Olympiad problems

2006 Grigore Moisil Urziceni, 1

Tags: group theory

Let be an element $ e $ from a group $ (G,\cdot ) , $ and let be the operation $ *:G^2\longrightarrow G $ defined as $ x*y=x\cdot a\cdot y . $ Prove that $ (G,*) $ is a group and is isomorphic with $ (G,\cdot ) . $

2023 Silk Road, 3

Tags: prime number , quadratic , quadratic residue , directed graph , number theory

Let $p$ be a prime number. We construct a directed graph of $p$ vertices, labeled with integers from $0$ to $p-1$. There is an edge from vertex $x$ to vertex $y$ if and only if $x^2+1\equiv y \pmod{p}$. Let $f(p)$ denotes the length of the longest directed cycle in this graph. Prove that $f(p)$ can attain arbitrarily large values.

2016 Korea Junior Math Olympiad, 8

Tags: combinatorics

One moving point in the coordinate plane can move right or up one position. $N$ is a number of all paths : paths that moving point starts from $(0, 0)$, without passing $(1, 0), (2, 1), . . . , (n, n-1)$ and moves $2n$ times to $(n, n)$. $a_k$ is a number of special paths : paths include in $N$, but $k$th moves to the right, $k+1$th moves to the up. find $$\frac{1}{N} (a_1+a_2+ . . . + a_{2n-1})$$

2021 Germany Team Selection Test, 2

Tags: number theory

For each prime $p$, construct a graph $G_p$ on $\{1,2,\ldots p\}$, where $m\neq n$ are adjacent if and only if $p$ divides $(m^{2} + 1-n)(n^{2} + 1-m)$. Prove that $G_p$ is disconnected for infinitely many $p$

1999 IMO Shortlist, 6

Tags: homothety , circumcircle , geometry

Two circles $\Omega_{1}$ and $\Omega_{2}$ touch internally the circle $\Omega$ in M and N and the center of $\Omega_{2}$ is on $\Omega_{1}$. The common chord of the circles $\Omega_{1}$ and $\Omega_{2}$ intersects $\Omega$ in $A$ and $B$. $MA$ and $MB$ intersects $\Omega_{1}$ in $C$ and $D$. Prove that $\Omega_{2}$ is tangent to $CD$.

2020 Purple Comet Problems, 2

Tags: geometry

The diagram below shows a $18\times 35$ rectangle with eight points marked that divide each side into three equal parts. Four triangles are removed from each of the corners of the rectangle leaving the shaded region. Find the area of this shaded region. [img]https://cdn.artofproblemsolving.com/attachments/2/1/e0fd592d589a1f5d3324a637743d0e9d6e3480.png[/img]

Russian TST 2022, P1

Tags: geometry

A convex 51-gon is given. For each of its vertices and each diagonal that does not contain this vertex, we mark in red a point symmetrical to the vertex relative to the middle of the diagonal. Prove that strictly inside the polygon there are no more than 20400 red dots. [i]Proposed by P. Kozhevnikov[/i]

2016 Fall CHMMC, 5

Tags: 3d geometry , probability , geometry

Suppose you have $27$ identical unit cubes colored such that $3$ faces adjacent to a vertex are red and the other $3$ are colored blue. Suppose further that you assemble these $27$ cubes randomly into a larger cube with $3$ cubes to an edge (in particular, the orientation of each cube is random). The probability that the entire cube is one solid color can be written as $\frac{1}{2^n}$ for some positive integer $n$. Find $n$.

2022 China National Olympiad, 3

Tags: number theory

Find all positive integers $a$ such that there exists a set $X$ of $6$ integers satisfying the following conditions: for every $k=1,2,\ldots ,36$ there exist $x,y\in X$ such that $ax+y-k$ is divisible by $37$.

2021 Durer Math Competition Finals, 4

Tags: digit , number theory

What is the number of $4$-digit numbers that contains exactly $3$ different digits that have consecutive value? Such numbers are for instance $5464$ or $2001$. Two digits in base $10$ are consecutive if their difference is $1$.

2014 Gulf Math Olympiad, 3

Tags: geometry , equal segments , right triangle , semicircle , angle

(i) $ABC$ is a triangle with a right angle at $A$, and $P$ is a point on the hypotenuse $BC$. The line $AP$ produced beyond $P$ meets the line through $B$ which is perpendicular to $BC$ at $U$. Prove that $BU = BA$ if, and only if, $CP = CA$. (ii) $A$ is a point on the semicircle $CB$, and points $X$ and $Y$ are on the line segment $BC$. The line $AX$, produced beyond $X$, meets the line through $B$ which is perpendicular to $BC$ at $U$. Also the line $AY$, produced beyond $Y$, meets the line through $C$ which is perpendicular to $BC$ at $V$. Given that $BY = BA$ and $CX = CA$, determine the angle $\angle VAU$.

