Olympiad problems

2008 F = Ma, 16

Tags:

A [i]massless[/i] spring with spring constant $k$ is vertically mounted so that bottom end is firmly attached to the ground, and the top end free. A ball with mass $m$ falls vertically down on the top end of the spring, becoming attached, so that the ball oscillates vertically on the spring. What equation describes the acceleration a of the ball when it is at a height $y$ above the [i]original[/i] position of the top end of the spring? Let down be negative, and neglect air resistance; $g$ is the magnitude of the acceleration of free fall. (a) $a=mv^2/y+g$ (b) $a=mv^2/k-g$ (c) $a=(k/m)y-g$ (d) $a=-(k/m)y+g$ (e) $a=-(k/m)y-g$

PEN H Problems, 26

Solve in integers the following equation \[n^{2002}=m(m+n)(m+2n)\cdots(m+2001n).\]

1993 Korea - Final Round, 2

Tags: incenter , geometry

Let be given a triangle $ABC$ with $BC = a, CA = b, AB = c$. Find point $P$ in the plane for which $aAP^{2}+bBP^{2}+cCP^{2}$ is minimum, and compute this minimum.

2013 National Olympiad First Round, 29

Tags: geometry , circumcircle , geometric transformation , reflection

Let $O$ be the circumcenter of triangle $ABC$ with $|AB|=5$, $|BC|=6$, $|AC|=7$. Let $A_1$, $B_1$, $C_1$ be the reflections of $O$ over the lines $BC$, $AC$, $AB$, respectively. What is the distance between $A$ and the circumcenter of triangle $A_1B_1C_1$? $ \textbf{(A)}\ 6 \qquad\textbf{(B)}\ \sqrt {29} \qquad\textbf{(C)}\ \dfrac {19}{2\sqrt 6} \qquad\textbf{(D)}\ \dfrac {35}{4\sqrt 6} \qquad\textbf{(E)}\ \sqrt {\dfrac {35}3} $

2015 IMO Shortlist, G1

Tags: triangle , geometry

Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ = AH$.

Durer Math Competition CD 1st Round - geometry, 2011.C4

Tags: geometry , area

Given a grid rectangle of size $2010 \times 1340$. A grid point is called [i]fair [/i] if the $2$ axis-parallel lines passing through it from the upper left and lower right corners of the large rectangle cut out a rectangle of equal area (such a point is shown in the figure). How many fair grid points lie inside the rectangle? [img]https://cdn.artofproblemsolving.com/attachments/1/b/21d4fb47c94b774994ac1c3aae7690bb98c7ae.png[/img]

2000 Moldova National Olympiad, Problem 3

Tags: number theory

Suppose that $m,n\ge2$ are integers such that $m+n-1$ divides $m^2+n^2-1$. Prove that the number $m+n-1$ is not prime.

2023 BMT, 9

Tags: algebra

The boxes in the expression below are filled with the numbers $3$, $4$, $5$, $6$, $7$, and $8$, so that each number is used exactly once. What is the least possible value of the expression? $$\square \times \square +\square \times \square -\square \times \square$$

2014 India IMO Training Camp, 3

Tags: circumcircle , isosceles triangle , angle chasing , inversion , geometry

Let $ABC$ be a triangle with $\angle B > \angle C$. Let $P$ and $Q$ be two different points on line $AC$ such that $\angle PBA = \angle QBA = \angle ACB $ and $A$ is located between $P$ and $C$. Suppose that there exists an interior point $D$ of segment $BQ$ for which $PD=PB$. Let the ray $AD$ intersect the circle $ABC$ at $R \neq A$. Prove that $QB = QR$.

2018 Romania Team Selection Tests, 2

Tags: geometry , mixtilinear incircle , incenter , circumcircle

Let $ABC$ be a triangle, let $I$ be its incenter, let $\Omega$ be its circumcircle, and let $\omega$ be the $A$- mixtilinear incircle. Let $D,E$ and $T$ be the intersections of $\omega$ and $AB,AC$ and $\Omega$, respectively, let the line $IT$ cross $\omega$ again at $P$, and let lines $PD$ and $PE$ cross the line $BC$ at $M$ and $N$ respectively. Prove that points $D,E,M,N$ are concyclic. What is the center of this circle?

2020-21 IOQM India, 10

Tags: statistics , median , mean , algebra

Five students take a test on which any integer score from $0$ to $100$ inclusive is possible. What is the largest possible difference between the median and the mean of the scores? [i](The median of a set of scores is the middlemost score when the data is arranged in increasing order. It is exactly the middle score when there are an odd number of scores and it is the average of the two middle scores when there are an even number of scores.)[/i]

2005 Federal Math Competition of S&M, Problem 1

Tags: power mean , divisibility , algebra , number theory

Let $a$ and $b$ be positive integers and $K=\sqrt{\frac{a^2+b^2}2}$, $A=\frac{a+b}2$. If $\frac KA$ is a positive integer, prove that $a=b$.

