Olympiad problems

2004 Pre-Preparation Course Examination, 5

Let $ A\equal{}\{A_1,\dots,A_m\}$ be a family distinct subsets of $ \{1,2,\dots,n\}$ with at most $ \frac n2$ elements. Assume that $ A_i\not\subset A_j$ and $ A_i\cap A_j\neq\emptyset$ for each $ i,j$. Prove that: \[ \sum_{i\equal{}1}^m\frac1{\binom{n\minus{}1}{|A_i|\minus{}1}}\leq1\]

1977 All Soviet Union Mathematical Olympiad, 237

Tags: concurrent , hexagon , geometry

a) Given a circle with two inscribed triangles $T_1$ and $T_2$. The vertices of $T_1$ are the midpoints of the arcs with the ends in the vertices of $T_2$. Consider a hexagon -- the intersection of $T_1$ and $T_2$. Prove that its main diagonals are parallel to $T_1$ sides and are intersecting in one point. b) The segment, that connects the midpoints of the arcs $AB$ and $AC$ of the circle circumscribed around the $ABC$ triangle, intersects $[AB]$ and $[AC]$ sides in $D$ and $K$ points. Prove that the points $A,D,K$ and $O$ -- the centre of the circle -- are the vertices of a diamond.

2023 Centroamerican and Caribbean Math Olympiad, 3

Tags: inequalities

Let $a,\ b$ and $c$ be positive real numbers such that $a b+b c+c a=1$. Show that $$ \frac{a^3}{a^2+3 b^2+3 a b+2 b c}+\frac{b^3}{b^2+3 c^2+3 b c+2 c a}+\frac{c^3}{c^2+3 a^2+3 c a+2 a b}>\frac{1}{6\left(a^2+b^2+c^2\right)^2} . $$

2003 India IMO Training Camp, 5

Tags: geometry , geometric transformation , reflection , combinatorics unsolved , combinatorics

On the real number line, paint red all points that correspond to integers of the form $81x+100y$, where $x$ and $y$ are positive integers. Paint the remaining integer point blue. Find a point $P$ on the line such that, for every integer point $T$, the reflection of $T$ with respect to $P$ is an integer point of a different colour than $T$.

Kyiv City MO Juniors Round2 2010+ geometry, 2022.8.4

Tags: geometry

Points $D, E, F$ are selected on sides $BC, CA, AB$ correspondingly of triangle $ABC$ with $\angle C = 90^\circ$ such that $\angle DAB = \angle CBE$ and $\angle BEC = \angle AEF$. Show that $DB = DF$. [i](Proposed by Mykhailo Shtandenko)[/i]

2000 Singapore Senior Math Olympiad, 2

Tags: factorial , Diophantine equation , diophantine , number theory

Prove that there exist no positive integers $m$ and $n$ such that $m > 5$ and $(m - 1)! + 1 = m^n$.

1965 Putnam, B5

Tags: Putnam

Consider collections of unordered pairs of $V$ different objects $a$, $b$, $c$, $\ldots$, $k$. Three pairs such as $ab$, $bc$, $ab$ are said to form a triangle. Prove that, if $4E\leq V^2$, it is possible to choose $E$ pairs so that no triangle is formed.

2012 Today's Calculation Of Integral, 780

Tags: calculus , integration , geometry , circumcircle , 3D geometry , sphere , geometric transformation

Let $n\geq 3$ be integer. Given a regular $n$-polygon $P$ with side length 4 on the plane $z=0$ in the $xyz$-space.Llet $G$ be a circumcenter of $P$. When the center of the sphere $B$ with radius 1 travels round along the sides of $P$, denote by $K_n$ the solid swept by $B$. Answer the following questions. (1) Take two adjacent vertices $P_1,\ P_2$ of $P$. Let $Q$ be the intersection point between the perpendicular dawn from $G$ to $P_1P_2$, prove that $GQ>1$. (2) (i) Express the area of cross section $S(t)$ in terms of $t,\ n$ when $K_n$ is cut by the plane $z=t\ (-1\leq t\leq 1)$. (ii) Express the volume $V(n)$ of $K_n$ in terms of $n$. (3) Denote by $l$ the line which passes through $G$ and perpendicular to the plane $z=0$. Express the volume $W(n)$ of the solid by generated by a rotation of $K_n$ around $l$ in terms of $n$. (4) Find $\lim_{n\to\infty} \frac{V(n)}{W(n)} .$

2022 Auckland Mathematical Olympiad, 1

Tags: algebra

Each of the $10$ dwarfs either always tells the truth or always lies. It is known that each of them loves exactly one type of ice cream: vanilla, chocolate or fruit. First, Snow White asked those who like the vanilla ice cream to raise their hands, and everyone raised their hands, then those who like chocolate ice cream - and half of the dwarves raised their hands, then those who like the fruit ice cream - and only one dwarf raised his hand. How many of the gnomes are truthful?

