Olympiad problems

2017 AMC 8, 7

Tags:

Let $Z$ be a 6-digit positive integer, such as 247247, whose first three digits are the same as its last three digits taken in the same order. Which of the following numbers must also be a factor of $Z$? $\textbf{(A) }11\qquad\textbf{(B) }19\qquad\textbf{(C) }101\qquad\textbf{(D) }111\qquad\textbf{(E) }1111$

2010 Contests, 2

Tags: geometric transformation , reflection , parallelogram , homothety , ratio , geometry

Given a triangle $ABC$, let $A',B',C'$ be the perpendicular feet dropped from the centroid $G$ of the triangle $ABC$ onto the sides $BC,CA,AB$ respectively. Reflect $A',B',C'$ through $G$ to $A'',B'',C''$ respectively. Prove that the lines $AA'',BB'',CC''$ are concurrent.

2021 IMO Shortlist, G1

Tags: geometry

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.

2023 AMC 8, 20

Tags:

Two integers are inserted into the list $3,3,8,11,28$ to double it's range. The mode and median remain unchanged. What is the maximum possible sum of two additional numbers? $\textbf{(A) } 56\qquad \textbf{(B) } 57 \qquad \textbf{(C) } 58 \qquad \textbf{(D) } 60 \qquad \textbf{(E) } 61$

2018 China Northern MO, 6

Tags: geometry , circumcircle

Let $H$ be the orthocenter of triangle $ABC$. Let $D$ and $E$ be points on $AB$ and $AC$ such that $DE$ is parallel to $CH$. If the circumcircle of triangle $BDH$ passes through $M$, the midpoint of $DE$, then prove that $\angle ABM=\angle ACM$

2009 Germany Team Selection Test, 1

Tags: geometry

Let $ ABCD$ be a chordal/cyclic quadrilateral. Consider points $ P,Q$ on $ AB$ and $ R,S$ on $ CD$ with \[ \overline{AP}: \overline{PB} \equal{} \overline{CS}: \overline{SD}, \quad \overline{AQ}: \overline{QB} \equal{} \overline{CR}: \overline{RD}.\] How to choose $ P,Q,R,S$ such that $ \overline{PR} \cdot \overline{AB} \plus{} \overline{QS} \cdot \overline{CD}$ is minimal?

2008 Moldova Team Selection Test, 3

Tags: trigonometry , geometry

In triangle $ ABC$ the bisector of $ \angle ACB$ intersects $ AB$ at $ D$. Consider an arbitrary circle $ O$ passing through $ C$ and $ D$, so that it is not tangent to $ BC$ or $ CA$. Let $ O\cap BC \equal{} \{M\}$ and $ O\cap CA \equal{} \{N\}$. a) Prove that there is a circle $ S$ so that $ DM$ and $ DN$ are tangent to $ S$ in $ M$ and $ N$, respectively. b) Circle $ S$ intersects lines $ BC$ and $ CA$ in $ P$ and $ Q$ respectively. Prove that the lengths of $ MP$ and $ NQ$ do not depend on the choice of circle $ O$.

2010 China Team Selection Test, 3

Tags: floor function , inequalities , algorithm , combinatorics

Let $n_1,n_2, \cdots, n_{26}$ be pairwise distinct positive integers satisfying (1) for each $n_i$, its digits belong to the set $\{1,2\}$; (2) for each $i,j$, $n_i$ can't be obtained from $n_j$ by adding some digits on the right. Find the smallest possible value of $\sum_{i=1}^{26} S(n_i)$, where $S(m)$ denotes the sum of all digits of a positive integer $m$.

2013 Federal Competition For Advanced Students, Part 2, 3

Tags: geometry

A square and an equilateral triangle are inscribed in a same circle. The seven vertices form a convex heptagon $S$ inscribed in the circle ($S$ might be a hexagon if two vertices coincide). For which positions of the triangle relative to the square does $S$ have the largest and smallest area, respectively?

2018 Estonia Team Selection Test, 3

Tags: sum , algebra , inequalities , max , min

Given a real number $c$ and an integer $m, m \ge 2$. Real numbers $x_1, x_2,... , x_m$ satisfy the conditions $x_1 + x_2 +...+ x_m = 0$ and $\frac{x^2_1 + x^2_2 + ...+ x^2_m}{m}= c$. Find max $(x_1, x_2,..., x_m)$ if it is known to be as small as possible.

2006 Estonia Math Open Junior Contests, 6

Tags: geometry , 3d geometry , quadratic , algebra

Find all real numbers with the following property: the difference of its cube and its square is equal to the square of the difference of its square and the number itself.

