Found problems: 85335
1985 Iran MO (2nd round), 1
Inscribe in the triangle $ABC$ a triangle with minimum perimeter.
2024 Belarus Team Selection Test, 3.2
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that for any reals $x \neq y$ the following equality is true:
$$f(x+y)^2=f(x+y)+f(x)+f(y)$$
[i]D. Zmiaikou[/i]
2012 CHMMC Spring, 3
Three different faces of a regular dodecahedron are selected at random and painted. What is the probability that there is at least one pair of painted faces that share an edge?
KoMaL A Problems 2019/2020, A. 776
Let $k > 1$ be a fixed odd number, and for non-negative integers $n$ let
$$f_n=\sum_{\substack{0\leq i\leq n\\ k\mid n-2i}}\binom{n}{i}.$$
Prove that $f_n$ satisfy the following recursion:
$$f_{n}^2=\sum_{i=0}^{n} \binom{n}{i}f_{i}f_{n-i}.$$
2016 India IMO Training Camp, 2
Given that $n$ is a natural number such that the leftmost digits in the decimal representations of $2^n$ and $3^n$ are the same, find all possible values of the leftmost digit.
2019 Balkan MO Shortlist, A1
Let $a_0$ be an arbitrary positive integer. Consider the infinite sequence $(a_n)_{n\geq 1}$, defined inductively as follows: given $a_0, a_1, ..., a_{n-1}$ define the term $a_n$ as the smallest positive integer such that $a_0+a_1+...+a_n$ is divisible by $n$. Prove that there exist a positive integer a positive integer $M$ such that $a_{n+1}=a_n$ for all $n\geq M$.
2023 CMIMC Integration Bee, 11
\[\int_{-1}^1 \frac{1}{(8 + x^2)\sqrt{1-x^2}} \,\mathrm dx\]
[i]Proposed by Vlad Oleksenko[/i]
1998 Harvard-MIT Mathematics Tournament, 8
Given any two positive real numbers $x$ and $y$, then $x\Diamond y$ is a positive real number defined in terms of $x$ and $y$ by some fixed rule. Suppose the operation $x\Diamond y$ satisfies the equations $(x\cdot y)\Diamond y=x(y\Diamond y)$ and $(x\Diamond 1)\Diamond x=x\Diamond 1$ for all $x,y>0$. Given that $1\Diamond 1=1$, find $19\Diamond 98$.
2012 NIMO Problems, 1
Compute the average of the integers $2, 3, 4, \dots, 2012$.
[i]Proposed by Eugene Chen[/i]
2013-2014 SDML (Middle School), 3
Simplify $\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}$.
2021 Saint Petersburg Mathematical Olympiad, 4
Given are $n$ points with different abscissas in the plane. Through every pair points is drawn a parabola - a graph of a square trinomial with leading coefficient equal to $1$. A parabola is called $good$ if there are no other marked points on it, except for the two through which it is drawn, and there are no marked points above it (i.e. inside it). What is the greatest number of $good$ parabolas?
2020-2021 OMMC, 3
The intersection of two squares with perimeter $8$ is a rectangle with diagonal length $1$. Given that the distance between the centers of the two squares is $2$, the perimeter of the rectangle can be expressed as $P$. Find $10P$.
2011 Belarus Team Selection Test, 1
$AB$ and $CD$ are two parallel chords of a parabola. Circle $S_1$ passing through points $A,B$ intersects circle $S_2$ passing through $C,D$ at points $E,F$. Prove that if $E$ belongs to the parabola, then $F$ also belongs to the parabola.
I.Voronovich
2015 Belarus Team Selection Test, 3
Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.)
[i]Proposed by Hong Kong[/i]
2016 Indonesia TST, 1
Determine all real numbers $x$ which satisfy
\[ x = \sqrt{a - \sqrt{a+x}} \]
where $a > 0$ is a parameter.
2021 Bosnia and Herzegovina Team Selection Test, 2
Let $p > 2$ be a prime number. Prove that there is a permutation $k_1, k_2, ..., k_{p-1}$ of numbers $1,2,...,p-1$ such that the number $1^{k_1}+2^{k_2}+3^{k_3}+...+(p-1)^{k_{p-1}}$ is divisible by $p$.
