Olympiad problems

Novosibirsk Oral Geo Oly VII, 2022.4

Tags: geometry , rectangle

Fold the next seven corners into a rectangle. [img]https://cdn.artofproblemsolving.com/attachments/b/b/2b8b9d6d4b72024996a66d41f865afb91bb9b7.png[/img]

2013 Lusophon Mathematical Olympiad, 1

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If Xiluva puts two oranges in each basket, four oranges are in excess. If she puts five oranges in each basket, one basket is in excess. How many oranges and baskets has Xiluva?

2024 TASIMO, 1

Tags: geometry , incenter

Let $ABC$ be a triangle with $AB<AC$ and incenter $I.$ A point $D$ lies on segment $AC$ such that $AB=AD,$ and the line $BI$ intersects $AC$ at $E.$ Suppose the line $CI$ intersects $BD$ at $F,$ and $G$ lies on segment $DI$ such that $FD=FG.$ Prove that the lines $AG$ and $EF$ intersect on the circumcircle of triangle $CEI.$ \\ Proposed by Avan Lim Zenn Ee, Malaysia

2017 Baltic Way, 5

Tags: algebra , functional equation , easy functional equation

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$f(x^2y)=f(xy)+yf(f(x)+y)$$ for all real numbers $x$ and $y$.

2017 Harvard-MIT Mathematics Tournament, 5

Tags: number theory , diophantine equation

Find the number of ordered triples of positive integers $(a, b, c)$ such that \[6a + 10b + 15c = 3000.\]

2000 Poland - Second Round, 1

Tags: fraction , algebra , rational number

Decide, whether every positive rational number can present in the form $\frac{a^2 + b^3}{c^5 + d^7}$, where $a, b, c, d$ are positive integers.

2023 India EGMO TST, P4

Tags: algebra , function , identity , periodic function

Let $f, g$ be functions $\mathbb{R} \rightarrow \mathbb{R}$ such that for all reals $x,y$, $$f(g(x) + y) = g(x + y)$$ Prove that either $f$ is the identity function or $g$ is periodic. [i]Proposed by Pranjal Srivastava[/i]

2007 Estonia National Olympiad, 2

Tags: geometry , 3d geometry , combinatorics

A 3-dimensional chess board consists of $ 4 \times 4 \times 4$ unit cubes. A rook can step from any unit cube K to any other unit cube that has a common face with K. A bishop can step from any unit cube K to any other unit cube that has a common edge with K, but does not have a common face. One move of both a rook and a bishop consists of an arbitrary positive number of consecutive steps in the same direction. Find the average number of possible moves for either piece, where the average is taken over all possible starting cubes K.

2011 Saudi Arabia IMO TST, 2

Tags: geometry , concurrency , excircle , excenter

In triangle $ABC$, let $I_a$ $,I_b$, $I_c$ be the centers of the excircles tangent to sides $BC$, $CA$, $AB$, respectively. Let $P$ and $Q$ be the tangency points of the excircle of center $I_a$ with lines $AB$ and $AC$. Line $PQ$ intersects $I_aB$ and $I_aC$ at $D$ and $E$. Let $A_1$ be the intersection of $DC$ and $BE$. In an analogous way we define points $B_1$ and $C_1$. Prove that $AA_1$, $BB_1$ , $CC_1$ are concurrent.

1978 IMO Shortlist, 5

Tags: arithmetic sequence , algebra , factorization , partition , number theory

For every integer $d \geq 1$, let $M_d$ be the set of all positive integers that cannot be written as a sum of an arithmetic progression with difference $d$, having at least two terms and consisting of positive integers. Let $A = M_1$, $B = M_2 \setminus \{2 \}, C = M_3$. Prove that every $c \in C$ may be written in a unique way as $c = ab$ with $a \in A, b \in B.$

2017 Junior Balkan Team Selection Tests - Moldova, Problem 3

Tags: geometry , projective geometry

Let $ABC$ be a triangle inscribed in a semicircle with center $O$ and diameter $BC.$ Two tangent lines to the semicircle at $A$ and $B$ intersect at $D.$ Prove that $DC$ goes through the midpoint of the altitude $AH$ of triangle $ABC.$

1984 AIME Problems, 11

Tags: probability , counting , distinguishability , symmetry

A gardener plants three maple trees, four oak trees, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let $\frac{m}{n}$ in lowest terms be the probability that no two birch trees are next to one another. Find $m + n$.

