Olympiad problems

2023 South East Mathematical Olympiad, 3

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 $AD$.

2018 MIG, 8

Tags: 2018 Individual

The set of natural numbers are arranged as so: $$\begin{array}{ccccccccc} & & & & 1 & & & &\\ & & & 2 & 3 & 4 & & &\\ & & 5 & 6 & 7 & 8 & 9 &\\ & 10 & 11 & 12 & 13 & 14 & 15 & 16 &\\ 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 & 25\\ & & & & \vdots & & & & \end{array}$$ so that each row has $2$ more numbers in it, and the rows are centered. What is the number under $49$? $\textbf{(A) }60\qquad\textbf{(B) }61\qquad\textbf{(C) }62\qquad\textbf{(D) }63\qquad\textbf{(E) }64$

LMT Speed Rounds, 2010.4

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Determine the largest positive integer that is a divisor of all three of $A=2^{2010}\times3^{2010}, B=3^{2010}\times5^{2010},$ and $C=5^{2010}\times2^{2010}.$

2022 Purple Comet Problems, 7

Tags: Purple Comet

The value of $$\left(1-\frac{1}{2^2-1}\right)\left(1-\frac{1}{2^3-1}\right)\left(1-\frac{1}{2^4-1}\right)\dots\left(1-\frac{1}{2^{29}-1}\right)$$ can be written as $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $2m - n.$

Ukraine Correspondence MO - geometry, 2019.11

Tags: geometry , perpendicular , arc midpoint , Ukraine Correspondence

Let $O$ be the center of the circle circumscribed around the acute triangle $ABC$, and let $N$ be the midpoint of the arc $ABC$ of this circle. On the sides $AB$ and $BC$ mark points $D$ and $E$ respectively, such that the point $O$ lies on the segment $DE$. The lines $DN$ and $BC$ intersect at the point $P$, and the lines $EN$ and $AB$ intersect at the point $Q$. Prove that $PQ \perp AC$.

2007 Sharygin Geometry Olympiad, 1

Tags: geometry , Equilateral , Equilateral Triangle , isosceles , Isosceles Triangle , angles

A triangle is cut into several (not less than two) triangles. One of them is isosceles (not equilateral), and all others are equilateral. Determine the angles of the original triangle.

2002 Poland - Second Round, 3

Tags: inequalities , inequalities proposed

Find all positive integers $n$ such that for all real numbers $x_1,x_2,\ldots ,x_n,y_1,y_2,\ldots ,y_n$ the following inequality holds: \[ x_1x_2\ldots x_n+y_1y_2\ldots y_n\le\sqrt{x_1^2+y_1^2}\cdot\sqrt{x_2^2+y_2^2}\cdot \cdots \sqrt{x_n^2+y_n^2}\cdot \]

Kvant 2023, M2747

Tags: geometry , 3D geometry , tetrahedron

In the tetrahedron $ABCD,$ on the continuation of the edges $AB, AC$ and $AD$, three points were marked for point $A{},$ located from $A{}$ at a distance equal to the semi-perimeter of the triangle $BCD.$ Similarly, we marked three points corresponding to vertices $B, C$ and $D.$ Prove that if there is a sphere touching all the edges of the tetrahedron $ABCD$, then the marked 12 points lie on the same sphere. [i]Proposed by V. Alexandrov[/i]

2011 Albania Team Selection Test, 1

Tags: analytic geometry , conics , parabola , function , algebra unsolved , algebra

The given parabola $y=ax^2+bx+c$ doesn't intersect the $X$-axis and passes from the points $A(-2,1)$ and $B(2,9)$. Find all the possible values of the $x$ coordinates of the vertex of this parabola.

