Found problems: 85335
2018 Taiwan TST Round 1, 2
In a plane, we are given $ 100 $ circles with radius $ 1 $ so that the area of any triangle whose vertices are circumcenters of those circles is at most $ 100 $. Prove that one may find a line that intersects at least $ 10 $ circles.
1999 China Team Selection Test, 3
For every permutation $ \tau$ of $ 1, 2, \ldots, 10$, $ \tau \equal{} (x_1, x_2, \ldots, x_{10})$, define $ S(\tau) \equal{} \sum_{k \equal{} 1}^{10} |2x_k \minus{} 3x_{k \minus{} 1}|$. Let $ x_{11} \equal{} x_1$. Find
[b]I.[/b] The maximum and minimum values of $ S(\tau)$.
[b]II.[/b] The number of $ \tau$ which lets $ S(\tau)$ attain its maximum.
[b]III.[/b] The number of $ \tau$ which lets $ S(\tau)$ attain its minimum.
2018 China Team Selection Test, 4
Suppose $A_1,A_2,\cdots ,A_n \subseteq \left \{ 1,2,\cdots ,2018 \right \}$ and $\left | A_i \right |=2, i=1,2,\cdots ,n$, satisfying that $$A_i + A_j, \; 1 \le i \le j \le n ,$$ are distinct from each other. $A + B = \left \{ a+b|a\in A,\,b\in B \right \}$. Determine the maximal value of $n$.
2006 JBMO ShortLists, 7
Determine all numbers $ \overline{abcd}$ such that $ \overline{abcd}\equal{}11(a\plus{}b\plus{}c\plus{}d)^2$.
2015 Postal Coaching, Problem 4
Let $ABCD$ be a convex quadrilateral. Construct equilateral triangles $AQB$, $BRC$, $CSD$ and $DPA$ externally on the sides $AB$, $BC$, $CD$ and $DA$ respectively. Let $K, L, M, N$ be the mid-points of $P Q, QR, RS, SP$. Find the maximum value of
$$\frac{KM + LN}{AC + BD}$$
.
1989 AMC 12/AHSME, 11
Hi guys,
I was just reading over old posts that I made last year ( :P ) and saw how much the level of Getting Started became harder. To encourage more people from posting, I decided to start a Problem of the Day. This is how I'll conduct this:
1. In each post (not including this one since it has rules, etc) everyday, I'll post the problem. I may post another thread after it to give hints though.
2. Level of problem.. This is VERY important. All problems in this thread will be all AHSME or problems similar to this level. No AIME. Some AHSME problems, however, that involve tough insight or skills will not be posted. The chosen problems will be usually ones that everyone can solve after working. Calculators are allowed when you solve problems but it is NOT necessary.
3. Response.. All you have to do is simply solve the problem and post the solution. There is no credit given or taken away if you get the problem wrong. This isn't like other threads where the number of problems you get right or not matters. As for posting, post your solutions here in this thread. Do NOT PM me. Also, here are some more restrictions when posting solutions:
A. No single answer post. It doesn't matter if you put hide and say "Answer is ###..." If you don't put explanation, it simply means you cheated off from some other people. I've seen several posts that went like "I know the answer" and simply post the letter. What is the purpose of even posting then? Huh?
B. Do NOT go back to the previous problem(s). This causes too much confusion.
C. You're FREE to give hints and post different idea, way or answer in some cases in problems. If you see someone did wrong or you don't understand what they did, post here. That's what this thread is for.
4. Main purpose.. This is for anyone who visits this forum to enjoy math. I rememeber when I first came into this forum, I was poor at math compared to other people. But I kindly got help from many people such as JBL, joml88, tokenadult, and many other people that would take too much time to type. Perhaps without them, I wouldn't be even a moderator in this forum now. This site clearly made me to enjoy math more and more and I'd like to do the same thing. That's about the rule.. Have fun problem solving!
Next post will contain the Day 1 Problem. You can post the solutions until I post one. :D
2001 India IMO Training Camp, 3
In a triangle $ABC$ with incircle $\omega$ and incenter $I$ , the segments $AI$ , $BI$ , $CI$ cut $\omega$ at $D$ , $E$ , $F$ , respectively. Rays $AI$ , $BI$ , $CI$ meet the sides $BC$ , $CA$ , $AB$ at $L$ , $M$ , $N$ respectively. Prove that:
\[AL+BM+CN \leq 3(AD+BE+CF)\]
When does equality occur?
1896 Eotvos Mathematical Competition, 2
Prove that the equations $$x^2-3xy+2y^2+x-y=0 \text{ and } x^2-2xy+y^2-5x+7y=0$$ imply the equation $xy-12x+15y=0$.
