Found problems: 85335
2010 Stanford Mathematics Tournament, 5
Alice sends a secret message to Bob using her RSA public key $n=400000001$. Eve wants to listen in on their conversation. But to do this, she needs Alice's private key, which is the factorization of $n$. Eve knows that $n=pq$, a product of two prime factors. Find $p$ and $q$.
2022 Poland - Second Round, 5
Let $n$ be an positive integer. We call $n$ $\textit{good}$ when there exists positive integer $k$ s.t. $n=k(k+1)$. Does there exist 2022 pairwise distinct $\textit{good}$ numbers s.t. their sum is also $\textit{good}$ number?
2008 Brazil Team Selection Test, 1
Let $AB$ be a chord, not a diameter, of a circle with center $O$. The smallest arc $AB$ is divided into three congruent arcs $AC$, $CD$, $DB$. The chord $AB$ is also divided into three equal segments $AC'$, $C'D'$, $D'B$. Let $P$ be the intersection point of between the lines $CC'$ and $DD'$. Prove that $\angle APB = \frac13 \angle AOB$.
2014 ELMO Shortlist, 6
Let $f_0$ be the function from $\mathbb{Z}^2$ to $\{0,1\}$ such that $f_0(0,0)=1$ and $f_0(x,y)=0$ otherwise. For each positive integer $m$, let $f_m(x,y)$ be the remainder when \[ f_{m-1}(x,y) + \sum_{j=-1}^{1} \sum_{k=-1}^{1} f_{m-1}(x+j,y+k) \] is divided by $2$.
Finally, for each nonnegative integer $n$, let $a_n$ denote the number of pairs $(x,y)$ such that $f_n(x,y) = 1$.
Find a closed form for $a_n$.
[i]Proposed by Bobby Shen[/i]
2006 South africa National Olympiad, 4
In triangle $ABC$, $AB=AC$ and $B\hat{A}C=100^\circ$. Let $D$ be on $AC$ such that $A\hat{B}D=C\hat{B}D$. Prove that $AD+DB=BC$.
2008 Harvard-MIT Mathematics Tournament, 1
A $ 3\times3\times3$ cube composed of $ 27$ unit cubes rests on a horizontal plane. Determine the number of ways of selecting two distinct unit cubes from a $ 3\times3\times1$ block (the order is irrelevant) with the property that the line joining the centers of the two cubes makes a $ 45^\circ$ angle with the horizontal plane.
1995 Abels Math Contest (Norwegian MO), 1b
Prove that if $(x+\sqrt{x^2 +1})(y+\sqrt{y^2 +1})= 1$ for real numbers $x,y$, then $x+y = 0$.
2008 Grigore Moisil Intercounty, 2
Determine the natural numbers a, b, c s.t. :
$ \frac{3a+2b}{6a}=\frac{8b+c}{10b}=\frac{3a+2c}{3c} $ and $ a^{2}+b^{2}+c^{2}=975 $
The challenge here is to come up with as basic solution as possible.
2011 Math Prize for Girls Olympiad, 3
Let $n$ be a positive integer such that $n + 1$ is divisible by 24. Prove that the sum of all the positive divisors of $n$ is divisible by 24.
1978 IMO Longlists, 52
Let $p$ be a prime and $A = \{a_1, \ldots , a_{p-1} \}$ an arbitrary subset of the set of natural numbers such that none of its elements is divisible by $p$. Let us define a mapping $f$ from $\mathcal P(A)$ (the set of all subsets of $A$) to the set $P = \{0, 1, \ldots, p - 1\}$ in the following way:
$(i)$ if $B = \{a_{i_{1}}, \ldots , a_{i_{k}} \} \subset A$ and $\sum_{j=1}^k a_{i_{j}} \equiv n \pmod p$, then $f(B) = n,$
$(ii)$ $f(\emptyset) = 0$, $\emptyset$ being the empty set.
Prove that for each $n \in P$ there exists $B \subset A$ such that $f(B) = n.$
2020 USMCA, 21
Let $ABCDEF$ be a regular octahedron with unit side length, such that $ABCD$ is a square. Points $G, H$ are on segments $BE, DF$ respectively. The planes $AGD$ and $BCH$ divide the octahedron into three pieces, each with equal volume. Compute $BG$.
1999 Denmark MO - Mohr Contest, 1
In a coordinate system, a circle with radius $7$ and center is on the y-axis placed inside the parabola with equation $y = x^2$ , so that it just touches the parabola in two points. Determine the coordinate set for the center of the circle.
2019 Miklós Schweitzer, 5
Let $S \subset \mathbb{R}^d$ be a convex compact body with nonempty interior. Show that there is an $\alpha > 0$ such that if $S = \cap_{i \in I} H_i$, where $I$ is an index set and $(H_i)_{i \in I}$ are halfspaces, then for any $P \in \mathbb{R}^d$, there is an $i \in I$ for which $\mathrm{dist}(P, H_i) \ge \alpha \, \mathrm{dist}(P, S)$.
