Olympiad problems

2018 Oral Moscow Geometry Olympiad, 4

On the side $AB$ of the triangle $ABC$, point $M$ is selected. In triangle $ACM$ point $I_1$ is the center of the inscribed circle, $J_1$ is the center of excircle wrt side $CM$. In the triangle $BCM$ point $I_2$ is the center of the inscribed circle, $J_2$ is the center of excircle wrt side $CM$. Prove that the line passing through the midpoints of the segments $I_1I_2$ and $J_1J_2$ is perpendicular to $AB$.

2021 Philippine MO, 7

Tags: algebra , number theory , inequalities , Philippines , PMO

Let $a, b, c,$ and $d$ be real numbers such that $a \geq b \geq c \geq d$ and $$a+b+c+d = 13$$ $$a^2+b^2+c^2+d^2=43.$$ Show that $ab \geq 3 + cd$.

2018 Greece Junior Math Olympiad, 2

Tags: rectangle , combinatorics , Coloring

A $8\times 8$ board is given. Seven out of $64$ unit squares are painted black. Suppose that there exists a positive $k$ such that no matter which squares are black, there exists a rectangle (with sides parallel to the sides of the board) with area $k$ containing no black squares. Find the maximum value of $k$.

2022 Brazil Team Selection Test, 1

Tags: geometry , IMO Shortlist , AZE IMO TST , Hi

Let $ABCD$ be a parallelogram with $AC=BC.$ A point $P$ is chosen on the extension of ray $AB$ past $B.$ The circumcircle of $ACD$ meets the segment $PD$ again at $Q.$ The circumcircle of triangle $APQ$ meets the segment $PC$ at $R.$ Prove that lines $CD,AQ,BR$ are concurrent.

2024 Saint Petersburg Mathematical Olympiad, 1

Tags: combinatorics , algebra

The $100 \times 100$ table is filled with numbers from $1$ to $10 \ 000$ as shown in the figure. Is it possible to rearrange some numbers so that there is still one number in each cell, and so that the sum of the numbers does not change in all rectangles of three cells?

2015 NIMO Problems, 7

Tags: combinatorics , probability

In a $4\times 4$ grid of unit squares, five squares are chosen at random. The probability that no two chosen squares share a side is $\tfrac mn$ for positive relatively prime integers $m$ and $n$. Find $m+n$. [i]Proposed by David Altizio[/i]

1992 Chile National Olympiad, 7

Tags: combinatorics , Magic squares , Sum , board

$\bullet$ Determine a natural $n$ such that the constant sum $S$ of a magic square of $ n \times n$ (that is, the sum of its elements in any column, or the diagonal) differs as little as possible from $1992$. $\bullet$ Construct or describe the construction of this magic square.

Kyiv City MO Seniors 2003+ geometry, 2013.10.4

Tags: bisects segment , circles , tangent circles , geometry

The two circles ${{w} _ {1}}, \, \, {{w} _ {2}}$ touch externally at the point $Q$. The common external tangent of these circles is tangent to ${{w} _ {1}}$ at the point $B$, $BA$ is the diameter of this circle. A tangent to the circle ${{w} _ {2}} $ is drawn through the point $A$, which touches this circle at the point $C$, such that the points $B$ and $C$ lie in one half-plane relative to the line $AQ$. Prove that the circle ${{w} _ {1}}$ bisects the segment $C $. (Igor Nagel)

2007 Moldova National Olympiad, 10.3

Tags: algebra unsolved , algebra

Determine strictly positive real numbers $ a_{1},a_{2},...,a_{n}$ if for any $ n\in N^*$ takes place equality: $ a_{1}^2\plus{}a_{2}^2\plus{}...\plus{}a_{n}^2\equal{}a_{1}\plus{}a_{2}\plus{}...\plus{}a_{n}\plus{}\frac{n(n^2\plus{}6n\plus{}11)}{3}$

1999 ITAMO, 5

Tags: combinatorics , grid

There is a village of pile-built dwellings on a lake, set on the gridpoints of an $m \times n$ rectangular grid. Each dwelling is connected by exactly $p$ bridges to some of the neighboring dwellings (diagonal connections are not allowed, two dwellings can be connected by more than one bridge). Determine for which values $m,n, p$ it is possible to place the bridges so that from any dwelling one can reach any other dwelling.

1995 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 6

Tags:

How many 11-digit bank account numbers are there consisting of 1's and 2's only, and such that there are no two consecutive 1's? A. 64 B. 233 C. 1024 D. 1279 E. 1365

PEN A Problems, 109

Tags: Divisibility Theory

Find all positive integers $a$ and $b$ such that \[\frac{a^{2}+b}{b^{2}-a}\text{ and }\frac{b^{2}+a}{a^{2}-b}\] are both integers.

