Found problems: 85335
2010 Harvard-MIT Mathematics Tournament, 8
Let $f(n)=\displaystyle\sum_{k=2}^\infty \dfrac{1}{k^n\cdot k!}.$ Calculate $\displaystyle\sum_{n=2}^\infty f(n)$.
1988 National High School Mathematics League, 5
In rectangular coordinate system, define that if and only if both $x$-axis and $y$-axis of a point are integers, we call it itegral point. $I$ is set of all lines, $M$ is set of lines that pass exactly one intengral point, $N$ is set of lines that pass no itengral point, $P$ is set of lines that pass infinitely many itengral points. Then, how many conclusions are right?
(1)$M\cup N\cup P=I$.
(2)$N\neq\varnothing$.
(3)$M\neq\varnothing$.
(4)$P\neq\varnothing$.
$\text{(A)}1\qquad\text{(B)}2\qquad\text{(C)}3\qquad\text{(D)}4\qquad$
1957 AMC 12/AHSME, 44
In triangle $ ABC$, $ AC \equal{} CD$ and $ \angle CAB \minus{} \angle ABC \equal{} 30^\circ$. Then $ \angle BAD$ is:
[asy]defaultpen(linewidth(.8pt));
unitsize(2.5cm);
pair A = origin;
pair B = (2,0);
pair C = (0.5,0.75);
pair D = midpoint(C--B);
draw(A--B--C--cycle);
draw(A--D);
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,N);
label("$D$",D,NE);[/asy]$ \textbf{(A)}\ 30^\circ\qquad \textbf{(B)}\ 20^\circ\qquad \textbf{(C)}\ 22\frac {1}{2}^\circ\qquad \textbf{(D)}\ 10^\circ\qquad \textbf{(E)}\ 15^\circ$
2015 IMO Shortlist, A4
Let $\mathbb R$ be the set of real numbers. Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy the equation\[f(x+f(x+y))+f(xy)=x+f(x+y)+yf(x)\]for all real numbers $x$ and $y$.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
2007 Thailand Mathematical Olympiad, 13
Let $S = \{1, 2,..., 8\}$. How many ways are there to select two disjoint subsets of $S$?
2015 Peru Cono Sur TST, P4
In a small city there are $n$ bus routes, with $n > 1$, and each route has exactly $4$ stops. If any two routes have exactly one common stop, and each pair of stops belongs to exactly one route, find all possible values of $n$.
2010 Sharygin Geometry Olympiad, 2
Two points $A$ and $B$ are given. Find the locus of points $C$ such that triangle $ABC$ can be covered by a circle with radius $1$.
(Arseny Akopyan)
2019 Peru EGMO TST, 5
Define the sequence sequence $a_0,a_1, a_2,....,a_{2018}, a_{2019}$ of real numbers as follows:
$\bullet$ $a_0 = 1$.
$\bullet$ $a_{n + 1} = a_n - \frac{a_n^2}{2019}$ for $n = 0, 1, ...,2018$.
Prove that $a_{2019} < \frac12 <a_{2018}$.
1979 All Soviet Union Mathematical Olympiad, 283
Given $n$ points (in sequence)$ A_1, A_2, ... , A_n$ on a line. All the segments $A_1A_2$, $A_2A_3$,$ ...$, $A_{n-1}A_n$ are shorter than $1$. We need to mark $(k-1)$ points so that the difference of every two segments, with the ends in the marked points, is shorter than $1$. Prove that it is possible
a) for $k=3$,
b) for every $k$ less than $(n-1)$.
2024 USEMO, 3
Let $ABC$ be a triangle with incenter $I$. Two distinct points $P$ and $Q$ are chosen on the circumcircle of $ABC$ such that
\[ \angle API = \angle AQI = 45^\circ. \]
Lines $PQ$ and $BC$ meet at $S$. Let $H$ denote the foot of the altitude from $A$ to $BC$. Prove that $\angle AHI = \angle ISH$.
[i]Matsvei Zorka[/i]
1999 Estonia National Olympiad, 1
Prove that if $p$ is an odd prime, then $p^2(p^2 -1999)$ is divisible by $6$ but not by $12$.
2006 Princeton University Math Competition, 5
In the diagram shown, how many pathways are there from point $A$ to point $B$ if you are only allowed to travel due East, Southeast, or Southwest?
[img]https://cdn.artofproblemsolving.com/attachments/9/1/0a1219fb430c402fef4b7555ddff7c88fec47e.jpg[/img]
2019 Balkan MO Shortlist, G5
Let $ABC$ ($BC > AC$) be an acute triangle with circumcircle $k$ centered at $O$. The tangent to $k$ at $C$ intersects the line $AB$ at the point $D$. The circumcircles of triangles $BCD, OCD$ and $AOB$ intersect the ray $CA$ (beyond $A$) at the points $Q, P$ and $K$, respectively, such that $P \in (AK)$ and $K \in (PQ)$. The line $PD$ intersects the circumcircle of triangle $BKQ$ at the point $T$, so that $P$ and $T$ are in different halfplanes with respect to $BQ$. Prove that $TB = TQ$.
