Found problems: 85335
2016 Saudi Arabia Pre-TST, 2.4
Let $ABC$ be a non isosceles triangle with circumcircle $(O)$ and incircle $(I)$. Denote $(O_1)$ as the circle that external tangent to $(O)$ at $A'$ and also tangent to the lines $AB,AC$ at $A_b,A_c$ respectively. Define the circles $(O_2), (O_3)$ and the points $B', C', B_c , B_a, C_a, C_b$ similarly.
1. Denote J as the radical center of $(O_1), (O_2), (O_3) $and suppose that $JA'$ intersects $(O_1)$ at the second point $X, JB'$ intersects $(O_2)$ at the second point Y , JC' intersects $(O_3)$ at the second point $Z$. Prove that the circle $(X Y Z)$ is tangent to $(O_1), (O_2), (O_3)$.
2. Prove that $AA', BB', CC'$ are concurrent at the point $M$ and $3$ points $I,M,O$ are collinear.
2023 USAMO, 4
A positive integer $a$ is selected, and some positive integers are written on a board. Alice and Bob play the following game. On Alice's turn, she must replace some integer $n$ on the board with $n+a$, and on Bob's turn he must replace some even integer $n$ on the board with $n/2$. Alice goes first and they alternate turns. If on his turn Bob has no valid moves, the game ends.
After analyzing the integers on the board, Bob realizes that, regardless of what moves Alice makes, he will be able to force the game to end eventually. Show that, in fact, for this value of $a$ and these integers on the board, the game is guaranteed to end regardless of Alice's or Bob's moves.
1978 All Soviet Union Mathematical Olympiad, 252
Let $a_n$ be the closest to $\sqrt n$ integer. Find the sum $$1/a_1 + 1/a_2 + ... + 1/a_{1980}$$
2016 Tournament Of Towns, 7
A spherical planet has the equator of length $1$. On this planet, $N$ circular roads of length $1$ each are to be built and used for several trains each. The trains must have the same constant positive speed and never stop or collide. What is the greatest possible sum of lengths of all the trains? The trains are arcs of zero width with endpoints removed (so that if only endpoints of two arcs have coincided then it is not a collision). Solve the problem for :
(a) $N=3$ ([i]4 points)[/i]
(b) $N=4$ ([i]6 points)[/i]
[i]Alexandr Berdnikov [/i]
2015 HMNT, 2
Let $a$ and $b$ be real numbers randomly (and independently) chosen from the range $[0,1]$. Find the probability that $a, b$ and $1$ form the side lengths of an obtuse triangle.
2014 Contests, 2
Let us consider a triangle $\Delta{PQR}$ in the co-ordinate plane. Show for every function $f: \mathbb{R}^2\to \mathbb{R}\;,f(X)=ax+by+c$ where $X\equiv (x,y) \text{ and } a,b,c\in\mathbb{R}$ and every point $A$ on $\Delta PQR$ or inside the triangle we have the inequality:
\begin{align*} & f(A)\le \text{max}\{f(P),f(Q),f(R)\} \end{align*}
2004 Putnam, A3
Define a sequence $\{u_n\}_{n=0}^{\infty}$ by $u_0=u_1=u_2=1,$ and thereafter by the condition that
$\det\begin{vmatrix} u_n & u_{n+1} \\ u_{n+2} & u_{n+3} \end{vmatrix}=n!$
for all $n\ge 0.$ Show that $u_n$ is an integer for all $n.$ (By convention, $0!=1$.)
1994 IMC, 1
Let $f\in C^1[a,b]$, $f(a)=0$ and suppose that $\lambda\in\mathbb R$, $\lambda >0$, is such that
$$|f'(x)|\leq \lambda |f(x)|$$
for all $x\in [a,b]$. Is it true that $f(x)=0$ for all $x\in [a,b]$?
1956 Moscow Mathematical Olympiad, 332
Prove that the system of equations $\begin{cases} x_1 - x_2 = a \\
x_3 - x_4 = b \\
x_1 + x_2 + x_3 + x_4 = 1\end{cases}$ has at least one solution in positive numbers ($x_1 ,x_2 ,x_3 ,x_4>0$) if and only if $|a| + |b| < 1$.
2004 Purple Comet Problems, 5
The number $2.5081081081081 \ldots$ can be written as $m/n$ where $m$ and $n$ are natural numbers with no common factors. Find $m + n$.
