Olympiad problems

2022 IFYM, Sozopol, 2

Tags: algebra

Does there exist a solution in integers for the equation $a^2+b^2+c^2+d^2+e^2=abcde-78$ where $a,b,c,d,e>2022$?

1999 Harvard-MIT Mathematics Tournament, 4

Tags:

You are given 16 pieces of paper numbered $16, 15, \ldots , 2, 1$ in that order. You want to put them in the order $1, 2, \ldots , 15, 16$ switching only two adjacent pieces of paper at a time. What is the minimum number of switches necessary?

1976 Euclid, 4

Tags: remainder , polynomial division , algebra , polynomial

Source: 1976 Euclid Part B Problem 4 ----- The remainder when $f(x)=x^5-2x^4+ax^3-x^2+bx-2$ is divided by $x+1$ is $-7$. When $f(x)$ is divided by $x-2$ the remainder is $32$. Determine the remainder when $f(x)$ is divided by $x-1$.

2019 Azerbaijan Senior NMO, 3

Tags: diophantine equation , number theory

Find all $x;y\in\mathbb{Z}$ satisfying the following condition: $$x^3=y^4+9x^2$$

2010 China Girls Math Olympiad, 8

Tags: calculus , integration , inequalities , modular arithmetic , number theory

Determine the least odd number $a > 5$ satisfying the following conditions: There are positive integers $m_1,m_2, n_1, n_2$ such that $a=m_1^2+n_1^2$, $a^2=m_2^2+n_2^2$, and $m_1-n_1=m_2-n_2.$

2005 iTest, 12

Tags: geometry , 3d geometry , sphere , cube

A sphere sits inside a cubic box, tangent on all $6$ sides of the box. If a side of the box is $5$, and the volume of the sphere is $x\pi$ , find $x$.

2008 Korea Junior Math Olympiad, 2

Tags: inequalities

Let $x,y\in\mathbb{R}$ such that $x>2, y>3$. Find the minimum value of $\frac{(x+y)^2}{\sqrt{x^2-4}+\sqrt{y^2-9}}$

Kvant 2020, M2604

Tags: geometry

Let $ABC$ be a triangle with a right angle at $C$. Let $I$ be the incentre of triangle $ABC$, and let $D$ be the foot of the altitude from $C$ to $AB$. The incircle $\omega$ of triangle $ABC$ is tangent to sides $BC$, $CA$, and $AB$ at $A_1$, $B_1$, and $C_1$, respectively. Let $E$ and $F$ be the reflections of $C$ in lines $C_1A_1$ and $C_1B_1$, respectively. Let $K$ and $L$ be the reflections of $D$ in lines $C_1A_1$ and $C_1B_1$, respectively. Prove that the circumcircles of triangles $A_1EI$, $B_1FI$, and $C_1KL$ have a common point.

1991 China Team Selection Test, 2

Tags: combinatorics

For $i = 1,2, \ldots, 1991$, we choose $n_i$ points and write number $i$ on them (each point has only written one number on it). A set of chords are drawn such that: (i) They are pairwise non-intersecting. (ii) The endpoints of each chord have distinct numbers. If for all possible assignments of numbers the operation can always be done, find the necessary and sufficient condition the numbers $n_1, n_2, \ldots, n_{1991}$ must satisfy for this to be possible.

2014 Contests, 1

Tags:

For every $3$-digit natural number $n$ (leading digit of $n$ is nonzero), we consider the number $n_0$ obtained from $n$ eliminating all possible digits that are zero. For example, if $n = 207$, then $n_0 = 27$. Determine the number of three-digit positive integers $n$, for which $n_0$ is a divisor of $n$ different from $n$.

Kvant 2022, M2716

Tags: number theory , factorial

Find all pairs of natural numbers $(k, m)$ such that for any natural $n{}$ the product\[(n+m)(n+2m)\cdots(n+km)\]is divisible by $k!{}$. [i]Proposed by P. Kozhevnikov[/i]

1960 IMO, 7

Tags: geometry , trapezoid , locus , locus problems

An isosceles trapezoid with bases $a$ and $c$ and altitude $h$ is given. a) On the axis of symmetry of this trapezoid, find all points $P$ such that both legs of the trapezoid subtend right angles at $P$; b) Calculate the distance of $p$ from either base; c) Determine under what conditions such points $P$ actually exist. Discuss various cases that might arise.

2022 Math Prize for Girls Problems, 5

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Given a real number $a$, the [i]floor[/i] of $a$, written $\lfloor a \rfloor$, is the greatest integer less than or equal to $a$. For how many real numbers $x$ such that $1 \le x \le 20$ is \[ x^2 + \lfloor 2x \rfloor = \lfloor x^2 \rfloor + 2x \, ? \]

2010 USAJMO, 4

Tags: parabola , analytic geometry , nt , conic

A triangle is called a parabolic triangle if its vertices lie on a parabola $y = x^2$. Prove that for every nonnegative integer $n$, there is an odd number $m$ and a parabolic triangle with vertices at three distinct points with integer coordinates with area $(2^nm)^2$.

