Found problems: 85335
1949 Putnam, B5
let $(a_{n})$ be an arbitrary sequence of positive numbers. Show that
$$\limsup_{n\to \infty} \left(\frac{a_1 +a_{n+1}}{a_{n}}\right)^{n} \geq e.$$
2020 Saint Petersburg Mathematical Olympiad, 1.
A positive integer is called [i]hypotenuse[/i] if it can be represented as a sum of two squares of non-negative integers.
Prove that any natural number greater than $10$ is the difference of two hypotenuse numbers.
2022 Chile National Olympiad, 2
Let $ABC$ be a triangle such that $\angle CAB = 60^o$. Consider $D, E$ points on sides $AC$ and $AB$ respectively such that $BD$ bisects angle $\angle ABC$ , $CE$ bisects angle $\angle BCA$ and let $I$ be the intersection of them. Prove that $|ID| =|IE|$.
TNO 2008 Junior, 1
There are three number-transforming machines. We input the pair $(a_1, a_2)$, and the machine returns $(b_1, b_2)$. We denote this transformation as $(a_1, a_2) \to (b_1, b_2)$.
(a) The first machine can perform two transformations:
- $(a, b) \to (a - 1, b - 1)$
- $(a, b) \to (a + 13, b + 5)$
If the input pair is $(5,2)$, is it possible to obtain the pair $(20,22)$ after a series of transformations?
(b) The second machine can perform two transformations:
- $(a, b) \to (a - 1, b - 1)$
- $(a, b) \to (2a, 2b)$
If the input pair is $(15,10)$, is it possible to obtain the pair $(27,23)$ after a series of transformations?
(c) The third machine can perform two transformations:
- $(a, b) \to (a - 2, b + 2)$
- $(a, b) \to (2a - b + 1, 2b - 1 - a)$
If the input pair is $(5,8)$, is it possible to obtain the pair $(13,17)$ after a series of transformations?
1999 Finnish National High School Mathematics Competition, 1
Show that the equation $x^3 + 2y^2 + 4z = n$ has an integral solution $(x, y, z)$ for all integers $n.$
1998 Turkey MO (2nd round), 3
Some of the vertices of unit squares of an $n\times n$ chessboard are colored so that any $k\times k$ ( $1\le k\le n$) square consisting of these unit squares has a colored point on at least one of its sides. Let $l(n)$ denote the minimum number of colored points required to satisfy this condition. Prove that $\underset{n\to \infty }{\mathop \lim }\,\frac{l(n)}{{{n}^{2}}}=\frac{2}{7}$.
2023 ELMO Shortlist, C2
Alice is performing a magic trick. She has a standard deck of 52 cards, which she may order beforehand. She invites a volunteer to pick an integer \(0\le n\le 52\), and cuts the deck into a pile with the top \(n\) cards and a pile with the remaining \(52-n\). She then gives both piles to the volunteer, who riffles them together and hands the deck back to her face down. (Thus, in the resulting deck, the cards that were in the deck of size \(n\) appear in order, as do the cards that were in the deck of size \(52-n\).)
Alice then flips the cards over one-by-one from the top. Before flipping over each card, she may choose to guess the color of the card she is about to flip over. She stops if she guesses incorrectly. What is the maximum number of correct guesses she can guarantee?
[i]Proposed by Espen Slettnes[/i]
2018 Adygea Teachers' Geometry Olympiad, 2
It is known that in a right triangle:
a) The height drawn from the top of the right angle is the geometric mean of the projections of the legs on the hypotenuse;
b) the leg is the geometric mean of the hypotenuse and the projection of this leg to the hypotenuse.
Are the converse statements true? Formulate them and justify the answer.
Is it possible to formulate the criterion of a right triangle based on these statements? If possible, then how? If not, why?
2024 LMT Fall, 8
Let $a$ and $b$ be positive integers such that $10< \gcd(a,b) < 20$ and $220 < \text{lcm}(a,b) < 230$. Find the difference between the smallest and largest possible values of $ab$.
2019 Estonia Team Selection Test, 11
Given a circle $\omega$ with radius $1$. Let $T$ be a set of triangles good, if the following conditions apply:
(a) the circumcircle of each triangle in the set $T$ is $\omega$;
(b) The interior of any two triangles in the set $T$ has no common point.
Find all positive real numbers $t$, for which for each positive integer $n$ there is a good set of $n$ triangles, where the perimeter of each triangle is greater than $t$.
2018 Yasinsky Geometry Olympiad, 4
In the quadrilateral $ABCD$, the length of the sides $AB$ and $BC$ is equal to $1, \angle B= 100^o , \angle D= 130^o$ . Find the length of $BD$.
