Olympiad problems

2007 Princeton University Math Competition, 10

Tags: logarithm

Find the values of $a$ such that $\log (ax+1) = \log (x-a) + \log (2-x)$ has a unique real solution.

1998 IMO, 4

Tags: number theory , divisibility , polynomial

Determine all pairs $(x,y)$ of positive integers such that $x^{2}y+x+y$ is divisible by $xy^{2}+y+7$.

2002 Brazil National Olympiad, 4

Tags: combinatorics

For any non-empty subset $A$ of $\{1, 2, \ldots , n\}$ define $f(A)$ as the largest element of $A$ minus the smallest element of $A$. Find $\sum f(A)$ where the sum is taken over all non-empty subsets of $\{1, 2, \ldots , n\}$.

2014 USAMTS Problems, 2:

Tags: modular arithmetic , search , number theory , prime number

Find all triples $(x, y, z)$ such that $x, y, z, x - y, y - z, x - z$ are all prime positive integers.

2011 Pre-Preparation Course Examination, 1

Tags: function , geometry , 3d geometry , sphere , topology

[b]a)[/b] prove that for every compressed set $K$ in the space $\mathbb R^3$, the function $f:\mathbb R^3 \longrightarrow \mathbb R$ that $f(p)=inf\{|p-k|,k\in K\}$ is continuous. [b]b)[/b] prove that we cannot cover the sphere $S^2\subseteq \mathbb R^3$ with it's three closed sets, such that none of them contain two antipodal points.

2019 India PRMO, 29

Tags: median , angle bisector , geometry

In a triangle $ABC$, the median $AD$ (with $D$ on $BC$) and the angle bisector $BE$ (with $E$ on $AC$) are perpedicular to each other. If $AD = 7$ and $BE = 9$, find the integer nearest to the area of triangle $ABC$.

1983 Brazil National Olympiad, 6

Tags: geometry , 3d geometry , sphere , maximum

Show that the maximum number of spheres of radius $1$ that can be placed touching a fixed sphere of radius $1$ so that no pair of spheres has an interior point in common is between $12$ and $14$.

2016 Junior Regional Olympiad - FBH, 3

Tags: combinatorics

From three boys and three girls, every boy knows exactly two girls and every girl knows exactly two boys. Prove that we can arrange boys and girls in pairs such that in every pair people know each other

1998 Italy TST, 1

Tags: functional equation , functional , function , algebra

A real number $\alpha$ is given. Find all functions $f : R^+ \to R^+$ satisfying $\alpha x^2f\left(\frac{1}{x}\right) +f(x) =\frac{x}{x+1}$ for all $x > 0$.

2006 Kazakhstan National Olympiad, 2

Tags: algebra , polynomial , trinomial

Product of square trinomials $ x ^ 2 + a_1x + b_1 $, $ x ^ 2 + a_2x + b_2 $, $ \dots $, $ x ^ 2 + a_n x + b_n $ equals polynomial $ P (x) = x ^ {2n} + c_1x ^ {2n-1} + c_2x ^ {2n-2} + \dots + c_ {2n-1} x + c_ {2n} $, where the coefficients $ c_1 $, $ c_2 $, $ \dots $, $ c_ {2n} $ are positive. Prove that for some $ k $ ($ 1 \leq k \leq n $) the coefficients $ a_k $ and $ b_k $ are positive.

2006 Hong Kong TST., 2

Tags: function , functional equation , hong kong

The function $f(x,y)$, defined on the set of all non-negative integers, satisfies (i) $f(0,y)=y+1$ (ii) $f(x+1,0)=f(x,1)$ (iii) $f(x+1,y+1)=f(x,f(x+1,y))$ Find f(3,2005), f(4,2005)

1982 USAMO, 3

Tags: geometry , perimeter , function , trigonometry

If a point $A_1$ is in the interior of an equilateral triangle $ABC$ and point $A_2$ is in the interior of $\triangle{A_1BC}$, prove that \[\operatorname{I.Q.} (A_1BC) > \operatorname{I.Q.} (A_2BC),\] where the [i]isoperrimetric quotient[/i] of a figure $F$ is defined by \[\operatorname{I.Q.}(F) = \frac{\operatorname{Area}(F)}{[\operatorname{Perimeter}(F)]^2}.\]

2002 AMC 12/AHSME, 18

Tags: geometry , similar triangles , circles , tangency

Let $ C_1$ and $ C_2$ be circles defined by \[ (x \minus{} 10)^2 \plus{} y^2 \equal{} 36\]and \[ (x \plus{} 15)^2 \plus{} y^2 \equal{} 81,\]respectively. What is the length of the shortest line segment $ \overline{PQ}$ that is tangent to $ C_1$ at $ P$ and to $ C_2$ at $ Q$? $ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 20 \qquad \textbf{(D)}\ 21 \qquad \textbf{(E)}\ 24$

2018 Junior Balkan Team Selection Tests - Moldova, 3

Tags: inequality , inequalities

Let $a,b,c \in\mathbb{R^*_+}$.Prove the inequality $\frac{a^2+4}{b+c}+\frac{b^2+9}{c+a}+\frac{c^2+16}{a+b}\ge9$.

