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.
9,595
questions
0
votes
0
answers
33
views
Advice about the book "Problem from the book" Titu Andreescu
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
...
- 300
0
votes
0
answers
111
views
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 $A_0=A,A_1=F(A_0),A_2=F(A_1),\dots$. ...
- 382
6
votes
0
answers
155
views
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 ...
- 1,362
-2
votes
0
answers
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
...
- 1
0
votes
1
answer
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 ...
- 469
2
votes
1
answer
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 (...
- 51
0
votes
1
answer
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 ...
- 119
1
vote
1
answer
125
views
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, ...
- 168
0
votes
1
answer
55
views
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 ...
- 81
2
votes
3
answers
87
views
Prove that $\sum_{i=0}^{\lfloor n/2 \rfloor }\sum_{j=0}^{\lfloor n/2 \rfloor-i}{n \choose 2i}{n-2i \choose 2j}2^{n-2i-2j}=4^{n-1}+2^{n-1}$
I was working on problem 1 from the IMC 2020 and got the following expression for the solution:
$ \sum_{i=0}^{\lfloor n/2 \rfloor }\sum_{j=0}^{\lfloor n/2 \rfloor-i}{n \choose 2i}{n-2i \choose 2j}2^{n-...
- 31
-1
votes
0
answers
44
views
Proving lemmas on mathematical contests
This question isn't really about math, but rather about writing up problems in contests.
Do I need to prove famous lemmas like the LTE lemma? And where can I find out which lemmas I don't have to ...
0
votes
0
answers
53
views
Understanding a the solution of the apmo problem.
Prove that the equation $[ 6(6a^2 + 3b^2 + c^2) = 5n^2 ]$ has no solutions in integers except $a = b = c = n = 0$
The left side is divisible by 3, so 3 must divide n. So $5n^{2}-36a^{2}-18b^{2}$ is ...
- 546
0
votes
0
answers
166
views
2023 Putnam A5 : For a nonnegative integer $k$, let $f(k)$ be the number of ones in the base 3 representation of $k$. Find all complex numbers $z$
For a nonnegative integer $k$, let $f(k)$ be the number of ones in the base 3 representation of $k$. Find all complex numbers $z$ such that
$$
\sum_{k=0}^{3^{1010}-1}(-2)^{f(k)}(z+k)^{2023}=0
$$
Note: ...
- 315
0
votes
1
answer
60
views
Function $f$ is defined on positive integers, with $f(xy)=f(x)+f(y)$. If $f(40)=20$ and $f(10)=14$, the what is the value of $f(500)$? [closed]
UKMT Hardest Question (2015)
Function is defined on set of positive integers is such that
$$f(xy) = f(x) + f(y)$$
Given that $f(40) = 20$, $f(10) = 14$, what is the value of $f(500)$?
- 23
3
votes
1
answer
146
views
Proving $\sum_{i=0}^{n}\sum_{j=0}^{n}\binom{i+j}{i}\binom{n-i}{j}\binom{n-j}{i}\ =\ \sum_{k=0}^{n}\binom{2k}{k} $
Prove that $$\sum_{i=0}^{n}\sum_{j=0}^{n}\binom{i+j}{i}\binom{n-i}{j}\binom{n-j}{i}\ =\ \sum_{k=0}^{n}\binom{2k}{k} $$
This one I have no idea how to crack.
Induction doesn't seem to be a sensible ...
- 205
8
votes
1
answer
182
views
Two element subset of $\{1,2,\dots,100\}$ with sum of elements being a square
Prove that every 50-element subset of $\{1,2,\dots,100\}$ contains two
elements $a,b$ such that $a+b$ is a square of integer.
Any 50-element subset of the set $\{1,2,\dots,100\}$ has $\binom{50}{2}=...
- 205
0
votes
1
answer
77
views
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 ...
- 81
0
votes
1
answer
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. ...
- 81
1
vote
3
answers
144
views
Knowing $x^2-x-1$ is a factor of $p(x)=ax^5 + bx^4 + 1$ , find a,b.
Since $x^2-x-1$ is factor then $x^2-x-1=0$ and so $x^2=x+1$ is a solution.
I’m using $x^2$ so I express $p(x)$ in terms of it:
$$p(x)=a{x^2}{x^2}{x}+b{x^2}{x^2}+1$$
$$a{(x+1)}{(x+1)}{(x+1)(x-1)}+b{(x+...
