Found problems: 85335
2005 Switzerland - Final Round, 3
Prove for all $a_1, ..., a_n > 0$ the following inequality and determine all cases in where the equaloty holds:
$$\sum_{k=1}^{n}ka_k\le {n \choose 2}+\sum_{k=1}^{n}a_k^k.$$
2025 Chile TST IMO-Cono, 4
Let \( ABC \) be a triangle with \( AB < AC \). Let \( M \) be the midpoint of \( AC \), and let \( D \) be a point on segment \( AC \) such that \( DB = DC \). Let \( E \) be the point of intersection, different from \( B \), of the circumcircle of triangle \( ABM \) and line \( BD \). Define \( P \) and \( Q \) as the points of intersection of line \( BC \) with \( EM \) and \( AE \), respectively. Prove that \( P \) is the midpoint of \( BQ \).
1999 AMC 12/AHSME, 28
Let $ x_1$, $ x_2$, $ \dots$, $ x_n$ be a sequence of integers such that
(i) $ \minus{}1 \le x_i \le 2$, for $ i \equal{} 1,2,3,\dots,n$;
(ii) $ x_1 \plus{} x_2 \plus{} \cdots \plus{} x_n \equal{} 19$; and
(iii) $ x_1^2 \plus{} x_2^2 \plus{} \cdots \plus{} x_n^2 \equal{} 99$.
Let $ m$ and $ M$ be the minimal and maximal possible values of $ x_1^3 \plus{} x_2^3 \plus{} \cdots \plus{} x_n^3$, respectively. Then $ \frac{M}{m} \equal{}$
$ \textbf{(A)}\ 3\qquad
\textbf{(B)}\ 4\qquad
\textbf{(C)}\ 5\qquad
\textbf{(D)}\ 6\qquad
\textbf{(E)}\ 7$
1976 Bulgaria National Olympiad, Problem 2
Find all polynomials $p(x)$ satisfying the condition:
$$p(x^2-2x)=p(x-2)^2.$$
1985 National High School Mathematics League, 4
Which figure can be the images of equations $mx+ny^2=0$ and $mx^2+ny^2=1$$(m,n\neq0)$?
[img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvZC9kLzUzZTAwZjU1YzEyN2I3ZDJjNjcwNDQ2ZmQ5MDBmYWZlODAwNGU0LnBuZw==&rn=YWFhYWFhYWFhYWFhYWFhYWFhYWFhYWFhYWFhYWFhYWFhYWFhYS5wbmc=[/img]
2014 HMNT, 2
Let $f(x) = x^2 + 6x + 7$. Determine the smallest possible value of $f(f(f(f(x))))$ over all real numbers $x.$
Kyiv City MO 1984-93 - geometry, 1985.7.3
$O$ is the point of intersection of the diagonals of the convex quadrilateral $ABCD$. It is known that the areas of triangles $AOB, BOC, COD$ and $DOA$ are expressed in natural numbers. Prove that the product of these areas cannot end in $1985$.
2012 AIME Problems, 12
For a positive integer $p$, define the positive integer $n$ to be $p$-safe if $n$ differs in absolute value by more than $2$ from all multiples of $p$. For example, the set of $10$-safe numbers is $\{3, 4, 5, 6, 7, 13, 14, 15, 16, 17,23, \ldots \}$. Find the number of positive integers less than or equal to $10,000$ which are simultaneously $7$-safe, $11$-safe, and $13$-safe.ยท
1985 Tournament Of Towns, (106) 6
In triangle $ABC, AH$ is an altitude ($H$ is on $BC$) and $BE$ is a bisector ($E$ is on $AC$) . We are given that angle $BEA$ equals $45^o$ .Prove that angle $EHC$ equals $45^o$ .
(I. Sharygin , Moscow)
2024 Vietnam Team Selection Test, 1
Let $P(x) \in \mathbb{R}[x]$ be a monic, non-constant polynomial. Determine all continuous functions $f: \mathbb{R} \to \mathbb{R}$ such that
$$f(f(P(x))+y+2023f(y))=P(x)+2024f(y),$$
for all reals $x,y$.
2013 All-Russian Olympiad, 4
On each of the cards written in $2013$ by number, all of these $2013$ numbers are different. The cards are turned down by numbers. In a single move is allowed to point out the ten cards and in return will report one of the numbers written on them (do not know what). For what most $w$ guaranteed to be able to find $w$ cards for which we know what numbers are written on each of them?
2002 Tournament Of Towns, 3
[list]
[*] A test was conducted in class. It is known that at least $\frac{2}{3}$ of the problems were hard. Each such problems were not solved by at least $\frac{2}{3}$ of the students. It is also known that at least $\frac{2}{3}$ of the students passed the test. Each such student solved at least $\frac{2}{3}$ of the suggested problems. Is this possible?
