]{2022+x(...
user avatar
0 votes
1 answer
37 views

Limit of series equals improper integral - IMC 2001 [duplicate]

I would like to know what result is used to justify the second to last inequality here (where the limit becomes the improper integral). It is the official solution from IMC 2001 (https://www.imc-math....
user avatar
  • 312
0 votes
2 answers
94 views

Find all possible solutions of $a^2+b^2+ab=1011$ [duplicate]

Find all possible solutions of $$(x-y)^2+(y-z)^2+(z-x)^2=2022$$ We can simply take $$x-y=a, y-z=b, z-x=-(a+b)\implies a^2+b^2+(a+b)^2=2022\implies a^2+b^2+ab=1011.$$ We can use modular arithmetic and ...
user avatar
  • 1,276
2 votes
2 answers
84 views

Find $\sqrt{\frac{1}{2}-f(1)}+\dots+ \sqrt{\frac{1}{2}-f(99)}$

Define $f(n)=\sqrt[2]{n^4+\frac{1}{4}}-n^2.$ Find $$\sqrt{\frac{1}{2}-f(1)}+\dots+ \sqrt{\frac{1}{2}-f(99)}$$ I tried to simply $f(n).$ So rationalising, we get $$\sqrt{\frac{1}{2}-f(n)}\sqrt{\frac{1}{...
user avatar
  • 1,276
0 votes
0 answers
62 views

Determine all $x,y \in \Bbb Z$ such that $1+2^x+2^{2x+1}=y^2$ [duplicate]

Determine all $x,y \in \Bbb Z$ such that $$1+2^x+2^{2x+1}=y^2$$ I've managed to show that $(0,2)$ and $(0,-2)$ are solutions. I've also noted that if $(x,y)$ is a solution, then $(x,-y)$ is also a ...
user avatar
  • 35
0 votes
1 answer
37 views

algorithms applied to n-tuples of numbers to end up with $(0,0,...,0)$

I came to the following problem in Problem Solving Strategies by Arthur Engel on page 18 : Start with a sequence $S=(a,b,c,d) $ of positive integers and find the derived sequence $ S_1=T(S)=(|a-b|,|b-...
user avatar
0 votes
1 answer
76 views

some questions about IMO 1986 problem 3

To each vertex of a pentagon, we assign an integer $x_i$ with sum $s=\sum{x_i > 0}.$ If $x, y,z $ are the numbers assigned to three successive vertices and if $ y < 0$ , then we replace $(x, y,...
user avatar
4 votes
1 answer
64 views

Fractional part and greatest integer function

Here are a few questions on the fractional part and the greatest integer function. Find out $[\sqrt[3]{2022^2}-12\sqrt[3]{2022}]$ If $\{x\}=x-[x],$ find out $[255\cdot x\{x\}]$ for $x=\sqrt[3]{15015}....
user avatar
  • 1,276
-2 votes
0 answers
32 views

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}.
user avatar
  • 1
0 votes
0 answers
26 views

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, ...
user avatar
-3 votes
0 answers
92 views

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 ...
user avatar
  • 107
0 votes
1 answer
90 views

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,...
user avatar
  • 13
3 votes
1 answer
139 views

number of permutations maximizing a sum

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}...
user avatar
  • 883

1
2 3 4 5
167