Found problems: 85335
2012 Mathcenter Contest + Longlist, 7
The arithmetic function $\nu$ is defined by $$\nu (n) = \begin{cases}0, \,\,\,\,\, n=1 \\ k, \,\,\,\,\, n=p_1^{a_1} p_2^{a_2} ... p_k^{a_k}\end{cases}$$, where $n=p_1^{a_1} p_2^{a_2} ... p_k^{a_k}$ represents the prime factorization of the number. Prove that for any naturals $m,n$, $$\tau (n^m) = \sum_{d | n} m^{\nu (d)}.$$ [i](PP-nine)[/i]
2015 Federal Competition For Advanced Students, P2, 5
Let I be the incenter of triangle $ABC$ and let $k$ be a circle through the points $A$ and $B$. The circle intersects
* the line $AI$ in points $A$ and $P$
* the line $BI$ in points $B$ and $Q$
* the line $AC$ in points $A$ and $R$
* the line $BC$ in points $B$ and $S$
with none of the points $A,B,P,Q,R$ and $S$ coinciding and such that $R$ and $S$ are interior points of the line segments $AC$ and $BC$, respectively.
Prove that the lines $PS$, $QR$, and $CI$ meet in a single point.
(Stephan Wagner)
2011 Kosovo National Mathematical Olympiad, 4
A point $P$ is given in the square $ABCD$ such that $\overline{PA}=3$, $\overline{PB}=7$ and $\overline{PD}=5$. Find the area of the square.
2010 Contests, 2
All positive divisors of a positive integer $N$ are written on a blackboard. Two players $A$ and $B$ play the following game taking alternate moves. In the firt move, the player $A$ erases $N$. If the last erased number is $d$, then the next player erases either a divisor of $d$ or a multiple of $d$. The player who cannot make a move loses. Determine all numbers $N$ for which $A$ can win independently of the moves of $B$.
[i](4th Middle European Mathematical Olympiad, Individual Competition, Problem 2)[/i]
2019 HMNT, 4
To celebrate $2019$, Faraz gets four sandwiches shaped in the digits $2$, $0$, $1$, and $9$ at lunch. However, the four digits get reordered (but not ipped or rotated) on his plate and he notices that they form a $4$-digit multiple of $7$. What is the greatest possible number that could have been formed?
2021 Israel TST, 3
Let $ABC$ be an acute triangle with orthocenter $H$. Prove that there is a line $l$ which is parallel to $BC$ and tangent to the incircles of $ABH$ and $ACH$.
2008 May Olympiad, 4
Let $ABF$ be a right-angled triangle with $\angle AFB = 90$, a square $ABCD$ is externally to the triangle. If $FA = 6$, $FB = 8$ and $E$ is the circumcenter of the square $ABCD$, determine the value of $EF$
2005 Gheorghe Vranceanu, 3
Let be a continuous function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having a positive period $ T. $ Prove that:
$$ \lim_{n\to\infty } e^{-nT}\int_0^{nT} e^tf(t)dt=\frac{1}{e^T-1}\int_0^T e^tf(t)dt $$
2010 Tuymaada Olympiad, 2
Let $ABC$ be an acute triangle, $H$ its orthocentre, $D$ a point on the side $[BC]$, and $P$ a point such that $ADPH$ is a parallelogram.
Show that $\angle BPC > \angle BAC$.
2016 CHMMC (Fall), 10
For a positive integer $n$, let $p(n)$ denote the number of prime divisors of $n$, counting multiplicity (i.e. $p(12)=3$). A sequence $a_n$ is defined such that $a_0 = 2$ and for $n > 0$, $a_n = 8^{p(a_{n-1})} + 2$. Compute $$\sum_{n=0}^{\infty} \frac{a_n}{2^n}$$
1999 Tuymaada Olympiad, 1
50 knights of King Arthur sat at the Round Table. A glass of white or red wine stood before each of them. It is known that at least one glass of red wine and at least one glass of white wine stood on the table. The king clapped his hands twice. After the first clap every knight with a glass of red wine before him took a glass from his left neighbour. After the second clap every knight with a glass of white wine (and possibly something more) before him gave this glass to the left neughbour of his left neighbour. Prove that some knight was left without wine.
[i]Proposed by A. Khrabrov, incorrect translation from Hungarian[/i]
2017 Iran MO (3rd round), 1
There's a tape with $n^2$ cells labeled by $1,2,\ldots,n^2$. Suppose that $x,y$ are two distinct positive integers less than or equal to $n$. We want to color the cells of the tape such that any two cells with label difference of $x$ or $y$ have different colors. Find the minimum number of colors needed to do so.
2007 Singapore Junior Math Olympiad, 2
Equilateral triangles $ABE$ and $BCF$ are erected externally onthe sidess $AB$ and $BC$ of a parallelogram $ABCD$. Prove that $\vartriangle DEF$ is equilateral.