2004 Pre-Preparation Course Examination, 1
A network is a simple directed graph such that each edge $ e$ has two intger lower and upper capacities $ 0\leq c_l(e)\leq c_u(e)$. A circular flow on this graph is a function such that:
1) For each edge $ e$, $ c_l(e)\leq f(e)\leq c_u(e)$.
2) For each vertex $ v$: \[ \sum_{e\in v^\plus{}}f(e)\equal{}\sum_{e\in v^\minus{}}f(e)\]
a) Prove that this graph has a circular flow, if and only if for each partition $ X,Y$ of vertices of the network we have:
\[ \sum_{\begin{array}{c}{e\equal{}xy}\\{x\in X,y\in Y}\end{array}} c_l(e)\leq \sum_{\begin{array}{c}{e\equal{}yx}\\{y\in Y,x\in X}\end{array}} c_l(e)\]
b) Suppose that $ f$ is a circular flow in this network. Prove that there exists a circular flow $ g$ in this network such that $ g(e)\equal{}\lfloor f(e)\rfloor$ or $ g(e)\equal{}\lceil f(e)\rceil$ for each edge $ e$.
2019 Romania National Olympiad, 4
Find the natural numbers $x, y, z$ that verify the equation: $$2^x + 3 \cdot 11^y =7^z$$
2012 Argentina National Olympiad Level 2, 3
Let $ABC$ be a triangle with $\angle A= 105^\circ$ and $\angle B= 45^\circ$. Let $L$ be a point on side $BC$ such that $AL$ is the bisector of angle $\angle BAC$ and let $M$ be the midpoint of side $AC$. Suppose that lines $AL$ and $BM$ intersect at point $P$. Calculate the ratio $\dfrac{AP}{AL}$.
JOM 2025, 4
There are $n$ people arranged in a circle, and $n^{n^n}$ coins are distributed among them, where each person has at least $n^n$ coins. Each person is then assigned a random index number in $\{1,2,...n\}$ such that no two people have the same number. Then every minute, if $i$ is the number of minutes passed, the person with index number congruent to $i$ mod $n$ will give a coin to the person on his left or right. After some time, everyone has the same number of coins.
For what $n$ is this always possible, regardless of the original distribution of coins and index numbers?
[i](Proposed by Ho Janson)[/i]
2017 JBMO Shortlist, G3
Consider triangle $ABC$ such that $AB \le AC$. Point $D$ on the arc $BC$ of thecircumcirle of $ABC$ not containing point $A$ and point $E$ on side $BC$ are such that $\angle BAD = \angle CAE < \frac12 \angle BAC$ . Let $S$ be the midpoint of segment $AD$. If $\angle ADE = \angle ABC - \angle ACB$ prove that $\angle BSC = 2 \angle BAC$ .
2012 Saint Petersburg Mathematical Olympiad, 6
On the coordinate plane in the first quarter there are $100$ non-intersecting single unit segments parallel to the coordinate axes. These segments aremirrors (on both sides), they reflect the light according to the rule. "The angle of incidence is equal to the angle of reflection." (If you hit the edge of the mirror, the beam of light does not change its direction.) From the point lying in the unit circle with the center at the origin, a ray of light in the direction of the bisector of the first coordinate angle. Prove that, that this initial point can be chosen so that the ray is reflected from the mirrors not more than $150$ times.
2020 IMO Shortlist, A8
Let $R^+$ be the set of positive real numbers. Determine all functions $f:R^+$ $\rightarrow$ $R^+$ such that for all positive real numbers $x$ and $y:$
\[f(x+f(xy))+y=f(x)f(y)+1\]
[i]Ukraine[/i]
2019 Brazil National Olympiad, 3
Let $ABC$ be an acutangle triangle inscribed in a circle $\Gamma$ of center $O$. Let $D$ be the height of the
vertex $A$. Let E and F be points over $\Gamma$ such that $AE = AD = AF$. Let $P$ and $Q$ be the intersection points of the $EF $ with sides $AB$ and $AC$ respectively. Let $X$ be the second intersection point of $\Gamma$ with the circle circumscribed to the triangle $AP Q$. Show that the lines $XD$ and $AO $ meet at a point above sobre
2013 ELMO Shortlist, 8
We define the [i]Fibonacci sequence[/i] $\{F_n\}_{n\ge0}$ by $F_0=0$, $F_1=1$, and for $n\ge2$, $F_n=F_{n-1}+F_{n-2}$; we define the [i]Stirling number of the second kind[/i] $S(n,k)$ as the number of ways to partition a set of $n\ge1$ distinguishable elements into $k\ge1$ indistinguishable nonempty subsets.
