Olympiad problems

2006 Junior Balkan Team Selection Tests - Romania, 1

Tags:

Let $A=\{1,2,\ldots, 2006\}$. Find the maximal number of subsets of $A$ that can be chosen such that the intersection of any 2 such distinct subsets has 2004 elements.

2021 European Mathematical Cup, 1

Tags: inequality , algebra

We say that a quadruple of nonnegative real numbers $(a,b,c,d)$ is [i]balanced [/i]if $$a+b+c+d=a^2+b^2+c^2+d^2.$$ Find all positive real numbers $x$ such that $$(x-a)(x-b)(x-c)(x-d)\geq 0$$ for every balanced quadruple $(a,b,c,d)$. \\ \\ (Ivan Novak)

2009 Indonesia TST, 3

Tags: power of a point , radical axis , geometry

Let $ ABC$ be an acute triangle with $ \angle BAC\equal{}60^{\circ}$. Let $ P$ be a point in triangle $ ABC$ with $ \angle APB\equal{}\angle BPC\equal{}\angle CPA\equal{}120^{\circ}$. The foots of perpendicular from $ P$ to $ BC,CA,AB$ are $ X,Y,Z$, respectively. Let $ M$ be the midpoint of $ YZ$. a) Prove that $ \angle YXZ\equal{}60^{\circ}$ b) Prove that $ X,P,M$ are collinear.

2021 JHMT HS, 8

Tags: calculus , integration , logarithm

Find the unique integer $a > 1$ that satisfies \[ \int_{a}^{a^2} \left(\frac{1}{\ln x} - \frac{2}{(\ln x)^3}\right) dx = \frac{a}{\ln a}. \]

the 11th XMO, 3

Tags: prime number , number theory

Let $p$ is a prime and $p\equiv 2\pmod 3$. For $\forall a\in\mathbb Z$, if $$p\mid \prod\limits_{i=1}^p(i^3-ai-1),$$then $a$ is called a "GuGu" number. How many "GuGu" numbers are there in the set $\{1,2,\cdots ,p\}?$ (We are allowed to discuss now. It is after 00:00 Feb 14 Beijing Time)

2025 Philippine MO, P3

Tags: sequence , algebra , arithmetic sequence , floor function

Let $d$ be a positive integer. Define the sequence $a_1, a_2, a_3, \dots$ such that \[\begin{cases} a_1 = 1 \\ a_{n+1} = n\left\lfloor\frac{a_n}{n}\right\rfloor + d, \quad n \ge 1.\end{cases}\] Prove that there exists a positive integer $M$ such that $a_M, a_{M+1}, a_{M+2}, \dots$ is an arithmetic sequence.

2014 PUMaC Team, 3

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How many integer $x$ are there such that $\frac{x^2-6}{x-6}$ is a positive integer?

2021 VIASM Math Olympiad Test, Problem 1

Tags: geometry , rectangle , combinatorics

Given a $8$x$8$ square board a) Prove that: for any ways to color the board, we are always be able to find a rectangle consists of $8$ squares such that these squares are not colored. b) Prove that: we can color $7$ squares on the board such that for any rectangles formed by $\geq 9$ squares, there are at least $1$ colored square.

2004 Croatia National Olympiad, Problem 2

Tags: geometry

Prove that the medians from the vertices $A$ and $B$ of a triangle $ABC$ are orthogonal if and only if $BC^2+AC^2=5AB^2$.

1935 Moscow Mathematical Olympiad, 006

Tags: geometry , 3d geometry , pyramid , volume

The base of a right pyramid is a quadrilateral whose sides are each of length $a$. The planar angles at the vertex of the pyramid are equal to the angles between the lateral edges and the base. Find the volume of the pyramid.

2012 Stanford Mathematics Tournament, 8

Tags: quadratic

For real numbers $(x, y, z)$ satisfying the following equations, ﬁnd all possible values of $x+y+z$ $x^2y+y^2z+z^2x=-1$ $xy^2+yz^2+zx^2=5$ $xyz=-2$

2018 Mathematical Talent Reward Programme, SAQ: P 6

Tags: coloring

Let $d(n)$ be the number of divisors of $n,$ where $n$ is a natural number. Prove that the natural numbers can be colured by 2 colours in such way, that for any infinite increasing sequence $\left\{a_{1}, a_{2}, \cdots\right\}$ if $\left\{d\left(a_{1}\right), d\left(a_{2}\right), \cdots\right\}$ is an nonconstant geometric series then $\left\{a_{1}, a_{2}, \cdots\right\}$ does not bear same colour.

2022 Czech-Polish-Slovak Junior Match, 5

Tags: number theory , divisible

An integer $n\ge1$ is [i]good [/i] if the following property is satisfied: If a positive integer is divisible by each of the nine numbers $n + 1, n + 2, ..., n + 9$, this is also divisible by $n + 10$. How many good integers are $n\ge 1$?

