Olympiad problems

2025 Bulgarian Spring Mathematical Competition, 10.3

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

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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 $n(S(n)-1)=2010.$

2021 Science ON all problems, 4

Tags: geometry , angle chasing , circles , projective

$ABCD$ is a cyclic convex quadrilateral whose diagonals meet at $X$. The circle $(AXD)$ cuts $CD$ again at $V$ and the circle $(BXC)$ cuts $AB$ again at $U$, such that $D$ lies strictly between $C$ and $V$ and $B$ lies strictly between $A$ and $U$. Let $P\in AB\cap CD$.\\ \\ If $M$ is the intersection point of the tangents to $U$ and $V$ at $(UPV)$ and $T$ is the second intersection of circles $(UPV)$ and $(PAC)$, prove that $\angle PTM=90^o$.\\ \\ [i](Vlad Robu)[/i]

2005 China Team Selection Test, 3

Tags: modular arithmetic , number theory , binomial expansion , fermat number , power of 2

$n$ is a positive integer, $F_n=2^{2^{n}}+1$. Prove that for $n \geq 3$, there exists a prime factor of $F_n$ which is larger than $2^{n+2}(n+1)$.

2012 Stanford Mathematics Tournament, 3

Tags: geometry , 3d geometry

Express $\frac{2^3-1}{2^3+1}\times\frac{3^3-1}{3^3+1}\times\frac{4^3-1}{4^3+1}\times\dots\times\frac{16^3-1}{16^3+1}$ as a fraction in lowest terms.

2013 AMC 12/AHSME, 23

Tags: search , modular arithmetic

Bernardo chooses a three-digit positive integer $N$ and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer $S$. For example, if $N=749$, Bernardo writes the numbers 10,444 and 3,245, and LeRoy obtains the sum $S=13,689$. For how many choices of $N$ are the two rightmost digits of $S$, in order, the same as those of $2N$? ${ \textbf{(A)}\ 5\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 15\qquad\textbf{(D}}\ 20\qquad\textbf{(E)}\ 25 $

2014 Middle European Mathematical Olympiad, 8

Tags: modular arithmetic , quadratic , inequalities , geometry , 3d geometry , number theory

Determine all quadruples $(x,y,z,t)$ of positive integers such that \[ 20^x + 14^{2y} = (x + 2y + z)^{zt}.\]

2015 Nordic, 2

Tags: prime number , number theory

Find the primes ${p, q, r}$, given that one of the numbers ${pqr}$ and ${p + q + r}$ is ${101}$ times the other.

2021 Kyiv City MO Round 1, 8.2

Tags: number theory , digit

Oleksiy writes all the digits from $0$ to $9$ on the board, after which Vlada erases one of them. Then he writes $10$ nine-digit numbers on the board, each consisting of all the nine digits written on the board (they don't have to be distinct). It turned out that the sum of these $10$ numbers is a ten-digit number, all of whose digits are distinct. Which digit could have been erased by Vlada? [i]Proposed by Oleksii Masalitin[/i]

2024 Chile TST IMO, 1

Tags: combinatorics

Consider a set of $ n \geq 3 $ points in the plane where no three are collinear. Prove that the points can be labeled as $ P_1, P_2, \dots, P_n $ so that the angles $ \angle P_i P_{i+1} P_{i+2} $ are less than $ 90^\circ $ for all $ i $.

2002 Estonia National Olympiad, 2

Tags: acute , obtuse , orthocenter , angle , geometry

Let $ABC$ be a non-right triangle with its altitudes intersecting in point $H$. Prove that $ABH$ is an acute triangle if and only if $\angle ACB$ is obtuse.

2022 Grand Duchy of Lithuania, 4

Tags: number theory , perfect square

Find all triples of natural numbers $(a, b, c)$ for which the number $$2^a + 2^b + 2^c + 3$$ is the square of an integer.

2022-2023 OMMC FINAL ROUND, 7

Tags: geometry

In $\triangle ABC$, let its incircle touch $\overline{AC}$ and $\overline{AB}$ at $E$ and $F$ respectively. Let its $A$-excircle have center $I_A$ and touch $\overline{BC}$ at $K$. Let $P$ and $Q$ be points so that $BFPI_A$ and $CEQI_A$ are parallelograms. If $\overline{AI_A}$ and $\overline{PQ}$ intersect at $X$, prove $\overline{KX} \perp \overline{PQ}$.

MOAA Gunga Bowls, 2023.15

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Triangle $ABC$ has $AB = 5$, $BC = 7$, $CA = 8$. Let $M$ be the midpoint of $BC$ and let points $P$ and $Q$ lie on $AB$ and $AC$ respectively such that $MP \perp AB$ and $MQ \perp AC$. If $H$ is the orthocenter of $\triangle{APQ}$ then the area of $\triangle{HPM}$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ where $a$ and $c$ are relatively prime positive integers and $b$ is square-free. Find $a+b+c$. [i]Proposed by Harry Kim[/i]

2010 Harvard-MIT Mathematics Tournament, 9

Tags: geometry

Let $ABCD$ be a quadrilateral with an inscribed circle centered at $I$. Let $CI$ intersect $AB$ at $E$. If $\angle IDE=35^\circ$, $\angle ABC=70^\circ$, and $\angle BCD=60^\circ$, then what are all possible measures of $\angle CDA$?

2013 India PRMO, 9

Tags: incenter , circumcircle , angle , geometry

In a triangle $ABC$, let $H, I$ and $O$ be the orthocentre, incentre and circumcentre, respectively. If the points $B, H, I, C$ lie on a circle, what is the magnitude of $\angle BOC$ in degrees?

2024 MMATHS, 2

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Consider the recursive sequence defined by $a_{n+1}=a_n^n+1,$ with $a_1=0.$ What is the last digit of $a_{2024}$?

2023 Romania EGMO TST, P3

Tags: geometry , inequalities

Let $D{}$ be a point inside the triangle $ABC$. Let $E{}$ and $F{}$ be the projections of $D{}$ onto $AB$ and $AC$, respectively. The lines $BD$ and $CD$ intersect the circumcircle of $ABC$ the second time at $M{}$ and $N{}$, respectively. Prove that \[\frac{EF}{MN}\geqslant \frac{r}{R},\]where $r{}$ and $R{}$ are the inradius and circumradius of $ABC$, respectively.

2005 Thailand Mathematical Olympiad, 19

Tags: algebra , polynomial , divisible

Let $P(x)$ be a monic polynomial of degree $4$ such that for $k = 1, 2, 3$, the remainder when $P(x)$ is divided by $x - k$ is equal to $k$. Find the value of $P(4) + P(0)$.

2018 CMIMC Number Theory, 8

Tags: number theory

It is given that there exists a unique triple of positive primes $(p,q,r)$ such that $p<q<r$ and \[\dfrac{p^3+q^3+r^3}{p+q+r} = 249.\] Find $r$.

CIME I 2018, 11

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Let $N$ be the set $\{1, 2, \dots, 2018\}$. For each subset $A$ of $N$ with exactly $1009$ elements, define $$f(A)=\sum\limits_{i \in A} i \sum\limits_{j \in N, j \notin A} j.$$If $\mathbb{E}[f(A)]$ is the expected value of $f(A)$ as $A$ ranges over all the possible subsets of $N$ with exactly $1009$ elements, find the remainder when the sum of the distinct prime factors of $\mathbb{E}[f(A)]$ is divided by $1000$. [i]Proposed by [b]FedeX333X[/b][/i]