Olympiad problems

2014 Sharygin Geometry Olympiad, 5

In triangle $ABC$ $\angle B = 60^o, O$ is the circumcenter, and $L$ is the foot of an angle bisector of angle $B$. The circumcirle of triangle $BOL$ meets the circumcircle of $ABC$ at point $D \ne B$. Prove that $BD \perp AC$. (D. Shvetsov)

2022 AIME Problems, 1

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Adults made up $\frac5{12}$ of the crowd of people at a concert. After a bus carrying $50$ more people arrived, adults made up $\frac{11}{25}$ of the people at the concert. Find the minimum number of adults who could have been at the concert after the bus arrived.

2019 Nigerian Senior MO Round 4, 2

Tags: geometry , midpoint , perpendicular , circles

Let $K,L, M$ be the midpoints of $BC,CA,AB$ repectively on a given triangle $ABC$. Let $\Gamma$ be a circle passing through $B$ and tangent to the circumcircle of $KLM$, say at $X$. Suppose that $LX$ and $BC$ meet at $\Gamma$ . Show that $CX$ is perpendicular to $AB$.

2015 Balkan MO Shortlist, N6

Tags: combinatorics , number theory

Prove that among $20$ consecutive positive integers there is an integer $d$ such that for every positive integer $n$ the following inequality holds $$n \sqrt{d} \left\{n \sqrt {d} \right \} > \dfrac{5}{2}$$ where by $\left \{x \right \}$ denotes the fractional part of the real number $x$. The fractional part of the real number $x$ is defined as the difference between the largest integer that is less than or equal to $x$ to the actual number $x$. [i](Serbia)[/i]

2010 VTRMC, Problem 3

Tags: equation , algebra

Solve in $R$ the equation: $8x^3-4x^2-4x+1=0$

1998 All-Russian Olympiad Regional Round, 10.3

Tags: point , combinatorics , combinatorial geometry , geometry

Prove that from any finite set of points on the plane, you can remove a point from the bottom in such a way that the remaining set can be split into two parts of smaller diameter. (Diameter is the maximum distance between points of the set.) [hide=original wording]Докажите, что из любого конечного множества точек на плоскости можно так удалитьо дну точку, что оставшееся множество можно разбить на две части меньшего диаметра. (Диаметр—это максимальное расстояние между точками множества.)[/hide]

2019 USMCA, 8

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Find all pairs of positive integers $(m, n)$ such that $(2^m - 1)(2^n - 1)$ is a perfect square.

1997 French Mathematical Olympiad, Problem 3

Tags: 3d geometry , geometry

Let $C$ be a unit cube and let $p$ denote the orthogonal projection onto the plane. Find the maximum area of $p(C)$.

2010 Princeton University Math Competition, 3

Tags: digit , decimal representation , number theory

Find (with proof) all natural numbers $n$ such that, for some natural numbers $a$ and $b$, $a\ne b$, the digits in the decimal representations of the two numbers $n^a+1$ and $n^b+1$ are in reverse order.

2017 India IMO Training Camp, 3

Tags: linear algebra , matrix , combinatorics

Let $n \ge 1$ be a positive integer. An $n \times n$ matrix is called [i]good[/i] if each entry is a non-negative integer, the sum of entries in each row and each column is equal. A [i]permutation[/i] matrix is an $n \times n$ matrix consisting of $n$ ones and $n(n-1)$ zeroes such that each row and each column has exactly one non-zero entry. Prove that any [i]good[/i] matrix is a sum of finitely many [i]permutation[/i] matrices.

1995 India Regional Mathematical Olympiad, 3

Tags: number theory , modular arithmetic , sum of digits

Prove that among any $18$ consecutive three digit numbers there is at least one number which is divisible by the sum of its digits.

2009 India IMO Training Camp, 8

Tags: modular arithmetic , number theory

Let $ n$ be a natural number $ \ge 2$ which divides $ 3^n\plus{}4^n$.Prove That $ 7\mid n$.

2018 USAMO, 5

Tags: geometry , cyclic quadrilateral , miquel point , prism lemma , inversion

In convex cyclic quadrilateral $ABCD$, we know that lines $AC$ and $BD$ intersect at $E$, lines $AB$ and $CD$ intersect at $F$, and lines $BC$ and $DA$ intersect at $G$. Suppose that the circumcircle of $\triangle ABE$ intersects line $CB$ at $B$ and $P$, and the circumcircle of $\triangle ADE$ intersects line $CD$ at $D$ and $Q$, where $C,B,P,G$ and $C,Q,D,F$ are collinear in that order. Prove that if lines $FP$ and $GQ$ intersect at $M$, then $\angle MAC = 90^\circ$. [i]Proposed by Kada Williams[/i]

2008 China Team Selection Test, 1

Tags: geometry , circumcircle , incenter , geometric transformation , reflection , ratio , trigonometry

Let $ ABC$ be an acute triangle, let $ M,N$ be the midpoints of minor arcs $ \widehat{CA},\widehat{AB}$ of the circumcircle of triangle $ ABC,$ point $ D$ is the midpoint of segment $ MN,$ point $ G$ lies on minor arc $ \widehat{BC}.$ Denote by $ I,I_{1},I_{2}$ the incenters of triangle $ ABC,ABG,ACG$ respectively.Let $ P$ be the second intersection of the circumcircle of triangle $ GI_{1}I_{2}$ with the circumcircle of triangle $ ABC.$ Prove that three points $ D,I,P$ are collinear.