1964 AMC 12/AHSME, 19

Tags: AMC

If $2x-3y-z=0$ and $x+3y-14z=0$, $z \neq 0$, the numerical value of $\frac{x^2+3xy}{y^2+z^2}$ is: $ \textbf{(A)}\ 7\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ -20/17\qquad\textbf{(E)}\ -2 $

Kvant 2021, M2670

Tags: geometry , combinatorial geometry , Kvant

There are 100 points on the plane so that any 10 of them are vertices of a convex polygon. Does it follow from this that all these points are the vertices of a convex 100-gon? [i]From the folklore[/i]

2023 ISI Entrance UGB, 6

Tags: inequalities , recurrence relation

Let $\{u_n\}_{n \ge 1}$ be a sequence of real numbers defined as $u_1 = 1$ and \[ u_{n+1} = u_n + \frac{1}{u_n} \text{ for all $n \ge 1$.}\] Prove that $u_n \le \frac{3\sqrt{n}}{2}$ for all $n$.

2014 IMO Shortlist, G7

Tags: IMO Shortlist , geometry

Let $ABC$ be a triangle with circumcircle $\Omega$ and incentre $I$. Let the line passing through $I$ and perpendicular to $CI$ intersect the segment $BC$ and the arc $BC$ (not containing $A$) of $\Omega$ at points $U$ and $V$ , respectively. Let the line passing through $U$ and parallel to $AI$ intersect $AV$ at $X$, and let the line passing through $V$ and parallel to $AI$ intersect $AB$ at $Y$ . Let $W$ and $Z$ be the midpoints of $AX$ and $BC$, respectively. Prove that if the points $I, X,$ and $Y$ are collinear, then the points $I, W ,$ and $Z$ are also collinear. [i]Proposed by David B. Rush, USA[/i]

2014 Taiwan TST Round 1, 6

Tags: combinatorics , graph theory , IMO Shortlist

In some country several pairs of cities are connected by direct two-way flights. It is possible to go from any city to any other by a sequence of flights. The distance between two cities is defined to be the least possible numbers of flights required to go from one of them to the other. It is known that for any city there are at most $100$ cities at distance exactly three from it. Prove that there is no city such that more than $2550$ other cities have distance exactly four from it.

2015 Math Prize for Girls Problems, 7

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Let $n$ be a positive integer. In $n$-dimensional space, consider the $2^n$ points whose coordinates are all $\pm 1$. Imagine placing an $n$-dimensional ball of radius 1 centered at each of these $2^n$ points. Let $B_n$ be the largest $n$-dimensional ball centered at the origin that does not intersect the interior of any of the original $2^n$ balls. What is the smallest value of $n$ such that $B_n$ contains a point with a coordinate greater than 2?

2002 Korea - Final Round, 1

Tags: algebra , polynomial , inequalities , algebra unsolved

For $n \ge 3$, let $S=a_1+a_2+\cdots+a_n$ and $T=b_1b_2\cdots b_n$ for positive real numbers $a_1,a_2,\ldots,a_n, b_1,b_2 ,\ldots,b_n$, where the numbers $b_i$ are pairwise distinct. (a) Find the number of distinct real zeroes of the polynomial \[f(x)=(x-b_1)(x-b_2)\cdots(x-b_n)\sum_{j=1}^n \frac{a_j}{x-b_j}\] (b) Prove the inequality \[\frac1{n-1}\sum_{j=1}^n \left(1-\frac{a_j}{S}\right)b_j > \left(\frac{T}{S}\sum_{j=1}^{n} \frac{a_j}{b_j}\right)^{\frac1{n-1}}\]

2002 Iran MO (2nd round), 6

Tags: combinatorics proposed , combinatorics

Let $G$ be a simple graph with $100$ edges on $20$ vertices. Suppose that we can choose a pair of disjoint edges in $4050$ ways. Prove that $G$ is regular.

