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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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0 votes
3 answers
72 views

Probability Question on Stanford Math Tournament

On Day 1, Aaron draws a smiley face on the board. From then on, on each day he does the same thing as the previous day (draw a smiley face or not) with probability $\dfrac{2}{3}$. What’s the ...
Satish Ramanathan's user avatar
0 votes
1 answer
25 views

Tetrahedron analogue of a triangle Cevians property

It's a cute Olympaid geometry problem to prove that given a triangle $\triangle ABC$ with three Cevians $AA_1,BB_1, CC_1$ intersecting at an interior point $M$: $$ab+bc+ca+2abc = 1 \quad (1)$$ where $$...
dezdichado's user avatar
  • 14.1k
-1 votes
0 answers
39 views

neighbouring numbers [duplicate]

reading practice tests, i came across this question two fractions $\displaystyle{\frac{a}{b}, \frac{c}{d}}$ are called neighbours if $\displaystyle{|ad-bc|=1}$. prove that there is no reduced ...
thatpithere's user avatar
0 votes
1 answer
136 views

Seeking "900 Geometry Problems" Book – Any Leads on Its Whereabouts?

I have been on a quest to find a book titled "900 Geometry Problems" that I've heard a lot about. Geometry is a subject I am deeply passionate about, and from what I've gathered, this book ...
MathsGuy's user avatar
4 votes
1 answer
152 views

Creative Algebra Net Problem Solving Question

I came across a problem that I found pretty tedious and difficult to answer and I would appreciate any views or solutions for this question. The diagram shows the net of a cube. On each face there is ...
Jonathan Xu's user avatar
0 votes
1 answer
98 views

IMO 2024 p-3,Sequence of Counts - Are Odd or Even Terms Eventually Periodic?

Let $a_1, a_2, a_3, \dots$ be an infinite sequence of positive integers, and let $N$ be a positive integer. We define $a_n$ for $n > N$ as the number of times $a_{n-1}$ appears in the list $a_1, ...
Saucitom's user avatar
1 vote
1 answer
71 views

Putnam 2008 B2 -- Soft Question

The 2008 Putnam exam has the following question for B2: Let $F_0(x) = \ln x.$ For $n \geq 0$ and $x>0,$ let $F_{n+1} (x) = \int _0 ^xF_n(t) \, dt$. Evaluate $$\lim _{n \rightarrow \infty} \frac{n! ...
Eli Yablon's user avatar
1 vote
0 answers
45 views

Finding all completely multiplicative arithmetic function such that m+n|f(m)+f(n)

My attempt: Since f is completely multiplicative we have $f(1)=1$. $2n|2f(n)$ for every n, so $f(n)=kn$ for some n. $n+1|f(n)+1$ for every n,so for p prime, $p+1|kp+1\rightarrow p+1|k-1$, so $k=l(p+1)+...
Dailin Li's user avatar
-1 votes
0 answers
39 views

Advance Angle Chasing [closed]

In acute triangle ABC, with AB<AC, H is orthocenter, O is circumcenter. Let m(OBC)=$a$, and m(OCA)=$b$ What is the value of m(AHO) in $a$ and $b$
Lim Zhao Sen's user avatar
0 votes
1 answer
70 views

An interesting Hamiltonian path problem concerning the connectivity of subgraphs

Suppose there are $n$ players, $P_1,...,P_n$, requiring any $t<n$ players to be able to talk to each other, each two players can share a telephone line, when there is a logical connection path (...
X.H. Yue's user avatar
4 votes
2 answers
340 views

Show that $\frac{a^2}{b^2-2b+2024}+ \frac{b^2}{c^2-2c+2024}+ \frac{c^2}{a^2-2a+2024} \geq \frac{3}{2023}$

Let the real numbers $a,b,c \in \mathbb{R}$ with $a+b+c=3$. Show that: $\frac{a^2}{b^2-2b+2024}+ \frac{b^2}{c^2-2c+2024}+ \frac{c^2}{a^2-2a+2024} \geq \frac{3}{2023}$. My idea: First of all, I thought ...
IONELA BUCIU's user avatar
1 vote
0 answers
140 views

