Olympiad problems

2000 All-Russian Olympiad, 6

Tags: number theory

A perfect number, greater than $6$, is divisible by $3$. Prove that it is also divisible by $9$.

1997 Chile National Olympiad, 3

Tags: geometry , trapezoid , area

Let $ ABCD $ be a quadrilateral, whose diagonals intersect at $ O $. The triangles $ \triangle AOB $, $ \triangle BOC $, $ \triangle COD $ have areas $1, 2, 4$, respectively. Find the area of $ \triangle AOD $ and prove that $ ABCD $ is a trapezoid.

2002 France Team Selection Test, 2

Tags: incenter , circumcircle , trigonometry , geometry

Let $ ABC$ be a non-equilateral triangle. Denote by $ I$ the incenter and by $ O$ the circumcenter of the triangle $ ABC$. Prove that $ \angle AIO\leq\frac{\pi}{2}$ holds if and only if $ 2\cdot BC\leq AB\plus{}AC$.

2012 Junior Balkan Team Selection Tests - Romania, 3

Tags: geometry , ratio

Consider the triangle $ABC$ and the points $D \in (BC)$ and $M \in (AD)$. Lines $BM$ and $AC$ meet at $E$, lines $CM$ and $AB$ meet at $F$, and lines $EF$ and $AD$ meet at $N$. Prove that $$\frac{AN}{DN}=\frac{1}{2}\cdot \frac{AM}{DM}$$

2012 Today's Calculation Of Integral, 772

Tags: calculus , integration , analytic geometry , geometry , geometric transformation , rotation , 3d geometry

Given are three points $A(2,\ 0,\ 2),\ B(1,\ 1,\ 0),\ C(0,\ 0,\ 3)$ in the coordinate space. Find the volume of the solid of a triangle $ABC$ generated by a rotation about $z$-axis.

2003 Tournament Of Towns, 5

Tags: geometry , 3d geometry , tetrahedron , combinatorics

A paper tetrahedron is cut along some of so that it can be developed onto the plane. Could it happen that this development cannot be placed on the plane in one layer?

1967 IMO Longlists, 36

Tags: 3d geometry , sphere , tetrahedron , circumcircle , geometry

Prove this proposition: Center the sphere circumscribed around a tetrahedron which coincides with the center of a sphere inscribed in that tetrahedron if and only if the skew edges of the tetrahedron are equal.

2019 Stars of Mathematics, 3

Tags: geometry

Let $ABC$ be a triangle. Let $M$ be a variable point interior to the segment $AB$, and let $\gamma_B$ be the circle through $M$ and tangent at $B$ to $BC$. Let $P$ and $Q$ be the touch points of $\gamma_B$ and its tangents from $A$, and let $X$ be the midpoint of the segment $PQ$. Similarly, let $N$ be a variable point interior to the segment $AC$, and let $\gamma_C$ be the circle through $M$ and tangent at $C$ to $BC$. Let $R$ and $S$ be the touch points of $\gamma_C$ and its tangents from $A$, and let $Y$ be the midpoint of the segment $RS$. Prove that the line through the centers of the circles $AMN$ and $AXY$ passes through a fixed point.

2016 Mathematical Talent Reward Programme, MCQ: P 12

Tags: function , continuity

Let $f(x)=(x-1)(x-2)(x-3)$. Consider $g(x)=min\{f(x),f'(x)\}$. Then the number of points of discontinuity are [list=1] [*] 0 [*] 1 [*] 2 [*] More than 2 [/list]

2014 District Olympiad, 1

Tags: inequalities , algebra

Prove that: [list=a][*]$\displaystyle\left( \frac{1}{2}\right) ^{3}+\left( \frac{2}{3}\right)^{3}+\left( \frac{5}{6}\right) ^{3}=1$ [*]$3^{33}+4^{33}+5^{33}<6^{33}$[/list]

2015 Princeton University Math Competition, A4/B6

Tags:

A number is [i]interesting [/i]if it is a $6$-digit integer that contains no zeros, its first $3$ digits are strictly increasing, and its last $3$ digits are non-increasing. What is the average of all interesting numbers?

2008 Moldova National Olympiad, 12.8

Tags: integration , trigonometry , logarithm , real analysis

Evaluate $ \displaystyle I \equal{} \int_0^{\frac\pi4}\left(\sin^62x \plus{} \cos^62x\right)\cdot \ln(1 \plus{} \tan x)\text{d}x$.

