Found problems: 85335
2023 All-Russian Olympiad Regional Round, 10.8
The bisector of $\angle BAD$ of a parallelogram $ABCD$ meets $BC$ at $K$. The point $L$ lies on $AB$ such that $AL=CK$. The lines $AK$ and $CL$ meet at $M$. Let $(ALM)$ meet $AD$ after $D$ at $N$. Prove that $\angle CNL=90^{o}$
1977 Poland - Second Round, 5
Let the polynomials $ w_n $ be given by the formulas: $$
w_1(x) = x^2 - 1, \quad w_{n+1}(x) = w_n(x)^2 - 1, \quad (n = 1, 2, \ldots)$$ and let $a$ be a real number. How many different real solutions does the equation $ w_n(x) = a $ have?
1940 Putnam, B7
Which is greater
$$\sqrt{n}^{\sqrt{n+1}} \;\; \; \text{or}\;\;\; \sqrt{n+1}^{\sqrt{n}}$$
where $n>8?$
2002 AMC 10, 2
For the nonzero numbers $ a$, $ b$, and $ c$, define \[(a,b,c)\equal{}\frac{abc}{a\plus{}b\plus{}c}.\] Find $(2,4,6)$.
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 4 \qquad
\textbf{(D)}\ 6 \qquad
\textbf{(E)}\ 24$
2009 Today's Calculation Of Integral, 397
In $ xy$ plane, find the minimum volume of the solid by rotating the region boubded by the parabola $ y \equal{} x^2 \plus{} ax \plus{} b$ passing through the point $ (1,\ \minus{} 1)$ and the $ x$ axis about the $ x$ axis
1988 USAMO, 1
By a [i]pure repeating decimal[/i] (in base $10$), we mean a decimal $0.\overline{a_1\cdots a_k}$ which repeats in blocks of $k$ digits beginning at the decimal point. An example is $.243243243\cdots = \tfrac{9}{37}$. By a [i]mixed repeating decimal[/i] we mean a decimal $0.b_1\cdots b_m\overline{a_1\cdots a_k}$ which eventually repeats, but which cannot be reduced to a pure repeating decimal. An example is $.011363636\cdots = \tfrac{1}{88}$.
Prove that if a mixed repeating decimal is written as a fraction $\tfrac pq$ in lowest terms, then the denominator $q$ is divisible by $2$ or $5$ or both.
2020-IMOC, C4
$\definecolor{A}{RGB}{70,80,0}\color{A}\fbox{C4.}$ Show that for any positive integer $n \ge 3$ and some subset of $\lbrace{1, 2, . . . , n}\rbrace$ with size more than $\frac{n}2 + 1$, there exist three distinct elements $a, b, c$ in the subset such that $$\definecolor{A}{RGB}{255,70,255}\color{A} (ab)^2 + (bc)^2 + (ca)^2$$is a perfect square.
[i]Proposed by [/i][b][color=#419DAB]ltf0501[/color][/b].
[color=#3D9186]#1736[/color]
2021 Science ON grade VII, 3
Are there any real numbers $a,b,c$ such that $a+b+c=6$, $ab+bc+ca=9$ and $a^4+b^4+c^4=260$? What about if we let $a^4+b^4+c^4=210$?
[i] (Andrei Bâra)[/i]
2014 Dutch BxMO/EGMO TST, 3
In triangle $ABC$, $I$ is the centre of the incircle. There is a circle tangent
to $AI$ at $I$ which passes through $B$. This circle intersects $AB$ once more
in $P$ and intersects $BC$ once more in $Q$. The line $QI$ intersects $AC$ in $R$.
Prove that $|AR|\cdot |BQ|=|P I|^2$
1997 Pre-Preparation Course Examination, 1
Let $n$ be a positive integer. Prove that there exist polynomials$f(x)$and $g(x$) with integer coefficients such that
\[f(x)\left(x + 1 \right)^{2^n}+ g(x) \left(x^{2^n}+ 1 \right) = 2.\]
2016 District Olympiad, 1
Let be a pyramid having a square as its base and the projection of the top vertex to the base is the center of the square. Prove that two opposite faces are perpendicular if and only if the angle between two adjacent faces is $ 120^{\circ } . $
2024 Moldova Team Selection Test, 5
Consider a natural number $n \ge 3$. A convex polygon with $n$ sides is entirely placed inside a square with side length 1. Prove that we can always find three vertices of this polygon, the triangle formed by which has area not greater than $\frac{8}{n^2}$.
