Found problems: 85335
2011 Princeton University Math Competition, B4
A function $f:\{1,2, \ldots, n\} \to \{1, \ldots, m\}$ is [i]multiplication-preserving[/i] if $f(i)f(j) = f(ij)$ for all $1 \le i \le j \le ij \le n$, and [i]injective[/i] if $f(i) = f(j)$ only when $i = j$. For $n = 9, m = 88$, the number of injective, multiplication-preserving functions is $N$. Find the sum of the prime factors of $N$, including multiplicity. (For example, if $N = 12$, the answer would be $2 + 2 + 3 = 7$.)
1979 Chisinau City MO, 183
Prove the identity $\sin^3 a \cos 3a + \cos^3 a \sin 3a=\frac{3}{4}\sin 4a.$
2006 Silk Road, 1
Found all functions $f: \mathbb{R} \to \mathbb{R}$, such that for any $x,y \in \mathbb{R}$,
\[f(x^2+xy+f(y))=f^2(x)+xf(y)+y.\]
2015 IMO Shortlist, C3
For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.
MathLinks Contest 2nd, 4.1
The real polynomial $f \in R[X]$ has an odd degree and it is given that $f$ is co-prime with $g(x) = x^2 - x - 1$ and
$$f(x^2 - 1) = f(x)f(-x), \forall x \in R.$$
Prove that $f$ has at least two complex non-real roots.
2025 Romanian Master of Mathematics, 3
Fix an integer $n \geq 3$. Determine the smallest positive integer $k$ satisfying the following condition:
For any tree $T$ with vertices $v_1, v_2, \dots, v_n$ and any pairwise distinct complex numbers $z_1, z_2, \dots, z_n$, there is a polynomial $P(X, Y)$ with complex coefficients of total degree at most $k$ such that for all $i \neq j$ satisfying $1 \leq i, j \leq n$, we have $P(z_i, z_j) = 0$ if and only if there is an edge in $T$ joining $v_i$ to $v_j$.
Note, for example, that the total degree of the polynomial
$$
9X^3Y^4 + XY^5 + X^6 - 2
$$
is 7 because $7 = 3 + 4$.
[i]Proposed by Andrei Chiriță, Romania[/i]
1991 AMC 12/AHSME, 7
If $x = \frac{a}{b}$, $a \ne b$ and $b \ne 0$, then $\frac{a + b}{a - b} = $
$ \textbf{(A)}\ \frac{x}{x + 1}\qquad\textbf{(B)}\ \frac{x + 1}{x - 1}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ x - \frac{1}{x}\qquad\textbf{(E)}\ x + \frac{1}{x} $
2015 Math Prize for Girls Problems, 4
A [i]binary palindrome[/i] is a positive integer whose standard base 2 (binary) representation is a palindrome (reads the same backward or forward). (Leading zeroes are not permitted in the standard representation.) For example, 2015 is a binary palindrome, because in base 2 it is 11111011111. How many positive integers less than 2015 are binary palindromes?
2021 Azerbaijan IMO TST, 3
A magician intends to perform the following trick. She announces a positive integer $n$, along with $2n$ real numbers $x_1 < \dots < x_{2n}$, to the audience. A member of the audience then secretly chooses a polynomial $P(x)$ of degree $n$ with real coefficients, computes the $2n$ values $P(x_1), \dots , P(x_{2n})$, and writes down these $2n$ values on the blackboard in non-decreasing order. After that the magician announces the secret polynomial to the audience. Can the magician find a strategy to perform such a trick?
1965 AMC 12/AHSME, 6
If $ 10^{\log_{10}9} \equal{} 8x \plus{} 5$ then $ x$ equals:
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ \frac {1}{2} \qquad \textbf{(C)}\ \frac {5}{8} \qquad \textbf{(D)}\ \frac {9}{8} \qquad \textbf{(E)}\ \frac {2\log_{10}3 \minus{} 5}{8}$
2018 CMIMC Combinatorics, 5
Victor shuffles a standard 54-card deck then flips over cards one at a time onto a pile stopping after the first ace. However, if he ever reveals a joker he discards the entire pile, including the joker, and starts a new pile; for example, if the sequence of cards is 2-3-Joker-A, the pile ends with one card in it. Find the expected number of cards in the end pile.
1988 IMO Longlists, 47
In the convex pentagon $ ABCDE,$ the sides $ BC, CD, DE$ are equal. Moreover each diagonal of the pentagon is parallel to a side ($ AC$ is parallel to $ DE$, $ BD$ is parallel to $ AE$ etc.). Prove that $ ABCDE$ is a regular pentagon.
2009 Saint Petersburg Mathematical Olympiad, 4
From $2008 \times 2008$ square we remove one corner cell $1 \times 1$. Is number of ways to divide this figure to corners from $3$ cells odd or even ?