2011 Postal Coaching, 3

Tags: function , functional equation , algebra

Suppose $f : \mathbb{R} \longrightarrow \mathbb{R}$ be a function such that \[2f (f (x)) = (x^2 - x)f (x) + 4 - 2x\] for all real $x$. Find $f (2)$ and all possible values of $f (1)$. For each value of $f (1)$, construct a function achieving it and satisfying the given equation.

1931 Eotvos Mathematical Competition, 1

Tags: number theory

Let $p$ be a prime greater than $2$. Prove that $\frac{2}{p}$ can be expressed in exactly one way in the form $$\frac{1}{x}+\frac{1}{y}$$ where $x$ and $y$ are positive integers with $x > y$.

2011 Tournament of Towns, 2

Tags: geometry , right triangle

On side $AB$ of triangle $ABC$ a point $P$ is taken such that $AP = 2PB$. It is known that $CP = 2PQ$ where $Q$ is the midpoint of $AC$. Prove that $ABC$ is a right triangle.

2013 Israel National Olympiad, 2

Tags: combinatorics , number theory , maximum , set

Let $A=\{n\in\mathbb{Z}\mid 0<n<2013\}$. A subset $B\subseteq A$ is called [b]reduced[/b] if for any two numbers $x,y\in B$, we must have $x\cdot y \notin B$. For example, any subset containing the numbers $3,5,15$ cannot be reduced, and same for a subset containing $4,16$. [list=a] [*] Find the maximal size of a reduced subset of $A$. [*] How many reduced subsets are there with that maximal size? [/list]

2021 Israel TST, 2

Tags:

Find all functions $f:\mathbb{R}\to \mathbb{R}$ so that for any reals $x,y$ the following holds: \[f(x\cdot f(x+y))+f(f(y)\cdot f(x+y))=(x+y)^2\]

2013 Stanford Mathematics Tournament, 3

Tags:

Queen Jack likes a 5-card hand if and only if the hand contains only queens and jacks. Considering all possible 5-card hands that can come from a standard 52-card deck, how many hands does Queen Jack like?

2017 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: number theory , diophantine , diophantine equation

Let $a$ be a fixed positive integer. Find the largest integer $b$ such that $(x+a)(x+b)=x+a+b$, for some integer $x$.

1935 Eotvos Mathematical Competition, 2

Tags: geometry , symmetry

Prove that a finite point set cannot have more than one center of symmetry.

2025 Bulgarian Spring Mathematical Competition, 10.3

Tags: table , perfect square , algebra

In the cell $(i,j)$ of a table $n\times n$ is written the number $(i-1)n + j$. Determine all positive integers $n$ such that there are exactly $2025$ rows not containing a perfect square.

1973 AMC 12/AHSME, 33

Tags:

When one ounce of water is added to a mixture of acid and water, the new mixture is $ 20\%$ acid. When one ounce of acid is added to the new mixture, the result is $ 33\frac13\%$ acid. The percentage of acid in the original mixture is $ \textbf{(A)}\ 22\% \qquad \textbf{(B)}\ 24\% \qquad \textbf{(C)}\ 25\% \qquad \textbf{(D)}\ 30\% \qquad \textbf{(E)}\ 33\frac13 \%$

2014 Grand Duchy of Lithuania, 3

Tags: combinatorics , table , coloring

In a table $n\times n$ some unit squares are coloured black and the other unit squares are coloured white. For each pair of columns and each pair of rows the four squares on the intersections of these rows and columns must not all be of the same colour. What is the largest possible value of $n$?

2020 Jozsef Wildt International Math Competition, W59

Tags: inequalities

If $a_k>0~(k=1,2,\ldots,n)$ then prove that $$\sum_{\text{cyc}}\left(\frac{(a_1+a_2+\ldots+a_{n-1})^2}{a_n}+\frac{a_n^2}{a_1}\right)\ge\frac{n^2}2\sum_{k=1}^na_k$$ [i]Proposed by Mihály Bencze[/i]

2019 Cono Sur Olympiad, 2

Tags: number theory , digit

We say that a positive integer $M$ with $2n$ digits is [i]hypersquared[/i] if the following three conditions are met: [list] [*]$M$ is a perfect square. [*]The number formed by the first $n$ digits of $M$ is a perfect square. [*]The number formed by the last $n$ digits of $M$ is a perfect square and has exactly $n$ digits (its first digit is not zero). [/list] Find a hypersquared number with $2000$ digits.

2010 Contests, 1

Tags: modular arithmetic , number theory

Denote by $S(n)$ the sum of the digits of the positive integer $n$. Find all the solutions of the equation $n(S(n)-1)=2010.$