2016 Moldova Team Selection Test, 8
Let us have $n$ ( $n>3$) balls with different rays. On each ball it is written an integer number. Determine the greatest natural number $d$ such that for any numbers written on the balls, we can always find at least 4 different ways to choose some balls with the sum of the numbers written on them divisible by $d$.
1979 Poland - Second Round, 5
Prove that among every ten consecutive natural numbers there is one that is coprime to each of the other nine.
2006 Baltic Way, 19
Does there exist a sequence $a_1,a_2,a_3,\ldots $ of positive integers such that the sum of every $n$ consecutive elements is divisible by $n^2$ for every positive integer $n$?
2001 Korea - Final Round, 1
For given positive integers $n$ and $N$, let $P_n$ be the set of all polynomials $f(x)=a_0+a_1x+\cdots+a_nx^n$ with integer coefficients such that:
[list]
(a) $|a_j| \le N$ for $j = 0,1, \cdots ,n$;
(b) The set $\{ j \mid a_j = N\}$ has at most two elements.
[/list]
Find the number of elements of the set $\{f(2N) \mid f(x) \in P_n\}$.
2009 Ukraine National Mathematical Olympiad, 3
Point $O$ is inside triangle $ABC$ such that $\angle AOB = \angle BOC = \angle COA = 120^\circ .$ Prove that
\[\frac{AO^2}{BC}+\frac{BO^2}{CA}+\frac{CO^2}{AB} \geq \frac{AO+BO+CO}{\sqrt 3}.\]
2018 Singapore MO Open, 2
Let O be a point inside triangle ABC such that $\angle BOC$ is $90^\circ$ and $\angle BAO = \angle BCO$. Prove that $\angle OMN$ is $90$ degrees, where $M$ and $N$ are the midpoints of $\overline{AC}$ and $\overline{BC}$, respectively.
2004 China Team Selection Test, 3
Find all positive integer $ m$ if there exists prime number $ p$ such that $ n^m\minus{}m$ can not be divided by $ p$ for any integer $ n$.
1986 Spain Mathematical Olympiad, 1
Define the distance between real numbers $x$ and $y$ by $d(x,y) =\sqrt{([x]-[y])^2+(\{x\}-\{y\})^2}$ .
Determine (as a union of intervals) the set of real numbers whose distance from $3/2$ is less than $202/100$ .
2006 Iran MO (3rd Round), 2
$f: \mathbb R^{n}\longrightarrow\mathbb R^{m}$ is a non-zero linear map. Prove that there is a base $\{v_{1},\dots,v_{n}m\}$ for $\mathbb R^{n}$ that the set $\{f(v_{1}),\dots,f(v_{n})\}$ is linearly independent, after ommitting Repetitive elements.
2019 Swedish Mathematical Competition, 6
Is there an infinite sequence of positive integers $\{a_n\}_{n = 1}^{\infty}$ which contains each positive integer exactly once and is such that the number $a_n + a_{n + 1} $ is a perfect square for each $n$?
1991 Greece National Olympiad, 4
In how many ways can we construct a square with dimensions $3\times 3$ using $3$ white, $3$ green and $3$ red squares of dimensions $1\times 1$, such that in every horizontal and in every certical line, squares have different colours .
2017 Iran Team Selection Test, 5
In triangle $ABC$, arbitrary points $P,Q$ lie on side $BC$ such that $BP=CQ$ and $P$ lies between $B,Q$.The circumcircle of triangle $APQ$ intersects sides $AB,AC$ at $E,F$ respectively.The point $T$ is the intersection of $EP,FQ$.Two lines passing through the midpoint of $BC$ and parallel to $AB$ and $AC$, intersect $EP$ and $FQ$ at points $X,Y$ respectively.
Prove that the circumcircle of triangle $TXY$ and triangle $APQ$ are tangent to each other.
[i]Proposed by Iman Maghsoudi[/i]
2021 CCA Math Bonanza, T5
We say that a [i]special word[/i] is any sequence of letters [b]containing a vowel[/b]. How many ordered triples of special words $(W_1,W_2,W_3)$ have the property that if you concatenate the three words, you obtain a rearrangement of "aadvarks"?
For example, the number of triples of special words such that the concatenation is a rearrangement of ``adaa" is $6$, and all of the possible triples are:
[center]
(da,a,a),(ad,a,a),(a,da,a),(a,ad,a),(a,a,da),(a,a,ad).
[/center]
[i]2021 CCA Math Bonanza Team Round #5[/i]