Found problems: 85335
1985 AMC 12/AHSME, 29
In their base $ 10$ representation, the integer $ a$ consists of a sequence of $ 1985$ eights and the integer $ b$ consists of a sequence of $ 1985$ fives. What is the sum of the digits of the base 10 representation of $ 9ab$?
$ \textbf{(A)}\ 15880 \qquad \textbf{(B)}\ 17856 \qquad \textbf{(C)}\ 17865 \qquad \textbf{(D)}\ 17874 \qquad \textbf{(E)}\ 19851$
The Golden Digits 2024, P2
Let $ABC$ be a triangle and $P$ a point in its interior. Circle $\Gamma_A$ is considered such that it is tangent to rays $(PB$ and $(PC$. Define similarly $\Gamma_B$ and $\Gamma_C$. Let $\ell_A\neq PA$ be the other common internal tangent of $\Gamma_B$ and $\Gamma_C$. Prove that $\ell_A$, $\ell_B$ and $\ell_C$ meet at a point.
[i]Proposed by Andrei Vila[/i]
2020 Online Math Open Problems, 20
Reimu invented a new number base system that uses exactly five digits. The number $0$ in the decimal system is represented as $00000$, and whenever a number is incremented, Reimu finds the leftmost digit (of the five digits) that is equal to the ``units" (rightmost) digit, increments this digit, and sets all the digits to its right to 0. (For example, an analogous system that uses three digits would begin with $000$, $100$, $110$, $111$, $200$, $210$, $211$, $220$, $221$, $222$, $300$, $\ldots$.) Compute the decimal representation of the number that Reimu would write as $98765$.
[i]Proposed by Yannick Yao[/i]
2025 Romania National Olympiad, 2
Let $n$ be a positive integer, and $a,b$ be two complex numbers such that $a \neq 1$ and $b^k \neq 1$, for any $k \in \{1,2,\dots ,n\}$. The matrices $A,B \in \mathcal{M}_n(\mathbb{C})$ satisfy the relation $BA=a I_n + bAB$. Prove that $A$ and $B$ are invertible.
2023 OlimphÃada, 2
Let $ABCD$ be a quadrilateral circumscribed around a circle $\omega$ with center $I$. Assume $P$ and $Q$ are distinct points and are isogonal conjugates such that $P, Q$, and $I$ are collinear. Show that $ABCD$ is a kite, that is, it has two disjoint pairs of consecutive equal sides.
Estonia Open Junior - geometry, 1998.1.3
Two non intersecting circles with centers $O_1$ and $O_2$ are tangent to line $s$ at points $A_1$ and $A_2$, respectively, and lying on the same side of this line. Line $O_1O_2$ intersects the first circle at $B_1$ and the second at $B_2$. Prove that the lines $A_1B_1$ and $A_2B_2$ are perpendicular to each other.
1980 Vietnam National Olympiad, 1
Let $\alpha_{1}, \alpha_{2}, \cdots , \alpha_{ n}$ be numbers in the interval $[0, 2\pi]$ such that the number $\displaystyle\sum_{i=1}^n (1 + \cos \alpha_{ i})$ is an odd integer. Prove that
\[\displaystyle\sum_{i=1}^n \sin \alpha_i \ge 1\]
2010 Contests, 2
Find all natural numbers $ n > 1$ such that $ n^{2}$ does $ \text{not}$ divide $ (n \minus{} 2)!$.
2002 HKIMO Preliminary Selection Contest, 17
Let $a_0=2$ and for $n\geq 1$, $a_n=\frac{\sqrt3 a_{n-1}+1}{\sqrt3-a_{n-1}}$. Find the value of $a_{2002}$ in the form $p+q\sqrt3$ where $p$ and $q$ are rational numbers
2018 Bosnia And Herzegovina - Regional Olympiad, 1
if $a$, $b$ and $c$ are real numbers such that $(a-b)(b-c)(c-a) \neq 0$, prove the equality:
$\frac{b^2c^2}{(a-b)(a-c)}+\frac{c^2a^2}{(b-c)(b-a)}+\frac{a^2b^2}{(c-a)(c-b)}=ab+bc+ca$
2012 AMC 10, 23
A solid tetrahedron is sliced off a solid wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface face down. What is the height of this object?