[*] Previous problem with $\frac{2}{3}$ replaced by $\frac{3}{4}$.
[*] Previous problem with $\frac{2}{3}$ replaced by $\frac{7}{10}$.[/list]
2008 Croatia Team Selection Test, 3
Point $ M$ is taken on side $ BC$ of a triangle $ ABC$ such that the centroid $ T_c$ of triangle $ ABM$ lies on the circumcircle of $ \triangle ACM$ and the centroid $ T_b$ of $ \triangle ACM$ lies on the circumcircle of $ \triangle ABM$. Prove that the medians of the triangles $ ABM$ and $ ACM$ from $ M$ are of the same length.
2007 Bulgaria National Olympiad, 1
Let $k>1$ be a given positive integer. A set $S$ of positive integers is called [i]good[/i] if we can colour the set of positive integers in $k$ colours such that each integer of $S$ cannot be represented as sum of two positive integers of the same colour. Find the greatest $t$ such that the set $S=\{a+1,a+2,\ldots ,a+t\}$ is [i]good[/i] for all positive integers $a$.
[i]A. Ivanov, E. Kolev[/i]
2022 Bulgarian Autumn Math Competition, Problem 9.2
Given is the triangle $ABC$ such that $BC=13, CA=14, AB=15$ Prove that $B$, the incenter $J$ and the midpoints of $AB$ and $BC$ all lie on a circle
1990 ITAMO, 6
Some marbles are distributed over $2n + 1$ bags. Suppose that, whichever bag is removed, it is possible to divide the remaining bags into two groups of $n$ bags such that the number of marbles in each group is the same. Prove that all the bags contain the same number of marbles.
Bangladesh Mathematical Olympiad 2020 Final, #7
Tiham is trying to find [b]6[/b] digit positive integers$ PQRSTU$ (where $PQRSTU $are not necessarily distinct). But he only wants the numbers where the sum of the [b]3[/b] digit number$ PQR$, and the [b]3[/b] digit number $STU$ is divisible by [b]37[/b]. How many such numbers Tiham can find?
2010 Purple Comet Problems, 7
Find the sum of the digits in the decimal representation of the number $5^{2010} \cdot 16^{502}.$
1960 IMO Shortlist, 2
For what values of the variable $x$ does the following inequality hold: \[ \dfrac{4x^2}{(1-\sqrt{2x+1})^2}<2x+9 \ ? \]
1995 Poland - First Round, 4
A line tangent to the incircle of the equilateral triangle ABC intersects the sides AB and BC at points D and E respectively. Prove that
$\frac{AD}{DB}+\frac{AE}{EC} = 1$.
2007 iTest Tournament of Champions, 2
The area of triangle $ABC$ is $2007$. One of its sides has length $18$, and the tangent of the angle opposite that side is $2007/24832$. When the altitude is dropped to the side of length $18$, it cuts that side into two segments. Find the sum of the squares of those two segments.
2024 Argentina National Olympiad Level 2, 3
[b]a)[/b] Find an example of an infinite list of numbers of the form $a + n \cdot d$, with $n \geqslant 0$, where $a$ and $d$ are positive integers, such that no number in the list is equal to the $k$-th power of an integer, for all $k = 2, 3, 4, \dots$
[b]b)[/b] Find an example of an infinite list of numbers of the form $a + n \cdot d$, with $n \geqslant 0$, where $a$ and $d$ are positive integers, such that no number in the list is equal to the square of an integer, but the list contains infinitely many numbers that are equal to the cubes of positive integers.
2011 Stars Of Mathematics, 3
The checkered plane is painted black and white, after a chessboard fashion. A polygon $\Pi$ of area $S$ and perimeter $P$ consists of some of these unit squares (i.e., its sides go along the borders of the squares).
Prove the polygon $\Pi$ contains not more than $\dfrac {S} {2} + \dfrac {P} {8}$, and not less than $\dfrac {S} {2} - \dfrac {P} {8}$ squares of a same color.
(Alexander Magazinov)
1988 Spain Mathematical Olympiad, 2
We choose $n > 3$ points on a circle and number them $1$ to $ n$ in some order. We say that two non-adjacent points $A$ and $B$ are related if, in one of the arcs $AB$, all the points are marked with numbers less than those at $A,B$. Show that the number of pairs of related points is exactly $n-3$.
2017 Olympic Revenge, 4
Let $f:\mathbb{R}_{+}^{*}$$\rightarrow$$\mathbb{R}_{+}^{*}$ such that $f'''(x)>0$ for all $x$ $\in$ $\mathbb{R}_{+}^{*}$. Prove that:
$f(a^{2}+b^{2}+c^{2})+2f(ab+bc+ac)$ $\geq$ $f(a^{2}+2bc)+f(b^{2}+2ca)+f(c^{2}+2ab)$, for all $a,b,c$ $\in$ $\mathbb{R}_{+}^{*}$.