Found problems: 85335
1940 Moscow Mathematical Olympiad, 063
Points $A, B, C$ are vertices of an equilateral triangle inscribed in a circle. Point $D$ lies on the shorter arc $\overarc {AB}$ . Prove that $AD + BD = DC$.
2014 Korea National Olympiad, 4
Prove that there exists a function $f : \mathbb{N} \rightarrow \mathbb{N}$ that satisfies the following
(1) $\{f(n) : n\in\mathbb{N}\}$ is a finite set; and
(2) For nonzero integers $x_1, x_2, \ldots, x_{1000}$ that satisfy $f(\left|x_1\right|)=f(\left|x_2\right|)=\cdots=f(\left|x_{1000}\right|)$, then $x_1+2x_2+2^2x_3+2^3x_4+2^4x_5+\cdots+2^{999}x_{1000}\ne 0$.
2007 USAMO, 4
An [i]animal[/i] with $n$ [i]cells[/i] is a connected figure consisting of $n$ equal-sized cells[1].
A [i]dinosaur[/i] is an animal with at least $2007$ cells. It is said to be [i]primitive[/i] it its cells cannot be partitioned into two or more dinosaurs. Find with proof the maximum number of cells in a primitive dinosaur.
(1) Animals are also called [i]polyominoes[/i]. They can be defined inductively. Two cells are [i]adjacent[/i] if they share a complete edge. A single cell is an animal, and given an animal with $n$ cells, one with $n+1$ cells is obtained by adjoining a new cell by making it adjacent to one or more existing cells.
2019 Jozsef Wildt International Math Competition, W. 16
If $f : [a, b] \to (0,\infty)$; $0 < a \leq b$; $f$ derivable; $f'$ continuous then:$$\int \limits_{a}^{b}\frac{f'(x)\sqrt{f(x)}}{f^3(x) + 1}\leq \tan^{-1}\left(\frac{f(b)-f(a)}{1 + f(a)f(b)}\right)$$
LMT Team Rounds 2021+, A30
Ryan Murphy is playing poker. He is dealt a hand of $5$ cards. Given that the probability that he has a straight hand (the ranks are all consecutive; e.g. $3,4,5,6,7$ or $9,10,J,Q,K$) or $3$ of a kind (at least $3$ cards of the same rank; e.g. $5, 5, 5, 7, 7$ or $5, 5, 5, 7,K$) is $m/n$ , where $m$ and $n$ are relatively prime positive integers, find $m +n$.
[i]Proposed by Aditya Rao[/i]
1956 AMC 12/AHSME, 21
If each of two intersecting lines intersects a hyperbola and neither line is tangent to the hyperbola, then the possible number of points of intersection with the hyperbola is:
$ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 2\text{ or }3 \qquad\textbf{(C)}\ 2\text{ or }4 \qquad\textbf{(D)}\ 3\text{ or }4 \qquad\textbf{(E)}\ 2,3,\text{ or }4$
2002 Singapore Team Selection Test, 2
Let $n$ be a positive integer and $(x_1, x_2, ..., x_{2n})$, $x_i = 0$ or $1, i = 1, 2, ... , 2n$ be a sequence of $2n$ integers. Let $S_n$ be the sum $S_n = x_1x_2 + x_3x_4 + ... + x_{2n-1}x_{2n}$.
If $O_n$ is the number of sequences such that $S_n$ is odd and $E_n$ is the number of sequences such that $S_n$ is even, prove that $$\frac{O_n}{E_n}=\frac{2^n - 1}{2^n + 1}$$
2024 All-Russian Olympiad Regional Round, 10.5
The quadrilateral $ABCD$ has perpendicular diagonals that meet at $O$. The incenters of triangles $ABC, BCD, CDA, DAB$ form a quadrilateral with perimeter $P$. Show that the sum of the inradii of the triangles $AOB, BOC, COD, DOA$ is less than or equal to $\frac{P} {2}$.
2015 Chile National Olympiad, 5
A quadrilateral $ABCD$ is inscribed in a circle. Suppose that $|DA| =|BC|= 2$ and$ |AB| = 4$. Let $E $ be the intersection point of lines $BC$ and $DA$. Suppose that $\angle AEB = 60^o$ and that $|CD| <|AB|$. Calculate the radius of the circle.
LMT Theme Rounds, 2023F 1B
Evaluate $\dbinom{6}{0}+\dbinom{6}{1}+\dbinom{6}{4}+\dbinom{6}{3}+\dbinom{6}{4}+\dbinom{6}{5}+\dbinom{6}{6}$
[i]Proposed by Jonathan Liu[/i]
[hide=Solution]
[i]Solution.[/i] $\boxed{64}$
We have that $\dbinom{6}{4}=\dbinom{6}{2}$, so $\displaystyle\sum_{n=0}^{6} \dbinom{6}{n}=2^6=\boxed{64}.$
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1997 All-Russian Olympiad Regional Round, 8.5
Segments $AB$, $BC$ and $CA$ are, respectively, diagonals of squares $K_1$, $K_2$, $K3$. Prove that if triangle $ABC$ is acute, then it completely covered by squares $K_1$, $K_2$ and $K_3$.
