Found problems: 85335
2021 Korea National Olympiad, P2
For positive integers $n, k, r$, denote by $A(n, k, r)$ the number of integer tuples $(x_1, x_2, \ldots, x_k)$ satisfying the following conditions.
[list]
[*] $x_1 \ge x_2 \ge \cdots \ge x_k \ge 0$
[*] $x_1+x_2+ \cdots +x_k = n$
[*] $x_1-x_k \le r$
[/list]
For all positive integers $m, s, t$, prove that $$A(m, s, t)=A(m, t, s).$$
2010 Contests, 1
The real numbers $a$, $b$, $c$, $d$ satisfy simultaneously the equations
\[abc -d = 1, \ \ \ bcd - a = 2, \ \ \ cda- b = 3, \ \ \ dab - c = -6.\] Prove that $a + b + c + d \not = 0$.
1998 National High School Mathematics League, 15
Parabola $y^2=2px$, two fixed points $A(a,b),B(-a,0)(ab\neq0,b^2\neq 2pa)$. $M$ is a point on the parabola, $AM$ intersects the parabola at $M_1$, $BM$ intersects the parabola at $M_2$.
Prove: When $M$ changes, line $M_1M_2$ passes a fixed point, and find the fixed point.
2007 USAMO, 1
Let $n$ be a positive integer. Define a sequence by setting $a_{1}= n$ and, for each $k > 1$, letting $a_{k}$ be the unique integer in the range $0\leq a_{k}\leq k-1$ for which $a_{1}+a_{2}+...+a_{k}$ is divisible by $k$. For instance, when $n = 9$ the obtained sequence is $9,1,2,0,3,3,3,...$. Prove that for any $n$ the sequence $a_{1},a_{2},...$ eventually becomes constant.
1988 IMO Longlists, 54
Find the least natural number $ n$ such that, if the set $ \{1,2, \ldots, n\}$ is arbitrarily divided into two non-intersecting subsets, then one of the subsets contains 3 distinct numbers such that the product of two of them equals the third.
1961 Putnam, B6
Consider the function $y(x)$ satisfying the differential equation $y'' = -(1+\sqrt{x})y$ with $y(0)=1$ and $y'(0)=0.$ Prove that $y(x)$ vanishes exactly once on the interval $0< x< \pi \slash 2,$ and find a positive lower bound for the zero.
2024 239 Open Mathematical Olympiad, 1
Let $f:\mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$ be a continuous function such that $f(0)=0$ and $$f(x)+f(f(x))+f(f(f(x)))=3x$$ for all $x>0$. Show that $f(x)=x$ for all $x>0$.
1994 India Regional Mathematical Olympiad, 6
Let $AC$ and $BD$ be two chords of a circle with center $O$ such that they intersect at right angles inside the circle at the point $M$. Suppose $K$ and $L$ are midpoints of the chords $AB$ and $CD$ respectively. Prove that $OKML$ is a parallelogram.
1999 Harvard-MIT Mathematics Tournament, 4
A cross-section of a river is a trapezoid with bases $10$ and $16$ and slanted sides of length $5$. At this section the water is flowing at $\pi$ mph. A little ways downstream is a dam where the water flows through $4$ identical circular holes at $16$ mph. What is the radius of the holes?
2010 India National Olympiad, 5
Let $ ABC$ be an acute-angled triangle with altitude $ AK$. Let $ H$ be its ortho-centre and $ O$ be its circum-centre. Suppose $ KOH$ is an acute-angled triangle and $ P$ its circum-centre. Let $ Q$ be the reflection of $ P$ in the line $ HO$. Show that $ Q$ lies on the line joining the mid-points of $ AB$ and $ AC$.