- 75
1
vote
2
answers
147
views
A polynomial's coefficients are non-negative integers, $f(1)=6$ and $f(7)=3438$. What is the value of $f(3)$?
Here is the question:
A polynomial $f$ is given. All its coefficients are non-negative integers, $f(1) = 6$ and $f(7) = 3438$. What is the value of $f(3)$?
And here is the solution:
We do not know the ...
- 81
0
votes
2
answers
51
views
Monochromatic $4$-Cycle in Bipartite Complete Graph [closed]
Given a complete bipartite graph on $n$, $n$ vertices (call this $K_{n,n}$), we colour all its
edges using two colours, red and blue. What is the least value of $n$ such that for any colouring
of the ...
- 45
9
votes
2
answers
379
views
counting sequences of elements of the set {1,2,3,4} with given property
I found this problem in some internet olympiad-prep materials (don't remember the source of this file):
Find the number of $n$-element sequences $(a_1,a_2,\dots,a_n)$ such
that $a_1=1$, $a_n=4$ and $...
- 205
-1
votes
0
answers
25
views
Consider a set $S$ and a binary operation $*$, that is, for each $a,b \in S$, $a*b \in S$. Assume $(a*b)*a=b$ for all $a,b \in S$. [duplicate]
Prove that $a*(b*a)=b$ for all $a,b \in S$.
.........................................................................................................
What I have tried to do so far:
Assume $a=b*a$.
By ...
-4
votes
0
answers
57
views
LLM's for IMO-the 10m usd problem [closed]
Just saw that Alex Gerko has launched a $10M challenge for the first AI to win IMO Gold. Link:https://aimoprize.com/
Curious, are there currently any viable LLM's that are even remotely good at ...
4
votes
1
answer
116
views
IMO 1987/P1 - Combinatoric approach
I was was solving IMO 1987, Problem 1 and also found the first solution. However, I also tried a combinatorics approach but couldn't find any valid argument. My argument was as follows:
Consider each $...
- 335
0
votes
0
answers
92
views
Square root of a polynomial - Question in the exercise B4 of the 1995 Putnam competition
B4 exercise: Express $$x=\sqrt[8]{2207-\cfrac{1}{2207-\cfrac{1}{2207-\cfrac{1}{2207-\cfrac{1}{\ddots}}}}}$$
as $\dfrac{a+b\sqrt{c}}{d}$, where $a,b,c,d$ are integers.
In the following solution
https://...
- 1
1
vote
0
answers
124
views
Solving a system of non linear equations in four variables with some equations looking like a determinant
$$\begin{align} a+b+c+d &= 110 \\ \frac{a}{b} + \frac{c}{d} &= \frac{5}{4} \\ ad - bc &=-242 \\ (c-a)(b-d) &=- 605\end{align}$$
Solve for a,b,c,d
Origin of the question:
This crazy ...
- 11.2k
3
votes
4
answers
189
views
Thailand MO $f(x)f(y)f(x-y) = x^2f(y) - y^2f(x)$
This is a functional equation question from Thailand MO 2023.
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfies the equation
$$f(x)f(y)f(x-y) = x^2f(y) - y^2f(x)$$
for all $x,y \in \mathbb R$....
- 148
8
votes
2
answers
189
views
Proving the existence of integers $a, b, c$ such that $\left\lvert a + b\sqrt{2} + c\sqrt{3}\right\rvert < 10^{-5}$ [duplicate]
Prove that there exist integers $a, b, c$ such that $|a|, |b|, |c| \leq 1000$ where not all of them are zero and $\left\lvert a + b\sqrt{2} + c\sqrt{3}\right\rvert < 10^{-5}$
I am stuck on this ...
- 1,838
10
votes
4
answers
298
views
If $a+b+c+abc=4,$ prove $\frac{1}{\sqrt{a^2+4bc}}+\frac{1}{\sqrt{b^2+4ac}}+\frac{1}{\sqrt{c^2+4ba}}\ge \frac{5}{4}.$
Let $a,b,c\ge 0: ab+bc+ca>0$ and $a+b+c+abc=4.$ Prove that$$\color{black}{\frac{1}{\sqrt{a^2+4bc}}+\frac{1}{\sqrt{b^2+4ca}}+\frac{1}{\sqrt{c^2+4ba}}\ge \frac{5}{4}. }$$
Remark.
