Olympiad problems

1987 Brazil National Olympiad, 2

Given a point $p$ inside a convex polyhedron $P$. Show that there is a face $F$ of $P$ such that the foot of the perpendicular from $p$ to $F$ lies in the interior of $F$.

2018 Peru EGMO TST, 2

Tags: algebra , functional equation

Find all functions $f:\mathbb R \rightarrow \mathbb R$, such that $2xyf(x^2-y^2)=(x^2-y^2)f(x)f(2y)$

2010 Oral Moscow Geometry Olympiad, 1

Tags: geometry , angles , Equilateral

Two equilateral triangles $ABC$ and $CDE$ have a common vertex (see fig). Find the angle between straight lines $AD$ and $BE$. [img]https://1.bp.blogspot.com/-OWpqpAqR7Zw/Xzj_fyqhbFI/AAAAAAAAMao/5y8vCfC7PegQLIUl9PARquaWypr8_luAgCLcBGAsYHQ/s0/2010%2Boral%2Bmoscow%2Bgeometru%2B8.1.gif[/img]

2009 India IMO Training Camp, 3

Tags: AMC , USA(J)MO , USAMO , number theory proposed , number theory

Let $ a,b$ be two distinct odd natural numbers.Define a Sequence $ { < a_n > }_{n\ge 0}$ like following: $ a_1 \equal{} a \\ a_2 \equal{} b \\ a_n \equal{} \text{largest odd divisor of }(a_{n \minus{} 1} \plus{} a_{n \minus{} 2})$. Prove that there exists a natural number $ N$ such that $ a_n \equal{} gcd(a,b) \forall n\ge N$.

1977 Miklós Schweitzer, 8

Tags: function , real analysis , real analysis unsolved

Let $ p \geq 1$ be a real number and $ \mathbb{R}_\plus{}\equal{}(0, \infty)$. For which continuous functions $ g : \mathbb{R}_\plus{} \rightarrow \mathbb{R}_\plus{}$ are following functions all convex? \[ M_n(x)\equal{}\left[ \frac{\sum_{i\equal{}1}^n g(\frac{x_i}{x_{i\plus{}1}}) x_{i\plus{}1}^p}{\sum_{i\equal{}1}^n g(\frac{x_i}{x_{i\plus{}1}})} \right ]^\frac 1p ,\] \[ x\equal{}(x_1,\ldots, x_{n\plus{}1}) \in \mathbb{R}_\plus{} ^ {n\plus{}1} , \; n\equal{}1,2,\ldots\] [i]L. Losonczi[/i]

2024 Czech-Polish-Slovak Junior Match, 2

Tags: number theory , number theory proposed , Product , perfect cube , geometry , 3D geometry

How many non-empty subsets of $\{1,2,\dots,11\}$ are there with the property that the product of its elements is the cube of an integer?

1992 Austrian-Polish Competition, 1

Tags: Sum , positive , Divisors , number theory , Product , Even

For a natural number $n$, denote by $s(n)$ the sum of all positive divisors of n. Prove that for every $n > 1$ the product $s(n - 1)s(n)s(n + 1)$ is even.

2021 JBMO Shortlist, G2

Tags: Junior , Balkan , shortlist , 2021 , geometry , arcs

Let $P$ be an interior point of the isosceles triangle $ABC$ with $\hat{A} = 90^{\circ}$. If $$\widehat{PAB} + \widehat{PBC} + \widehat{PCA} = 90^{\circ},$$ prove that $AP \perp BC$. Proposed by [i]Mehmet Akif Yıldız, Turkey[/i]

2006 Finnish National High School Mathematics Competition, 4

Tags: geometry , geometry unsolved

Two medians of a triangle are perpendicular. Prove that the medians of the triangle are the sides of a right-angled triangle.

2022 AIME Problems, 13

Tags: AMC , AIME

Let $S$ be the set of all rational numbers that can be expressed as a repeating decimal in the form $0.\overline{abcd},$ where at least one of the digits $a, b, c, $ or $d$ is nonzero. Let $N$ be the number of distinct numerators when numbers in $S$ are written as fractions in lowest terms. For example, both $4$ and $410$ are counted among the distinct numerators for numbers in $S$ because $0.\overline{3636} = \frac{4}{11}$ and $0.\overline{1230} = \frac{410}{3333}.$ Find the remainder when $N$ is divided by $1000.$

2019 IMO Shortlist, G2

Tags: geometry , IMO Shortlist , TST

Let $ABC$ be an acute-angled triangle and let $D, E$, and $F$ be the feet of altitudes from $A, B$, and $C$ to sides $BC, CA$, and $AB$, respectively. Denote by $\omega_B$ and $\omega_C$ the incircles of triangles $BDF$ and $CDE$, and let these circles be tangent to segments $DF$ and $DE$ at $M$ and $N$, respectively. Let line $MN$ meet circles $\omega_B$ and $\omega_C$ again at $P \ne M$ and $Q \ne N$, respectively. Prove that $MP = NQ$. (Vietnam)

1998 All-Russian Olympiad Regional Round, 11.8

Tags: Combinatorial Number Theory , combinatorics , algebra , inequalities , number theory

A sequence $a_1,a_2,\cdots$ of positive integers contains each positive integer exactly once. Moreover for every pair of distinct positive integer $m$ and $n$, $\frac{1}{1998} < \frac{|a_n- a_m|}{|n-m|} < 1998$, show that $|a_n - n | <2000000$ for all $n$.

