Olympiad problems

1999 Tournament Of Towns, 4

Tags: combinatorics

Every $24$ hours , the minute hand of an ordinary clock completes $24$ revolutions while the hour hand completes $2$. Every $24$ hours , the minute hand of an Italian clock completes $24$ revolutions while the hour hand completes only $1$ . The minute hand of each clock is longer than the hour hand, and "zero hour" is located at the top of the clock's face. How many positions of the two hands can occur on an Italian clock within a $24$-hour period that are possible on an ordinary one? (Folklore)

1992 AMC 8, 8

Tags:

A store owner bought $1500$ pencils at $\$0.10$ each. If he sells them for $\$0.25$ each, how many of them must he sell to make a profit of exactly $\$100.00?$ $\text{(A)}\ 400 \qquad \text{(B)}\ 667 \qquad \text{(C)}\ 1000 \qquad \text{(D)}\ 1500 \qquad \text{(E)}\ 1900$

I Soros Olympiad 1994-95 (Rus + Ukr), 11.10

Tags: projection , geometry , 3d geometry , tetrahedron

Given a tetrahedron $A_1A_2A_3A_4$ (not necessarily regulart). We shall call a point $N$ in space [i]Serve point[/i], if it's six projection points on the six edges of the tetrahedron lie on one plane. This plane we denote it by $a (N)$ and call the [i]Serve plane[/i] of the point $N$. By $B_{ij}$ denote, respectively, the midpoint of the edges $A_1A_j$, $1\le i <j \le 4$. For each point $M$, denote by $M_{ij}$ the points symmetric to $M$ with respect to $B_{ij},$ $1\le i <j \le 4$. Prove that if all points $M_{ij}$ are Serve points, then the point $M$ belongs to all Serve planes $a (M_{ij})$, $1\le i <j \le 4$.

2009 Finnish National High School Mathematics Competition, 3

Tags: geometry

The circles $\mathcal{Y}_0$ and $\mathcal{Y}_1$ lies outside each other. Let $O_0$ be the center of $\mathcal{Y}_0$ and $O_1$ be the center of $\mathcal{Y}_1$. From $O_0$, draw the rays which are tangents to $\mathcal{Y}_1$ and similarty from $O_1$, draw the rays which are tangents to $\mathcal{Y}_0$. Let the intersection points of rays and circle $\mathcal{Y}_i$ be $A_i$ and $B_i$. Show that the line segments $A_0B_0$ and $A_1B_1$ have equal lengths.

2013 Bangladesh Mathematical Olympiad, 5

Tags: number theory

Higher Secondary P5 Let $x>1$ be an integer such that for any two positive integers $a$ and $b$, if $x$ divides $ab$ then $x$ either divides $a$ or divides $b$. Find with proof the number of positive integers that divide $x$.

2001 IMO Shortlist, 4

Tags: combinatorics , combinatorial number theory , partition , coloring , greedy algorithm

A set of three nonnegative integers $\{x,y,z\}$ with $x < y < z$ is called [i]historic[/i] if $\{z-y,y-x\} = \{1776,2001\}$. Show that the set of all nonnegative integers can be written as the union of pairwise disjoint historic sets.

2010 Indonesia TST, 1

Tags: inequalities

Let $ a$, $ b$, and $ c$ be non-negative real numbers and let $ x$, $ y$, and $ z$ be positive real numbers such that $ a\plus{}b\plus{}c\equal{}x\plus{}y\plus{}z$. Prove that \[ \dfrac{a^3}{x^2}\plus{}\dfrac{b^3}{y^2}\plus{}\dfrac{c^3}{z^2} \ge a\plus{}b\plus{}c.\] [i]Hery Susanto, Malang[/i]

2013 Harvard-MIT Mathematics Tournament, 24

Tags: hmmt , function , geometry

Given a point $p$ and a line segment $l$, let $d(p,l)$ be the distance between them. Let $A$, $B$, and $C$ be points in the plane such that $AB=6$, $BC=8$, $AC=10$. What is the area of the region in the $(x,y)$-plane formed by the ordered pairs $(x,y)$ such that there exists a point $P$ inside triangle $ABC$ with $d(P,AB)+x=d(P,BC)+y=d(P,AC)?$

2014 China Western Mathematical Olympiad, 6

Tags: ceiling function , inequalities

Let $n\ge 2$ is a given integer , $x_1,x_2,\ldots,x_n $ be real numbers such that $(1) x_1+x_2+\ldots+x_n=0 $, $(2) |x_i|\le 1$ $(i=1,2,\cdots,n)$. Find the maximum of Min$\{|x_1-x_2|,|x_2-x_3|,\cdots,|x_{n-1}-x_n|\}$.