For every positive integer $n$, let $t_n = \sum_{k=1}^{n} S(n,k) F_k$. Let $p\ge7$ be a prime. Prove that \[ t_{n+p^{2p}-1} \equiv t_n \pmod{p} \] for all $n\ge1$.
[i]Proposed by Victor Wang[/i]
2010 Contests, 1
For all natural $n$, an $n$-staircase is a figure consisting of unit squares, with one square in the first row, two squares in the second row, and so on, up to $n$ squares in the $n^{th}$ row, such that all the left-most squares in each row are aligned vertically.
Let $f(n)$ denote the minimum number of square tiles requires to tile the $n$-staircase, where the side lengths of the square tiles can be any natural number. e.g. $f(2)=3$ and $f(4)=7$.
(a) Find all $n$ such that $f(n)=n$.
(b) Find all $n$ such that $f(n) = n+1$.
1950 Miklós Schweitzer, 4
Find the polynomials $ f(x)$ having the following properties:
(i) $ f(0) \equal{} 1$, $ f'(0) \equal{} f''(0) \equal{} \cdots \equal{} f^{(n)}(0) \equal{} 0$
(ii) $ f(1) \equal{} f'(1) \equal{} f''(1) \equal{} \cdots \equal{} f^{(m)}(1) \equal{} 0$
2007 AMC 10, 20
A set of $ 25$ square blocks is arranged into a $ 5\times 5$ square. How many different combinations of $ 3$ blocks can be selected from that set so that no two are in the same row or column?
$ \textbf{(A)}\ 100\qquad
\textbf{(B)}\ 125\qquad
\textbf{(C)}\ 600\qquad
\textbf{(D)}\ 2300\qquad
\textbf{(E)}\ 3600$
1940 Putnam, B3
Let $p>0$ be a real constant. From any point $(a,b)$ in the cartesian plane, show that
i) Three normals, real or imaginary, can be drawn to the parabola $y^2=4px$.
ii) These are real and distinct if $4(2-p)^3 +27pb^2<0$.
iii) Two of them coincide if $(a,b)$ lies on the curve $27py^2=4(x-2p)^3$.
iv) All three coincide only if $a=2p$ and $b=0$.
2023 IMO, 4
Let $x_1,x_2,\dots,x_{2023}$ be pairwise different positive real numbers such that
\[a_n=\sqrt{(x_1+x_2+\dots+x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\dots+\frac{1}{x_n}\right)}\]
is an integer for every $n=1,2,\dots,2023.$ Prove that $a_{2023} \geqslant 3034.$
2021 China Team Selection Test, 6
Given positive integer $n$ and $r$ pairwise distinct primes $p_1,p_2,\cdots,p_r.$ Initially, there are $(n+1)^r$ numbers written on the blackboard: $p_1^{i_1}p_2^{i_2}\cdots p_r^{i_r} (0 \le i_1,i_2,\cdots,i_r \le n).$
Alice and Bob play a game by making a move by turns, with Alice going first. In Alice's round, she erases two numbers $a,b$ (not necessarily different) and write $\gcd(a,b)$. In Bob's round, he erases two numbers $a,b$ (not necessarily different) and write $\mathrm{lcm} (a,b)$. The game ends when only one number remains on the blackboard.
Determine the minimal possible $M$ such that Alice could guarantee the remaining number no greater than $M$, regardless of Bob's move.
2010 Harvard-MIT Mathematics Tournament, 10
Let $f(n)=\displaystyle\sum_{k=1}^n \dfrac{1}{k}$. Then there exists constants $\gamma$, $c$, and $d$ such that \[f(n)=\ln(x)+\gamma+\dfrac{c}{n}+\dfrac{d}{n^2}+O\left(\dfrac{1}{n^3}\right),\] where the $O\left(\dfrac{1}{n^3}\right)$ means terms of order $\dfrac{1}{n^3}$ or lower. Compute the ordered pair $(c,d)$.
2024 Serbia Team Selection Test, 5
The circles $k_1, k_2$, centered at $O_1, O_2$, meet at two points, one of which is $A$. Let $P, Q$ lie on $AO_1, AO_2$, respectively, so that $PQ \parallel O_1O_2$. The tangents from $P$ to $k_2$ touch it at $X, Y$ and the tangents from $Q$ to $k_1$ touch it at $Z, T$. Show that $X, Y, Z, T$ are collinear or concyclic.