Putnam 1987 Algebra Question

I've been trying to solve the Putnam 1987 A1 question: A$-$1 $\quad$ Curves $A,B,C$ and $D$ are defined in the plane as follows: \begin{equation*} A=\left\{(x,y):x^2-y^2=\frac{x}{x^2+y^2}\right\} \...
Riccardo Caiulo's user avatar
-4 votes
0 answers
50 views

Find min degree of required polynomial [closed]

Let $f$ be a polynomial with integer coefficients such that the greatest common divisor of all its coefficients is $1$. For any $n ∈ \mathbb N$, $f(n)$ is a multiple of $85$. Find the smallest ...
stickynote's user avatar
5 votes
1 answer
101 views

Reciprocals of positive integers in arithmetic progression

Question: Let $S$ be the set of the reciprocals of the first $2024$ positive integers and $T$ be the set of all subsets of $S$ whose elements form an arithmetic progression. What is the largest ...
Indecisive's user avatar
-2 votes
1 answer
62 views

What is the product of the solutions to the equation (√10)(x^(log (x))) = x^2? [closed]

I have spent a few hours on this question but I can't seem to grasp it. I have found that taking the log of both sides doesn't give me any progress. The log exponent identity didn't work. I also tried ...
shivank chintalpati's user avatar
1 vote
1 answer
98 views

Two numbers written on a board get replaced

Question: "Several (at least two) nonzero numbers are written on a board. One may erase any two numbers, say $a$ and $b$, and then write the numbers $a+\frac{b}{2}$ and $b−\frac{a}{2}$ instead. ...
mathisdagoat's user avatar
2 votes
4 answers
149 views

Prove $\sum_{i=1}^{30}\frac{1}{10+i} >\frac{13}{12}$

Prove $$\sum_{i=1}^{30}\frac{1}{10+i} >\frac{13}{12}$$ My effort: $$\begin{aligned} & \frac{1}{11}>\frac{1}{42} \\ & \frac{1}{12}>\frac{1}{42} \\ & \frac{1}{13}>\frac{1}{42} \\...
LifeIsMath's user avatar
11 votes
4 answers
2k views

How can I learn how to solve hard problems like this Example?

Preparing to the Math Olympiad , I found a book and decided to go through it as a summer project. I'm good at math but far from being an Olympian, yet I want to broaden my horizons - problems in this ...
Iruven's user avatar
  • 113
1 vote
0 answers
42 views

Stationary distributions in Markov chains

This question is with regards to HMMT Guts 2021/28 and 2018/27. Caroline starts with the number $1$, and every second she flips a fair coin; if it lands heads, she adds $1$ to her number, and if it ...
PunySoloist's user avatar
0 votes
1 answer
55 views

Higher Lemmas and Topics for the International Mathematics Olympiad

I am passionate about mathematics and want to reach IMO camp this year. I have learnt a lot of theorems and lemmas and have learned enough to clear the first 2 levels of IMO in my country.( I have ...
MathsGuy's user avatar
2 votes
1 answer
55 views

Finding the circumradius of a cyclic hexagon, given three non-consecutive sides and the fact that the midpoints of all sides are also cyclic

Cruel Geometry Question, Mock AIME-i 2015: Let $A_1A_2A_3A_4A_5A_6$ be a hexagon inscribed inside a circle of radius $r$. Furthermore, for each positive integer $1 \leq i \leq 6$ let $M_i$ be the ...
CLASH ROYAL's user avatar
1 vote
1 answer
90 views

When does $n$ divide $u_n$ if $u_1=1$, $u_n=(n-1)u_{n-1}+1$? [duplicate]

I'm working through the Background section to 'The Mathematical Olympiad Handbook' by A. Gardiner, OUP, 1997, and this appears on page 17: (***) Let $u_1=1$, $u_n=(n-1)u_{n-1}+1$. For which values of ...
John1970's user avatar
  • 441
1 vote
2 answers
499 views

Which of the following values can $ \frac{5x^{3}yz}{x^{5}+y^{5}+z^{5}} $ NOT take?

