Olympiad problems

2021 Romania National Olympiad, 2

Determine all non-trivial finite rings with am unit element in which the sum of all elements is invertible. [i]Mihai Opincariu[/i]

1988 Dutch Mathematical Olympiad, 4

Tags: geometry , geometric inequality , geometric inequalities , square

Given is an isosceles triangle $ABC$ with $AB = 2$ and $AC = BC = 3$. We consider squares where $A, B$ and $C$ lie on the sides of the square (so not on the extension of such a side). Determine the maximum and minimum value of the area of such a square. Justify the answer.

2019 Azerbaijan Senior NMO, 1

Tags: algebra , square root

Solve the following equation $$\sqrt{\frac{x^2}3-ax+a^2}+\sqrt{\frac{x^2}3-bx+b^2}=\sqrt{a^2-ab+b^2}$$ where $a;b\in\mathbb{R^+}$

2022 Bulgaria EGMO TST, 1

Tags: algebra , closure , binary operation , addition

The finite set $M$ of real numbers is such that among any three of its elements there are two whose sum is in $M$. What is the maximum possible cardinality of $M$? [hide=Remark about the other problems] Problem 2 is UK National Round 2022 P2, Problem 3 is UK National Round 2022 P4, Problem 4 is Balkan MO 2021 Shortlist N2 (the one with Bertrand), Problem 5 is IMO Shortlist 2021 A1 and Problem 6 is USAMO 2002/1. Hence neither of these will be posted here. [/hide]

2020 Harvard-MIT Mathematics Tournament, 6

Tags: hmmt

Alice writes $1001$ letters on a blackboard, each one chosen independently and uniformly at random from the set $S=\{a, b, c\}$. A move consists of erasing two distinct letters from the board and replacing them with the third letter in $S$. What is the probability that Alice can perform a sequence of moves which results in one letter remaining on the blackboard? [i]Proposed by Daniel Zhu.[/i]

2001 National High School Mathematics League, 9

Tags: 3d geometry , geometry

The length of edge of cube $ABCD-A_1B_1C_1D_1$ is $1$, then the distance between lines $A_1C_1$ and $BD_1$ is________.

2009 USAMTS Problems, 3

Tags:

Prove that if $a$ and $b$ are positive integers such that $a^2 + b^2$ is a multiple of $7^{2009}$, then $ab$ is a multiple of $7^{2010}$.

1969 All Soviet Union Mathematical Olympiad, 116

Tags: algebra

There is a wolf in the centre of a square field, and four dogs in the corners. The wolf can easily kill one dog, but two dogs can kill the wolf. The wolf can run all over the field, and the dogs -- along the fence (border) only. Prove that if the dog's speed is $1.5$ times more than the wolf's, than the dogs can prevent the wolf escaping.

2010 Malaysia National Olympiad, 5

Tags: number theory

Let $n$ be an integer greater than 1. If all digits of $97n$ are odd, find the smallest possible value of $n$.

2021 AMC 10 Spring, 11

Tags:

For which of the following integers $b$ is the base-$b$ number $2021_b - 221_b$ not divisible by $3$? $\textbf{(A) } 3 \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 8$

2013 Dutch BxMO/EGMO TST, 4

Tags: functional equation , determine all functions , algebra

Determine all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying \[f(x+yf(x))=f(xf(y))-x+f(y+f(x))\]

2025 Ukraine National Mathematical Olympiad, 9.1

Tags: algebra , system of equations

Solve the system of equations in reals: \[ \begin{cases} y = x^2 + 2x \\ z = y^2 + 2y \\ x = z^2 + 2z \end{cases} \] [i]Proposed by Mykhailo Shtandenko[/i]

2012 Kosovo Team Selection Test, 2

Tags: number theory

Find all three digit numbers, for which the sum of squares of each digit is $90$ .

