Found problems: 85335
2007 Harvard-MIT Mathematics Tournament, 25
Two real numbers $x$ and $y$ are such that $8y^4+4x^2y^2+4xy^2+2x^3+2y^2+2x=x^2+1$.
Find all possible values of $x+2y^2$
STEMS 2023 Math Cat A, 6
Define a positive integer $n$ to be a fake square if either $n = 1$ or $n$ can be written as a product of an even number of not necessarily distinct primes. Prove that for any even integer $k \geqslant 2$, there exist distinct positive integers $a_1$, $a_2, \cdots, a_k$ such that the polynomial $(x+a_1)(x+a_2) \cdots (x+a_k)$ takes ‘fake square’ values for all $x = 1,2,\cdots,2023$.
[i]Proposed by Prof. Aditya Karnataki[/i]
2016 Baltic Way, 3
For which integers $n = 1, \ldots , 6$ does the equation $$a^n + b^n = c^n + n$$ have a solution in integers?
2018 Saint Petersburg Mathematical Olympiad, 7
In $10\times 10$ square we choose $n$ cells. In every chosen cell we draw one arrow from the angle to opposite angle. It is known, that for any two arrows, or the end of one of them coincides with the beginning of the other, or
the distance between their ends is at least 2. What is the maximum possible value of $n$?
Russian TST 2019, P1
Let $n>1$ be a positive integer. Show that the number of residues modulo $n^2$ of the elements of the set $\{ x^n + y^n : x,y \in \mathbb{N} \}$ is at most $\frac{n(n+1)}{2}$.
[I]Proposed by N. Safaei (Iran)[/i]
1969 IMO Longlists, 57
Given triangle $ ABC $ with points $ M $ and $ N $ are in the sides $ AB $ and $ AC $ respectively.
If $ \dfrac{BM}{MA} +\dfrac{CN}{NA} = 1 $ , then prove that the centroid of $ ABC $ lies on $ MN $ .
2019 Thailand Mathematical Olympiad, 6
Determine all function $f:\mathbb{R}\to\mathbb{R}$ such that $xf(y)+yf(x)\leqslant xy$ for all $x,y\in\mathbb{R}$.
Croatia MO (HMO) - geometry, 2012.7
Let the points $M$ and $N$ be the intersections of the inscribed circle of the right-angled triangle $ABC$, with sides $AB$ and $CA$ respectively , and points $P$ and $Q$ respectively be the intersections of the ex-scribed circles opposite to vertices $B$ and $C$ with direction $BC$. Prove that the quadrilateral $MNPQ$ is a cyclic if and only if the triangle $ABC$ is right-angled with a right angle at the vertex $A$.
2005 QEDMO 1st, 14 (G4)
In the following, the abbreviation $g \cap h$ will mean the point of intersection of two lines $g$ and $h$.
Let $ABCDE$ be a convex pentagon. Let $A^{\prime}=BD\cap CE$, $B^{\prime}=CE\cap DA$, $C^{\prime}=DA\cap EB$, $D^{\prime}=EB\cap AC$ and $E^{\prime}=AC\cap BD$. Furthermore, let $A^{\prime\prime}=AA^{\prime}\cap EB$, $B^{\prime\prime}=BB^{\prime}\cap AC$, $C^{\prime\prime}=CC^{\prime}\cap BD$, $D^{\prime\prime}=DD^{\prime}\cap CE$ and $E^{\prime\prime}=EE^{\prime}\cap DA$.
Prove that:
\[ \frac{EA^{\prime\prime}}{A^{\prime\prime}B}\cdot\frac{AB^{\prime\prime}}{B^{\prime\prime}C}\cdot\frac{BC^{\prime\prime}}{C^{\prime\prime}D}\cdot\frac{CD^{\prime\prime}}{D^{\prime\prime}E}\cdot\frac{DE^{\prime\prime}}{E^{\prime\prime}A}=1. \]
Darij
2012 CHMMC Fall, 3
A particular graph has $6$ vertices, $12$ edges, and has the property that it contains no Eulerian path; a Eulerian path is a route from vertex to vertex along edges that traces each edge exactly once. Determine all the possible degrees of its vertices in no particular order. There are two solutions, and you need to get both to get credit for this problem.
2013 NIMO Problems, 7
For each integer $k\ge2$, the decimal expansions of the numbers $1024,1024^2,\dots,1024^k$ are concatenated, in that order, to obtain a number $X_k$. (For example, $X_2 = 10241048576$.) If \[ \frac{X_n}{1024^n} \] is an odd integer, find the smallest possible value of $n$, where $n\ge2$ is an integer.
[i]Proposed by Evan Chen[/i]
1986 Iran MO (2nd round), 4
Find all positive integers $n$ for which the number $1!+2!+3!+\cdots+n!$ is a perfect power of an integer.