2023 Assam Mathematics Olympiad, 4

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Real numbers $a, b, c$ satisfy $(2b - a)^2 + (2b - c)^2 = 2(2b^2 - ac)$. Prove that $a + c = 2b$.

2011 Baltic Way, 15

Tags: geometry

Let $ABCD$ be a convex quadrilateral such that $\angle ADB=\angle BDC$. Suppose that a point $E$ on the side $AD$ satisfies the equality \[AE\cdot ED + BE^2=CD\cdot AE.\] Show that $\angle EBA=\angle DCB$.

1962 Miklós Schweitzer, 1

Tags: advanced fields , algebra , polynomial , search

Let $ f$ and $ g$ be polynomials with rational coefficients, and let $ F$ and $ G$ denote the sets of values of $ f$ and $ g$ at rational numbers. Prove that $ F \equal{} G$ holds if and only if $ f(x) \equal{} g(ax \plus{} b)$ for some suitable rational numbers $ a\not \equal{} 0$ and $ b$. [i]E. Fried[/i]

2009 Germany Team Selection Test, 1

Tags: geometry , rectangle , combinatorics , combinatorial geometry , extremal combinatorics

In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a [i]box[/i]. Two boxes [i]intersect[/i] if they have a common point in their interior or on their boundary. Find the largest $ n$ for which there exist $ n$ boxes $ B_1$, $ \ldots$, $ B_n$ such that $ B_i$ and $ B_j$ intersect if and only if $ i\not\equiv j\pm 1\pmod n$. [i]Proposed by Gerhard Woeginger, Netherlands[/i]

2008 International Zhautykov Olympiad, 1

Tags: rhombus , geometry

Points $ K,L,M,N$ are repectively the midpoints of sides $ AB,BC,CD,DA$ in a convex quadrliateral $ ABCD$.Line $ KM$ meets dioganals $ AC$ and $ BD$ at points $ P$ and $ Q$,respectively.Line $ LN$ meets dioganals $ AC$ and $ BD$ at points $ R$ and $ S$,respectively. Prove that if $ AP\cdot PC\equal{}BQ\cdot QD$,then $ AR\cdot RC\equal{}BS\cdot SD$.

2010 Today's Calculation Of Integral, 587

Tags: calculus , integration , calculus computations

Evaluate $ \int_0^1 \frac{(x^2\plus{}3x)e^x\minus{}(x^2\minus{}3x)e^{\minus{}x}\plus{}2}{\sqrt{1\plus{}x(e^x\plus{}e^{\minus{}x})}}\ dx$.

2009 Puerto Rico Team Selection Test, 3

Tags: altitude , geometry , geometric inequality

Show that if $ h_A, h_B,$ and $ h_C$ are the altitudes of $ \triangle ABC$, and $ r$ is the radius of the incircle, then $$ h_A + h_B + h_C \ge 9r$$

2018 HMIC, 4

Tags: college , combinatorics

Find all functions $f: \mathbb{R}^+\to\mathbb{R}^+$ such that \[f(x+f(y+xy))=(y+1)f(x+1)-1\]for all $x,y\in\mathbb{R}^+$. ($\mathbb{R}^+$ denotes the set of positive real numbers.)

2010 LMT, 2

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Let points $A,B,$ and $C$ lie on a line such that $AB=1, BC=1,$ and $AC=2.$ Let $C_1$ be the circle centered at $A$ passing through $B,$ and let $C_2$ be the circle centered at $A$ passing through $C.$ Find the area of the region outside $C_1,$ but inside $C_2.$

2008 Purple Comet Problems, 9

Tags: geometry , 3d geometry

Find the sum of all the integers $N > 1$ with the properties that the each prime factor of $N $ is either $2, 3,$ or $5,$ and $N$ is not divisible by any perfect cube greater than $1.$

2002 China Team Selection Test, 2

Tags: inequalities , trigonometry , geometry

$ A_1$, $ B_1$ and $ C_1$ are the projections of the vertices $ A$, $ B$ and $ C$ of triangle $ ABC$ on the respective sides. If $ AB \equal{} c$, $ AC \equal{} b$, $ BC \equal{} a$ and $ AC_1 \equal{} 2t AB$, $ BA_1 \equal{} 2rBC$, $ CB_1 \equal{} 2 \mu AC$. Prove that: \[ \frac {a^2}{b^2} \cdot \left( \frac {t}{1 \minus{} 2t} \right)^2 \plus{} \frac {b^2}{c^2} \cdot \left( \frac {r}{1 \minus{} 2r} \right)^2 \plus{} \frac {c^2}{a^2} \cdot \left( \frac {\mu}{1 \minus{} 2\mu} \right)^2 \plus{} 16tr \mu \geq 1 \]

1925 Eotvos Mathematical Competition, 3

Tags: geometry , inradius , geometric inequalities , right triangle

Let $r$ be the radius of the inscribed circle of a right triangle $ABC$. Show that $r$ is less than half of either leg and less than one fourth of the hypotenuse.