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.
10,584 questions
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geometric problem for spiral similarity
We consider a quadrilateral $ABCD$ inscribed in a circle $\omega$. Let $P$ be a point inside $\omega$ and the following equalities are satisfied
$$\angle PAD = \angle PCB,\ \angle ADP = \angle CBP.$$ ...
- 5,246
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votes
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answer
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views
Minimum cardinality of the set of values for a sequence($a_1,a_2...a_{2025}$) with distinct cyclic ratios
Let $n = 2025$. We are given a sequence of positive integers $a_1, a_2, \dots, a_n$.
Let the cyclic ratios be defined as:
$$r_i = \frac{a_i}{a_{i+1}} \quad \text{for } 1 \le i \le n-1, \quad \text{and}...
2
votes
1
answer
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Construct AM–GM Proofs of Muirhead Inequalities (From Majorization to Explicit Weighted AM–GM Chains)
Motivation
I am experimenting with symbolic implementations of algorithms that, given a majorization relation between two exponent vectors, automatically generate Olympiad-style inequality proofs ...
- 6,087
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Prove that $BN=LC$. A geometry problem from the national round of math olympiad.
Problem: Let $ABC$ be an acute triangle and $D$ be the foot of the altitude from $A$ onto $BC$. A semicircle with diameter $BC$ intersects segments $AB, AC$ and $AD$ in the points $F, E$ and $X$, ...
- 643
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Can I get some help with IMO 1977 #6? [duplicate]
Let $f : \mathbb{N} \to \mathbb{N}$ be a function such that $f(n+1) > f(f(n))$ for all $n \in \mathbb{N}$. Prove that $f(n) = n$ for all $n$.
Can I get some help with this question? My approach is ...
2
votes
2
answers
97
views
The least positive integer ending in $7$ which quintuples if the $7$ is moved to the front
I wrote up an attempt at the first problem in "Problem Primer for Olympiad," which is:
Find the least number whose last digit is $7$ and which becomes $5$ times larger when this last digit ...
7
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257
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Minimum size of a sequence summing to $2013$ that guarantees a consecutive subset sum of $31$ (still wanted rigorous proof)
I am trying to solve the following problem on integer sequences and subset sums from a 2023 Shanghai high school entrance exam:
Let $A = (a_1, a_2, \dots, a_n)$ be a sequence of positive integers ...
2
votes
1
answer
151
views
Limit with floor sums reminiscent of the exponent of the central binomial coefficient
This problem comes from the 1976 Putnam exam.
Evaluate
$$
L=\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n
\left(
\left\lfloor\frac{2n}{k}\right\rfloor
-2\left\lfloor\frac{n}{k}\right\rfloor
\right),
$$
...
- 23
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Prove or disprove that $a+b+c \geq ab+bc+ca$, for positive $a$, $b$, $c$ satisfying $\sum_{cyc}\frac{1}{a+b+1}\geq1$ [closed]
Prove that for positive $a,b,c$, if
$$
\frac{1}{a+b+1} + \frac{1}{b+c+1} + \frac{1}{c+a+1} \geq 1
$$
then $$a+b+c \geq ab+bc+ca$$
My attempt: expanded everything and stuck at
$$
\begin{aligned}
2(a+b+...
- 37
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votes
3
answers
504
views
$f(x)$ is a cubic polynomial $f(k-1)f(k+1)<0$ is NOT true for any integer $k$. Find $f(8)$
$f(x)$ is a monic cubic polynomial and $f(k-1)f(k+1)<0$ is NOT true for any integer $k$.
Also $f'\left(-\frac{1}{4}\right)=-\frac{1}{4}$ and $f'\left(\frac{1}{4}\right)<0$.
Find value of $f(8)$
...
- 11.2k
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votes
3
answers
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views
Prove $AF, BC, GE$ are concurrent in a tangent–circumcircle configuration
Problem:
Let $\triangle ABC$ be inscribed in a circles. $D$ be the intersection of the tangents at $B$ and $C$. Let $AD$ intersect the circumcircle at $E$. Let $F$ be any point on the circumcircle on ...
- 643
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votes
1
answer
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views
How to prove that $f \equiv 0$ on $[1;2]$?
A real-valued function $f \in C[1;2]$ satisfies $\bigg | \int_{1}^{2} f(x) x^n dx \bigg |<2^{-1000n}$ for every positive integer $n$. How to prove that $f(x)=0$ for all $x \in [1;2]$?
My attempt: ...
- 593
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votes
3
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Prove that triangle CHL is isosceles in rectangle with specific construction
Problem Statement:
Let $ABCD$ be a rectangle with $BC > AB$. Take point $E$ on side $BC$ such that $DE = DA$. Draw perpendiculars from vertices $A$ and $C$ to line $DE$, with feet at points $Z$ and ...
4
votes
2
answers
120
views
Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that $f(y^2+1)+f(xy)=f(x+y)f(y)+1$ holds for all $x,y \in \mathbb{R}$
Problem: Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that $f(y^2+1)+f(xy)=f(x+y)f(y)+1$ holds for all $x,y \in \mathbb{R}$.
My approach: I have got $f(0)=0,1$. Taking $f(0)=1$, and setting $...
- 499
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Hints to prove Leibniz Theorem regarding the centroid of a triangle
Well for a quite hours I tried to solve Leibniz' theorem (as given in my textbook Pathfinder for Olympiad).
Let $G$ be the centroid of triangle $ABC$, and $P$ is an arbitrary point. Prove that
$$PA^2+...