2022/2023 Tournament of Towns, P5

Tags: weight , combinatorics

There is a single coin on each square of a $5 \times 5$ board. All the coins look the same. Two of them are fakes and have equal weight. Genuine coins are heavier than fake ones and also weigh the same. The fake coins are on the squares sharing just one vertice. Is it possible to determine for sure a) 13 genuine coins; b) 15 genuine coins; and c) 17 genuine coins in a single weighing on a balance with no unit weights? [i]Rustem Zhenodarov, Alexandr Gribalko, Sergey Tokarev[/i]

Geometry Mathley 2011-12, 7.1

Tags: geometry , circumcircle

Let $ABCD$ be a cyclic quadrilateral. Suppose that $E$ is the intersection of $AB$ and $CD, F$ is the intersection of $AD$ and $CB, I$ is the intersection of $AC$ and $BD$. The circumcircles $(FAB), (FCD)$ meet $FI$ at $K, L$. Prove that $EK = EL$ Nguyễn Minh Hà

2023 Sharygin Geometry Olympiad, 10

Tags: geometry , angle bisector , tangent circle

Altitudes $BE$ and $CF$ of an acute-angled triangle $ABC$ meet at point $H$. The perpendicular from $H$ to $EF$ meets the line $\ell$ passing through $A$ and parallel to $BC$ at point $P$. The bisectors of two angles between $\ell$ and $HP$ meet $BC$ at points $S$ and $T$. Prove that the circumcircles of triangles $ABC$ and $PST$ are tangent.

2022 Assara - South Russian Girl's MO, 7

Tags: coloring , combinatorics , geometry , 3d geometry

In a $7\times 7\times 7$ cube, the unit cubes are colored white, black and gray colors so that for any two colors the number of cubes of these two colors are different. In this case, $N$ parallel rows of $7$ cubes were found, each of which there are more white cubes than gray and than black. Likewise, there were $N$ parallel rows of $7$ cubes, each of which contained gray there are more cubes than white and than black, and there are also N parallel rows of $7$ cubes, each of which contains more black cubes than white ones and than gray ones. What is the largest $N$ for which this is possible?

2013 Denmark MO - Mohr Contest, 2

Tags: geometry , circumcircle , semicircle , rectangle , area

The figure shows a rectangle, its circumscribed circle and four semicircles, which have the rectangle’s sides as diameters. Prove that the combined area of the four dark gray crescentshaped regions is equal to the area of the light gray rectangle. [img]https://1.bp.blogspot.com/-gojv6KfBC9I/XzT9ZMKrIeI/AAAAAAAAMVU/NB-vUldjULI7jvqiFWmBC_Sd8QFtwrc7wCLcBGAsYHQ/s0/2013%2BMohr%2Bp3.png[/img]

2006 China Team Selection Test, 3

Tags: polynomial , algebra

Find all second degree polynomial $d(x)=x^{2}+ax+b$ with integer coefficients, so that there exists an integer coefficient polynomial $p(x)$ and a non-zero integer coefficient polynomial $q(x)$ that satisfy: \[\left( p(x) \right)^{2}-d(x) \left( q(x) \right)^{2}=1, \quad \forall x \in \mathbb R.\]

2004 Iran MO (2nd round), 1

Tags: trigonometry , geometry

$ABC$ is a triangle and $\angle A=90^{\circ}$. Let $D$ be the meet point of the interior bisector of $\angle A$ and $BC$. And let $I_a$ be the $A-$excenter of $\triangle ABC$. Prove that: \[\frac{AD}{DI_a}\leq\sqrt{2}-1.\]

2008 International Zhautykov Olympiad, 3

Tags: combinatorics

Let $ A \equal{} \{(a_1,\dots,a_8)|a_i\in\mathbb{N}$ , $ 1\leq a_i\leq i \plus{} 1$ for each $ i \equal{} 1,2\dots,8\}$.A subset $ X\subset A$ is called sparse if for each two distinct elements $ (a_1,\dots,a_8)$,$ (b_1,\dots,b_8)\in X$,there exist at least three indices $ i$,such that $ a_i\neq b_i$. Find the maximal possible number of elements in a sparse subset of set $ A$.

VII Soros Olympiad 2000 - 01, 11.6

Tags: geometry , tangent circle

A circle is tangent internally by $6$ circles so that each one is tangent internally to two adjacent ones and the radii of opposite circles are pairwise equal. Prove that the sum of the radii of the inner circles is equal to the diameter of the given circle.

2024 Malaysian IMO Training Camp, 5

Tags: number theory

Do there exist infinitely many triplets of positive integers $(a, b, c)$ such that the following two conditions hold: 1. $\gcd(a, b, c) = 1$; 2. $a+b+c, a^2+b^2+c^2$ and $abc$ are all perfect squares? [i](Proposed by Ivan Chan Guan Yu)[/i]

2014 Contests, 3

Tags: algebra , inequality

For positive real numbers $a,b,c$ with $abc=1$ prove that $\left(a+\frac{1}{b}\right)^{2}+\left(b+\frac{1}{c}\right)^{2}+\left(c+\frac{1}{a}\right)^{2}\geq 3(a+b+c+1)$