Consider the expression: $$ \frac{5x^{3}yz}{x^{5}+y^{5}+z^{5}},$$ which is defined for $x > 0$, $y> 0$, and $z> 0$. Which of the following values cannot be taken by the expression? $A)1 \quad ...
Tamercan's user avatar
1 vote
2 answers
62 views

Positive reals that satisfies $\sum_{i=1}^{5}a_{i}=20$ and $\sum_{i=1}^{5}a^{2}_{i}=100$. Find min and max of $\max\{a_{1},a_{2},a_{3},a_{4},a_{5}\}$

Let $a_{1},a_{2},a_{3},a_{4},a_{5}$ be non-negative real numbers that satisfy: $$\sum_{i=1}^{5}a_{i}=20$$ and$$\sum_{i=1}^{5}a^{2}_{i}=100$$ How can I find the maximum and the minimum of $\max\{a_{1},...
JAB's user avatar
  • 321
3 votes
2 answers
245 views

Stanford Math Tournament Calculus Problem: Probability that a frog with uniformly distributed step length can pass an abyss on real line

A frog hops on the real number line, starting from the origin: Each second, it moves right uniformly at random a distance between $0$ and $1$. There is an abyss between $1$ and $5/4$, and if the frog ...
Satish Ramanathan's user avatar
0 votes
1 answer
97 views

Prove that the function $f(x,y,z)=1+x^2+y^2+z^2+2xyz-2xy-2xz-2yz$ is strictly positive outside of some bounded set. [closed]

Consider the function $$f(x,y,z)=1+x^2+y^2+z^2+2xyz-2xy-2xz-2yz,$$ where $x,y,z\geq 0$. Prove that there exists a bounded set in $\mathbb{R}^3$ such that $f\geq \frac{1}{2}$ outside of this set. Note ...
Ryan Hendricks's user avatar
7 votes
5 answers
321 views

Solve $\,n^{21}= 37\ldots 2719\ (53$ digits) for natural $n$

This is a problem $4$ of the "Bulgaria International Mathematics Competition $2023$": The $53$-digit number $37,984,318,966,591,152,105,649,545,470,741,788,308,402,068,827,142,719$ can be ...
Soheil's user avatar
  • 6,794
5 votes
3 answers
450 views

Evaluating $\frac1{m^2}\sum_{k=1}^m\sum_{l=1}^m\sin(rA_k)\sin(rA_l)\cos(rA_k-rA_l)$ for natural $r\leq m$, and $A_k=\frac{2k-1}{2m}\pi$

IMO 1969 Longlist problem 38: Let $r$ and $m$ ($r \leq m$) be natural numbers and $A_k = \frac{2k-1}{2m}\pi$. Evaluate $$\frac{1}{m^2}\sum_{k=1}^{m}\sum_{l=1}^{m}\sin(rA_k)\sin(rA_l)\cos(rA_k - rA_l)$...
koiboi's user avatar
  • 356
0 votes
1 answer
38 views

Proving Symmedian intersects intersection of tangents

I'm going through Evan Chen's "Euclidean Geometry in Math Olympiads" and I've come to Chapter 4's section on Symmedians. Proposition 4.24 says: Let $X$ be the intersection of the tangents to ...
PabloGamerX's user avatar
-2 votes
2 answers
80 views

AMC 10 Math Question About Formula For A Sequence... [closed]

Find the value of $ \frac{{1^2 + 1 \cdot 2 + 2^2}}{{1^3 \cdot 2^3}} + \frac{{2^2 + 2 \cdot 3 + 3^2}}{{2^3 \cdot 3^3}} + \cdots + \frac{{10^2 + 10 \cdot 11 + 11^2}}{{10^3 \cdot 11^3}} $ This is a ...
GalacticWood's user avatar
0 votes
1 answer
107 views

How to maximize the area of the triangle? [duplicate]