2011 Croatia Team Selection Test, 4
Find all pairs of integers $x,y$ for which
\[x^3+x^2+x=y^2+y.\]
2022 Princeton University Math Competition, A7
For a positive integer $n \ge 1,$ let $a_n=\lfloor \sqrt[3]{n}+\tfrac{1}{2}\rfloor.$ Given a positive integer $N \ge 1,$ let $\mathcal{F}_N$ denote the set of positive integers $n \ge 1$ such that $a_n \le N.$ Let $S_N = \sum_{n \in \mathcal{F}_N} \tfrac{1}{a_n^2}.$ As $N$ goes to infinity, the quantity $S_N - 3N$ tends to $\tfrac{a\pi^2}{b}$ for relatifvely prime positive integers $a,b.$ Given that $\sum_{k=1}^{\infty} \tfrac{1}{k^2} = \tfrac{\pi^2}{6},$ find $a+b.$
1997 ITAMO, 1
An infinite rectangular stripe of width $3$ cm is folded along a line. What is the minimum possible area of the region of overlapping?
2020 Princeton University Math Competition, A4/B6
Given two positive integers $a \ne b$, let $f(a, b)$ be the smallest integer that divides exactly one of $a, b$, but not both. Determine the number of pairs of positive integers $(x, y)$, where $x \ne y$, $1\le x, y, \le 100$ and $\gcd(f(x, y), \gcd(x, y)) = 2$.
1990 IMO Longlists, 8
Let $a, b, c$ be the side lengths and $P$ be area of a triangle, respectively. Prove that
\[(a^2+b^2+c^2-4\sqrt 3 P) (a^2+b^2+c^2) \geq 2 \left(a^2(b - c)^2 + b^2(c - a)^2 + c^2(a - b)^2\right).\]
2024 German National Olympiad, 4
Let $k>2$ be a positive integer such that the $k$-digit number $n_k=133\dots 3$, consisting of a digit $1$ followed by $k-1$ digits $3$ is prime. Show that $24 \mid k(k+2)$.
2019 Simon Marais Mathematical Competition, A2
Consider the operation $\ast$ that takes pair of integers and returns an integer according to the rule
$$a\ast b=a\times (b+1).$$
[list=a]
[*]For each positive integer $n$, determine all permutations $a_1,a_2,\dotsc , a_n$ of the set $\{ 1,2,\dotsc ,n\}$ that maximise the value of $$(\cdots ((a_1\ast a_2)\ast a_3) \ast \cdots \ast a_{n-1})\ast a_n.$$[/*]
[*]For each positive integer $n$, determine all permutations $b_1,b_2,\dotsc , b_n$ of the set $\{ 1,2,\dotsc ,n\}$ that maximise the value of $$b_1\ast (b_2\ast (b_3\ast \cdots \ast (b_{n-1}\ast b_n)\cdots )).$$[/*]
[/list]
1990 China Team Selection Test, 3
In set $S$, there is an operation $'' \circ ''$ such that $\forall a,b \in S$, a unique $a \circ b \in S$ exists. And
(i) $\forall a,b,c \in S$, $(a \circ b) \circ c = a \circ (b \circ c)$.
(ii) $a \circ b \neq b \circ a$ when $a \neq b$.
Prove that:
a.) $\forall a,b,c \in S$, $(a \circ b) \circ c = a \circ c$.
b.) If $S = \{1,2, \ldots, 1990\}$, try to define an operation $'' \circ ''$ in $S$ with the above properties.
Putnam 1938, A3
A particle moves in the Euclidean plane. At time $t$ (taking all real values) its coordinates are $x = t^3 - t$ and $y = t^4 + t.$ Show that its velocity has a maximum at $t = 0,$ and that its path has an inflection at $t = 0.$
2024 Mongolian Mathematical Olympiad, 2
Let $ABC$ be an acute-angled triangle and let $E$ and $F$ be the feet of the altitudes from $B$ and $C$ to the sides $AC$ and $AB$ respectively. Suppose $AD$ is the diameter of the circle $ABC$. Let $M$ be the midpoint of $BC$. Let $K$ be the imsimilicenter of the incircles of the triangles $BMF$ and $CME$. Prove that the points $K, M, D$ are collinear.
[i]Proposed by Bilegdembrel Bat-Amgalan.[/i]
2012 Turkey Junior National Olympiad, 3
Let $a, b, c$ be positive real numbers satisfying $a^3+b^3+c^3=a^4+b^4+c^4$. Show that
\[ \frac{a}{a^2+b^3+c^3}+\frac{b}{a^3+b^2+c^3}+\frac{c}{a^3+b^3+c^2} \geq 1 \]
1971 Spain Mathematical Olympiad, 5
Prove that whatever the complex number $z$ is, it is true that
$$(1 + z^{2^n})(1-z^{2^n})= 1- z^{2^{n+1}}.$$
Writing the equalities that result from giving $n$ the values $0, 1, 2, . . .$ and multiplying them, show that for $|z| < 1$ holds
$$\frac{1}{1-z}= \lim_{k\to \infty}(1 + z)(1 + z^2)(1 + z^{2^2})...(1 + z^{2^k}).$$