Olympiad problems

1998 May Olympiad, 3

Tags: coloring , combinatorics

Given a $4 \times 4$ grid board with each square painted a different color, you want to cut it into two pieces of equal area by making a single cut along the grid lines. In how many ways can it be done?

2012 Kazakhstan National Olympiad, 2

Tags: circumcircle , symmetry , cyclic quadrilateral , angle bisector , perpendicular bisector , geometry

Let $ABCD$ be an inscribed quadrilateral, in which $\angle BAD<90$. On the rays $AB$ and $AD$ are selected points $K$ and $L$, respectively, such that$ KA = KD, LA = LB$. Let $N$ - the midpoint of $AC$.Prove that if $\angle BNC=\angle DNC $,so $\angle KNL=\angle BCD $

2016 Latvia Baltic Way TST, 1

Tags: combinatorics

$2016$ numbers written on the board: $\frac{1}{2016}, \frac{2}{2016}, \frac{3}{2016}, ..., \frac{2016}{2016}$. In one move, it is allowed to choose any two numbers $a$ and $b$ written on the board, delete them, and write the number $3ab - 2a - 2b + 2$ instead. Determine what number will remain written on the board after $2015$ moves.

MOAA Gunga Bowls, 2023.2

Tags:

Harry wants to put $5$ identical blue books, $3$ identical red books, and $1$ white book on his bookshelf. If no two adjacent books may be the same color, how many distinct arrangements can Harry make? [i]Proposed by Anthony Yang[/i]

Putnam 1939, B2

Tags:

Evaluate $\int_{1}^{3} ( (x - 1)(3 - x) )^{\dfrac{-1}{2}} dx$ and $\int_{1}^{\infty} (e^{x+1} + e^{3-x})^{-1} dx.$

2017 Azerbaijan Junior National Olympiad, P4

Tags: geometry , rhombus , trapezoid

A Rhombus and an Isosceles trapezoid that has same area is drawn in the same circle's outside. Compare their acute angles \\ (explain your answer)

2023 Girls in Mathematics Tournament, 2

Tags: algebra

Let $a,b,c$ real numbers such that $a^n+b^n= c^n$ for three positive integers consecutive of $n$. Prove that $abc= 0$

2021 MMATHS, 4

Tags:

Let triangle $ABC$ with incenter $I$ and circumcircle $\Gamma$ satisfy $AB = 6\sqrt{3}, BC = 14,$ and $CA = 22$. Construct points $P$ and $Q$ on rays $BA$ and $CA$ such that $BP = CQ = 14$. Lines $PI$ and $QI$ meet the tangents from $B$ and $C$ to $\Gamma$, respectively, at points $X$ and $Y$. If $XY$ can be expressed as $a\sqrt{b} - c$ for positive integers $a,b,c$ with $c$ squarefree, find $a + b + c$. [i]Proposed by Andrew Wu[/i]

2010 Today's Calculation Of Integral, 596

Tags: calculus , integration , trigonometry , calculus computations

Find the minimum value of $\int_0^{\frac{\pi}{2}} |a\sin 2x-\cos ^ 2 x|dx\ (a>0).$ 2009 Shimane University entrance exam/Medicine

1998 Tournament Of Towns, 3

Tags: combinatorics , chessboard

On an $8 \times 8$ chessboard, $17$ cells are marked. Prove that one can always choose two cells among the marked ones so that a Knight will need at least three moves to go from one of the chosen cells to the other. (R Zhenodarov)

2009 Brazil Team Selection Test, 3

Tags: number theory , modular arithmetic , sequence , divisibility

Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$. [i]Proposed by Mohsen Jamaali, Iran[/i]

2020 Saint Petersburg Mathematical Olympiad, 5.

Tags: geometric inequality , inequalities

The altitudes $BB_1$ and $CC_1$ of the acute triangle $\triangle ABC$ intersect at $H$. The circle centered at $O_b$ passes through points $A,C_1$, and the midpoint of $BH$. The circle centered at $O_c$ passes through $A,B_1$ and the midpoint of $CH$. Prove that $B_1 O_b +C_1O_c > \frac{BC}{4}$

2021 Regional Competition For Advanced Students, 3

Tags: number theory , game , combinatorics

The numbers $1, 2, ..., 2020$ and $2021$ are written on a blackboard. The following operation is executed: Two numbers are chosen, both are erased and replaced by the absolute value of their difference. This operation is repeated until there is only one number left on the blackboard. (a) Show that $2021$ can be the final number on the blackboard. (b) Show that $2020$ cannot be the final number on the blackboard. (Karl Czakler)

2016 Dutch BxMO TST, 4

Tags: combinatorics , group theory

The Facebook group Olympiad training has at least five members. There is a certain integer $k$ with following property: [i]for each $k$-tuple of members there is at least one member of this $k$-tuple friends with each of the other $k - 1$.[/i] (Friendship is mutual: if $A$ is friends with $B$, then also $B$ is friends with $A$.) (a) Suppose $k = 4$. Can you say with certainty that the Facebook group has a member that is friends with each of the other members? (b) Suppose $k = 5$. Can you say with certainty that the Facebook group has a member that is friends with each of the other members?