1949-56 Chisinau City MO, 18

Tags: algebra , trinomial

Prove that if the numbers $a, b, c$ are the lengths of the sides of some nondegenerate triangle, then the equation $$b^2x^2 + (b^2 + c^2 - a^2) x + c^2 = 0$$ has imaginary roots.

2003 Cuba MO, 1

Tags: number theory , Digits , prime

Given the following list of numbers: $$1990, 1991, 1992, ..., 2002, 2003, 2003, 2003, ..., 2003$$ where the number $2003$ appears $12$ times. Is it possible to write these numbers in some order so that the $100$-digit number that we get is prime?

2021 India National Olympiad, 3

Tags: geometry , construction , INMO

Betal marks $2021$ points on the plane such that no three are collinear, and draws all possible segments joining these. He then chooses any $1011$ of these segments, and marks their midpoints. Finally, he chooses a segment whose midpoint is not marked yet, and challenges Vikram to construct its midpoint using [b]only[/b] a straightedge. Can Vikram always complete this challenge? [i]Note.[/i] A straightedge is an infinitely long ruler without markings, which can only be used to draw the line joining any two given distinct points. [i]Proposed by Prithwijit De and Sutanay Bhattacharya[/i]

1940 Moscow Mathematical Olympiad, 067

Tags: Compare , algebra , factorial

Which is greater: $300!$ or $100^{300}$?

2012 Balkan MO Shortlist, G4

Tags: geometry , incenter , circumcircle , Cyclic , Circumcenter , circles

Let $M$ be the point of intersection of the diagonals of a cyclic quadrilateral $ABCD$. Let $I_1$ and $I_2$ are the incenters of triangles $AMD$ and $BMC$, respectively, and let $L$ be the point of intersection of the lines $DI_1$ and $CI_2$. The foot of the perpendicular from the midpoint $T$ of $I_1I_2$ to $CL$ is $N$, and $F$ is the midpoint of $TN$. Let $G$ and $J$ be the points of intersection of the line $LF$ with $I_1N$ and $I_1I_2$, respectively. Let $O_1$ be the circumcenter of triangle $LI_1J$, and let $\Gamma_1$ and $\Gamma_2$ be the circles with diameters $O_1L$ and $O_1J$, respectively. Let $V$ and $S$ be the second points of intersection of $I_1O_1$ with $\Gamma_1$ and $\Gamma_2$, respectively. If $K$ is point where the circles $\Gamma_1$ and $\Gamma_2$ meet again, prove that $K$ is the circumcenter of the triangle $SVG$.

2021 Alibaba Global Math Competition, 5

Tags: calculus , complex analysis , Ode , differential equation

For the complex-valued function $f(x)$ which is continuous and absolutely integrable on $\mathbb{R}$, define the function $(Sf)(x)$ on $\mathbb{R}$: $(Sf)(x)=\int_{-\infty}^{+\infty}e^{2\pi iux}f(u)du$. (a) Find the expression for $S(\frac{1}{1+x^2})$ and $S(\frac{1}{(1+x^2)^2})$. (b) For any integer $k$, let $f_k(x)=(1+x^2)^{-1-k}$. Assume $k\geq 1$, find constant $c_1$, $c_2$ such that the function $y=(Sf_k)(x)$ satisfies the ODE with second order: $xy''+c_1y'+c_2xy=0$.

1956 Czech and Slovak Olympiad III A, 3

Tags: system of equations , absolute value , algebra , national olympiad

Find all real pairs $x,y$ such that \begin{align*} x-|y+1|&=1, \\ x^2+y&=10. \end{align*}

1986 Poland - Second Round, 3

Tags: geometry , 3D geometry , sphere , tetrahedron , geometric inequality

Let S be a sphere cirucmscribed on a regular tetrahedron with an edge length greater than 1. The sphere $ S $ is represented as the sum of four sets. Prove that one of these sets includes points $ P $, $ Q $ such that the length of the segment $ PQ $ exceeds 1.

2001 CentroAmerican, 3

Tags: modular arithmetic

In a circumference of a circle, $ 10000$ points are marked, and they are numbered from $ 1$ to $ 10000$ in a clockwise manner. $ 5000$ segments are drawn in such a way so that the following conditions are met: 1. Each segment joins two marked points. 2. Each marked point belongs to one and only one segment. 3. Each segment intersects exactly one of the remaining segments. 4. A number is assigned to each segment that is the product of the number assigned to each end point of the segment. Let $ S$ be the sum of the products assigned to all the segments. Show that $ S$ is a multiple of $ 4$.

2004 All-Russian Olympiad, 4

Tags: number theory , least common multiple , number theory unsolved

Is there a natural number $ n > 10^{1000}$ which is not divisible by 10 and which satisfies: in its decimal representation one can exchange two distinct non-zero digits such that the set of prime divisors does not change.

2015 AMC 12/AHSME, 12

Tags: conics , parabola , analytic geometry , geometry , AMC 10 , AMC

The parabolas $y=ax^2-2$ and $y=4-bx^2$ intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area $12$. What is $a+b$? $\textbf{(A) }1\qquad\textbf{(B) }1.5\qquad\textbf{(C) }2\qquad\textbf{(D) }2.5\qquad\textbf{(E) }3$

1973 Canada National Olympiad, 3

Tags:

Prove that if $p$ and $p+2$ are prime integers greater than 3, then 6 is a factor of $p+1$.

Today's calculation of integrals, 892

Tags: calculus , integration , trigonometry , logarithms , calculus computations

Evaluate $\int_0^{\frac{\pi}{2}} \frac{\sin x-\cos x}{1+\cos x}\ dx.$