Note: The numbers $k_1, k_2, ..., k_{p-1}$ are a permutation of the numbers $1,2,...,p-1$ if each of of numbers $1,2,...,p-1$ appears exactly once among the numbers $k_1, k_2, ..., k_{p-1}$.
2012 Romania Team Selection Test, 4
Let $S$ be a set of positive integers, each of them having exactly $100$ digits in base $10$ representation. An element of $S$ is called [i]atom[/i] if it is not divisible by the sum of any two (not necessarily distinct) elements of $S$. If $S$ contains at most $10$ atoms, at most how many elements can $S$ have?
KoMaL A Problems 2021/2022, A. 817
Let $ABC$ be a triangle. Let $T$ be the point of tangency of the circumcircle of triangle $ABC$ and the $A$-mixtilinear incircle. The incircle of triangle $ABC$ has center $I$ and touches sides $BC,CA$ and $AB$ at points $D,E$ and $F,$ respectively. Let $N$ be the midpoint of line segment $DF.$ Prove that the circumcircle of triangle $BTN,$ line $TI$ and the perpendicular from $D$ to $EF$ are concurrent.
[i]Proposed by Diaconescu Tashi, Romania[/i]
2014 AMC 8, 22
A $2$-digit number is such that the product of the digits plus the sum of the digits is equal to the number. What is the units digit of the number?
$\textbf{(A) }1\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }7\qquad \textbf{(E) }9$
2023 Assam Mathematics Olympiad, 3
Find the number of integer solutions of $||x| - 2023| < 2020$.
1984 Poland - Second Round, 6
The sequence $(x_n)$ is defined by formulas
$$
x_1=c,\; x_{n+1} = cx_n + \sqrt{(c^2-1)(x_n^2-1)} \quad\text{ for }\quad n=1,2,\ldots$$
Prove that if $ c $ is a natural number, then all numbers $ x_n $ are natural.
1998 ITAMO, 5
Suppose $a_1,a_2,a_3,a_4$ are distinct integers and $P(x)$ is a polynomial with integer coefficients satisfying $P(a_1) = P(a_2) = P(a_3) = P(a_4) = 1$.
(a) Prove that there is no integer $n$ such that $P(n) = 12$.
(b) Do there exist such a polynomial and $a_n$ integer $n$ such that $P(n) = 1998$?
1956 Miklós Schweitzer, 8
[b]8.[/b] Let $(a_n)_{n=1}^{\infty}$ be a sequence of positive numbers and suppose that $\sum_{n=1}^{\infty} a_n^2$ is divergent. Let further $0<\epsilon<\frac{1}{2}$. Show that there exists a sequence $(b_n)_{n=1}^{\infty}$ of positive numbers such that $\sum_{n=1}^{\infty}b_n^2$ is convergent and
$\sum_{n=1}^{N}a_n b_n >(\sum_{n=1}^{N}a_n^2)^{\frac{1}{2}-\epsilon}$
for every positive integer $N$. [b](S. 8)[/b]
1975 Miklós Schweitzer, 8
Prove that if \[ \sum_{n=1}^m a_n \leq Na_m \;(m=1,2,...)\] holds for a sequence $ \{a_n \}$ of nonnegative real numbers with some positive integer $ N$, then $ \alpha_{i+p} \geq p \alpha_i$ for $ i,p=1,2,...,$ where \[ \alpha_i= \sum_{n=(i-1)N+1}^{iN} a_n \;(i=1,2,...)\ .\]
[i]L. Leindler[/i]
2013 Hanoi Open Mathematics Competitions, 15
Denote by $Q$ and $N^*$ the set of all rational and positive integer numbers, respectively.
Suppose that $\frac{ax + b}{cx + d} \in Q$ for every $x \in N^*$:
Prove that there exist integers $A,B,C,D$ such that $\frac{ax + b}{cx + d}= \frac{Ax + B}{Cx+D}$ for all $x \in N^* $