Found problems: 85335
2009 China Team Selection Test, 2
Given an integer $ n\ge 2$, find the maximal constant $ \lambda (n)$ having the following property: if a sequence of real numbers $ a_{0},a_{1},a_{2},\cdots,a_{n}$ satisfies $ 0 \equal{} a_{0}\le a_{1}\le a_{2}\le \cdots\le a_{n},$ and $ a_{i}\ge\frac {1}{2}(a_{i \plus{} 1} \plus{} a_{i \minus{} 1}),i \equal{} 1,2,\cdots,n \minus{} 1,$ then $ (\sum_{i \equal{} 1}^n{ia_{i}})^2\ge \lambda (n)\sum_{i \equal{} 1}^n{a_{i}^2}.$
1984 Poland - Second Round, 5
Calculate the lower bound of the areas of convex hexagons whose vertices all have integer coordinates.
2011 AMC 12/AHSME, 10
Rectangle $ABCD$ has $AB=6$ and $BC=3$. Point $M$ is chosen on side $AB$ so that $\angle AMD = \angle CMD$. What is the degree measure of $\angle AMD$?
$ \textbf{(A)}\ 15 \qquad
\textbf{(B)}\ 30 \qquad
\textbf{(C)}\ 45 \qquad
\textbf{(D)}\ 60 \qquad
\textbf{(E)}\ 75 $
1938 Moscow Mathematical Olympiad, 042
How many positive integers smaller than $1000$ and not divisible by $5$ and by $7$ are there?
1992 Kurschak Competition, 1
Define for $n$ given positive reals the [i]strange mean[/i] as the sum of the squares of these numbers divided by their sum. Decide which of the following statements hold for $n=2$:
a) The strange mean is never smaller than the third power mean.
b) The strange mean is never larger than the third power mean.
c) The strange mean, depending on the given numbers, can be larger or smaller than the third power mean.
Which statement is valid for $n=3$?
2018 Taiwan TST Round 1, 6
Given six points $ A, B, C, D, E, F $ such that $ \triangle BCD \stackrel{+}{\sim} \triangle ECA \stackrel{+}{\sim} \triangle BFA $ and let $ I $ be the incenter of $ \triangle ABC. $ Prove that the circumcenter of $ \triangle AID, \triangle BIE, \triangle CIF $ are collinear.
[i]Proposed by Telv Cohl[/i]
2003 Switzerland Team Selection Test, 4
Find the largest natural number $n$ that divides $a^{25} -a$ for all integers $a$.
2000 Chile National Olympiad, 3
A number $N_k$ is defined as [i]periodic[/i] if it is composed in number base $N$ of a repeated $k$ times . Prove that $7$ divides to infinite periodic numbers of the set $N_1, N_2, N_3,...$
2019 Tuymaada Olympiad, 7
$N$ cells chosen on a rectangular grid. Let $a_i$ is number of chosen cells in $i$-th row, $b_j$ is number of chosen cells in $j$-th column. Prove that
$$ \prod_{i} a_i! \cdot \prod_{j} b_j! \leq N! $$
2008 All-Russian Olympiad, 6
A magician should determine the area of a hidden convex $ 2008$-gon $ A_{1}A_{2}\cdots A_{2008}$. In each step he chooses two points on the perimeter, whereas the chosen points can be vertices or points dividing selected sides in selected ratios. Then his helper divides the polygon into two parts by the line through these two points and announces the area of the smaller of the two parts. Show that the magician can find the area of the polygon in $ 2006$ steps.
2003 BAMO, 4
An integer $n > 1$ has the following property: for every (positive) divisor $d$ of $n, d + 1$ is a divisor of $n + 1$.
Prove that $n$ is prime.
2007 Purple Comet Problems, 6
Find the sum of all the positive integers that are divisors of either $96$ or $180$.
1968 AMC 12/AHSME, 30
Convex polygons $P_1$ and $P_2$ are drawn in the same plane with $n_1$ and $n_2$ sides, respectively, $n_1 \le n_2$. If $P_1$ and $P_2$ do not have any line segment in common, then the maximum number of intersections of $P_1$ and $P_2$ is:
$\textbf{(A)}\ 2n_1 \qquad\textbf{(B)}\ 2n_2 \qquad\textbf{(C)}\ n_1n_2 \qquad\textbf{(D)}\ n_1+n_2 \qquad\textbf{(E)}\ \text{none of these} $
1990 China National Olympiad, 2
Let $x$ be a natural number. We call $\{x_0,x_1,\dots ,x_l\}$ a [i]factor link [/i]of $x$ if the sequence $\{x_0,x_1,\dots ,x_l\}$ satisfies the following conditions:
(1) $x_0=1, x_l=x$;
(2) $x_{i-1}<x_i, x_{i-1}|x_i, i=1,2,\dots,l$ .
Meanwhile, we define $l$ as the length of the [i]factor link [/i] $\{x_0,x_1,\dots ,x_l\}$. Denote by $L(x)$ and $R(x)$ the length and the number of the longest [i]factor link[/i] of $x$ respectively. For $x=5^k\times 31^m\times 1990^n$, where $k,m,n$ are natural numbers, find the value of $L(x)$ and $R(x)$.