2022 Taiwan Mathematics Olympiad, 4
Two babies A and B are playing a game with $2022$ bottles of milk. Each bottle has a maximum capacity of $200$ml, and initially each bottle holds $30$ml of milk.
Starting from A, they take turns and do one of the following:
(1) Pick a bottle with at least $100$ml of milk, and drink half of it.
(2) Pick two bottles with less than $100$ml of milk, pour the milk of one bottle into the other one, and toss away the empty bottle.
Whoever cannot do any operations loses the game. Who has a winning strategy?
[i]
Proposed by Chu-Lan Kao and usjl[/i]
2012 IMAC Arhimede, 1
Let $a_1,a_2,..., a_n$ be different integers and let $(b_1,b_2,..., b_n),(c_1,c_2,..., c_n)$ be two of their permutations, different from the identity. Prove that
$$(|a_1-b_1|+|a_2-b_2|+...+|a_n-b_n| , |a_1-c_1|+|a_2-c_2|+...+|a_n-c_n| ) \ge 2$$
where $(x,y)$ denotes the greatest common divisor of the numbers $x,y$
2024 Korea Summer Program Practice Test, 1
Find all polynomials $P$ with integer coefficients such that $P(P(x))-x$ is irreducible over $\mathbb{Z}[x]$.
2004 Junior Balkan Team Selection Tests - Moldova, 2
Let $n \in N^*$ . Let $a_1, a_2..., a_n$ be real such that $a_1 + a_2 +...+ a_n \ge 0$.
Prove the inequality $\sqrt{a_1^2+1}+\sqrt{a_2^2+1}+...+\sqrt{a_1^2+1}\ge \sqrt{2n(a_1 + a_2 +...+ a_n )}$.
2004 Nicolae Coculescu, 4
Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a continuous function having a primitive $ F $ having the property that $ f-F $ is positive globally. Calculate $ \lim_{x\to\infty } f(x) . $
[i]Florian Dumitrel[/i]
2020 Stars of Mathematics, 3
Determine the primes $p$ for which the numbers $2\lfloor p/k\rfloor - 1, \ k = 1,2,\ldots, p,$ are all quadratic residues modulo $p.$
[i]Vlad Matei[/i]
2019 Switzerland Team Selection Test, 9
Let $ABC$ be an acute triangle with $AB<AC$. $E,F$ are foots of the altitudes drawn from $B,C$ respectively. Let $M$ be the midpoint of segment $BC$. The tangent at $A$ to the circumcircle of $ABC$ cuts $BC$ in $P$ and $EF$ cuts the parallel to $BC$ from $A$ at $Q$. Prove that $PQ$ is perpendicular to $AM$.
2015 Geolympiad Spring, 3
Let $ABC$ be an acute triangle with orthocenter $H$, incenter $I$, and excenters $I_A, I_B, I_C$. Show that $II_A * II_B * II_C \ge 8 AH * BH * CH$.
2016 Swedish Mathematical Competition, 2
Determine whether the inequality $$ \left|\sqrt{x^2+2x+5}-\sqrt{x^2-4x+8}\right|<3$$ is valid for all real numbers $x$.
1991 IMTS, 1
What is the smallest integer multiple of 9997, other than 9997 itself, which contains only odd digits?
2005 Baltic Way, 9
A rectangle is divided into $200\times 3$ unit squares. Prove that the number of ways of splitting this rectangle into rectangles of size $1\times 2$ is divisible by $3$.
2011 Purple Comet Problems, 6
Working alone, the expert can paint a car in one day, the amateur can paint a car in two days, and the beginner can paint a car in three days. If the three painters work together at these speeds to paint three cars, it will take them $\frac{m}{n}$ days where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
2001 Grosman Memorial Mathematical Olympiad, 5
Triangle $ABC$ in the plane $\Pi$ is called [i]good [/i] if it has the following property:
For any point $D$ in space outside the plane $\Pi$, it is possible to construct a triangle with sides of lengths $CD,BD,AD$. Find all good triangles
2012 Saint Petersburg Mathematical Olympiad, 2
We have big multivolume encyclopaedia about dogs on the shelf in alphabetical order, each volume in its specially selected place. Near each place there is an instruction that prescribes one of four actions: to rearrange
this volume is one or two places left or right. If you simultaneously run all instructions, volumes will be placed in the same places in another order. The cynologist Dima performs all the instructions every morning. Once he discovered,
that the volume of "Bichons" stands still, which was initially occupied by the volume of "Terriers". Prove ,
that after some time the volume of "Mudies" will stand on the original place of the volume
"Poodles".