2023 Grosman Mathematical Olympiad, 3
Find all pairs of polynomials $p$, $q$ with complex coefficients so that
\[p(x)\cdot q(x)=p(q(x)).\]
2005 Vietnam National Olympiad, 2
Find all triples of natural $ (x,y,n)$ satisfying the condition:
\[ \frac {x! \plus{} y!}{n!} \equal{} 3^n
\]
Define $ 0! \equal{} 1$
1985 AMC 8, 6
A ream of paper containing $ 500$ sheets is $ 5$ cm thick. Approximately how many sheets of this type of paper would there be in a stack $ 7.5$ cm high?
\[ \textbf{(A)}\ 250 \qquad
\textbf{(B)}\ 550 \qquad
\textbf{(C)}\ 667 \qquad
\textbf{(D)}\ 750 \qquad
\textbf{(E)}\ 1250
\]
1998 All-Russian Olympiad Regional Round, 8.4
A set of $n\ge 9$ points is given on the plane. For any 9 it points can be selected from all circles so that all these points end up on selected circles. Prove that all n points lie on two circles
2024 LMT Fall, 11
Let $\phi=\tfrac{1+\sqrt 5}{2}$. Find
\[\left(4+\phi^{\frac12}\right)\left(4-\phi^{\frac12}\right)\left(4+i\phi^{-\frac12}\right)\left(4-i\phi^{-\frac12}\right).\]
2023 Miklós Schweitzer, 2
Let $G_0, G_1,\ldots$ be infinite open subsets of a Hausdorff space. Prove that there exist some infinite pairwise disjoint open sets $V_0,V_1,\ldots$ and some indices $n_0<n_1<\cdots$ such that $V_i\subseteq G_{n_i}$ for every $i\geqslant 0.$
2017 Bosnia And Herzegovina - Regional Olympiad, 2
Prove that numbers $1,2,...,16$ can be divided in sequence such that sum of any two neighboring numbers is perfect square
2008 Mathcenter Contest, 1
Let $x,y,z$ be a positive real numbers. Prove that $$\frac {x}{\sqrt {x + y}} + \frac {y}{\sqrt {y + z}} + \frac { z}{\sqrt {z + x}}\geq\sqrt [4]{\frac {27(yz + zx + xy)}{4}}$$
[i](dektep)[/i]
1992 Polish MO Finals, 1
The functions $f_0, f_1, f_2, ...$ are defined on the reals by $f_0(x) = 8$ for all $x$, $f_{n+1}(x) = \sqrt{x^2 + 6f_n(x)}$. For all $n$ solve the equation $f_n(x) = 2x$.
1976 IMO Shortlist, 1
Let $ABC$ be a triangle with bisectors $AA_1,BB_1, CC_1$ ($A_1 \in BC$, etc.) and $M$ their common point. Consider the triangles $MB_1A, MC_1A,MC_1B,MA_1B,MA_1C,MB_1C$, and their inscribed circles. Prove that if four of these six inscribed circles have equal radii, then $AB = BC = CA.$
2022 Saudi Arabia IMO TST, 3
Let $A,B,C,D$ be points on the line $d$ in that order and $AB = CD$. Denote $(P)$ as some circle that passes through $A, B$ with its tangent lines at $A, B$ are $a,b$. Denote $(Q)$ as some circle that passes through $C, D$ with its tangent lines at $C, D$ are $c,d$. Suppose that $a$ cuts $c, d$ at $K, L$ respectively and $b$ cuts $c, d$ at $M, N$ respectively. Prove that four points $K, L, M,N$ belong to a same circle $(\omega)$ and the common external tangent lines of circles $(P)$, $(Q)$ meet on $(\omega)$.
2017 Tuymaada Olympiad, 7
An equilateral triangle with side $20$ is divided by there series of parallel lines into $400$ equilateral triangles with side $1$. What maximum number of these small triangles can be crossed (internally) by one line?
Tuymaada 2017 Q7 Juniors
2010 May Olympiad, 2
Let $ABCD$ be a rectangle and the circle of center $D$ and radius $DA$, which cuts the extension of the side $AD$ at point $P$. Line $PC$ cuts the circle at point $Q$ and the extension of the side $AB$ at point $R$. Show that $QB = BR$.
1996 IberoAmerican, 2
Three tokens $A$, $B$, $C$ are, each one in a vertex of an equilateral triangle of side $n$. Its divided on equilateral triangles of side 1, such as it is shown in the figure for the case $n=3$
Initially, all the lines of the figure are painted blue. The tokens are moving along the lines painting them of red, following the next two rules:
[b](1) [/b]First $A$ moves, after that $B$ moves, and then $C$, by turns. On each turn, the token moves over exactly one line of one of the little triangles, form one side to the other.
[b](2)[/b] Non token moves over a line that is already painted red, but it can rest on one endpoint of a side that is already red, even if there is another token there waiting its turn.
Show that for every positive integer $n$ it is possible to paint red all the sides of the little triangles.
2020 CHKMO, 1
Given that ${a_n}$ and ${b_n}$ are two sequences of integers defined by
\begin{align*}
a_1=1, a_2=10, a_{n+1}=2a_n+3a_{n-1} & ~~~\text{for }n=2,3,4,\ldots, \\
b_1=1, b_2=8, b_{n+1}=3b_n+4b_{n-1} & ~~~\text{for }n=2,3,4,\ldots.
\end{align*}
Prove that, besides the number $1$, no two numbers in the sequences are identical.