2018 JBMO Shortlist, NT4

Tags: number theory

Prove that there exist infinitely many positive integers $n$ such that $\frac{4^n+2^n+1}{n^2+n+1}$ is a positive integer.

2024 AIME, 15

Tags: 3d geometry , prism , inequalities , geometry

Let $\mathcal{B}$ be the set of rectangular boxes that have volume $23$ and surface area $54$. Suppose $r$ is the least possible radius of a sphere that can fit any element of $\mathcal{B}$ inside it. Then $r^{2}$ can be expressed as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

2025 Sharygin Geometry Olympiad, 18

Tags: geometry

Let $ABCD$ be a quadrilateral such that the excircles $\omega_{1}$ and $\omega_{2}$ of triangles $ABC$ and $BCD$ touching their sides $AB$ and $BD$ respectively touch the extension of $BC$ at the same point $P$. The segment $AD$ meets $\omega_{2}$ at point $Q$, and the line $AD$ meets $\omega_{1}$ at $R$ and $S$. Prove that one of angles $RPQ$ and $SPQ$ is right Proposed by: I.Kukharchuk

2020 USA EGMO Team Selection Test, 6

Tags: algebra , polynomial , integer polynomial , periodic sequence , modular arithmetic

Find the largest integer $N \in \{1, 2, \ldots , 2019 \}$ such that there exists a polynomial $P(x)$ with integer coefficients satisfying the following property: for each positive integer $k$, $P^k(0)$ is divisible by $2020$ if and only if $k$ is divisible by $N$. Here $P^k$ means $P$ applied $k$ times, so $P^1(0)=P(0), P^2(0)=P(P(0)),$ etc.

2020 MBMT, 3

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Square $ABCD$ has a side length of 1. Point $E$ lies on the interior of $ABCD$, and is on the line $\overleftrightarrow{AC}$ such that the length of $\overline{AE}$ is 1. Find the shortest distance from point $E$ to a side of square $ABCD$. [i]Proposed by Chris Tong[/i]

2022 Moldova Team Selection Test, 7

Tags: number theory

Let $f:\mathbb{N} \rightarrow \mathbb{N},$ $f(n)=n^2-69n+2250$ be a function. Find the prime number $p$, for which the sum of the digits of the number $f(p^2+32)$ is as small as possible.

1990 Romania Team Selection Test, 9

Tags: distance , combinatorial geometry , geometry

The distance between any two of six given points in the plane is at least $1$. Prove that the distance between some two points is at least $\sqrt{\frac{5+\sqrt5}{2}}$

2016 CMIMC, 9

Tags: computer science

Ryan has three distinct eggs, one of which is made of rubber and thus cannot break; unfortunately, he doesn't know which egg is the rubber one. Further, in some 100-story building there exists a floor such that all normal eggs dropped from below that floor will not break, while those dropped from at or above that floor will break and cannot be dropped again. What is the minimum number of times Ryan must drop an egg to determine the floor satisfying this property?

2008 Portugal MO, 6

Tags: combinatorics

Let $n$ be a natural number larger than $2$. Vanessa has $n$ piles of jade stones, and all the piles have a different number of stones. Vanessa can distribute the stones from any pile by the other piles and stay with $n-1$ piles with the same number of stones. She also can distribute the stones from any two piles by the other piles and stay with $n-2$ piles with the same number of stones. Find the smallest possible number of jade's stones that the pile with the largest number of stones can have.

2009 Philippine MO, 2

Tags: diophantine , number theory

[b](a)[/b] Find all pairs $(n,x)$ of positive integers that satisfy the equation $2^n + 1 = x^2$. [b](b)[/b] Find all pairs $(n,x)$ of positive integers that satisfy the equation $2^n = x^2 + 1$.

2016 Costa Rica - Final Round, N3

Tags: diophantine equation , diophantine , number theory

Find all natural values of $n$ and $m$, such that $(n -1)2^{n - 1} + 5 = m^2 + 4m$.

2017 BAMO, 5

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Call a number $T$ [i]persistent[/i] if the following holds: Whenever $a,b,c,d$ are real numbers different from $0$ and $1$ such that $$a+b+c+d = T$$ and $$\frac{1}{a}+\frac{1}{b} +\frac{1}{c}+\frac{1}{d} = T,$$ we also have $$\frac{1}{1 - a}+\frac{1}{1-b}+\frac{1}{1-c}+\frac{1}{1-d}= T.$$ (a) If $T$ is persistent, prove that $T$ must be equal to $2$. (b) Prove that $2$ is persistent. Note: alternatively, we can just ask “Show that there exists a unique persistent number, and determine its value”.

2006 Hong Kong TST., 4

Tags: inequalities

Let x,y,z be positive real numbers such that $x+y+z=1$. For positive integer n, define $S_n = x^n+y^n+z^n$ Furthermore, let $P=S_2 S_{2005}$ and $Q=S_3 S_{2004}$. (a) Find the smallest possible value of Q. (b) If $x,y,z$ are pairwise distinct, determine whether P or Q is larger.