Found problems: 85335
2014 Lusophon Mathematical Olympiad, 5
Find all quadruples of positive integers $(k,a,b,c)$ such that $2^k=a!+b!+c!$ and $a\geq b\geq c$.
1999 National High School Mathematics League, 3
$n$ is a given positive integer, such that it’s possible to weigh out the mass of any product weighing $1,2,3,\cdots ,n\text{g}$ with a counter balance and $k$ counterweights, whose weights are positive integers.
[b](a)[/b] Find $f(n)$: the minumum value of $k$.
[b](b)[/b] Find all possible number of $n,$ such that the mass of $f(n)$ counterweights is uniquely determined.
1999 Flanders Math Olympiad, 2
Let $[mn]$ be a diameter of the circle $C$ and $[AB]$ a chord with given length on this circle. $[AB]$ neither coincides nor is perpendicular to $[MN]$.
Let $C,D$ be the orthogonal projections of $A$ and $B$ on $[MN]$ and $P$ the midpoint of $[AB]$.
Prove that $\angle CPD$ does not depend on the chord $[AB]$.
2009 Balkan MO Shortlist, A7
Let $n\geq 2$ be a positive integer and
\begin{align*} P(x) = c_0 X^n + c_1 X^{n-1} + \ldots + c_{n-1} X +c_n \end{align*}
be a polynomial with integer coefficients, such that $\mid c_n \mid$ is a prime number and
\begin{align*} |c_0| + |c_1| + \ldots + |c_{n-1}| < |c_n| \end{align*}
Prove that the polynomial $P(X)$ is irreducible in the $\mathbb{Z}[x]$
2014 NIMO Summer Contest, 13
Let $\alpha$ and $\beta$ be nonnegative integers. Suppose the number of strictly increasing sequences of integers $a_0,a_1,\dots,a_{2014}$ satisfying $0 \leq a_m \leq 3m$ is $2^\alpha (2\beta + 1)$. Find $\alpha$.
[i]Proposed by Lewis Chen[/i]
2024 APMO, 1
Let $ABC$ be an acute triangle. Let $D$ be a point on side $AB$ and $E$ be a point on side $AC$ such that lines $BC$ and $DE$ are parallel. Let $X$ be an interior point of $BCED$. Suppose rays $DX$ and $EX$ meet side $BC$ at points $P$ and $Q$, respectively, such that both $P$ and $Q$ lie between $B$ and $C$. Suppose that the circumcircles of triangles $BQX$ and $CPX$ intersect at a point $Y \neq X$. Prove that the points $A, X$, and $Y$ are collinear.
2024 AMC 12/AHSME, 12
Suppose $z$ is a complex number with positive imaginary part, with real part greater than $1$, and with $|z| = 2$. In the complex plane, the four points $0$, $z$, $z^{2}$, and $z^{3}$ are the vertices of a quadrilateral with area $15$. What is the imaginary part of $z$?
$\textbf{(A)}~\displaystyle\frac{3}{4}\qquad\textbf{(B)}~1\qquad\textbf{(C)}~\displaystyle\frac{4}{3}\qquad\textbf{(D)}~\displaystyle\frac{3}{2}\qquad\textbf{(E)}~\displaystyle\frac{5}{3}$
2015 Romania National Olympiad, 2
Consider a natural number $ n $ for which it exist a natural number $ k $ and $ k $ distinct primes so that $ n=p_1\cdot p_2\cdots p_k. $
[b]a)[/b] Find the number of functions $ f:\{ 1, 2,\ldots , n\}\longrightarrow\{ 1,2,\ldots ,n\} $ that have the property that $ f(1)\cdot f(2)\cdots f\left( n \right) $ divides $ n. $
[b]b)[/b] If $ n=6, $ find the number of functions $ f:\{ 1, 2,3,4,5,6\}\longrightarrow\{ 1,2,3,4,5,6\} $ that have the property that $ f(1)\cdot f(2)\cdot f(3)\cdot f(4)\cdot f(5)\cdot f(6) $ divides $ 36. $
1983 Putnam, A2
The shorthand of a clock has the length 3, the longhand has the length 4. Determine the distance between the endpoints of the hands at the time, where their distance increases the most.
1983 IMO Longlists, 43
Given a square $ABCD$, let $P, Q, R$, and $S$ be four variable points on the sides $AB, BC, CD$, and $DA$, respectively. Determine the positions of the points $P, Q, R$, and $S$ for which the quadrilateral $PQRS$ is a parallelogram, a rectangle, a square, or a trapezoid.
1991 Nordic, 4
Let $f(x)$ be a polynomial with integer coefficients. We assume that there exists a positive integer $k$ and $k$ consecutive integers $n, n+1, ... , n+k -1$ so that none of the numbers $f(n), f(n+ 1),... , f(n + k - 1)$ is divisible by $k$.
Show that the zeroes of $f(x)$ are not integers.
