Olympiad problems

2016 Israel Team Selection Test, 4

Tags: combinatorics

A regular 60-gon is given. What is the maximum size of a subset of its vertices containing no isosceles triangles?

2018 AMC 12/AHSME, 12

Tags: geometry , angle bisector

Side $\overline{AB}$ of $\triangle ABC$ has length $10$. The bisector of angle $A$ meets $\overline{BC}$ at $D$, and $CD = 3$. The set of all possible values of $AC$ is an open interval $(m,n)$. What is $m+n$? $ \textbf{(A) }16 \qquad \textbf{(B) }17 \qquad \textbf{(C) }18 \qquad \textbf{(D) }19 \qquad \textbf{(E) }20 \qquad $

2009 Stanford Mathematics Tournament, 9

Tags: parabola , calculus , algebra , polynomial , derivative , quadratic , conic

Find the shortest distance between the point $(6,12)$ and the parabola given by the equation $x=\frac{y^2}{2}$

2015 Hanoi Open Mathematics Competitions, 13

Tags: algebra , rational , equation

Give rational numbers $x, y$ such that $(x^2 + y^2 - 2) (x + y)^2 + (xy + 1)^2 = 0 $ Prove that $\sqrt{1 + xy}$ is a rational number.

2012 District Olympiad, 1

Tags: limit , real analysis , sequence

Consider the sequence $ \left( x_n \right)_{n\ge 1} $ having $ x_1>1 $ and satisfying the equation $$ x_1+x_2+\cdots +x_{n+1} =x_1x_2\cdots x_{n+1} ,\quad\forall n\in\mathbb{N} . $$ Show that this sequence is convergent and find its limit.

1998 Romania Team Selection Test, 3

Tags: inequalities , induction , divisibility theory

Find all positive integers $(x, n)$ such that $x^{n}+2^{n}+1$ divides $x^{n+1}+2^{n+1}+1$.

2014 Online Math Open Problems, 27

Tags: ratio , probability , expected value , number theory , relatively prime , algebra

A frog starts at $0$ on a number line and plays a game. On each turn the frog chooses at random to jump $1$ or $2$ integers to the right or left. It stops moving if it lands on a nonpositive number or a number on which it has already landed. If the expected number of times it will jump is $\tfrac{p}{q}$ for relatively prime positive integers $p$ and $q$, find $p+q$. [i]Proposed by Michael Kural[/i]

2005 Czech And Slovak Olympiad III A, 6

Tags: coloring , monotone , sequence , combinatorics , permutation

Decide whether for every arrangement of the numbers $1,2,3, . . . ,15$ in a sequence one can color these numbers with at most four different colors in such a way that the numbers of each color form a monotone subsequence.

1998 Turkey MO (2nd round), 3

Tags: ramsey theory , combinatorics

The points of a circle are colored by three colors. Prove that there exist infinitely many isosceles triangles inscribed in the circle whose vertices are of the same color.

Kvant 2023, M2735

Tags: geometry , circles

Let $AB$ be a diameter of the circle $\Omega$ with center $O{}$. The points $C, D, X$ and $Y{}$ are chosen on $\Omega$ so that the segments $CX$ and $DX$ intersect the segment $AB$ at points symmetric with respect to $O{}$, and $XY\parallel AB$. Let the lines $AB{}$ and $CD{}$ intersect at the point $E$. Prove that the tangent to $\Omega$ through $Y{}$ passes through $E{}$.

2024 China Team Selection Test, 22

Tags: geometry

$ABC$ is an isosceles triangle, with $AB=AC$. $D$ is a moving point such that $AD\parallel BC$, $BD>CD$. Moving point $E$ is on the arc of $BC$ in circumcircle of $ABC$ not containing $A$, such that $EB<EC$. Ray $BC$ contains point $F$ with $\angle ADE=\angle DFE$. If ray $FD$ intersects ray $BA$ at $X$, and intersects ray $CA$ at $Y$, prove that $\angle XEY$ is a fixed angle.