2010 AMC 12/AHSME, 13

Tags: trigonometry , function

In $ \triangle ABC, \ \cos(2A \minus{} B) \plus{} \sin(A\plus{}B) \equal{} 2$ and $ AB\equal{}4.$ What is $ BC?$ $ \textbf{(A)}\ \sqrt{2} \qquad \textbf{(B)}\ \sqrt{3} \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 2\sqrt{2} \qquad \textbf{(E)}\ 2\sqrt{3}$

1985 Traian Lălescu, 1.2

Tags: function , algebra , polynomial

Find the first degree polynomial function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that satisfy the equation $$ f(x-1)=-3x-5-f(2), $$ for all real numbers $ x. $

2002 Moldova Team Selection Test, 3

Tags: geometry , incenter , similar triangles , arc

A triangle $ABC$ is inscribed in a circle $G$. Points $M$ and $N$ are the midpoints of the arcs $BC$ and $AC$ respectively, and $D$ is an arbitrary point on the arc $AB$ (not containing $C$). Points $I_1$ and $I_2$ are the incenters of the triangles $ADC$ and $BDC$, respectively. If the circumcircle of triangle $DI_1I_2$ meets $G$ again at $P$, prove that triangles $PNI_1$ and $PMI_2$ are similar.

2010 Iran MO (3rd Round), 4

Tags: floor function , ceiling function , combinatorics

suppose that $\mathcal F\subseteq X^{(K)}$ and $|X|=n$. we know that for every three distinct elements of $\mathcal F$ like $A,B$ and $C$ we have $A\cap B \not\subset C$. a)(10 points) Prove that : \[|\mathcal F|\le \dbinom{k}{\lfloor\frac{k}{2}\rfloor}+1\] b)(15 points) if elements of $\mathcal F$ do not necessarily have $k$ elements, with the above conditions show that: \[|\mathcal F|\le \dbinom{n}{\lceil\frac{n-2}{3}\rceil}+2\]

2015 Hanoi Open Mathematics Competitions, 15

Tags: maximum , inequalities , algebra

Let the numbers $a, b,c$ satisfy the relation $a^2+b^2+c^2 \le 8$. Determine the maximum value of $M = 4(a^3 + b^3 + c^3) - (a^4 + b^4 + c^4)$

Novosibirsk Oral Geo Oly IX, 2021.6

Tags: geometry , rectangle

Two congruent rectangles are located as shown in the figure. Find the area of the shaded part. [img]https://cdn.artofproblemsolving.com/attachments/2/e/10b164535ab5b3a3b98ce1a0b84892cd11d76f.png[/img]

2021 BMT, 1

Tags: combinatorics

Towa has a hand of three different red cards and three different black cards. How many ways can Towa pick a set of three cards from her hand that uses at least one card of each color?

1997 Baltic Way, 5

Tags: number theory

In a sequence $u_0,u_1,\ldots $ of positive integers, $u_0$ is arbitrary, and for any non-negative integer $n$, \[ u_{n+1}=\begin{cases}\frac{1}{2}u_n & \text{for even }u_n \\ a+u_n & \text{for odd }u_n \end{cases} \] where $a$ is a fixed odd positive integer. Prove that the sequence is periodic from a certain step.

1980 IMO, 12

Tags: circumcircle , geometry

There is a triangle $ABC$. Its circumcircle and its circumcentre are given. Show how the orthocentre of $ABC$ may be constructed using only a straightedge (unmarked ruler). [The straightedge and paper may be assumed large enough for the construction to be completed]

2023 South East Mathematical Olympiad, 3

Tags: geometry , angle bisector

In $\triangle {ABC}$, ${D}$ is on the internal angle bisector of $\angle BAC$ and $\angle ADB=\angle ACD$. $E, F$ is on the external angle bisector of $\angle BAC$, such that $AE=BE$ and $AF=CF$. The circumcircles of $\triangle ACE$ and $\triangle ABF$ intersects at ${A}$ and ${K}$ and $A'$ is the reflection of ${A}$ with respect to $BC$. Prove that: if $AD=BC$, then the circumcenter of $\triangle AKA'$ is on line $AD$.

2017 Peru Iberoamerican Team Selection Test, P2

Tags: number theory

Determine if there exists a positive integer $n$ such that $n^2+11$ is a prime number and $n+4$ is a perfect cube.