Found problems: 85335
2013 Chile TST Ibero, 2
Let $a \in \mathbb{N}$ such that $a + n^2$ can be expressed as the sum of two squares for all $n \in \mathbb{N}$. Prove that $a$ is the square of a natural number.
1970 IMO Longlists, 30
Let $u_1, u_2, \ldots, u_n, v_1, v_2, \ldots, v_n$ be real numbers. Prove that
\[1+ \sum_{i=1}^n (u_i+v_i)^2 \leq \frac 43 \Biggr( 1+ \sum_{i=1}^n u_i^2 \Biggl) \Biggr( 1+ \sum_{i=1}^n v_i^2 \Biggl) .\]
2014 Irish Math Olympiad, 2
Prove that for $N>1$ that $(N^{2})^{2014} - (N^{11})^{106}$ is divisible by $N^6 + N^3 +1$
Is this just a proof by induction or is there a more elegant method? I don't think calculating $N = 2$ was expected.
2015 Saudi Arabia Pre-TST, 3.1
Let $ABC$ be a triangle, $I$ its incenter, and $D$ a point on the arc $BC$ of the circumcircle of $ABC$ not containing $A$. The bisector of the angle $\angle ADB$ intesects the segment $AB$ at $E$. The bisector of the angle $\angle CDA$ intesects the segment $AC$ at $F$. Prove that the points $E, F,I$ are collinear.
(Malik Talbi)
2019 Online Math Open Problems, 3
Let $k$ be a positive real number. Suppose that the set of real numbers $x$ such that $x^2+k|x| \leq 2019$ is an interval of length $6$. Compute $k$.
[i]Proposed by Luke Robitaille[/i]
2012 Indonesia TST, 3
Given a convex quadrilateral $ABCD$, let $P$ and $Q$ be points on $BC$ and $CD$ respectively such that $\angle BAP = \angle DAQ$. Prove that the triangles $ABP$ and $ADQ$ have the same area if the line connecting their orthocenters is perpendicular to $AC$.
1979 Polish MO Finals, 1
Let be given a set $\{r_1,r_2,...,r_k\}$ of natural numbers that give distinct remainders when divided by a natural number $m$. Prove that if $k > m/2$, then for every integer $n$ there exist indices $i$ and $j$ (not necessarily distinct) such that $r_i +r_j -n$ is divisible by $m$.
2018 AMC 12/AHSME, 10
A list of $2018$ positive integers has a unique mode, which occurs exactly $10$ times. What is the least number of distinct values that can occur in the list?
$\textbf{(A)}\ 202\qquad\textbf{(B)}\ 223\qquad\textbf{(C)}\ 224\qquad\textbf{(D)}\ 225\qquad\textbf{(E)}\ 234$
2015 Estonia Team Selection Test, 12
Call an $n$-tuple $(a_1, . . . , a_n)$ [i]occasionally periodic [/i] if there exist a nonnegative integer $i$ and a positive integer $p$ satisfying $i + 2p \le n$ and $a_{i+j} = a_{i+p+j}$ for every $j = 1, 2, . . . , p$. Let $k$ be a positive integer. Find the least positive integer $n$ for which there exists an $n$-tuple $(a_1, . . . , a_n)$ with elements from set $\{1, 2, . . . , k\}$, which is not occasionally periodic but whose arbitrary extension $(a_1, . . . , a_n, a_{n+1})$ is occasionally periodic for any $a_{n+1} \in \{1, 2, . . . , k\}$.
2019 New Zealand MO, 1
A positive integer is called sparkly if it has exactly 9 digits, and for any n between 1 and 9 (inclusive), the nth digit is a positive multiple of n. How many positive integers are sparkly?
2008 Croatia Team Selection Test, 4
Let $ S$ be the set of all odd positive integers less than $ 30m$ which are not multiples of $ 5$, where $ m$ is a given positive integer. Find the smallest positive integer $ k$ such that each $ k$-element subset of $ S$ contains two distinct numbers, one of which divides the other.