2012 AMC 10, 19

Tags: algebra , linear equation , AMC , AMC10 , AMC12

Paula the painter and her two helpers each paint at constant, but different, rates. They always start at $\text{8:00 AM}$, and all three always take the same amount of time to eat lunch. On Monday the three of them painted $50\%$ of a house, quitting at $\text{4:00 PM}$. On Tuesday, when Paula wasn't there, the two helpers painted only $24\%$ of the house and quit at $\text{2:12 PM}$. On Wednesday Paula worked by herself and finished the house by working until $\text{7:12 PM}$. How long, in minutes, was each day's lunch break? $ \textbf{(A)}\ 30 \qquad\textbf{(B)}\ 36 \qquad\textbf{(C)}\ 42 \qquad\textbf{(D)}\ 48 \qquad\textbf{(E)}\ 60 $

2022 Iberoamerican, 4

Tags: combinatorics

Let $n> 2$ be a positive integer. Given is a horizontal row of $n$ cells where each cell is painted blue or red. We say that a block is a sequence of consecutive boxes of the same color. Arepito the crab is initially standing at the leftmost cell. On each turn, he counts the number $m$ of cells belonging to the largest block containing the square he is on, and does one of the following: If the square he is on is blue and there are at least $m$ squares to the right of him, Arepito moves $m$ squares to the right; If the square he is in is red and there are at least $m$ squares to the left of him, Arepito moves $m$ cells to the left; In any other case, he stays on the same square and does not move any further. For each $n$, determine the smallest integer $k$ for which there is an initial coloring of the row with $k$ blue cells, for which Arepito will reach the rightmost cell.

1980 AMC 12/AHSME, 27

Tags: geometry , 3D geometry

The sum $\sqrt[3] {5+2\sqrt{13}}+\sqrt[3]{5-2\sqrt{13}}$ equals $\text{(A)} \ \frac 32 \qquad \text{(B)} \ \frac{\sqrt[3]{65}}{4} \qquad \text{(C)} \ \frac{1+\sqrt[6]{13}}{2} \qquad \text{(D)} \ \sqrt[3]{2} \qquad \text{(E)} \ \text{none of these}$

2013 CHMMC (Fall), 8

Tags: combinatorics

Two kids $A$ and $B$ play a game as follows: from a box containing $n$ marbles ($n > 1$), they alternately take some marbles for themselves, such that: 1. $A$ goes first. 2. The number of marbles taken by $A$ in his first turn, denoted by $k$, must be between $1$ and $n - 1$, inclusive. 3. The number of marbles taken in a turn by any player must be between $1$ and $k$, inclusive. The winner is the one who takes the last marble. Determine all natural numbers $n$ for which $A$ has a winning strategy

2020 USMCA, 21

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Let $ABCDEF$ be a regular octahedron with unit side length, such that $ABCD$ is a square. Points $G, H$ are on segments $BE, DF$ respectively. The planes $AGD$ and $BCH$ divide the octahedron into three pieces, each with equal volume. Compute $BG$.

1995 Poland - First Round, 2

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A number is called a palindromic number if its decimal representation read from the left to the right is the same as read from the right to the left. Let $(x_n)$ be the increasing sequence of all palindromic numbers. Determine all primes, which are divisors of at least one of the differences $x_{n+1} - x_n$.

2012 Turkey Junior National Olympiad, 2

Tags: geometry , circumcircle , ratio , trapezoid , symmetry , geometric transformation , reflection

In a convex quadrilateral $ABCD$, the diagonals are perpendicular to each other and they intersect at $E$. Let $P$ be a point on the side $AD$ which is different from $A$ such that $PE=EC.$ The circumcircle of triangle $BCD$ intersects the side $AD$ at $Q$ where $Q$ is also different from $A$. The circle, passing through $A$ and tangent to line $EP$ at $P$, intersects the line segment $AC$ at $R$. If the points $B, R, Q$ are concurrent then show that $\angle BCD=90^{\circ}$.

2009 Today's Calculation Of Integral, 491

Tags: calculus , integration , trigonometry , geometry , calculus computations

Let $ f(x)\equal{}\sin 3x\plus{}\cos x,\ g(x)\equal{}\cos 3x\plus{}\sin x.$ (1) Evaluate $ \int_0^{2\pi} \{f(x)^2\plus{}g(x)^2\}\ dx$. (2) Find the area of the region bounded by two curves $ y\equal{}f(x)$ and $ y\equal{}g(x)\ (0\leq x\leq \pi).$