2013 Dutch IMO TST, 3
Fix a triangle $ABC$. Let $\Gamma_1$ the circle through $B$, tangent to edge in $A$. Let $\Gamma_2$ the circle through C tangent to edge $AB$ in $A$. The second intersection of $\Gamma_1$ and $\Gamma_2$ is denoted by $D$. The line $AD$ has second intersection $E$ with the circumcircle of $\vartriangle ABC$. Show that $D$ is the midpoint of the segment $AE$.
MIPT student olimpiad spring 2023, 4
Is it true that if two linear subspaces $V$ and $W$ of a Hilbert space are closed, then their sum $V+W$ is also closed?
2017 Mathematical Talent Reward Programme, SAQ: P 2
Let $a$, $b$, $c$ be positive reals such that $a+b+c=3$. Show that $$\sqrt{\frac{a}{b+c}} + \sqrt{\frac{b}{c+a}} + \sqrt{\frac{c}{a+b}} \leq \frac{6}{\sqrt(a+b)(b+c)(c+a)}$$
2009 Mathcenter Contest, 2
Find the locus of points $P$ in the plane of a square $ABCD$ such that $$\max\{ PA,\ PC\}=\frac12(PB+PD).$$
[i](Anonymous314)[/i]
2003 All-Russian Olympiad, 4
Ana and Bora are each given a sufficiently long paper strip, one with letter $A$ written , and the other with letter $B$. Every minute, one of them (not necessarily one after another) writes either on the left or on the right to the word on his/her strip the word written on the other strip. Prove that the day after, one will be able to cut word on Ana's strip into two words and exchange their places, obtaining a palindromic word.
KoMaL A Problems 2024/2025, A. 902
In triangle $ABC$, interior point $D$ is chosen such that triangle $BCD$ is equilateral. Let $E$ be the isogonal conjugate of point $D$ with respect to triangle $ABC$. Define point $P$ on the ray $AB$ such that $AP=BE$. Similarly, define point $Q$ on the ray $AC$ such that $AQ=CE$. Prove that line $AD$ bisects segment $PQ$.
[i]Proposed by Áron Bán-Szabó, Budapest[/i]
1973 All Soviet Union Mathematical Olympiad, 178
The real numbers $a,b,c$ satisfy the condition:
for all $x$, such that for $ -1 \le x \le 1$, the inequality $$| ax^2 + bx + c | \le 1$$ is held. Prove that for the same $x$ , $$| cx^2 + bx + a | \le 2$$
2023 SAFEST Olympiad, 2
There are $n!$ empty baskets in a row, labelled $1, 2, . . . , n!$. Caesar
first puts a stone in every basket. Caesar then puts 2 stones in every second basket.
Caesar continues similarly until he has put $n$ stones into every nth basket. In
other words, for each $i = 1, 2, . . . , n,$ Caesar puts $i$ stones into the baskets labelled
$i, 2i, 3i, . . . , n!.$
Let $x_i$ be the number of stones in basket $i$ after all these steps. Show that
$n! \cdot n^2 \leq \sum_{i=1}^{n!} x_i^2 \leq n! \cdot n^2 \cdot \sum_{i=1}^{n} \frac{1}{i} $
1993 Baltic Way, 14
A square is divided into $16$ equal squares, obtaining the set of $25$ different vertices. What is the least number of vertices one must remove from this set, so that no $4$ points of the remaining set are the vertices of any square with sides parallel to the sides of the initial square?
2014 Paraguay Mathematical Olympiad, 1
Consider a square of side length $12$ centimeters. Irina draws another square that has $8$ centimeters more of perimeter than the original square. What is the area of the square drawn by Irina?
2022 Costa Rica - Final Round, 1
Let $\Gamma$ be a circle with center $O$. Consider the points $A$, $B$, $C$, $D$, $E$ and $F$ in $\Gamma$, with $D$ and $E$ in the (minor) arc $BC$ and $C$ in the (minor) arc $EF$, such that $DEFO$ is a rhombus and $\vartriangle ABC$ It is equilateral. Show that $\overleftrightarrow{BD}$ and $\overleftrightarrow{CE}$ are perpendicular.
2001 Vietnam Team Selection Test, 3
Let a sequence $\{a_n\}$, $n \in \mathbb{N}^{*}$ given, satisfying the condition
\[0 < a_{n+1} - a_n \leq 2001\]
for all $n \in \mathbb{N}^{*}$
Show that there are infinitely many pairs of positive integers $(p, q)$ such that $p < q$ and $a_p$ is divisor of $a_q$.
2021 Thailand TST, 3
Let $ABC$ be a triangle with $AB < AC$, incenter $I$, and $A$ excenter $I_{A}$. The incircle meets $BC$ at $D$. Define $E = AD\cap BI_{A}$, $F = AD\cap CI_{A}$. Show that the circumcircle of $\triangle AID$ and $\triangle I_{A}EF$ are tangent to each other