$ \textbf{(A)}\ \dfrac{\sqrt{3}}{3}\qquad\textbf{(B)}\ \dfrac{2\sqrt{2}}{3}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \dfrac{2\sqrt{3}}{3}\qquad\textbf{(E)}\ \sqrt{2} $
2014 Contests, 2
For the integer $n>1$, define $D(n)=\{ a-b\mid ab=n, a>b>0, a,b\in\mathbb{N} \}$. Prove that for any integer $k>1$, there exists pairwise distinct positive integers $n_1,n_2,\ldots,n_k$ such that $n_1,\ldots,n_k>1$ and $|D(n_1)\cap D(n_2)\cap\cdots\cap D(n_k)|\geq 2$.
1982 IMO Longlists, 26
Let $(a_n)_{n\geq0}$ and $(b_n)_{n \geq 0}$ be two sequences of natural numbers. Determine whether there exists a pair $(p, q)$ of natural numbers that satisfy
\[p < q \quad \text{ and } \quad a_p \leq a_q, b_p \leq b_q.\]
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]
2011 Sharygin Geometry Olympiad, 23
Given are triangle $ABC$ and line $\ell$ intersecting $BC, CA$ and $AB$ at points $A_1, B_1$ and $C_1$ respectively. Point $A'$ is the midpoint of the segment between the projections of $A_1$ to $AB$ and $AC$. Points $B'$ and $C'$ are defined similarly.
(a) Prove that $A', B'$ and $C'$ lie on some line $\ell'$.
(b) Suppose $\ell$ passes through the circumcenter of $\triangle ABC$. Prove that in this case $\ell'$ passes through the center of its nine-points circle.
[i]M. Marinov and N. Beluhov[/i]
2010 Stanford Mathematics Tournament, 1
Find the reflection of the point $(11, 16, 22)$ across the plane $3x+4y+5z=7$.
2002 Iran MO (3rd Round), 15
Let A be be a point outside the circle C, and AB and AC be the two tangents from A to this circle C. Let L be an arbitrary tangent to C that cuts AB and AC in P and Q. A line through P parallel to AC cuts BC in R. Prove that while L varies, QR passes through a fixed point. :)
1985 Austrian-Polish Competition, 7
Find an upper bound for the ratio
$$\frac{x_1x_2+2x_2x_3+x_3x_4}{x_1^2+x_2^2+x_3^2+x_4^2}$$
over all quadruples of real numbers $(x_1,x_2,x_3,x_4)\neq (0,0,0,0)$.
[i]Note.[/i] The smaller the bound, the better the solution.
1998 Tournament Of Towns, 4
For some positive numbers $A, B, C$ and $D$, the system of equations
$$\begin{cases} x^2 + y^2 = A \\ |x| + |y| = B \end{cases}$$
has $m$ solutions, while the system of equations
$$\begin{cases} x^2 + y^2 +z^2= X\\ |x| + |y| +|z| = D \end{cases}$$
has $n$ solutions. If $m > n > 1$, find $m$ and $n$.
( G Galperin)
1991 Arnold's Trivium, 61
What is the largest value of $t$ for which the solution of the problem
\[\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=\sin x,\; u|_{t=0}=0\]
can be extended to the interval $[0,t)$.
1972 All Soviet Union Mathematical Olympiad, 166
Each of the $9$ straight lines divides the given square onto two quadrangles with the areas ratio as $2:3$. Prove that there exist three of them intersecting in one point
2009 Hong Kong TST, 3
Prove that there are infinitely many primes $ p$ such that the total number of solutions mod $ p$ to the equation $ 3x^{3}\plus{}4y^{4}\plus{}5z^{3}\minus{}y^{4}z \equiv 0$ is $ p^2$
2009 India IMO Training Camp, 2
Let us consider a simle graph with vertex set $ V$. All ordered pair $ (a,b)$ of integers with $ gcd(a,b) \equal{} 1$, are elements of V.
$ (a,b)$ is connected to $ (a,b \plus{} kab)$ by an edge and to $ (a \plus{} kab,b)$ by another edge for all integer k.
Prove that for all $ (a,b)\in V$, there exists a path fromm $ (1,1)$ to $ (a,b)$.
MathLinks Contest 3rd, 1
Let $a, b, c$ be positive reals. Prove that $$\sqrt{abc}(\sqrt{a} +\sqrt{b} +\sqrt{c}) + (a + b + c)^2 \ge 4 \sqrt{3abc(a + b + c)}.$$
2014 ELMO Shortlist, 4
Let $ABCD$ be a quadrilateral inscribed in circle $\omega$. Define $E = AA \cap CD$, $F = AA \cap BC$, $G = BE \cap \omega$, $H = BE \cap AD$, $I = DF \cap \omega$, and $J = DF \cap AB$. Prove that $GI$, $HJ$, and the $B$-symmedian are concurrent.
[i]Proposed by Robin Park[/i]