2019 China Northern MO, 5

Tags: geometry

Two circles $O_1$ and $O_2$ intersect at $A,B$. Bisector of outer angle $\angle O_1AO_2$ intersects $O_1$ at $C$, $O_2$ at $D$. $P$ is a point on $\odot(BCD)$, $CP\cap O_1=E,DP\cap O_2=F$. Prove that $PE=PF$.

1973 AMC 12/AHSME, 21

Tags:

The number of sets of two or more consecutive positive integers whose sum is 100 is $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

2008 Putnam, B1

Tags: analytic geometry , geometry , circumcircle , graphing lines , slope , geometric transformation

What is the maximum number of rational points that can lie on a circle in $ \mathbb{R}^2$ whose center is not a rational point? (A [i]rational point[/i] is a point both of whose coordinates are rational numbers.)

2004 Romania National Olympiad, 1

Tags:

Find all non-negative integers $n$ such that there are $a,b \in \mathbb Z$ satisfying $n^2=a+b$ and $n^3=a^2+b^2$. [i]Lucian Dragomir[/i]

2024 Switzerland Team Selection Test, 5

Tags: arithmetic sequence , number theory

Let $a_1, \dots, a_n, b_1, \dots, b_n$ be $2n$ positive integers such that the $n+1$ products \[a_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n\] form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.

2000 Putnam, 4

Tags: function , trigonometry , induction

Let $f(x)$ be a continuous function such that $f(2x^2-1)=2xf(x)$ for all $x$. Show that $f(x)=0$ for $-1\le x \le 1$.

2023 ELMO Shortlist, A3

Tags: algebra

Does there exist an infinite sequence of integers $a_0$, $a_1$, $a_2$, $\ldots$ such that $a_0\ne0$ and, for any integer $n\ge0$, the polynomial \[P_n(x)=\sum_{k=0}^na_kx^k\] has $n$ distinct real roots? [i]Proposed by Amol Rama and Espen Slettnes[/i]

2011 Kosovo National Mathematical Olympiad, 2

Tags: function

It is given the function $f: \left(\mathbb{R} - \{0\}\right) \to \mathbb{R}$ such that $f(x)=x+\frac{1}{x}$. Is this function injective ? Justify your answer.

2008 AMC 12/AHSME, 18

Tags: geometry , 3d geometry , tetrahedron , pyramid , calculus , analytic geometry , area of a triangle

Triangle $ ABC$, with sides of length $ 5$, $ 6$, and $ 7$, has one vertex on the positive $ x$-axis, one on the positive $ y$-axis, and one on the positive $ z$-axis. Let $ O$ be the origin. What is the volume of tetrahedron $ OABC$? $ \textbf{(A)}\ \sqrt{85} \qquad \textbf{(B)}\ \sqrt{90} \qquad \textbf{(C)}\ \sqrt{95} \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ \sqrt{105}$

2007 China Team Selection Test, 2

Tags: geometry inequality

Let $ x_1, \ldots, x_n$ be $ n>1$ real numbers satisfying $ A\equal{}\left |\sum^n_{i\equal{}1}x_i\right |\not \equal{}0$ and $ B\equal{}\max_{1\leq i<j\leq n}|x_j\minus{}x_i|\not \equal{}0$. Prove that for any $ n$ vectors $ \vec{\alpha_i}$ in the plane, there exists a permutation $ (k_1, \ldots, k_n)$ of the numbers $ (1, \ldots, n)$ such that \[ \left |\sum_{i\equal{}1}^nx_{k_i}\vec{\alpha_i}\right | \geq \dfrac{AB}{2A\plus{}B}\max_{1\leq i\leq n}|\alpha_i|.\]

2019 Saudi Arabia JBMO TST, 3

Tags: number theory

Determine all primes $p$, for which there exist positive integers $m, n$, such that $p=m^2+n^2$ and $p|m^3+n^3+8mn$.

PEN J Problems, 6

Tags: divisor function , modular arithmetic , quadratic , number theory , relatively prime , prime factorization

Show that if $m$ and $n$ are relatively prime positive integers, then $\phi( 5^m -1) \neq 5^{n}-1$.

Estonia Open Junior - geometry, 2016.2.5

Tags: point , geometry , distance

On the plane three different points $P, Q$, and $R$ are chosen. It is known that however one chooses another point $X$ on the plane, the point $P$ is always either closer to $X$ than the point $Q$ or closer to $X$ than the point $R$. Prove that the point $P$ lies on the line segment $QR$.

2019 Kyiv Mathematical Festival, 3

Tags: geometry

Let $ABC$ be an isosceles triangle in which $\angle BAC=120^\circ,$ $D$ be the midpoint of $BC,$ $DE$ be the altitude of triangle $ADC,$ and $M$ be the midpoint of $DE.$ Prove that $BM=3AM.$