2020 HK IMO Preliminary Selection Contest, 16

Tags: geometry

$\Delta ABC$ is right-angled at $B$, with $AB=1$ and $BC=3$. $E$ is the foot of perpendicular from $B$ to $AC$. $BA$ and $BE$ are produced to $D$ and $F$ respectively such that $D$, $F$, $C$ are collinear and $\angle DAF=\angle BAC$. Find the length of $AD$.

2007 Middle European Mathematical Olympiad, 1

Tags: inequalities , inequalities proposed

Let $ a,b,c,d$ be real numbers which satisfy $ \frac{1}{2}\leq a,b,c,d\leq 2$ and $ abcd\equal{}1$. Find the maximum value of \[ \left(a\plus{}\frac{1}{b}\right)\left(b\plus{}\frac{1}{c}\right)\left(c\plus{}\frac{1}{d}\right)\left(d\plus{}\frac{1}{a}\right).\]

2012 ELMO Shortlist, 10

Tags: geometry , conics , circumcircle , projective geometry , algebra proposed , algebra

Let $A_1A_2A_3A_4A_5A_6A_7A_8$ be a cyclic octagon. Let $B_i$ by the intersection of $A_iA_{i+1}$ and $A_{i+3}A_{i+4}$. (Take $A_9 = A_1$, $A_{10} = A_2$, etc.) Prove that $B_1, B_2, \ldots , B_8$ lie on a conic. [i]David Yang.[/i]

2025 Benelux, 1

Tags: algebra , functional equation , BxMO

Does there exist a function $f:\mathbb{R}\to \mathbb{R}$ such that $$f(x^2+f(y))=f(x)^2-y$$ for all $x, y\in \mathbb{R}$?

2005 Romania National Olympiad, 2

Tags: function , LaTeX , algebra proposed , algebra

Find all functions $f:\mathbb{R}\to\mathbb{R}$ for which \[ x(f(x+1)-f(x)) = f(x), \] for all $x\in\mathbb{R}$ and \[ | f(x) - f(y) | \leq |x-y| , \] for all $x,y\in\mathbb{R}$. [i]Mihai Piticari[/i]

2024-25 IOQM India, 30

Tags: geometry , perimeter

Let $ABC$ be a right-angled triangle with $\angle B = 90^{\circ}$. Let the length of the altitude $BD$ be equal to $12$. What is the minimum possible length of $AC$, given that $AC$ and the perimeter of triangle $ABC$ are integers?

2018 Puerto Rico Team Selection Test, 3

Tags: number theory , divisible , sivides

Let $A$ be a set of $m$ positive integers where $m\ge 1$. Show that there exists a nonempty subset $B$ of $A$ such that the sum of all the elements of $B$ is divisible by $m$.

2018 Purple Comet Problems, 26

Tags: algebra

Let $a, b$, and $c$ be real numbers. Let $u = a^2 + b^2 + c^2$ and $v = 2ab + 2bc + 2ca$. Suppose $2018u = 1001v + 1024$. Find the maximum possible value of $35a - 28b - 3c$.

2015 Regional Olympiad of Mexico Center Zone, 3

Tags: combinatorics , grid , Coloring

A board of size $2015 \times 2015$ is covered with sub-boards of size $2 \times 2$, each of which is painted like chessboard. Each sub-board covers exactly $4$ squares of the board and each square of the board is covered with at least one square of a sub-board (the painted of the sub-boards can be of any shape). Prove that there is a way to cover the board in such a way that there are exactly $2015$ black squares visible. What is the maximum number of visible black squares?

2024 LMT Fall, 5

Tags: speed

Find the area of the quadrilateral with vertices at $(0,0), (2,0), (20,24), (0,2)$ in that order.

2020 Novosibirsk Oral Olympiad in Geometry, 5

Tags: geometry , perpendicular bisector , midpoint

Line $\ell$ is perpendicular to one of the medians of the triangle. The median perpendiculars to the sides of this triangle intersect the line $\ell$ at three points. Prove that one of them is the midpoint of the segment formed by the other two.

2001 AMC 10, 14

Tags:

A charity sells 140 benefit tickets for a total of $ \$2001$. Some tickets sell for full price (a whole dollar amount), and the rest sells for half price. How much money is raised by the full-price tickets? $ \textbf{(A)} \ \$782 \qquad \textbf{(B)} \ \$986 \qquad \textbf{(C)} \ \$1158 \qquad \textbf{(D)} \ \$1219 \qquad \textbf{(E)} \ \$1449$

2013 ELMO Problems, 2

Tags: inequalities , logarithms , function , calculus , derivative , inequalities unsolved , 3-variable inequality

Let $a,b,c$ be positive reals satisfying $a+b+c = \sqrt[7]{a} + \sqrt[7]{b} + \sqrt[7]{c}$. Prove that $a^a b^b c^c \ge 1$. [i]Proposed by Evan Chen[/i]