2009 Today's Calculation Of Integral, 473
For nonzero real numbers $ r,\ l$ and the positive constant number $ c$, consider the curve on the $ xy$ plane : $ y \equal{} \left\{ \begin{array}{ll} x^2 & (0\leq x\leq r)\quad \\
r^2 & (r\leq x\leq l \plus{} r)\quad \\
(x \minus{} l \minus{} 2r)^2 & (l \plus{} r\leq x\leq l \plus{} 2r)\quad \end{array} \right.$
Denote $ V$ the volume of the solid by revolvering the curve about the $ x$ axis. Let $ r,\ l$ vary in such a way that $ r^2 \plus{} l \equal{} c$. Find the values of $ r,\ l$ which gives the maxmimum volume.
1969 IMO Shortlist, 68
$(USS 5)$ Given $5$ points in the plane, no three of which are collinear, prove that we can choose $4$ points among them that form a convex quadrilateral.
2010 Puerto Rico Team Selection Test, 1
The circles in the figure have their centers at $C$ and $D$ and intersect at $A$ and $B$. Let $\angle ACB =60$, $\angle ADB =90^o$ and $DA = 1$ . Find the length of $CA$.
[img]https://cdn.artofproblemsolving.com/attachments/0/1/950a55984283091d15083fadcf35e8b95cb229.png[/img]
1980 AMC 12/AHSME, 23
Line segments drawn from the vertex opposite the hypotenuse of a right triangle to the points trisecting the hypotenuse have lengths $\sin x$ and $\cos x$, where $x$ is a real number such that $0<x<\frac{\pi}2$. The length of the hypotenuse is
$\text{(A)} \ \frac 43 \qquad \text{(B)} \ \frac 32 \qquad \text{(C)} \ \frac{3\sqrt{5}}{5} \qquad \text{(D)} \ \frac{2\sqrt{5}}{3} \qquad \text{(E)} \ \text{not uniquely determined}$
2021 Swedish Mathematical Competition, 4
Give examples of a function $f : R \to R$ that satisfies $0 < f(x) < f(x + f(x)) <\sqrt2 x$, for all positive $x$,
and show that there is no function $f : R \to R$ that satisfies $x < f(x + f(x)) <\sqrt2 f(x)$, for all positive $x$.
2021 AMC 12/AHSME Spring, 7
Let $N=34 \cdot 34 \cdot 63 \cdot 270$. What is the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N$?
$\textbf{(A) }1:16 \qquad \textbf{(B) }1:15 \qquad \textbf{(C) }1:14 \qquad \textbf{(D) }1:8 \qquad \textbf{(E) }1:3$
1995 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 7
In a class, the teacher discovers that every pupil has exactly three friends in the class, that two friends never have a common friend, and that every pair of two pupils who are not friends they have exactly one common friend. How many pupils are there in the class?
A. 6
B. 9
C. 10
D. 12
E. 17
2009 Harvard-MIT Mathematics Tournament, 10
Given a rearrangement of the numbers from $1$ to $n$, each pair of consecutive elements $a$ and $b$ of the sequence can be either increasing (if $a < b$) or decreasing (if $b < a$). How many rearrangements of the numbers from $1$ to $n$ have exactly two increasing pairs of consecutive elements? Express your answer in terms of $n$.
2017 Saudi Arabia IMO TST, 2
Find all $f:\mathbb{R}\to\mathbb{R}$ satisfying:
$$f(xf(y)-y)+f(xy-x)+f(x+y)=2xy,\quad\forall x,y\in\mathbb{R}.$$
2021 Canadian Mathematical Olympiad Qualification, 7
If $A, B$ and $C$ are real angles such that
$$\cos (B-C)+\cos (C-A)+\cos (A-B)=-3/2,$$
find
$$\cos (A)+\cos (B)+\cos (C)$$
2019 Ukraine Team Selection Test, 1
In a triangle $ABC$, $\angle ABC= 60^o$, point $I$ is the incenter. Let the points $P$ and $T$ on the sides $AB$ and $BC$ respectively such that $PI \parallel BC$ and $TI \parallel AB$ , and points $P_1$ and $T_1$ on the sides $AB$ and $BC$ respectively such that $AP_1 = BP$ and $CT_1 = BT$. Prove that point $I$ lies on segment $P_1T_1$.
(Anton Trygub)
Novosibirsk Oral Geo Oly IX, 2023.3
An isosceles triangle $ABC$ with base $AC$ is given. On the rays $CA$, $AB$ and $BC$, the points $D, E$ and $F$ were marked, respectively, in such a way that $AD = AC$, $BE = BA$ and $CF = CB$. Find the sum of the angles $\angle ADB$, $\angle BEC$ and $\angle CFA$.