2007 Indonesia TST, 3
Let $ a_1,a_2,a_3,\dots$ be infinite sequence of positive integers satisfying the following conditon: for each prime number $ p$, there are only finite number of positive integers $ i$ such that $ p|a_i$. Prove that that sequence contains a sub-sequence $ a_{i_1},a_{i_2},a_{i_3},\dots$, with $ 1 \le i_1<i_2<i_3<\dots$, such that for each $ m \ne n$, $ \gcd(a_{i_m},a_{i_n})\equal{}1$.
2013 Iran Team Selection Test, 10
On each edge of a graph is written a real number,such that for every even tour of this graph,sum the edges with signs alternatively positive and negative is zero.prove that one can assign to each of the vertices of the graph a real number such that sum of the numbers on two adjacent vertices is the number on the edge between them.(tour is a closed path from the edges of the graph that may have repeated edges or vertices)
2020 Switzerland - Final Round, 5
Find all the positive integers $a, b, c$ such that $$a! \cdot b! = a! + b! + c!$$
2022 Sharygin Geometry Olympiad, 10.3
A line meets a segment $AB$ at point $C$. Which is the maximal number of points $X$ of this line such that one of angles $AXC$ and $BXC$ is equlal to a half of the second one?
2019 Paraguay Mathematical Olympiad, 3
Let $\overline{ABCD}$ be a $4$-digit number. What is the smallest possible positive value of $\overline{ABCD}- \overline{DCBA}$?
2021 Balkan MO Shortlist, G1
Let $ABC$ be a triangle with $AB < AC < BC$. On the side $BC$ we consider points $D$
and $E$ such that $BA = BD$ and $CE = CA$. Let $K$ be the circumcenter of triangle $ADE$ and
let $F$, $G$ be the points of intersection of the lines $AD$, $KC$ and $AE$, $KB$ respectively. Let $\omega_1$ be
the circumcircle of triangle $KDE$, $\omega_2$ the circle with center $F$ and radius $FE$, and $\omega_3$ the circle
with center $G$ and radius $GD$.
Prove that $\omega_1$, $\omega_2$, and $\omega_3$ pass through the same point and that this point of intersection lies on the line $AK$.
2024 SEEMOUS, P1
Let $(x_n)_{n\geq 1}$ be the sequence defined by $x_1\in (0,1)$ and $x_{n+1}=x_n-\frac{x_n^2}{\sqrt{n}}$ for all $n\geq 1$. Find the values of $\alpha\in\mathbb{R}$ for which the series $\sum_{n=1}^{\infty}x_n^{\alpha}$ is convergent.
2024 HMNT, 22
Suppose that $a$ and $b$ are positive integers such that $\gcd(a^3 - b^3,(a-b)^3)$ is not divisible by any perfect square except $1.$ Given that $1 \le a-b \le 50,$ compute the number of possible values of $a-b$ across all such $a,b.$
2023 Bulgarian Autumn Math Competition, 11.1
A quadruplet of distinct positive integers $(a, b, c, d)$ is called $k$-good if the following conditions hold:
1. Among $a, b, c, d$, no three form an arithmetic progression.
2. Among $a+b, a+c, a+d, b+c, b+d, c+d$, there are $k$ of them, forming an arithmetic progression.
$a)$ Find a $4$-good quadruplet.
$b)$ What is the maximal $k$, such that there is a $k$-good quadruplet?
2008 National Olympiad First Round, 6
A positive integer $n$ is called a good number if every integer multiple of $n$ is divisible by $n$ however its digits are rearranged. How many good numbers are there?
$
\textbf{(A)}\ 3
\qquad\textbf{(B)}\ 4
\qquad\textbf{(C)}\ 6
\qquad\textbf{(D)}\ 12
\qquad\textbf{(E)}\ \text{Infinitely many}
$
Geometry Mathley 2011-12, 7.3
Let $ABCD$ be a tangential quadrilateral. Let $AB$ meet $CD$ at $E, AD$ intersect $BC$ at $F$. Two arbitrary lines through $E$ meet $AD,BC$ at $M,N, P,Q$ respectively ($M,N \in AD$, $P,Q \in BC$). Another arbitrary pair of lines through $F$ intersect $AB,CD$ at $X, Y,Z, T$ respectively ($X, Y \in AB$,$Z, T \in CD$). Suppose that $d_1, d_2$ are the second tangents from $E$ to the incircles of triangles $FXY, FZT,d_3, d_4$ are the second tangents from $F$ to the incircles of triangles $EMN,EPQ$. Prove that the four lines $d_1, d_2, d_3, d_4$ meet each other at four points and these intersections make a tangential quadrilateral.
Nguyễn Văn Linh
1988 IMO Longlists, 71
The quadrilateral $A_1A_2A_3A_4$ is cyclic, and its sides are $a_1 = A_1A_2, a_2 = A_2A_3, a_3 = A_3A_4$ and $a_4 = A_4A_1.$ The respective circles with centres $I_i$ and radii $r_i$ are tangent externally to each side $a_i$ and to the sides $a_{i+1}$ and $a_{i-1}$ extended. ($a_0 = a_4$). Show that \[ \prod^4_{i=1} \frac{a_i}{r_i} = 4 \cdot (\csc (A_1) + \csc (A_2) )^2. \]