2011 China Girls Math Olympiad, 2
The diagonals $AC,BD$ of the quadrilateral $ABCD$ intersect at $E$. Let $M,N$ be the midpoints of $AB,CD$ respectively. Let the perpendicular bisectors of the segments $AB,CD$ meet at $F$. Suppose that $EF$ meets $BC,AD$ at $P,Q$ respectively. If $MF\cdot CD=NF\cdot AB$ and $DQ\cdot BP=AQ\cdot CP$, prove that $PQ\perp BC$.
2022 Kosovo & Albania Mathematical Olympiad, 3
Is it possible to partition $\{1, 2, 3, \ldots, 28\}$ into two sets $A$ and $B$ such that both of the following conditions hold simultaneously:
(i) the number of odd integers in $A$ is equal to the number of odd integers in $B$;
(ii) the difference between the sum of squares of the integers in $A$ and the sum of squares of the integers in $B$ is $16$?
1967 Swedish Mathematical Competition, 4
The sequence $a_1, a_2, a_3, ...$ of positive reals is such that $\sum a_i$ diverges.
Show that there is a sequence $b_1, b_2, b_3, ...$ of positive reals such that $\lim b_n = 0$ and $\sum a_ib_i$ diverges.
1999 AMC 12/AHSME, 6
What is the sum of the digits of the decimal form of the product $ 2^{1999}\cdot 5^{2001}$?
$ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 10$
2016 Ecuador NMO (OMEC), 3
Let $A, B, C, D$ be four different points on a line $\ell$, such that $AB = BC = CD$. In one of the semiplanes determined by the line $\ell$, the points $P$ and $Q$ are chosen in such a way that the triangle $CPQ$ is equilateral with its vertices named clockwise. Let $M$ and $N$ be two points on the plane such that the triangles $MAP$ and $NQD$ are equilateral (the vertices are also named clockwise). Find the measure of the angle $\angle MBN$.
2003 Pan African, 2
The circumference of a circle is arbitrarily divided into four arcs. The midpoints of the arcs are connected by segments. Show that two of these segments are perpendicular.
2013 China Northern MO, 1
Find the largest positive integer $n$ ($n \ge 3$), so that there is a convex $n$-gon, the tangent of each interior angle is an integer.
2010 Laurențiu Panaitopol, Tulcea, 3
Let be a complex number $ z $ having the property that $ \Re \left( z^n \right) >\Im \left( z^n \right) , $ for any natural numbers $ n. $
Show that $ z $ is a positive real number.
[i]Laurențiu Panaitopol[/i]
2022 HMNT, 29
Consider the set $S$ of all complex numbers $z$ with nonnegative real and imaginary part such that $$|z^2+2| \le |z|.$$ Across all $z \in S,$ compute the minimum possible value of $\tan(\theta),$ where $\theta$ is the angle formed between $z$ and the real axis.
2020 SIME, 14
Let $P(x) = x^3 - 3x^2 + 3$. For how many positive integers $n < 1000$ does there not exist a pair $(a, b)$ of positive integers such that the equation
\[ \underbrace{P(P(\dots P}_{a \text{ times}}(x)\dots))=\underbrace{P(P(\dots P}_{b \text{ times}}(x)\dots))\]
has exactly $n$ distinct real solutions?
2003 AMC 12-AHSME, 11
A square and an equilateral triangle have the same perimeter. Let $ A$ be the area of the circle circumscribed about the square and $ B$ be the area of the circle circumscribed about the triangle. Find $ A/B$.
$ \textbf{(A)}\ \frac{9}{16} \qquad
\textbf{(B)}\ \frac{3}{4} \qquad
\textbf{(C)}\ \frac{27}{32} \qquad
\textbf{(D)}\ \frac{3\sqrt{6}}{8} \qquad
\textbf{(E)}\ 1$
2021 Iranian Geometry Olympiad, 5
Let $A_1, A_2, . . . , A_{2021}$ be $2021$ points on the plane, no three collinear and $$\angle A_1A_2A_3 + \angle A_2A_3A_4 +... + \angle A_{2021}A_1A_2 = 360^o,$$ in which by the angle $\angle A_{i-1}A_iA_{i+1}$ we mean the one which is less than $180^o$ (assume that $A_{2022} =A_1$ and $A_0 = A_{2021}$). Prove that some of these angles will add up to $90^o$.
[i]Proposed by Morteza Saghafian - Iran[/i]
2016 Costa Rica - Final Round, F1
Let $a, b$ and $c$ be real numbers, and let $f (x) = ax^2 + bx + c$ and $g (x) = cx^2 + bx + a$ functions such that $| f (-1) | \le 1$, $| f (0) | \le 1$ and $| f (1) | \le 1$. Show that if $-1 \le x \le 1$, then $| f (x) | \le \frac54$ and $| g (x) | \le 2$.
2018 IFYM, Sozopol, 6
Find all sets $(a, b, c)$ of different positive integers $a$, $b$, $c$, for which:
[b]*[/b] $2a - 1$ is a multiple of $b$;
[b]*[/b] $2b - 1$ is a multiple of $c$;
[b]*[/b] $2c - 1$ is a multiple of $a$.