2022 Princeton University Math Competition, 12

Tags: analytic geometry

Observe the set $S =\{(x, y) \in Z^2 : |x| \le 5$ and $-10 \le y\le 0\}$. Find the number of points $P$ in $S$ such that there exists a tangent line from $P$ to the parabola $y = x^2 + 1$ that can be written in the form $y = mx + b$, where $m$ and $b$ are integers.

2002 Pan African, 2

Tags: geometry , circumcircle , perpendicular bisector

$\triangle{AOB}$ is a right triangle with $\angle{AOB}=90^{o}$. $C$ and $D$ are moving on $AO$ and $BO$ respectively such that $AC=BD$. Show that there is a fixed point $P$ through which the perpendicular bisector of $CD$ always passes.

2016 Iran MO (3rd Round), 1

Tags: number theory , algebra , polynomial

Let $F$ be a subset of the set of positive integers with at least two elements and $P(x)$ be a polynomial with integer coefficients such that for any two distinct elements of $F$ like $a$ and $b$, the following two conditions hold [list] [*] $a+b \in F$, and [*] $\gcd(P(a),P(b))=1$. [/list] Prove that $P(x)$ is a constant polynomial.

2010 Today's Calculation Of Integral, 643

Tags: calculus , integration , trigonometry , calculus computations

Evaluate \[\int_0^{\pi} \frac{x}{\sqrt{1+\sin ^ 3 x}}\{(3\pi \cos x+4\sin x)\sin ^ 2 x+4\}dx.\] Own

2024 Irish Math Olympiad, P8

Tags: inequalities

Let $a,b,c$ be positive real numbers with $a \leq c$ and $b \leq c$. Prove that $$ (a +10b)(b +22c)(c +7a) \geq 2024 abc.$$

IV Soros Olympiad 1997 - 98 (Russia), 11.9

Tags: combinatorics , geometry , 3d geometry , combinatorial geometry , pyramid

Cut pyramid $ABCD$ into $8$ equal and similar pyramids, if: a) $AB = BC = CD$, $\angle ABC =\angle BCD = 90^o$, dihedral angle at edge $BC$ is right b) all plane angles at vertex $B$ are right and $AB = BC = BD\sqrt2$. Note. Whether there are other types of triangular pyramids that can be cut into any number similar to the original pyramids (their number is not necessarily $8$ and the pyramids are not necessarily equal to each other) is currently unknown

1954 Moscow Mathematical Olympiad, 284

Tags: geometry , 3d geometry , symmetry , pyramid

How many planes of symmetry can a triangular pyramid have?

2005 Junior Balkan MO, 1

Tags: quadratic

Find all positive integers $x,y$ satisfying the equation \[ 9(x^2+y^2+1) + 2(3xy+2) = 2005 . \]

2022 Bolivia IMO TST, P1

Tags: algebra , gauss identity

Find all possible values of $\frac{1}{x}+\frac{1}{y}$, if $x,y$ are real numbers not equal to $0$ that satisfy $$x^3+y^3+3x^2y^2=x^3y^3$$

2021 Korea Winter Program Practice Test, 8

Tags: number theory , polynomial , algebra

$P$ is an monic integer coefficient polynomial which has no integer roots. deg$P=n$ and define $A$ $:=${$v_2(P(m))|m\in Z, v_2(P(m)) \ge 1$}. If $|A|=n$, show that all of the elements of $A$ is smaller than $\frac{3}{2}n^2$.

Estonia Open Senior - geometry, 2012.1.3

Tags: right triangle , median , circumcenter , geometry

Let $ABC$ be a triangle with median AK. Let $O$ be the circumcenter of the triangle $ABK$. a) Prove that if $O$ lies on a midline of the triangle $ABC$, but does not coincide with its endpoints, then $ABC$ is a right triangle. b) Is the statement still true if $O$ can coincide with an endpoint of the midsegment?

2005 Croatia National Olympiad, 1

Tags: algebra , equation , polynomial

Let $a \not = 0, b, c$ be real numbers. If $x_{1}$ is a root of the equation $ax^{2}+bx+c = 0$ and $x_{2}$ a root of $-ax^{2}+bx+c = 0$, show that there is a root $x_{3}$ of $\frac{a}{2}\cdot x^{2}+bx+c = 0$ between $x_{1}$ and $x_{2}$.

2005 Turkey MO (2nd round), 6