1952 AMC 12/AHSME, 21

Tags:

The sides of a regular polygon of $ n$ sides, $ n > 4$, are extended to form a star. The number of degrees at each point of the star is: $ \textbf{(A)}\ \frac {360}{n} \qquad\textbf{(B)}\ \frac {(n \minus{} 4)180}{n} \qquad\textbf{(C)}\ \frac {(n \minus{} 2)180}{n}$ $ \textbf{(D)}\ 180 \minus{} \frac {90}{n} \qquad\textbf{(E)}\ \frac {180}{n}$

2007 Stanford Mathematics Tournament, 15

Tags: integration , trigonometry , logarithm

Evaluate $\int_{0}^{\infty}\frac{\tan^{-1}(\pi x)-\tan^{-1}x}{x}dx$

2019 Moldova Team Selection Test, 12

Tags: number theory , prime number

Let $p\ge 5$ be a prime number. Prove that there exist positive integers $m$ and $n$ with $m+n\le \frac{p+1}{2}$ for which $p$ divides $2^n\cdot 3^m-1.$

PEN S Problems, 6

Tags: algebra , polynomial , induction , complex number

Suppose that $x$ and $y$ are complex numbers such that \[\frac{x^{n}-y^{n}}{x-y}\] are integers for some four consecutive positive integers $n$. Prove that it is an integer for all positive integers $n$.

2024 Canadian Junior Mathematical Olympiad, 2

Tags:

Let $I_n=\sum_{i=1}^n\sum_{j=1}^n\sum_{k=1}^n \min \left( \frac{1}{i}, \frac{1}{j}, \frac{1}{k} \right)$ and let $H_n=1+\frac{1}{2}+\ldots \frac{1}{n}$ Find $I_n-H_n$ in terms of $n$ (Paraphrased)

1990 Baltic Way, 14

Tags:

Do there exist $1990$ pairwise coprime positive integers such that all sums of two or more of these numbers are composite numbers?

2006 VTRMC, Problem 7

Tags: geometry , 3d geometry , sphere

Three spheres each of unit radius have centers $P,Q,R$ with the property that the center of each sphere lies on the surface of the other two spheres. Let $C$ denote the cylinder with cross-section $PQR$ (the triangular lamina with vertices $P,Q,R$) and axis perpendicular to $PQR$. Let $M$ denote the space which is common to the three spheres and the cylinder $C$, and suppose the mass density of $M$ at a given point is the distance of the point from $PQR$. Determine the mass of $M$.

2007 Tournament Of Towns, 3

Tags: modular arithmetic

Anna's number is obtained by writing down $20$ consecutive positive integers, one after another in arbitrary order. Bob's number is obtained in the same way, but with $21$ consecutive positive integers. Can they obtain the same number?

India EGMO 2024 TST, 1

Tags: euler , geometry , circumcircle

Let $ABC$ be a triangle with circumcentre $O$ and centroid $G$. Let $M$ be the midpoint of $BC$ and $N$ the reflection of $M$ across $O$. Prove that $NO = NA$ if and only if $\angle AOG = 90^{\circ}$. [i]Proposed by Pranjal Srivastava[/i]

2010 Contests, 3

Tags: geometry , 3d geometry , trigonometry , vector , linear algebra , matrix , inequalities

What is the biggest shadow that a cube of side length $1$ can have, with the sun at its peak? Note: "The biggest shadow of a figure with the sun at its peak" is understood to be the biggest possible area of the orthogonal projection of the figure on a plane.

2006 Moldova MO 11-12, 3

Tags: combinatorics

On each of the 2006 cards a natural number is written. Cards are placed arbitrarily in a row. 2 players take in turns a card from any end of the row until all the cards are taken. After that each player calculates sum of the numbers written of his cards. If the sum of the first player is not less then the sum of the second one then the first player wins. Show that there's a winning strategy for the first player.

2014 Swedish Mathematical Competition, 4

Tags: combinatorial geometry , square , geometric inequalities , geometry

A square is cut into a finitely number of triangles in an arbitrary way. Show the sum of the diameters of the inscribed circles in these triangles is greater than the side length of the square.

2021 Moldova Team Selection Test, 1

Tags: algebra , polynomial

Let $P(x) = x^3 + ax^2 + bx + 1$ be a polynomial with real coefficients and three real roots $\rho_1$, $\rho_2$, $\rho_3$ such that $|\rho_1| < |\rho_2| < |\rho_3|$. Let $A$ be the point where the graph of $P(x)$ intersects $yy'$ and the point $B(\rho_1, 0)$, $C(\rho_2, 0)$, $D(\rho_3, 0)$. If the circumcircle of $\vartriangle ABD$ intersects $yy'$ for a second time at $E$, find the minimum value of the length of the segment $EC$ and the polynomials for which this is attained. [i]Brazitikos Silouanos, Greece[/i]

2007 Today's Calculation Of Integral, 207

Tags: calculus , integration , calculus computations , logarithm

Evaluate the following definite integral. \[\int_{e^{e}}^{e^{e+1}}\left\{\frac{1}{\ln x \cdot\ln (\ln x)}+\ln (\ln (\ln x))\right\}dx\]