1967 IMO Longlists, 25

Tags: geometry , 3d geometry , sphere , angle

Three disks of diameter $d$ are touching a sphere in their centers. Besides, every disk touches the other two disks. How to choose the radius $R$ of the sphere in order that axis of the whole figure has an angle of $60^\circ$ with the line connecting the center of the sphere with the point of the disks which is at the largest distance from the axis ? (The axis of the figure is the line having the property that rotation of the figure of $120^\circ$ around that line brings the figure in the initial position. Disks are all on one side of the plane, passing through the center of the sphere and orthogonal to the axis).

2010 Indonesia TST, 1

Tags: trapezoid , circumcircle , geometry

Let $ ABCD$ be a trapezoid such that $ AB \parallel CD$ and assume that there are points $ E$ on the line outside the segment $ BC$ and $ F$ on the segment $ AD$ such that $ \angle DAE \equal{} \angle CBF$. Let $ I,J,K$ respectively be the intersection of line $ EF$ and line $ CD$, the intersection of line $ EF$ and line $ AB$, and the midpoint of segment $ EF$. Prove that $ K$ is on the circumcircle of triangle $ CDJ$ if and only if $ I$ is on the circumcircle of triangle $ ABK$. [i]Utari Wijayanti, Bandung[/i]

2012 ELMO Shortlist, 6

Tags: floor function , algebra , polynomial , vieta , quadratic , inequalities , number theory

Prove that if $a$ and $b$ are positive integers and $ab>1$, then \[\left\lfloor\frac{(a-b)^2-1}{ab}\right\rfloor=\left\lfloor\frac{(a-b)^2-1}{ab-1}\right\rfloor.\]Here $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$. [i]Calvin Deng.[/i]

1982 AMC 12/AHSME, 29

Tags: inequalities , calculus , symmetry , function , algebra , domain

Let $ x$,$ y$, and $ z$ be three positive real numbers whose sum is $ 1$. If no one of these numbers is more than twice any other, then the minimum possible value of the product $ xyz$ is $ \textbf{(A)}\ \frac{1}{32}\qquad \textbf{(B)}\ \frac{1}{36}\qquad \textbf{(C)}\ \frac{4}{125}\qquad \textbf{(D)}\ \frac{1}{127}\qquad \textbf{(E)}\ \text{none of these}$

2012 BAMO, 3

Tags: combinatorics

Two infinite rows of evenly-spaced dots are aligned as in the figure below. Arrows point from every dot in the top row to some dot in the lower row in such a way that: [list][*]No two arrows point at the same dot. [*]Now arrow can extend right or left by more than 1006 positions.[/list] [img]https://cdn.artofproblemsolving.com/attachments/7/6/47abf37771176fce21bce057edf0429d0181fb.png[/img] Show that at most 2012 dots in the lower row could have no arrow pointing to them.

1973 USAMO, 5

Tags: 3d geometry , number theory , prime number , arithmetic sequence , geometry

Show that the cube roots of three distinct prime numbers cannot be three terms (not necessarily consecutive) of an arithmetic progression.

2004 Romania Team Selection Test, 10

Tags: induction , algebra

Prove that for all positive integers $n,m$, with $m$ odd, the following number is an integer \[ \frac 1{3^mn}\sum^m_{k=0} { 3m \choose 3k } (3n-1)^k. \]

1954 AMC 12/AHSME, 26

Tags: geometry , similar triangles

The straight line $ \overline{AB}$ is divided at $ C$ so that $ AC\equal{}3CB$. Circles are described on $ \overline{AC}$ and $ \overline{CB}$ as diameters and a common tangent meets $ AB$ produced at $ D$. Then $ BD$ equals: $ \textbf{(A)}\ \text{diameter of the smaller circle} \\ \textbf{(B)}\ \text{radius of the smaller circle} \\ \textbf{(C)}\ \text{radius of the larger circle} \\ \textbf{(D)}\ CB\sqrt{3}\\ \textbf{(E)}\ \text{the difference of the two radii}$

1971 Putnam, B4

Tags:

A "spherical ellipse" with foci $A,B$ on a given sphere is defined as the set of all points $P$ on the sphere such that $\overset{\Large\frown}{PA}+\overset{\Large\frown}{PB}=$ constant. Here $\overset{\Large\frown}{PA}$ denotes the shortest distance on the sphere between $P$ and $A$. Determine the entire class of real spherical ellipses which are circles.

2022 Sharygin Geometry Olympiad, 10.6

Tags: concurrency , geometry

Let $O, I$ be the circumcenter and the incenter of triangle $ABC$, $P$ be an arbitrary point on segment $OI$, $P_A$, $P_B$, and $P_C$ be the second common points of lines $PA$, $PB$, and $PC$ with the circumcircle of triangle $ABC$. Prove that the bisectors of angles $BP_AC$, $CP_BA$, and $AP_CB$ concur at a point lying on $OI$.

2006 Stanford Mathematics Tournament, 16

Tags: number theory , least common multiple

Points $ A_1$, $ A_2$, $ ...$ are placed on a circle with center $ O$ such that $ \angle OA_n A_{n\plus{}1}\equal{}35^\circ$ and $ A_n\neq A_{n\plus{}2}$ for all positive integers $ n$. What is the smallest $ n>1$ for which $ A_n\equal{}A_1$?