Found problems: 85335
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
$$\frac{1}{1-z}= \lim_{k\to \infty}(1 + z)(1 + z^2)(1 + z^{2^2})...(1 + z^{2^k}).$$
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 $${{(p+q)n}\choose{n}} \sum \limits_{k=0}^n (-1)^k {{n}\choose{k}} {{(p+q-1)n}\choose{pn-k}}= {{(p+q)n}\choose{pn}} \sum \limits_{k=0}^{\left [\frac{n}{2} \right ]} (-1)^k {{pn}\choose{k}} {{(q-p)n}\choose{n-2k}} $$