This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 85335

2007 District Olympiad, 3
Tags: limit , floor function , real analysis
Let $a,b\in \mathbb{R}$. Evaluate: \[\lim_{n\to \infty}\left(\sqrt{a^2n^2+bn}-an\right)\] Consider the sequence $(x_n)_{n\ge 1}$, defined by $x_n=\sqrt{n}-\lfloor \sqrt{n}\rfloor$. Denote by $A$ the set of the points $x\in \mathbb{R}$, for which there is a subsequence of $(x_n)_{n\ge 1}$ tending to $x$. a) Prove that $\mathbb{Q}\cap [0,1]\subset A$. b) Find $A$.

2007 France Team Selection Test, 2
Tags: function , induction , functional equation , algebra
Find all functions $f: \mathbb{Z}\rightarrow\mathbb{Z}$ such that for all $x,y \in \mathbb{Z}$: \[f(x-y+f(y))=f(x)+f(y).\]

2020 HK IMO Preliminary Selection Contest, 10
Tags: algebra , absolute value
Let $k$ be an integer. If the equation $(x-1)|x+1|=x+\frac{k}{2020}$ has three distinct real roots, how many different possible values of $k$ are there?

KoMaL A Problems 2024/2025, A. 887
Tags: combinatorial geometry , grid , geometry
A non self-intersecting polygon is given in a Cartesian coordinate system such that its perimeter contains no lattice points, and its vertices have no integer coordinates. A point is called semi-integer if exactly one of its coordinates is an integer. Let $P_1, P_2,\ldots, P_k$ denote the semi-integer points on the perimeter of the polygon. Let ni denote the floor of the non-integer coordinate of $P_i$. Prove that integers $n_1,n_2,\ldots ,n_k$ can be divided into two groups with the same sum. [i]Proposed by Áron Bán-Szabó, Budapest[/i]

2010 Postal Coaching, 4
Tags: ratio , geometry
Let $C_1 , C_2$ be two circles in the plane intersecting at two distinct points. Let $P$ be the midpoint of a variable chord $AB$ of $C_2$ with the property that the circle on $AB$ as diameter meets $C_1$ at a point $T$ such that $P T$ is tangent to $C_1$ . Find the locus of $P$ .

2022 IFYM, Sozopol, 5
Tags: geometry
Let $\Delta ABC$ be an acute scalene triangle with $AC<BC$, an orthocenter $H$ and altitudes $AE$, $BF$. The points $E'$ and $F'$ are symmetrical to $E$ and $F$ with respect to $A$ and $B$ respectively. Point $O$ is the center of the circumscribed circle of $ABC$ and $M$ is the midpoint of $AB$. Let $N$ be the midpoint of $OM$. Prove that the tangent through $H$ to the circumscribed circle of $\Delta E'HF'$ is perpendicular to line $CN$.

2013 Sharygin Geometry Olympiad, 3
Tags: geometry , inequality , area
Each sidelength of a convex quadrilateral $ABCD$ is not less than $1$ and not greater than $2$. The diagonals of this quadrilateral meet at point $O$. Prove that $S_{AOB}+ S_{COD} \le 2(S_{AOD}+ S_{BOC})$.

2015 India PRMO, 15
Tags: number theory , digit , prime number
$15.$ Let $n$ be the largest integer that is the product of exactly $3$ distinct prime numbers, $x,y,$ and $10x+y,$ where $x$ and $y$ are digits. What is the sum of digits of $n ?$

2023 Harvard-MIT Mathematics Tournament, 13
Tags: guts
Suppose $a, b, c,$ and $d$ are pairwise distinct positive perfect squares such that $a^b = c^d.$ Compute the smallest possible value of $a + b + c + d.$

2014 Singapore Senior Math Olympiad, 9
Tags: absolute value
Find the number of real numbers which satisfy the equation $x|x-1|-4|x|+3=0$. $ \textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4 $