1996 Kurschak Competition, 3
Let $n$ and $k$ be arbitrary non-negative integers. Suppose we have drawn $2kn+1$ (different) diagonals of a convex $n$-gon. Show that there exists a broken line formed by $2k+1$ of these diagonals that passes through no point more than once. Prove also that this is not necessarily true when we draw only $kn$ diagonals.
PEN S Problems, 20
Let $n$ be a positive integer that is not a perfect cube. Define real numbers $a$, $b$, $c$ by \[a=\sqrt[3]{n}, \; b=\frac{1}{a-\lfloor a\rfloor}, \; c=\frac{1}{b-\lfloor b\rfloor}.\] Prove that there are infinitely many such integers $n$ with the property that there exist integers $r$, $s$, $t$, not all zero, such that $ra+sb+tc=0$.
2009 Today's Calculation Of Integral, 464
Evaluate $ \int_1^e \frac {(1 \plus{} 2x^2)\ln x}{\sqrt {1 \plus{} x^2}}\ dx$.
1994 Putnam, 5
For each $\alpha\in \mathbb{R}$ define $f_{\alpha}(x)=\lfloor{\alpha x}\rfloor$. Let $n\in \mathbb{N}$. Show there exists a real $\alpha$ such that for $1\le \ell \le n$ :
\[ f_{\alpha}^{\ell}(n^2)=n^2-\ell=f_{\alpha^{\ell}}(n^2).\]
Here $f^{\ell}_{\alpha}(x)=(f_{\alpha}\circ f_{\alpha}\circ \cdots \circ f_{\alpha})(x)$ where the composition is carried out $\ell$ times.
1974 Chisinau City MO, 82
Is there a moment in a day when three hands - hour, minute and second - of a clock running correctly form angles of $120^o$ in pairs?
1974 Swedish Mathematical Competition, 4
Find all polynomials $p(x)$ such that $p(x^2) = p(x)^2$ for all $x$. Hence find all polynomials $q(x)$ such that
\[
q\left(x^2 - 2x\right) = q\left(x-2\right)^2
\]
2009 IMAR Test, 4
Given any $n$ positive integers, and a sequence of $2^n$ integers (with terms among them), prove there exists a subsequence made of consecutive terms, such that the product of its terms is a perfect square. Also show that we cannot replace $2^n$ with any lower value (therefore $2^n$ is the threshold value for this property).
2022 International Zhautykov Olympiad, 3
In parallelogram $ABCD$ with acute angle $A$ a point $N$ is chosen on the segment $AD$, and a point $M$ on the segment $CN$ so that $AB = BM = CM$. Point $K$ is the reflection of $N$ in line $MD$. The line $MK$ meets the segment $AD$ at point $L$. Let $P$ be the common point of the circumcircles of $AMD$ and $CNK$ such that $A$ and $P$ share the same side of the line $MK$. Prove that $\angle CPM = \angle DPL$.
1975 Chisinau City MO, 100
Solve the equation: $\sin ^3x+\sin ^32x+\sin ^33x=(\sin x + \sin 2x + \sin 3x)^3$.
2018 Switzerland - Final Round, 1
The cells of an $8\times 8$ chessboard are all coloured in white. A move consists in inverting the colours of a rectangle $1 \times 3$ horizontal or vertical (the white cells become black and conversely). Is it possible to colour all the cells of the chessboard in black in a finite number of moves ?
2012 China National Olympiad, 3
Find the smallest positive integer $k$ such that, for any subset $A$ of $S=\{1,2,\ldots,2012\}$ with $|A|=k$, there exist three elements $x,y,z$ in $A$ such that $x=a+b$, $y=b+c$, $z=c+a$, where $a,b,c$ are in $S$ and are distinct integers.
[i]Proposed by Huawei Zhu[/i]
1998 Romania National Olympiad, 1
Find the integer numbers $a, b, c$ such that the function $f: R \to R$, $f(x) = ax^2 +bx + c$ satisfies the equalities : $$f(f(1) ))= f (f(2 ) )= f(f (3 ))$$
2023 Flanders Math Olympiad, 3
The vertices of a regular $4$-gon, $6$-gon and $12$-goncan be brought together in one point to form a complete angle of $360^o$ (see figure). [center][img]https://cdn.artofproblemsolving.com/attachments/b/1/e9245179b7e0f5acb98b226bdc6db87fd72ad5.png[/img] [/center]
Determine all triples $a, b, c \in N$ with $a < b < c$ for which the angles of a regular $a$-gon, $b$-gon and $c$-gon together also form $360^o$ .
2011 Dutch BxMO TST, 1
All positive integers are coloured either red or green, such that the following conditions are satisfi ed:
- There are equally many red as green integers.
- The sum of three (not necessarily distinct) red integers is red.
- The sum of three (not necessarily distinct) green integers is green.
Find all colourings that satisfy these conditions.