2001 Cuba MO, 7
Prove that the equation $x^{19} + x^{17} = x^{16 }+ x^7 + a$ for any $a \in R$ has at least two imaginary roots
2023 Princeton University Math Competition, A3 / B5
Let $\vartriangle ABC$ be a triangle with $AB = 13$, $BC = 14$, and $CA = 15$. Let $D$, $E$, and $F$ be the midpoints of $AB$, $BC$, and $CA$ respectively. Imagine cutting $\vartriangle ABC$ out of paper and then folding $\vartriangle AFD$ up along $FD$, folding $\vartriangle BED$ up along $DE$, and folding $\vartriangle CEF$ up along $EF$ until $A$, $B$, and $C$ coincide at a point $G$. The volume of the tetrahedron formed by vertices $D$, $E$, $F$, and $G$ can be expressed as $\frac{p\sqrt{q}}{r}$ , where $p$, $q$, and $r$ are positive integers, $p$ and $r$ are relatively prime, and $q$ is square-free. Find $p + q + r$.
1997 Canadian Open Math Challenge, 11
In an isosceles right-angled triangle AOB, points P; Q and S are chosen on sides OB, OA, and AB respectively such that a square PQRS is formed as shown. If the lengths of OP and OQ are a and b respectively, and the area of PQRS is 2 5 that of triangle AOB, determine a : b.
[asy]
pair A = (0,3);
pair B = (0,0);
pair C = (3,0);
pair D = (0,1.5);
pair E = (0.35,0);
pair F = (1.2,1.8);
pair J = (0.17,0);
pair Y = (0.17,0.75);
pair Z = (1.6,0.2);
draw(A--B);
draw(B--C);
draw(C--A);
draw(D--F--Z--E--D);
draw("$O$", B, dir(180));
draw("$B$", A, dir(45));
draw("$A$", C, dir(45));
draw("$Q$", E, dir(45));
draw("$P$", D, dir(45));
draw("$R$", Z, dir(45));
draw("$S$", F, dir(45));
draw("$a$", Y, dir(210));
draw("$b$", J, dir(100));
[/asy]
2001 Argentina National Olympiad, 1
Sergio thinks of a positive integer $S$, less than or equal to $100$. Iván must guess the number that Sergio thought of, using the following procedure: in each step, he chooses two positive integers $A$ and $B$ less than $100$, and asks Sergio what is the greatest common factor between $A+ S$ and $B$. Give a sequence of seven steps that ensures Iván guesses the number $S$ that Sergio thought of.
Clarification:In each step, Sergio correctly answers Iván's question.
2023 Ukraine National Mathematical Olympiad, 8.5
Do there exist $10$ real numbers, not all of which are equal, each of which is equal to the square of the sum of the remaining $9$ numbers?
[i]Proposed by Bogdan Rublov[/i]
2008 JBMO Shortlist, 8
Show that $(x + y + z) \big(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\big) \ge 4 \big(\frac{x}{xy+1}+\frac{y}{yz+1}+\frac{z}{zx+1}\big)^2$ , for all real positive numbers $x, y $ and $z$.
1998 Baltic Way, 8
Let $P_k(x)=1+x+x^2+\ldots +x^{k-1}$. Show that
\[ \sum_{k=1}^n \binom{n}{k} P_k(x)=2^{n-1} P_n \left( \frac{x+1}{2} \right) \]
for every real number $x$ and every positive integer $n$.
2020 Iran Team Selection Test, 1
A weighted complete graph with distinct positive wights is given such that in every triangle is [i]degenerate [/i] that is wight of an edge is equal to sum of two other. Prove that one can assign values to the vertexes of this graph such that the wight of each edge is the difference between two assigned values of the endpoints.
[i]Proposed by Morteza Saghafian [/i]
1994 AMC 12/AHSME, 23
In the $xy$-plane, consider the L-shaped region bounded by horizontal and vertical segments with vertices at $(0,0), (0,3), (3,3), (3,1), (5,1)$ and $(5,0)$. The slope of the line through the origin that divides the area of this region exactly in half is
[asy]
size(200);
Label l;
l.p=fontsize(6);
xaxis("$x$",0,6,Ticks(l,1.0,0.5),EndArrow);
yaxis("$y$",0,4,Ticks(l,1.0,0.5),EndArrow);
draw((0,3)--(3,3)--(3,1)--(5,1)--(5,0)--(0,0)--cycle,black+linewidth(2));[/asy]
$ \textbf{(A)}\ \frac{2}{7} \qquad\textbf{(B)}\ \frac{1}{3} \qquad\textbf{(C)}\ \frac{2}{3} \qquad\textbf{(D)}\ \frac{3}{4} \qquad\textbf{(E)}\ \frac{7}{9} $
2018 Hanoi Open Mathematics Competitions, 1
Let $a, b$, and $c$ be distinct positive integers such that $a + 2b + 3c < 12$.
Which of the following inequalities must be true?
A. $a + b + c < 7$
B. $a- b + c < 4$
C. $b + c- a < 3$
D. $a + b- c <5 $
E. $5a + 3b + c < 27$
2007 Alexandru Myller, 3
The convex pentagon $ ABCDE $ has the following properties:
$ \text{(i)} AB=BC $
$ \text{(ii)} \angle ABE+\angle CBD =\angle DBE $
$ \text{(iii)} \angle AEB +\angle BDC=180^{\circ} $
Prove that the orthocenter of $ BDE $ lies on $ AC. $