Olympiad problems

2005 Singapore Senior Math Olympiad, 1

The digits of a $3$-digit number are interchanged so that none of the digits retain their original position. The difference of the two numbers is a $2$-digit number and is a perfect square. Find the difference.

Taiwan TST 2015 Round 1, 1

Tags: combinatorics

Prove that for any set containing $2047$ positive integers, there exists $1024$ positive integers in the set such that the sum of these positive integers is divisible by $1024$.

2009 ISI B.Stat Entrance Exam, 8

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Find the number of ways in which three numbers can be selected from the set $\{1,2,\cdots ,4n\}$, such that the sum of the three selected numbers is divisible by $4$.

2022 Taiwan TST Round 2, 2

Tags: combinatorics , chessboard , square grid

A $100 \times100$ chessboard has a non-negative real number in each of its cells. A chessboard is [b]balanced[/b] if and only if the numbers sum up to one for each column of cells as well as each row of cells. Find the largest positive real number $x$ so that, for any balanced chessboard, we can find $100$ cells of it so that these cells all have number greater or equal to $x$, and no two of these cells are on the same column or row. [i]Proposed by CSJL.[/i]

2016 Germany Team Selection Test, 2

Tags: functional equation

Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\] holds for all $x,y\in\mathbb{Z}$.

Russian TST 2019, P3

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Let $S$ be the set of sequences of length 2018 whose terms are in the set $\{1, 2, 3, 4, 5, 6, 10\}$ and sum to 3860. Prove that the cardinality of $S$ is at most \[2^{3860} \cdot \left(\frac{2018}{2048}\right)^{2018}.\]

2017 CHMMC (Fall), 5

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Felix picks four points uniformly at random inside a unit circle $\mathcal{C}$. He then draws the four possible triangles which can be formed using these points as vertices. Finally, he randomly chooses of the six possible pairs of the triangles he just drew. What is the probability that the center of the circle $\mathcal{C}$ is contained in the union of the interiors of the two triangles that Felix chose?

2025 Euler Olympiad, Round 1, 9

Tags: geometry

Three circles with radii $1$, $2$, and $3$ are pairwise tangent to each other. Find the radius of the circle that is externally tangent to all three of these circles. [i]Proposed by Tamar Turashvili, Georgia [/i]

2023 Malaysian IMO Team Selection Test, 2

Tags: algebra

Let $a_1, a_2, \cdots, a_n$ be a sequence of real numbers with $a_1+a_2+\cdots+a_n=0$. Define the score $S(\sigma)$ of a permutation $\sigma=(b_1, \cdots b_n)$ of $(a_1, \cdots a_n)$ to be the minima of the sum $$(x_1-b_1)^2+\cdots+(x_n-b_n)^2$$ over all real numbers $x_1\le \cdots \le x_n$. Prove that $S(\sigma)$ attains the maxima over all permutations $\sigma$, if and only if for all $1\le k\le n$, $$b_1+b_2+\cdots+b_k\ge 0.$$ [i]Proposed by Anzo Teh Zhao Yang[/i]

1979 IMO Shortlist, 20

Tags: algebra , maximization , optimization , lagrange , calculus

Given the integer $n > 1$ and the real number $a > 0$ determine the maximum of $\sum_{i=1}^{n-1} x_i x_{i+1}$ taken over all nonnegative numbers $x_i$ with sum $a.$

2022 JBMO Shortlist, N2

Tags: lcm , shortlist , inequality , number theory

Let $a < b < c < d < e$ be positive integers. Prove that $$\frac{1}{[a, b]} + \frac{1}{[b, c]} + \frac{1}{[c, d]} + \frac{2}{[d, e]} \le 1$$ where $[x, y]$ is the least common multiple of $x$ and $y$ (e.g., $[6, 10] = 30$). When does equality hold?

2006 Princeton University Math Competition, 6

Tags: polynomial , algebra

Suppose that $P(x)$ is a polynomial with the property that there exists another polynomial $Q(x)$ to satisfy $P(x)Q(x)=P(x^2)$. $P(x)$ and $Q(x)$ may have complex coefficients. If $P(x)$ is a quintic with distinct complex roots $r_1,\dots,r_5$, find all possible values of $|r_1|+\dots+|r_5|$.

