Found problems: 85335
2018 Yasinsky Geometry Olympiad, 2
Let $ABCD$ be a parallelogram, such that the point $M$ is the midpoint of the side $CD$ and lies on the bisector of the angle $\angle BAD$. Prove that $\angle AMB = 90^o$.
2006 Switzerland Team Selection Test, 3
Find all the functions $f : \mathbb{R} \to \mathbb{R}$ satisfying for all $x,y \in \mathbb{R}$ $f(f(x)-y^2) = f(x)^2 - 2f(x)y^2 + f(f(y))$.
1981 Tournament Of Towns, (012) 1
We will say that two pyramids touch each other by faces if they have no common interior points and if the intersection of a face of one of them with a face of the other is either a triangle or a polygon. Is it possible to place $8$ tetrahedra in such a way that every two of them touch each other by faces?
(A Andjans, Riga)
1968 Miklós Schweitzer, 6
Let $ \Psi\equal{}\langle A;...\rangle$ be an arbitrary, countable algebraic structure (that is, $ \Psi$ can have an arbitrary number of finitary operations and relations). Prove that $ \Psi$ has as many as continuum automorphisms if and only if for any finite subset $ A'$ of $ A$ there is an automorphism $ \pi_{A'}$ of $ \Psi$ different from the identity automorphism and such that \[ (x) \pi_{A'}\equal{}x\] for every $ x \in A'$.
[i]M. Makkai[/i]
2003 Junior Balkan Team Selection Tests - Moldova, 3
The quadrilateral $ABCD$ with perpendicular diagonals is inscribed in the circle with center $O$, the points $M,N$ are the midpoints of $[BC]$ and $[CD]$ respectively. Find the ratio of areas of the figures $OMCN$ and $ABCD$
1950 Polish MO Finals, 3
Prove that if the two altitudes of a tetrahedron intersect, then the other two atltitudes intersect also.
2022 Saint Petersburg Mathematical Olympiad, 2
$12$ schoolchildren are engaged in a circle of patriotic songs, each of them knows a few songs (maybe none). We will say that a group of schoolchildren can sing a song if at least one member of the group knows it. Supervisor the circle noticed that any group of $10$ circle members can sing exactly $20$ songs, and any group of $8$ circle members - exactly $16$ songs. Prove that the group of all $12$ circle members can sing exactly $24$ songs.
2022 China Second Round, 4
Find the smallest positive integer $k$ with the following property: if each cell of a $100\times 100$ grid is dyed with one color and the number of cells of each color is not more than $104$, then there is a $k\times1$ or $1\times k$ rectangle that contains cells of at least three different colors.
2019 Kosovo Team Selection Test, 5
$a,b,c,d$ are fixed positive real numbers. Find the maximum value of the function $f: \mathbb{R^{+}}_{0} \rightarrow \mathbb{R}$ $f(x)=\frac{a+bx}{b+cx}+\frac{b+cx}{c+dx}+\frac{c+dx}{d+ax}+\frac{d+ax}{a+bx}, x \geq 0$
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]