Found problems: 85335
2004 Indonesia MO, 3
Given triangle $ ABC$ with $ C$ a right angle, show that the diameter of the incenter is $ a\plus{}b\minus{}c$, where $ a\equal{}BC$, $ b\equal{}CA$, and $ c\equal{}AB$.
2011 NIMO Summer Contest, 2
The sum of three consecutive integers is $15$. Determine their product.
1967 IMO Longlists, 47
Prove the following inequality:
\[\prod^k_{i=1} x_i \cdot \sum^k_{i=1} x^{n-1}_i \leq \sum^k_{i=1}
x^{n+k-1}_i,\] where $x_i > 0,$ $k \in \mathbb{N}, n \in
\mathbb{N}.$
2016 Math Prize for Girls Olympiad, 4
Let $d(n)$ be the number of positive divisors of a positive integer $n$. Let $\mathbb{N}$ be the set of all positive integers. Say that a bijection $F$ from $\mathbb{N}$ to $\mathbb{N}$ is [i]divisor-friendly[/i] if $d(F(mn)) = d(F(m)) d(F(n))$ for all positive integers $m$ and $n$. (Note: A bijection is a one-to-one, onto function.) Does there exist a divisor-friendly bijection? Prove or disprove.
2023 Romania EGMO TST, P4
Consider fractions $\frac{a}{b}$ where $a$ and $b$ are positive integers.
(a) Prove that for every positive integer $n$, there exists such a fraction $\frac{a}{b}$ such that $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}+1$.
(b) Show that there are infinitely many positive integers $n$ such that no such fraction $\frac{a}{b}$ satisfies $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}$.
2015 India Regional MathematicaI Olympiad, 1
Let ABC be a triangle. Let B' and C' denote the reflection of B and C in the internal angle bisector of angle A. Show that the triangles ABC and AB'C' have the same incenter.
1961 AMC 12/AHSME, 26
For a given arithmetic series the sum of the first $50$ terms is $200$, and the sum of the next $50$ terms is $2700$. The first term in the series is:
${{ \textbf{(A)}\ -1221 \qquad\textbf{(B)}\ -21.5 \qquad\textbf{(C)}\ -20.5 \qquad\textbf{(D)}\ 3 }\qquad\textbf{(E)}\ 3.5 } $
2024 5th Memorial "Aleksandar Blazhevski-Cane", P1
This year, some contestants at the Memorial Contest ABC are friends with each other (friendship is always mutual). For each contestant $X$, let $t(X)$ be the total score that this contestant achieved in previous years before this contest. It is known that the following statements are true:
$1)$ For any two friends $X'$ and $X''$, we have $t(X') \neq t(X''),$
$2)$ For every contestant $X$, the set $\{ t(Y) : Y \text{ is a friend of } X \}$ consists of consecutive integers.
The organizers want to distribute the contestants into contest halls in such a way that no two friends are in the same hall. What is the minimal number of halls they need?
2021 Oral Moscow Geometry Olympiad, 5
The trapezoid is inscribed in a circle. Prove that the sum of distances from any point of the circle to the midpoints of the lateral sides are not less than the diagonal of the trapezoid.
2019 India Regional Mathematical Olympiad, 2
Let $ABC$ be a triangle with circumcircle $\Omega$ and let $G$ be the centroid of triangle $ABC$. Extend $AG, BG$ and $CG$ to meet the circle $\Omega$ again in $A_1, B_1$ and $C_1$. Suppose $\angle BAC = \angle A_1 B_1 C_1, \angle ABC = \angle A_1 C_1 B_1$ and $ \angle ACB = B_1 A_1 C_1$. Prove that $ABC$ and $A_1 B_1 C_1$ are equilateral triangles.
1994 Dutch Mathematical Olympiad, 3
$ (a)$ Prove that every multiple of $ 6$ can be written as a sum of four cubes.
$ (b)$ Prove that every integer can be written as a sum of five cubes.
2006 Taiwan National Olympiad, 3
$a_1, a_2, ..., a_{95}$ are positive reals. Show that
$\displaystyle \sum_{k=1}^{95}{a_k} \le 94+ \prod_{k=1}^{95}{\max{\{1,a_k\}}}$
2019 Hanoi Open Mathematics Competitions, 11
Find all integers $x$ and $y$ satisfying the following equation $x^2 - 2xy + 5y^2 + 2x - 6y - 3 = 0$.
2014 Argentine National Olympiad, Level 3, 2.
Given several numbers, one of them, $a$, is chosen and replaced by the three numbers $\frac{a}{3}, \frac{a}{3}, \frac{a}{3}$. This process is repeated with the new set of numbers, and so on. Originally, there are $1000$ ones, and we apply the process several times. A number $m$ is called [i]good[/i] if there are $m$ or more numbers that are the same after each iteration, no matter how many or what operations are performed. Find the largest possible good number.