2014 Middle European Mathematical Olympiad, 1

Tags: inequalities

Determine the lowest possible value of the expression \[ \frac{1}{a+x} + \frac{1}{a+y} + \frac{1}{b+x} + \frac{1}{b+y} \] where $a,b,x,$ and $y$ are positive real numbers satisfying the inequalities \[ \frac{1}{a+x} \ge \frac{1}{2} \] \[\frac{1}{a+y} \ge \frac{1}{2} \] \[ \frac{1}{b+x} \ge \frac{1}{2} \] \[ \frac{1}{b+y} \ge 1. \]

2010 Balkan MO Shortlist, N2

Tags: calculus , integration , number theory

Solve the following equation in positive integers: $x^{3} = 2y^{2} + 1 $

2002 Austria Beginners' Competition, 2

Tags: floor function , algebra

Prove that there are no $x\in\mathbb{R}^+$ such that $$x^{\lfloor x \rfloor }=\frac92.$$

2022 IOQM India, 5

Tags: geometry , parallelogram

In parallelogram $ABCD$, the longer side is twice the shorter side. Let $XYZW$ be the quadrilateral formed by the internal bisectors of the angles of $ABCD$. If the area of $XYZW$ is $10$, find the area of $ABCD$

2014 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: algebra , equation , real number

Find all real solutions of the equation: $$x=\frac{2z^2}{1+z^2}$$ $$y=\frac{2x^2}{1+x^2}$$ $$z=\frac{2y^2}{1+y^2}$$

2016 Harvard-MIT Mathematics Tournament, 4

Tags: hmmt

A positive integer is written on each corner of a square such that numbers on opposite vertices are relatively prime while numbers on adjacent vertices are not relatively prime. What is the smallest possible value of the sum of these $4$ numbers?

Estonia Open Senior - geometry, 2009.1.5

Tags: geometry , geometric inequality , inradius

Let any point $D$ be chosen on the side $BC$ of the triangle $ABC$. Let the radii of the incircles of the triangles $ABC, ABD$ and $ACD$ be $r_1, r_2$ and $r_3$. Prove that $r_1 <r_2 + r_3$.

2021 Iberoamerican, 2

Tags: geometry , circumcircle

Consider an acute-angled triangle $ABC$, with $AC>AB$, and let $\Gamma$ be its circumcircle. Let $E$ and $F$ be the midpoints of the sides $AC$ and $AB$, respectively. The circumcircle of the triangle $CEF$ and $\Gamma$ meet at $X$ and $C$, with $X\neq C$. The line $BX$ and the tangent to $\Gamma$ through $A$ meet at $Y$. Let $P$ be the point on segment $AB$ so that $YP = YA$, with $P\neq A$, and let $Q$ be the point where $AB$ and the parallel to $BC$ through $Y$ meet each other. Show that $F$ is the midpoint of $PQ$.

2018 Hanoi Open Mathematics Competitions, 4

Tags: equation , algebra

Find the number of distinct real roots of the following equation $x^2 +\frac{9x^2}{(x + 3)^2} = 40$. A. $0$ B. $1$ C. $2$ D. $3$ E. $4$

2024 Baltic Way, 13

Tags: geometric transformation , reflection , geometry

Let $ABC$ be an acute triangle with orthocentre $H$. Let $D$ be a point outside the circumcircle of triangle $ABC$ such that $\angle ABD=\angle DCA$. The reflection of $AB$ in $BD$ intersects $CD$ at $X$. The reflection of $AC$ in $CD$ intersects $BD$ at $Y$. The lines through $X$ and $Y$ perpendicular to $AC$ and $AB$, respectively, intersect at $P$. Prove that points $D$, $P$ and $H$ are collinear.

2019 Simurgh, 3

Tags: graph , combinatorics , symmetry

We call a graph symmetric, if we can put its vertices on the plane such that if the edges are segments, the graph has a reflectional symmetry with respect to a line not passing through its vertices. Find the least value of $K$ such that the edges of every graph with $100$ vertices, can be divided into $K$ symmetric subgraphs.

2016 CCA Math Bonanza, I3

Tags:

Amanda has the list of even numbers $2, 4, 6, \dots 100$ and Billy has the list of odd numbers $1, 3, 5, \dots 99$. Carlos creates a list by adding the square of each number in Amanda's list to the square of the corresponding number in Billy's list. Daisy creates a list by taking twice the product of corresponding numbers in Amanda's list and Billy's list. What is the positive difference between the sum of the numbers in Carlos's list and the sum of the numbers in Daisy's list? [i]2016 CCA Math Bonanza Individual #3[/i]

2022 ISI Entrance Examination, 5

Tags: number theory

For any positive integer $n$, and $i=1,2$, let $f_{i}(n)$ denote the number of divisors of $n$ of the form $3 k+i$ (including $1$ and $n$ ). Define, for any positive integer $n$, $$f(n)=f_{1}(n)-f_{2}(n)$$ Find the value of $f\left(5^{2022}\right)$ and $f\left(21^{2022}\right)$.