2006 Italy TST, 3

Tags: function , induction , algebra , functional equation

Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that for all integers $m,n$, \[f(m - n + f(n)) = f(m) + f(n).\]

2021 Taiwan TST Round 3, G

Tags: geometry , circumcircle

Let $ABC$ be a triangle with $AB<AC$, and let $I_a$ be its $A$-excenter. Let $D$ be the projection of $I_a$ to $BC$. Let $X$ be the intersection of $AI_a$ and $BC$, and let $Y,Z$ be the points on $AC,AB$, respectively, such that $X,Y,Z$ are on a line perpendicular to $AI_a$. Let the circumcircle of $AYZ$ intersect $AI_a$ again at $U$. Suppose that the tangent of the circumcircle of $ABC$ at $A$ intersects $BC$ at $T$, and the segment $TU$ intersects the circumcircle of $ABC$ at $V$. Show that $\angle BAV=\angle DAC$. [i]Proposed by usjl.[/i]

2000 Stanford Mathematics Tournament, 1

Tags:

If $ a\equal{}2b\plus{}c$, $ b\equal{}2c\plus{}d$, $ 2c\equal{}d\plus{}a\minus{}1$, $ d\equal{}a\minus{}c$, what is $ b$?

2014 May Olympiad, 5

Tags: weight , combinatorics

Given $6$ balls: $2$ white, $2$ green, $2$ red, it is known that there is a white, a green and a red that weigh $99$ g each and that the other balls weigh $101$ g each. Determine the weight of each ball using two times a two-plate scale . Clarification: A two-pan scale only reports if the left pan weighs more than, equal to or less than the right.

2011 Today's Calculation Of Integral, 756

Tags: calculus , integration , geometry , analytic geometry , calculus computations

Let $a$ be real number. A circle $C$ touches the line $y=-x$ at the point $(a, -a)$ and passes through the point $(0,\ 1).$ Denote by $P$ the center of $C$. When $a$ moves, find the area of the figure enclosed by the locus of $P$ and the line $y=1$.

2012 AIME Problems, 2

Tags: arithmetic sequence

The terms of an arithmetic sequence add to $715$. The first term of the sequence is increased by $1$, the second term is increased by $3$, the third term is increased by $5$, and in general, the $k$th term is increased by the $k$th odd positive integer. The terms of the new sequence add to $836$. Find the sum of the first, last, and middle terms of the original sequence.

2018 Azerbaijan IMO TST, 2

Tags: number theory

Determine all integers $ n\geq 2$ having the following property: for any integers $a_1,a_2,\ldots, a_n$ whose sum is not divisible by $n$, there exists an index $1 \leq i \leq n$ such that none of the numbers $$a_i,a_i+a_{i+1},\ldots,a_i+a_{i+1}+\ldots+a_{i+n-1}$$ is divisible by $n$. Here, we let $a_i=a_{i-n}$ when $i >n$. [i]Proposed by Warut Suksompong, Thailand[/i]

1991 Czech And Slovak Olympiad IIIA, 5

Tags: combinatorics

In a group of mathematicians everybody has at least one friend (friendship is a symmetric relation). Show that there is a mathematician all of whose friends have average number of friends not smaller than the average number of friends in the whole group.

2005 Sharygin Geometry Olympiad, 9.3

Tags: midpoint , locus , geometry , arc

Given a circle and points $A, B$ on it. Draw the set of midpoints of the segments, one of the ends of which lies on one of the arcs $AB$, and the other on the second.

1996 Estonia Team Selection Test, 1

Tags: number theory , divisibility , perfect cube

Suppose that $x,y$ and $\frac{x^2+y^2+6}{xy}$ are positive integers . Prove that $\frac{x^2+y^2+6}{xy}$ is a perfect cube.

2017 QEDMO 15th, 2

Tags: matrix , algebra

Let $A, B, X$ be real $n\times n$ matrices for which $AXB + A + B = 0$ holds. Prove that $AXB = BXA$.

2014 Vietnam National Olympiad, 1

Tags: symmetry , geometry , gauss , trigonometry , circumcircle , radical axis , perpendicular bisector

Given a circle $(O)$ and two fixed points $B,C$ on $(O),$ and an arbitrary point $A$ on $(O)$ such that the triangle $ABC$ is acute. $M$ lies on ray $AB,$ $N$ lies on ray $AC$ such that $MA=MC$ and $NA=NB.$ Let $P$ be the intersection of $(AMN)$ and $(ABC),$ $P\ne A.$ $MN$ intersects $BC$ at $Q.$ a) Prove that $A,P,Q$ are collinear. b) $D$ is the midpoint of $BC.$ Let $K$ be the intersection of $(M,MA)$ and $(N,NA),$ $K\ne A.$ $d$ is the line passing through $A$ and perpendicular to $AK.$ $E$ is the intersection of $d$ and $BC.$ $(ADE)$ intersects $(O)$ at $F,$ $F\ne A.$ Prove that $AF$ passes through a fixed point.