KoMaL A Problems 2020/2021, A. 792
Let $p\geq 3$ be a prime number and $0\leq r\leq p-3.$ Let $x_1,x_2,\ldots,x_{p-1+r}$ be integers satisfying \[\sum_{i=1}^{p-1+r}x_i^k\equiv r \bmod{p}\]for all $1\leq k\leq p-2.$ What are the possible remainders of numbers $x_2,x_2,\ldots,x_{p-1+r}$ modulo $p?$
[i]Proposed by Dávid Matolcsi, Budapest[/i]
2007 Junior Balkan Team Selection Tests - Moldova, 8
a) Calculate the product $$\left(1+\frac{1}{2}\right) \left(1+\frac{1}{3}\right) \left(1+\frac{1}{4}\right)... \left(1+\frac{1}{2006}\right) \left(1+\frac{1}{2007}\right)$$
b) Let the set $$A =\left\{\frac{1}{2}, \frac{1}{3},\frac{1}{4}, ...,\frac{1}{2006}, \frac{1}{2007}\right\}$$
Determine the sum of all products of $2$, of $4$, of $6$,... , of $2004$ ¸and of $ 2006$ different elements of the set $A$.
2002 Junior Balkan Team Selection Tests - Moldova, 11
Simultaneously from the same point of a circular route and in the same direction for two hours two bodies move evenly. The first body performs a complete rotation three minutes faster than the second body and exceeds it every $9$ minutes and $20$ seconds. Whenever the first body will overtake the other the second exactly at the starting point?
2009 Tournament Of Towns, 4
Consider an in finite sequence consisting of distinct positive integers such that each term (except the rst one) is either an arithmetic mean or a geometric mean of two neighboring terms. Does it necessarily imply that starting at some point the sequence becomes either arithmetic progression or a geometric progression?
2008 AIME Problems, 6
A triangular array of numbers has a first row consisting of the odd integers $ 1,3,5,\ldots,99$ in increasing order. Each row below the first has one fewer entry than the row above it, and the bottom row has a single entry. Each entry in any row after the top row equals the sum of the two entries diagonally above it in the row immediately above it. How many entries in the array are multiples of $ 67$?
[asy]size(200);
defaultpen(fontsize(10));
label("1", origin);
label("3", (2,0));
label("5", (4,0));
label("$\cdots$", (6,0));
label("97", (8,0));
label("99", (10,0));
label("4", (1,-1));
label("8", (3,-1));
label("12", (5,-1));
label("196", (9,-1));
label(rotate(90)*"$\cdots$", (6,-2));[/asy]
2019 BAMO, D/2
Initially, all the squares of an $8\times 8$ grid are white. You start by choosing one of the squares and coloring it gray. After that, you may color additional squares gray one at a time, but you may only color a square gray if it has exactly $1$ or $3$ gray neighbors at that moment (where a neighbor is a square sharing an edge).
For example, the configuration below (of a smaller $3\times 4$ grid) shows a situation where six squares have been colored gray so far. The squares that can be colored at the next step are marked with a dot.
Is it possible to color all the squares gray? Justify your answer.
[img]https://cdn.artofproblemsolving.com/attachments/1/c/d50ab269f481e4e516dace06a991e6b37f2a85.png[/img]
2010 Princeton University Math Competition, 8
Matt is asked to write the numbers from 1 to 10 in order, but he forgets how to count. He writes a permutation of the numbers $\{1, 2, 3\ldots , 10\}$ across his paper such that:
[list]
[*]The leftmost number is 1.
[*]The rightmost number is 10.
[*]Exactly one number (not including 1 or 10) is less than both the number to its immediate left and the number to its immediate right.[/list]
How many such permutations are there?
2017 CMIMC Individual Finals, 3
In a certain game, the set $\{1, 2, \dots, 10\}$ is partitioned into equally-sized sets $A$ and $B$. In each of five consecutive rounds, Alice and Bob simultaneously choose an element from $A$ or $B$, respectively, that they have not yet chosen; whoever chooses the larger number receives a point, and whoever obtains three points wins the game. Determine the probability that Alice is guaranteed to win immediately after the set is initially partitioned.
2009 QEDMO 6th, 7
Albatross and Frankinfueter both own a circle. Frankinfueter also has stolen from Prof. Trugweg a ruler. Before that, Trugweg had two points with a distance of $1$ drawn his (infinitely large) board. For a natural number $n$, let A $(n)$ be the number of the construction steps that Albatross needs at least to create two points with a distance of $n$ to construct. Similarly, Frankinfueter needs at least $F(n)$ steps for this.
How big can $\frac{A (n)}{F (n)}$ become?
There are only the following three construction steps:
a) Mark an intersection of two straight lines, two circles or a straight line with one circle.
b) Pierce at a marked point $P$ and draw a circle around $P$ through one marked point .
c) Draw a straight line through two marked points (this implies possession of a ruler ahead!).
Ukrainian TYM Qualifying - geometry, 2014.8
In the triangle $ABC$ on the ray $BA$ mark the point $K$ so that $\angle BCA= \angle KCA$ , and on the median $BM$ mark the point $T$ so that $\angle CTK=90^o$ . Prove that $\angle MTC=\angle MCB$ .
2005 China Team Selection Test, 1
Let $k$ be a positive integer. Prove that one can partition the set $\{ 0,1,2,3, \cdots ,2^{k+1}-1 \}$ into two disdinct subsets $\{ x_1,x_2, \cdots, x_{2k} \}$ and $\{ y_1, y_2, \cdots, y_{2k} \}$ such that $\sum_{i=1}^{2^k} x_i^m =\sum_{i=1}^{2^k} y_i^m$ for all $m \in \{ 1,2, \cdots, k \}$.