Let $W$ be a focus of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. $P$ and $Q$ are two points on the ellipse such that for any position of $P$ and $Q$ on the ellipse, perimeter of $\Delta PQW$ is ...
whatamidoing's user avatar
  • 2,879
1 vote
1 answer
80 views

Prove $f'(r_{+}) + f'(r_{-}) < 0$ for roots of $1 - \frac{M}{r^2} + \frac{Q}{r^4} - r^2 = 0$

Problem. Let $M, Q > 0$ be given. Let $$f(r) := 1 - \frac{M}{r^2} + \frac{Q}{r^4} - r^2, \quad r > 0.$$ If $f$ has three distinct positive real roots $r_c > r_{+} > r_{-} > 0$, prove ...
River Li's user avatar
  • 40.3k
2 votes
0 answers
53 views

Find $m \in\mathbb{R}$ such that $mX^4 + (1-3m)X^3 + 3(m-1)X^2 + (3-m)X - 1 = 0$ has only one root which is strictly greater than $1$.

Find $m \in\mathbb{R}$ such that $mX^4 + (1-3m)X^3 + 3(m-1)X^2 + (3-m)X - 1 = 0$ has only one root which is strictly greater than $1$. I'm having trouble in solving this type of problems. I tried ...
charlibrat's user avatar
5 votes
1 answer
107 views

Is this proof correct for Putnam 1986 B4?

I have been practicing on some old Putnam questions, and I attempted to solve this 1986 Putnam B4: For a positive real number $r$ define $G(r)$ to be the minimum value of $|r-\sqrt{m^2+n^2}|$ for all ...
Riccardo Caiulo's user avatar
2 votes
2 answers
120 views

Maximize $f(x)=(1-x)^5(1+x)(1+2x)^2$

For which value of $x$ is the product $(1-x)^5(1+x)(1+2x)^2$ a maximum, and what is this value? This is easy with calculus, but how would you do it without calculus? $f(x)=(1-x)^5(1+x)(1+2x)^2 \geq 0$ ...
Mike Bertrand's user avatar
8 votes
2 answers
398 views

Prove $\sum\limits_{\mathrm{cyc}} \sqrt{a+b} \ge \sum\limits_{\mathrm{cyc}} \sqrt{a + bc}$ for $ab + bc + ca + abc =4$

Problem: Given non-negative real numbers $a,b,c$ satisfying $ab+bc+ca+abc=4.$ Prove that $$\color{blue}{\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\ge \sqrt{a+bc}+\sqrt{b+ca}+\sqrt{c+ab}.}$$ When does equality ...
Danh Trung's user avatar
5 votes
1 answer
426 views

Finding the real roots of an octic polynomial (degree eight). [duplicate]

I have been spending some time on a question which I encountered in the PYQs of a small maths competition from $2$ years ago. It is from the topic quadratic equations and polynomials, it however deals ...
Sarvagya's user avatar
2 votes
0 answers
115 views

Given positive $a,b,c$, show that $\frac{1}{4a^2-ab+4b^2}+\frac{1}{4b^2-bc+4c^2}+ \frac{1}{4c^2-ca+4a^2}\geq \frac{9}{7(a^2+b^2+c^2)}$

This is question 41 of this inequality exercise sheet, and a solution is provided. However, the author's solution is purposefully cumbersome, showing how complicated brute force solution can get. Here ...
rosemary 2.0's user avatar
0 votes
0 answers
45 views

Consecutive multiplication of natural numbers problem [duplicate]

Prove that the product of any three consecutive natural numbers is not a perfect square. If there were four numbers, I know how to solve the problem, as I would somehow mention the number $ n(n+1)(n+2)...
user avatar
1 vote
0 answers
41 views

An inequality supposedly using Titu's Lemma aka Sedrakayan's Inequality

If $$(a,b,c)\in\mathbb{R}^{+}$$ and $$a+b+c=ab+bc+ac,$$ how do you prove that $$\frac{a^{2}}{b}+\frac{b^{2}}{c}+\frac{c^{2}}{a}+\frac{3}{a^{2}+b^{2}+c^{2}}\geq4?$$ Apparently, this inequality involves ...
Jeremiah Tan's user avatar
2 votes
1 answer
111 views