2008 Spain Mathematical Olympiad, 3

Tags: geometry , trapezoid , rectangle , combinatorics unsolved , combinatorics

Every point in the plane is coloured one of seven distinct colours. Is there an inscribed trapezoid whose vertices are all of the same colour?

2018 Caucasus Mathematical Olympiad, 8

Tags: inequalities

Let $a, b, c$ be the lengths of sides of a triangle. Prove the inequality $$(a+b)\sqrt{ab}+(a+c)\sqrt{ac}+(b+c)\sqrt{bc} \geq (a+b+c)^2/2.$$

2017 AIME Problems, 3

Tags: AMC , AIME , AIME II

A triangle has vertices $A(0,0)$, $B(12,0)$, and $C(8,10)$. The probability that a randomly chosen point inside the triangle is closer to vertex $B$ than to either vertex $A$ or vertex $C$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

2003 USA Team Selection Test, 6

Tags: geometry , circumcircle , geometric transformation , reflection , incenter , rhombus , trapezoid

Let $\overline{AH_1}, \overline{BH_2}$, and $\overline{CH_3}$ be the altitudes of an acute scalene triangle $ABC$. The incircle of triangle $ABC$ is tangent to $\overline{BC}, \overline{CA},$ and $\overline{AB}$ at $T_1, T_2,$ and $T_3$, respectively. For $k = 1, 2, 3$, let $P_i$ be the point on line $H_iH_{i+1}$ (where $H_4 = H_1$) such that $H_iT_iP_i$ is an acute isosceles triangle with $H_iT_i = H_iP_i$. Prove that the circumcircles of triangles $T_1P_1T_2$, $T_2P_2T_3$, $T_3P_3T_1$ pass through a common point.

1987 All Soviet Union Mathematical Olympiad, 443

Tags: Heptagon , geometry

Given a regular heptagon $A_1...A_7$. Prove that $$\frac{1}{|A_1A_5|} + \frac{1}{|A_1A_3| }= \frac{1}{|A_1A_7|}$$.

2010 Purple Comet Problems, 23

Tags: symmetry , quadratics , number theory , relatively prime , Pythagorean Theorem , geometry , algebra

A disk with radius $10$ and a disk with radius $8$ are drawn so that the distance between their centers is $3$. Two congruent small circles lie in the intersection of the two disks so that they are tangent to each other and to each of the larger circles as shown. The radii of the smaller circles are both $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [asy] size(150); defaultpen(linewidth(1)); draw(circle(origin,10)^^circle((3,0),8)^^circle((5,15/4),15/4)^^circle((5,-15/4),15/4)); [/asy]

2001 Junior Balkan Team Selection Tests - Moldova, 7

Tags: combinatorics

Noah has on his ark $4$ large coffins in which to place $8$ animals. It is known that for any animal there are at most $5$ animals with which it is incompatible (those can't live together). Show that: a) Noah can place the animals in the cages according to their compatibility. b) Noah can place two animals in each cage.

1969 All Soviet Union Mathematical Olympiad, 118

Tags: inequalities , algebra

Given positive numbers $a,b,c,d$. Prove that the set of inequalities $$a+b<c+d$$ $$(a+b)(c+d)<ab+cd$$ $$(a+b)cd<ab(c+d)$$ contain at least one wrong.

2015 Junior Regional Olympiad - FBH, 4

Tags: Fraction , Digits

Which number we need to substract from numerator and add to denominator of $\frac{\overline{28a3}}{7276}$ such that we get fraction equal to $\frac{2}{7}$

2020 IberoAmerican, 4

Tags: number theory

Show that there exists a set $\mathcal{C}$ of $2020$ distinct, positive integers that satisfies simultaneously the following properties: $\bullet$ When one computes the greatest common divisor of each pair of elements of $\mathcal{C}$, one gets a list of numbers that are all distinct. $\bullet$ When one computes the least common multiple of each pair of elements of $\mathcal{C}$, one gets a list of numbers that are all distinct.

2015 AoPS Mathematical Olympiad, 8

Tags: function , number theory

Consider the function $f(x)=5x^4-12x^3+30x^2-12x+5$. Let $f(x_1)=p$, wher $x_1$ and $p$ are non-negative integers, and $p$ is prime. Find with proof the largest possible value of $p$. [i]Proposed by tkhalid[/i]

2005 Sharygin Geometry Olympiad, 5

Tags: geometry , Locus , orthocenter , Circumcenter , Centroid , circumcircle

There are two parallel lines $p_1$ and $p_2$. Points $A$ and $B$ lie on $p_1$, and $C$ on $p_2$. We will move the segment $BC$ parallel to itself and consider all the triangles $AB'C '$ thus obtained. Find the locus of the points in these triangles: a) points of intersection of heights, b) the intersection points of the medians, c) the centers of the circumscribed circles.