2004 Nordic, 1

Tags: algebra

Twenty-seven balls labelled from $1$ to $27$ are distributed in three bowls: red, blue, and yellow. What are the possible values of the number of balls in the red bowl if the average labels in the red, blue and yellow bowl are $15$, $3$, and $18$, respectively?

2003 Costa Rica - Final Round, 5

Tags: geometric transformation , rotation , combinatorics , geometry

Each of the squares of an $8 \times 8$ board can be colored white or black. Find the number of colorings of the board such that every $2 \times 2$ square contains exactly 2 black squares and 2 white squares.

2015 NIMO Problems, 5

Tags: algebra , polynomial , complex number

Let $a, b, c, d, e,$ and $f$ be real numbers. Define the polynomials \[ P(x) = 2x^4 - 26x^3 + ax^2 + bx + c \quad\text{ and }\quad Q(x) = 5x^4 - 80x^3 + dx^2 + ex + f. \] Let $S$ be the set of all complex numbers which are a root of [i]either[/i] $P$ or $Q$ (or both). Given that $S = \{1,2,3,4,5\}$, compute $P(6) \cdot Q(6).$ [i]Proposed by Michael Tang[/i]

2000 Italy TST, 4

Tags: combinatorics

On a mathematical competition $ n$ problems were given. The final results showed that: (i) on each problem, exactly three contestants scored $ 7$ points; (ii) for each pair of problems, exactly one contestant scored $ 7$ points on both problems. Prove that if $ n \geq 8$, then there is a contestant who got $ 7$ points on each problem. Is this statement necessarily true if $ n \equal{} 7$?

2023 Canadian Junior Mathematical Olympiad, 2

Tags: geometry , angle bisector , circumcircle

An acute triangle is a triangle that has all angles less that $90^{\circ}$ ($90^{\circ}$ is a Right Angle). Let $ABC$ be a right-angled triangle with $\angle ACB =90^{\circ}.$ Let $CD$ be the altitude from $C$ to $AB,$ and let $E$ be the intersection of the angle bisector of $\angle ACD$ with $AD.$ Let $EF$ be the altitude from $E$ to $BC.$ Prove that the circumcircle of $BEF$ passes through the midpoint of $CE.$

2022 Germany Team Selection Test, 1

Tags: combinatorics , graph theory , greatest common divisor

Let $S$ be an infinite set of positive integers, such that there exist four pairwise distinct $a,b,c,d \in S$ with $\gcd(a,b) \neq \gcd(c,d)$. Prove that there exist three pairwise distinct $x,y,z \in S$ such that $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.

1993 China National Olympiad, 2

Tags: inequalities

Given a natural number $k$ and a real number $a (a>0)$, find the maximal value of $a^{k_1}+a^{k_2}+\cdots +a^{k_r}$, where $k_1+k_2+\cdots +k_r=k$ ($k_i\in \mathbb{N} ,1\le r \le k$).

2018 District Olympiad, 3

Tags: inequalities , logarithm

Let $a, b, c$ be strictly positive real numbers such that $1 < b \le c^2 \le a^{10}$, and \[\log_ab + 2\log_bc + 5\log_ca = 12.\] Prove that \[2\log_ac + 5\log_cb + 10\log_ba \ge 21.\]

2020 Mediterranean Mathematics Olympiad, 4

Tags: geometry , collinear

Let $P,Q,R$ be three points on a circle $k_1$ with $|PQ|=|PR|$ and $|PQ|>|QR|$. Let $k_2$ be the circle with center in $P$ that goes through $Q$ and $R$. The circle with center $Q$ through $R$ intersects $k_1$ in another point $X\ne R$ and intersects $k_2$ in another point $Y\ne R$. The two points $X$ and $R$ lie on different sides of the line through $PQ$. Show that the three points $P$, $X$, $Y$ lie on a common line.

2010 AMC 10, 5

Tags: geometry

The area of a circle whose circumference is $ 24\pi$ is $ k\pi$. What is the value of $ k$? $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 36 \qquad \textbf{(E)}\ 144$

2012 German National Olympiad, 3

Tags: geometry , circumcircle , tangent , triangle , circles

Let $ABC$ a triangle and $k$ a circle such that: (1) The circle $k$ passes through $A$ and $B$ and touches the line $AC.$ (2) The tangent to $k$ at $B$ intersects the line $AC$ in a point $X\ne C.$ (3) The circumcircle $\omega$ of $BXC$ intersects $k$ in a point $Q\ne B.$ (4) The tangent to $\omega$ at $X$ intersects the line $AB$ in a point $Y.$ Prove that the line $XY$ is tangent to the circumcircle of $BQY.$