Found problems: 85335
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
$$\frac{1}{1-z}= \lim_{k\to \infty}(1 + z)(1 + z^2)(1 + z^{2^2})...(1 + z^{2^k}).$$
2012 Canadian Mathematical Olympiad Qualification Repechage, 6
Determine whether there exist two real numbers $a$ and $b$ such that both $(x-a)^3+ (x-b)^2+x$ and $(x-b)^3 + (x-a)^2 +x$ contain only real roots.
Brazil L2 Finals (OBM) - geometry, 2002.5
Let $ABC$ be a triangle inscribed in a circle of center $O$ and $P$ be a point on the arc $AB$, that does not contain $C$. The perpendicular drawn fom $P$ on line $BO$ intersects $AB$ at $S$ and $BC$ at $T$. The perpendicular drawn from $P$ on line $AO$ intersects $AB$ at $Q$ and $AC$ at $R$. Prove that:
a) $PQS$ is an isosceles triangle
b) $PQ^2=QR= ST$
2010 ELMO Shortlist, 4
Let $r$ and $s$ be positive integers. Define $a_0 = 0$, $a_1 = 1$, and $a_n = ra_{n-1} + sa_{n-2}$ for $n \geq 2$. Let $f_n = a_1a_2\cdots a_n$. Prove that $\displaystyle\frac{f_n}{f_kf_{n-k}}$ is an integer for all integers $n$ and $k$ such that $0 < k < n$.
[i]Evan O' Dorney.[/i]
2019 HMNT, 3
Katie has a fair $2019$-sided die with sides labeled $1, 2,..., 2019$. After each roll, she replaces her $n$-sided die with an $(n+1)$-sided die having the $n$ sides of her previous die and an additional side with the number she just rolled. What is the probability that Katie's $2019^{th}$ roll is a $ 2019$?
2015 Estonia Team Selection Test, 7
Prove that for every prime number $p$ and positive integer $a$, there exists a natural number $n$ such that $p^n$ contains $a$ consecutive equal digits.
2005 Oral Moscow Geometry Olympiad, 3
In triangle $ABC$, points $K ,P$ are chosen on the side $AB$ so that $AK = BL$, and points $M,N$ are chosen on the side $BC$ so that $CN = BM$. Prove that $KN + LM \ge AC$.
(I. Bogdanov)
2024 HMNT, 32
Let $ABC$ be an acute triangle and $D$ be the foot of altitude from $A$ to $BC.$ Let $X$ and $Y$ be points on the segment $BC$ such that $\angle{BAX} = \angle{YAC}, BX = 2, XY = 6,$ and $YC = 3.$ Given that $AD = 12,$ compute $BD.$
2023 USAMO, 1
In an acute triangle $ABC$, let $M$ be the midpoint of $\overline{BC}$. Let $P$ be the foot of the perpendicular from $C$ to $AM$. Suppose that the circumcircle of triangle $ABP$ intersects line $BC$ at two distinct points $B$ and $Q$. Let $N$ be the midpoint of $\overline{AQ}$. Prove that $NB=NC$.
[i]Proposed by Holden Mui[/i]
2017-2018 SDML (Middle School), 1
Let $N = \frac{1}{3} + \frac{3}{5} + \frac{5}{7} + \frac{7}{9} + \frac{9}{11}$. What is the greatest integer which is less than $N$?
1962 AMC 12/AHSME, 17
If $ a \equal{} \log_8 225$ and $ b \equal{} \log_2 15,$ then $ a$, in terms of $ b,$ is:
$ \textbf{(A)}\ \frac{b}{2} \qquad
\textbf{(B)}\ \frac{2b}{3}\qquad
\textbf{(C)}\ b \qquad
\textbf{(D)}\ \frac{3b}{2} \qquad
\textbf{(E)}\ 2b$
2010 Switzerland - Final Round, 6
Find all functions $ f: \mathbb{R}\mapsto\mathbb{R}$ such that for all $ x$, $ y$ $ \in\mathbb{R}$,
\[ f(f(x))\plus{}f(f(y))\equal{}2y\plus{}f(x\minus{}y)\]
holds.
2009 Jozsef Wildt International Math Competition, W. 8
If $n,p,q \in \mathbb{N}, p<q $ then $${{(p+q)n}\choose{n}} \sum \limits_{k=0}^n (-1)^k {{n}\choose{k}} {{(p+q-1)n}\choose{pn-k}}= {{(p+q)n}\choose{pn}} \sum \limits_{k=0}^{\left [\frac{n}{2} \right ]} (-1)^k {{pn}\choose{k}} {{(q-p)n}\choose{n-2k}} $$
2013 Greece National Olympiad, 1
Let the sequence of real numbers $(a_n),n=1,2,3...$ with $a_1=2$ and $a_n=\left(\frac{n+1}{n-1} \right)\left(a_1+a_2+...+a_{n-1} \right),n\geq 2$.
Find the term $a_{2013}$.
Putnam 1938, B3
A horizontal disk diameter $3$ inches rotates once every $15$ seconds. An insect starts at the southernmost point of the disk facing due north. Always facing due north, it crawls over the disk at $1$ inch per second. Where does it again reach the edge of the disk?
2018 USA TSTST, 2
In the nation of Onewaynia, certain pairs of cities are connected by one-way roads. Every road connects exactly two cities (roads are allowed to cross each other, e.g., via bridges), and each pair of cities has at most one road between them. Moreover, every city has exactly two roads leaving it and exactly two roads entering it.
We wish to close half the roads of Onewaynia in such a way that every city has exactly one road leaving it and exactly one road entering it. Show that the number of ways to do so is a power of $2$ greater than $1$ (i.e.\ of the form $2^n$ for some integer $n \ge 1$).
[i]Victor Wang[/i]
2022 JBMO Shortlist, G2
Let $ABC$ be a triangle with circumcircle $k$. The points $A_1, B_1,$ and $C_1$ on $k$ are the midpoints of arcs $\widehat{BC}$ (not containing $A$), $\widehat{AC}$ (not containing $B$), and $\widehat{AB}$ (not containing $C$), respectively. The pairwise distinct points $A_2, B_2,$ and $C_2$ are chosen such that the quadrilaterals $AB_1A_2C_1, BA_1B_2C_1,$ and $CA_1C_2B_1$ are parallelograms. Prove that $k$ and the circumcircle of triangle $A_2B_2C_2$ have a common center.
[b]Comment.[/b] Point $A_2$ can also be defined as the reflection of $A$ with respect to the midpoint of $B_1C_1$, and analogous definitions can be used for $B_2$ and $C_2$.
2014 Contests, 1
Prove that for $n\ge 2$ the following inequality holds:
$$\frac{1}{n+1}\left(1+\frac{1}{3}+\ldots +\frac{1}{2n-1}\right) >\frac{1}{n}\left(\frac{1}{2}+\ldots+\frac{1}{2n}\right).$$
2014 AMC 10, 3
Randy drove the first third of his trip on a gravel road, the next 20 miles on pavement, and the remaining one-fifth on a dirt road. In miles, how long was Randy's trip?
$ \textbf {(A) } 30 \qquad \textbf {(B) } \frac{400}{11} \qquad \textbf {(C) } \frac{75}{2} \qquad \textbf {(D) } 40 \qquad \textbf {(E) } \frac{300}{7} $
2009 Turkey Team Selection Test, 1
For which $ p$ prime numbers, there is an integer root of the polynominal $ 1 \plus{} p \plus{} Q(x^1)\cdot\ Q(x^2)\ldots\ Q(x^{2p \minus{} 2})$ such that $ Q(x)$ is a polynominal with integer coefficients?