2016 Hanoi Open Mathematics Competitions, 9
Let rational numbers $a, b, c$ satisfy the conditions $a + b + c = a^2 + b^2 + c^2 \in Z$.
Prove that there exist two relative prime numbers $m, n$ such that $abc =\frac{m^2}{n^3}$ .
1992 Tournament Of Towns, (355) 4
A table has $m$ rows and $n$ columns. The following permutations of its $mn$ elements are permitted: an arbitrary permutation leaving each element in the same row (a“horizontal move”) and an arbitrary permutation leaving each element in the same column (a “vertical move”). Find the number $k$ such that any permutation of $mn$ elements can be obtained by $k$ permitted moves but there exists a permutation that cannot be achieved in less than $k$ moves.
(A Andjans, Riga0
2024 Auckland Mathematical Olympiad, 11
It is known that for quadratic polynomials $P(x)=x^2+ax+b$ and $Q(x)=x^2+cx+d$ the equation $P(Q(x))=Q(P(x))$ does not have real roots. Prove that $b \neq d$.
2007 Indonesia TST, 4
Determine all pairs $ (n,p)$ of positive integers, where $ p$ is prime, such that $ 3^p\minus{}np\equal{}n\plus{}p$.
2014 AMC 10, 16
Four fair six-sided dice are rolled. What is the probability that at least three of the four dice show the same value?
$ \textbf{(A) } \frac{1}{36} \qquad\textbf{(B) } \frac{7}{72} \qquad\textbf{(C) } \frac{1}{9} \qquad\textbf{(D) }\frac{5}{36}\qquad\textbf{(E) }\frac{1}{6} \qquad $
1972 IMO Longlists, 8
We are given $3n$ points $A_1,A_2, \ldots , A_{3n}$ in the plane, no three of them collinear. Prove that one can construct $n$ disjoint triangles with vertices at the points $A_i.$
2025 EGMO, 3
Let $ABC$ be an acute triangle. Points $B, D, E$, and $C$ lie on a line in this order and satisfy $BD = DE = EC$. Let $M$ and $N$ be the midpoints of $AD$ and $AE$, respectively. Suppose triangle $ADE$ is acute, and let $H$ be its orthocentre. Points $P$ and $Q$ lie on lines $BM$ and $CN$, respectively, such that $D, H, M,$ and $P$ are concyclic and pairwise different, and $E, H, N,$ and $Q$ are concyclic and pairwise different. Prove that $P, Q, N,$ and $M$ are concyclic.
2006 Switzerland - Final Round, 2
Let $ABC$ be an equilateral triangle and let $D$ be an inner point of the side $BC$. A circle is tangent to $BC$ at $D$ and intersects the sides $AB$ and $AC$ in the inner points $M, N$ and $P, Q$ respectively. Prove that $|BD| + |AM| + |AN| = |CD| + |AP| + |AQ|$.
2017 AMC 10, 9
A radio program has a quiz consisting of $3$ multiple-choice questions, each with $3$ choices. A contestant wins if he or she gets $2$ or more of the questions right. The contestant answers randomly to each question. What is the probability of winning?
$\textbf{(A) } \frac{1}{27}\qquad \textbf{(B) } \frac{1}{9}\qquad \textbf{(C) } \frac{2}{9}\qquad \textbf{(D) } \frac{7}{27}\qquad \textbf{(E) } \frac{1}{2}$
2020 Online Math Open Problems, 21
Among all ellipses with center at the origin, exactly one such ellipse is tangent to the graph of the curve $x^3 - 6x^2y + 3xy^2 + y^3 + 9x^2 - 9xy + 9y^2 = 0$ at three distinct points. The area of this ellipse is $\frac{k\pi\sqrt{m}}{n}$, where $k,m,$ and $n$ are positive integers such that $\gcd(k,n)=1$ and $m$ is not divisible by the square of any prime. Compute $100k+10m+n$.
[i]Proposed by Jaedon Whyte[/i]
2006 Pre-Preparation Course Examination, 6
Show that the product of every $k$ consecutive members of the Fibonacci sequence is divisible by $f_1f_2\ldots f_k$ (where $f_0=0$ and $f_1=1$).
2023 Centroamerican and Caribbean Math Olympiad, 3
Let $a,\ b$ and $c$ be positive real numbers such that $a b+b c+c a=1$. Show that
$$
\frac{a^3}{a^2+3 b^2+3 a b+2 b c}+\frac{b^3}{b^2+3 c^2+3 b c+2 c a}+\frac{c^3}{c^2+3 a^2+3 c a+2 a b}>\frac{1}{6\left(a^2+b^2+c^2\right)^2} .
$$