May Olympiad L1 - geometry, 2021.4

Tags: geometry

Facundo and Luca have been given a cake that is shaped like the quadrilateral in the figure. [img]https://cdn.artofproblemsolving.com/attachments/3/2/630286edc1935e1a8dd9e704ed4c813c900381.png[/img] They are going to make two straight cuts on the cake, thus obtaining $4$ portions in the shape of a quadrilateral. Then Facundo will be left with two portions that do not share any side, the other two will be for Luca. Show how they can cut the cuts so that both children get the same amount of cake. Justify why cutting in this way achieves the objective.

2002 IMO Shortlist, 4

Tags: algebra , number theory , equation , infinitely many solutions , vieta jumping , pell equation

Is there a positive integer $m$ such that the equation \[ {1\over a}+{1\over b}+{1\over c}+{1\over abc}={m\over a+b+c} \] has infinitely many solutions in positive integers $a,b,c$?

2024 Greece National Olympiad, 2

Tags: geometry

Let $ABC$ be a triangle with $AB<AC<BC$ with circumcircle $\Gamma_1$. The circle $\Gamma_2$ has center $D$ lying on $\Gamma_1$ and touches $BC$ at $E$ and the extension of $AB$ at $F$. Let $\Gamma_1$ and $\Gamma_2$ meet at $K, G$ and the line $KG$ meets $EF$ and $CD$ at $M, N$. Show that $BCNM$ is cyclic.

2015 CCA Math Bonanza, T3

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A tortoise is given an $80$-second head start in a race. When Achilles catches up to where the tortoise was when he (Achilles) began running, he finds that while he is now $40$ meters ahead of the starting line, the tortoise is now $5$ meters ahead of him. At this point, how long will it be, in seconds, before Achilles passes the tortoise? [i]2015 CCA Math Bonanza Team Round #3[/i]

1993 Baltic Way, 14

Tags: combinatorics

A square is divided into $16$ equal squares, obtaining the set of $25$ different vertices. What is the least number of vertices one must remove from this set, so that no $4$ points of the remaining set are the vertices of any square with sides parallel to the sides of the initial square?

VMEO III 2006 Shortlist, N5

Tags: number theory , diophantine equation , diophantine

Find all triples of integers $(x, y, z)$ such that $x^4 + 5y^4 = z^4$.

2023 Philippine MO, 2

Tags: number theory

Find all primes $p$ such that $\dfrac{2^{p+1}-4}{p}$ is a perfect square.

2012 IMO Shortlist, N1

Tags: quadratic , number theory

Call admissible a set $A$ of integers that has the following property: If $x,y \in A$ (possibly $x=y$) then $x^2+kxy+y^2 \in A$ for every integer $k$. Determine all pairs $m,n$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is the set of all integers. [i]Proposed by Warut Suksompong, Thailand[/i]

2020 Czech and Slovak Olympiad III A, 4

Tags: number theory , perfect square

Positive integers $a, b$ satisfy equality $b^2 = a^2 + ab + b$. Prove that $b$ is a square of a positive integer. (Patrik Bak)

2016 Tuymaada Olympiad, 2

Tags: combinatorics

A cube stands on one of the squares of an infinite chessboard. On each face of the cube there is an arrow pointing in one of the four directions parallel to the sides of the face. Anton looks on the cube from above and rolls it over an edge in the direction pointed by the arrow on the top face. Prove that the cube cannot cover any $5\times 5$ square.

2006 Croatia Team Selection Test, 4

Tags: inequalities , number theory

Find all natural solutions of $3^{x}= 2^{x}y+1.$

2005 IMC, 6

Tags: group theory , abstract algebra

6) $G$ group, $G_{m}$ and $G_{n}$ commutative subgroups being the $m$ and $n$ th powers of the elements in $G$. Prove $G_{gcd(m,n)}$ is commutative.

2006 Junior Balkan Team Selection Tests - Romania, 2

Tags: geometry , rectangle , circles , circumcenter , right angle , tangent circle

Let $C (O)$ be a circle (with center $O$ ) and $A, B$ points on the circle with $\angle AOB = 90^o$. Circles $C_1 (O_1)$ and $C_2 (O_2)$ are tangent internally with circle $C$ at $A$ and $B$, respectively, and, also, are tangent to each other. Consider another circle $C_3 (O_3)$ tangent externally to the circles $C_1, C_2$ and tangent internally to circle $C$, located inside angle $\angle AOB$. Show that the points $O, O_1, O_2, O_3$ are the vertices of a rectangle.