My teacher ...
- 758
1
vote
0
answers
33
views
Prove that a convex quadrilateral $ABCD$ with certain properties is a cyclic quadrilateral [closed]
$ABCD$ is a convex quadrilateral such that $\angle ABD = \angle DBC$, $AD=CD$ and $AB \neq BC$. Prove that $ABCD$ is cyclic.
-1
votes
0
answers
55
views
Modular Arithmetic in Math Competition [duplicate]
I came across this problem and a few problems similar to this one that uses modular arithmetic in the solutions. I know the basics of modular arithmetic but I don't know how and why it is apply in ...
- 81
4
votes
1
answer
142
views
BMO1 number theory question on fibonacci sequence and divisibility
This is question 2 from the 1983 British Maths Olympaid
The fibonacci sequence $f_{n}$ is defined by $f_{1} = 1, f_{2} = 1,$ and $f_{n} = f_{n-1} + f_{n-2}, n > 2$
prove that there are integers a,...
- 169
1
vote
2
answers
87
views
Diagonals in an inscribed quadrilateral can simultaneously be angle bisectors of triangles involving their midpoints
This is follow-up of an interesting question asked some days ago on this site, that the asker has erased 24 hours later.
The initial question was (as reflected in my title) in terms of an inscribed ...
- 79k
1
vote
4
answers
120
views
Maximum value of $(1-a)(1-b)+(1-p)(1-q)$
Given that real numbers $a,b,p,q$ satisfy $$a^2+b^2=p^2+q^2=2$$
Find the maximum value of $E=(1-a)(1-b)+(1-p)(1-q)$.
My try:
I have chosen $a=\sqrt{2}\sin x, b=\sqrt{2}\cos x, p=\sqrt{2}\sin y, q=\...
- 12.9k
1
vote
2
answers
296
views
How long does it take to get to "Olympiad-Bronze-level" of math problem solving ability from no competition experience? [closed]
Is it possible to go from almost zero math competition experience to "Olympiad-Bronze-level" within a span of ~3 years?
I am currently double-majoring in Maths and Mathematical Statistics, ...
2
votes
0
answers
185
views
Prove $ 2\sqrt{2} \le \vert a+b+c \vert + \vert b+c-a \vert + \vert c+a-b \vert + \vert a+b-c \vert \le 4 $ [closed]
Prove that for all $a,b,c \in \mathbb{R} $ and $a^2 + b^2 + c^2 = 1 $ the following inequalities are true:
$$ 2\sqrt{2} \le \vert a+b+c \vert + \vert b+c-a \vert + \vert c+a-b \vert + \vert a+b-c \...
- 81
0
votes
1
answer
59
views
$a_{m+n}+a_{m-n}=\frac{1}{2}(a_{2m}+a_{2n}), a_1=1$, 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 ...
- 13
1
vote
2
answers
57
views
Find the area of the quadrilateral $ABDC$
If $A=(2,1), B(8,1), C(4,3), D(6,6)$ then find the area of the quadrilateral $ABDC$.
My Attempt:
Area of quadrilateral= area of triangle ABD + area of triangle ADC.
Area of triangle ABD= $\frac12|2(1-...
- 8,108
-1
votes
0
answers
68
views
Finding all real solutions to a system of equations
The following problem came from an exercise for olympiad mathematics
Find all real solutions to the system $\left\{\begin{array}{lll} x+y+z=0 \\ \\ x^{3}+y^{3}+z^{3}=18 \\ \\ x^{7}+y^{7}+z^{7}=2058 \...
- 1
1
vote
1
answer
190
views
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 ...
- 2,242
0
votes
0
answers
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 ...
- 85
1
vote
2
answers
94
views
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 ...
- 5,084
0
votes
0
answers
31
views
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 ...
- 1
0
votes
1
answer
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?...
- 546
-1
votes
1
answer
109
views
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 ...
- 169
1
vote
0
answers
59
views
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 ...
- 546
1
vote
0
answers
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 ...
- 546
2
votes
1
answer
263
views
Partition a stable (Middle School Math)
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 ...
- 129
0
votes
2
answers
124
views
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 $...
- 386