Putnam 1993 probability question

Two real numbers $x$ and $y$ are chosen at random in the interval $(0, 1)$ with respect to the uniform distribution. What is the probability that the closest integer to $x/y$ is even? Express the ...
Tobias Hermans's user avatar
5 votes
4 answers
212 views

Functional equation with $f(x+y)=f(x)f(1-y)+f(1-x)f(y)$

Let $f:\Bbb R\to\Bbb R$ such that $f(x+y)=f(x)f(1-y)+f(1-x)f(y)$, $f(x)$ is strictly increasing on $[0,1]$. Find the solution set to $f(x)\ge\frac12$. This is a problem form a mock test with no ...
youthdoo's user avatar
  • 1,475
2 votes
3 answers
90 views

Prove the inequality $(ab + 1)(a + b) \geq 4ab$ for any non-negative real numbers $a$ and $b$. When does equality hold?

Prove the inequality $$(ab + 1)(a + b) \geq 4ab$$ for any non-negative real numbers $a$ and $b$. When does equality hold? Attempt: To prove the inequality $$(ab + 1)(a + b) \geq 4ab$$ for any non-...
user avatar
3 votes
3 answers
193 views

Calculate the length of segment $AD$.

Given a triangle $ABC$ and its circumscribed circle, point $E \in BC$. Let $D$ be the intersection of the circle and line $AE$ (see the figure). Also, let $|AB| = |AC| = 12$ and $|AE| = 8$. Calculate ...
user avatar
0 votes
2 answers
84 views

Proving $\sqrt{x - \frac{1}{2}} + \frac{x + 1}{4} > \sqrt{2x - 1 + \left(\frac{x + 1}{4}\right)^2}$

Solve the inequality and justify each step in the solution: $$\sqrt{x - \frac{1}{2}} + \frac{x + 1}{4} > \sqrt{2x - 1 + \left(\frac{x + 1}{4}\right)^2}$$ I have no idea how to solve this ineqaulity,...
user avatar
3 votes
4 answers
177 views

$ \frac{xyz}{(x+y)(y+z)(x+z)} \leq \frac{1}{8} $

Show that for positive real numbers $ x $, $ y $, and $ z $, the following inequality holds: $ \frac{xyz}{(x+y)(y+z)(x+z)} \leq \frac{1}{8} $ Attempt: I know that $\frac{x+y}{2} \geq \sqrt{xy}$, ...
user avatar
-1 votes
2 answers
58 views

surjectivity of a function in a functional equation [closed]

Full statement I don’t understand this implication in the first line: $f((xf(0)+f(x))=f(0)+x$ means $f$ is surjective.
Cool Gamer's user avatar
1 vote
1 answer
68 views

Let $(F_n)$ and $(L_n)$ be the sequences of Fibonacci and Lucas numbers, indexed by natural numbers. Further, let $\phi = \frac{1 + \sqrt{5}}{2}$.

Let $(F_n)$ and $(L_n)$ be the sequences of Fibonacci and Lucas numbers, indexed by natural numbers. Further, let $\phi = \frac{1 + \sqrt{5}}{2}$. (a) Show that for every natural number $n$, it holds ...
user avatar
1 vote
2 answers
46 views

Find all triplets $(a, b, c)$ of integers for which the following holds: $a^2 = bc + 1$ and $b^2 = ca + 1$. [duplicate]

Find all triplets $(a, b, c)$ of integers for which the following holds: $a^2 = bc + 1$ and $b^2 = ca + 1$. Attempt: First, I subtracted the two equations and obtained $(a-b)(a+b) = c(a-b)(-1)$. Now, ...
user avatar
4 votes
2 answers
737 views

How many six-digit numbers are there where the third digit is equal to the second last digit, ...

How many six-digit numbers are there where the third digit is equal to the second last digit, the digit in the ten-thousands place is equal to the digit in the hundreds place, and the product of all ...
user avatar

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