2021-IMOC, C4
There is a city with many houses, where the houses are connected by some two-way roads. It is known that for any two houses $A,B$, there is exactly one house $C$ such that both $A,B$ are connected to $C$. Show that for any two houses not connected directly by a road, they have the same number of roads adjacent to them.
[i]ST[/i]
2019 Bangladesh Mathematical Olympiad, 9
Let $ABCD$ is a convex quadrilateral.The internal angle bisectors of $\angle {BAC}$ and $\angle {BDC}$ meets at $P$.$\angle {APB}=\angle {CPD}$.Prove that $AB+BD=AC+CD$.
LMT Speed Rounds, 2010.3
Start with a positive integer. Double it, subtract $4,$ halve it, then subtract the original integer to get $x.$ What is $x?$
2016 EGMO TST Turkey, 1
Prove that
\[ x^4y+y^4z+z^4x+xyz(x^3+y^3+z^3) \geq (x+y+z)(3xyz-1) \]
for all positive real numbers $x, y, z$.
Kvant 2019, M2586
A polygon is given in which any two adjacent sides are perpendicular. We call its two vertices non-friendly if the bisectors of the polygon emerging from these vertices are perpendicular. Prove that for any vertex the number of vertices that are not friends with it is even.
2001 JBMO ShortLists, 10
A triangle $ABC$ is inscribed in the circle $\mathcal{C}(O,R)$. Let $\alpha <1$ be the ratio of the radii of the circles tangent to $\mathcal{C}$, and both of the rays $(AB$ and $(AC$. The numbers $\beta <1$ and $\gamma <1$ are defined analogously. Prove that $\alpha + \beta + \gamma =1$.
2006 Moldova National Olympiad, 11.7
Let $n\in\mathbb{N}^*$. $2n+3$ points on the plane are given so that no 3 lie on a line and no 4 lie on a circle. Is it possible to find 3 points so that the interior of the circle passing through them would contain exactly $n$ of the remaining points.
2015 Paraguay Mathematical Olympiad, 4
The sidelengths of a triangle are natural numbers multiples of $7$, smaller than $40$. How many triangles satisfy these conditions?
2007 Tournament Of Towns, 3
Give a construction by straight-edge and compass of a point $C$ on a line $\ell$ parallel to a segment $AB$, such that the product $AC \cdot BC$ is minimum.
1966 IMO, 6
Let $ ABC$ be a triangle, and let $ P$, $ Q$, $ R$ be three points in the interiors of the sides $ BC$, $ CA$, $ AB$ of this triangle. Prove that the area of at least one of the three triangles $ AQR$, $ BRP$, $ CPQ$ is less than or equal to one quarter of the area of triangle $ ABC$.
[i]Alternative formulation:[/i] Let $ ABC$ be a triangle, and let $ P$, $ Q$, $ R$ be three points on the segments $ BC$, $ CA$, $ AB$, respectively. Prove that
$ \min\left\{\left|AQR\right|,\left|BRP\right|,\left|CPQ\right|\right\}\leq\frac14\cdot\left|ABC\right|$,
where the abbreviation $ \left|P_1P_2P_3\right|$ denotes the (non-directed) area of an arbitrary triangle $ P_1P_2P_3$.
2011 ISI B.Stat Entrance Exam, 2
Consider three positive real numbers $a,b$ and $c$. Show that there cannot exist two distinct positive integers $m$ and $n$ such that both $a^m+b^m=c^m$ and $a^n+b^n=c^n$ hold.
2024 Thailand TST, 3
Determine all functions $f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ such that, for all positive integers $a$ and $b$,
\[
f^{bf(a)}(a+1)=(a+1)f(b).
\]
1969 Spain Mathematical Olympiad, 4
A circle of radius $R$ is divided into $8$ equal parts. The points of division are denoted successively by $A, B, C, D, E, F , G$ and $H$. Find the area of the square formed by drawing the chords $AF$ , $BE$, $CH$ and $DG$.
2022 European Mathematical Cup, 3
Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that
$$ f(x^3) + f(y)^3 + f(z)^3 = 3xyz $$
for all real numbers $x$, $y$ and $z$ with $x+y+z=0$.