2019 Serbia Team Selection Test, P5

Tags: diophantine equation , number theory

Solve the equation in nonnegative integers:\\ $2^x=5^y+3$

2024 Romania Team Selection Tests, P1

Tags: number theory

Determine all ordered pairs $(a,p)$ of positive integers, with $p$ prime, such that $p^a+a^4$ is a perfect square. [i]Proposed by Tahjib Hossain Khan, Bangladesh[/i]

1997 Vietnam National Olympiad, 3

Tags: function , number theory , relatively prime , algebra

Find the number of functions $ f: \mathbb N\rightarrow\mathbb N$ which satisfying: (i) $ f(1) \equal{} 1$ (ii) $ f(n)f(n \plus{} 2) \equal{} f^2(n \plus{} 1) \plus{} 1997$ for every natural numbers n.

2025 Sharygin Geometry Olympiad, 10

Tags: geometry

An acute-angled triangle with one side equal to the altitude from the opposite vertex is cut from paper. Construct a point inside this triangle such that the square of the distance from it to one of the vertices equals the sum of the squares of distances to to the remaining two vertices. No instruments are available, it is allowed only to fold the paper and to mark the common points of folding lines. Proposed by: M.Evdokimov

2015 Regional Olympiad of Mexico Southeast, 6

Tags: algebra

If we separate the numbers $1,2,3,4,\dots, 100$ in two lists with $$a_1<a_2<\cdots<a_{50}$$ and $$b_1>b_2>\cdots>b_{50}$$ Prove that, no matter how we do the separation, $$\vert a_1-b_1\vert +\vert a_2-b_2\vert+\cdots +\vert a_{50}-b_{50}\vert=2500$$

2015 NIMO Summer Contest, 4

Tags: algebra , exponent

Let $P$ be a function defined by $P(t)=a^t+b^t$, where $a$ and $b$ are complex numbers. If $P(1)=7$ and $P(3)=28$, compute $P(2)$. [i] Proposed by Justin Stevens [/i]

1999 Harvard-MIT Mathematics Tournament, 6

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Matt has somewhere between $1000$ and $2000$ pieces of paper he's trying to divide into piles of the same size (but not all in one pile or piles of one sheet each). He tries $2$, $3$, $4$, $5$, $6$, $7$, and $8$ piles but ends up with one sheet left over each time. How many piles does he need?

2017 NIMO Summer Contest, 14

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Let $x, y, z$ be real numbers such that $x+y+z=-2$ and \[\begin{aligned} & (x^2+xy+y^2)(y^2+yz+z^2) \\ &+ (y^2+yz+z^2)(z^2+zx+x^2) \\ &+ (z^2+zx+x^2)(x^2+xy+y^2) \\ & = 625+ \tfrac34(xy+yz+zx)^2. \end{aligned}\] Compute $|xy+yz+zx|$. [i]Proposed by Michael Tang[/i]

2019 Hong Kong TST, 4

Tags: combinatorics

We choose 100 points in the coordinate plane. Let $N$ be the number of triples $(A,B,C)$ of distinct chosen points such that $A$ and $B$ have the same $y$-coordinate, and $B$ and $C$ have the same $x$-coordinate. Find the greatest value that $N$ can attain considering all possible ways to choose the points.

1995 Singapore Team Selection Test, 2

Tags: equal angles , angle , geometry

$ABC$ is a triangle with $\angle A > 90^o$ . On the side $BC$, two distinct points $P$ and $Q$ are chosen such that $\angle BAP = \angle PAQ$ and $BP \cdot CQ = BC \cdot PQ$. Calculate the size of $\angle PAC$.

2021 Taiwan TST Round 2, A

Tags: algebra , inequality , inequalities

[i]Version 1[/i]. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x . \] [i]Version 2[/i]. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x . \]

2013 239 Open Mathematical Olympiad, 5

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A squirrel has infinitely many nuts; one nut of each of the masses $1g, 2g, 3g, \ldots$. The squirrel took $100$ bags, in each put a finite number of nuts, after which wrote on each bag the total mass of the nuts inside it. Prove that it is possible to create bags of the same mass using no more than $500$ nuts.

Kvant 2021, M2679

Tags: game , combinatorics

The number 7 is written on a board. Alice and Bob in turn (Alice begins) write an additional digit in the number on the board: it is allowed to write the digit at the beginning (provided the digit is nonzero), between any two digits or at the end. If after someone’s turn the number on the board is a perfect square then this person wins. Is it possible for a player to guarantee the win? [i]Alexandr Gribalko[/i]