Olympiad problems

2023 Denmark MO - Mohr Contest, 2

Tags: algebra

The numbers $1, 2, 3, . . . , 16$ must be placed in the $16$ squares in such a way that the sum of the numbers in each of the four rows and columns is the same. What is the smallest possible sum of the four numbers in the corner squares? [img]https://cdn.artofproblemsolving.com/attachments/c/2/fad1837625fd71e8ea333f9f9477f0bd120e05.png[/img]

May Olympiad L2 - geometry, 2022.3

Tags: angle , geometry

Let $ABCD$ be a square, $E$ a point on the side $CD$, and $F$ a point inside the square such that that triangle $BFE$ is isosceles and $\angle BFE = 90^o$ . If $DF=DE$, find the measure of angle $\angle FDE$.

1996 All-Russian Olympiad, 3

Tags: number theory

Let $x, y, p, n$, and $k$ be positive integers such that $x^n + y^n = p^k$. Prove that if $n > 1$ is odd, and $p$ is an odd prime, then $n$ is a power of $p$. [i]A. Kovaldji, V. Senderov[/i]

2012 CHMMC Spring, 2

Tags: 3d geometry , octahedron , geometric inequality , geometry

A convex octahedron in Cartesian space contains the origin in its interior. Two of its vertices are on the $x$-axis, two are on the $y$-axis, and two are on the $z$-axis. One triangular face $F$ has side lengths $\sqrt{17}$, $\sqrt{37}$, $\sqrt{52}$. A second triangular face $F_0$ has side lengths $\sqrt{13}$, $\sqrt{29}$, $\sqrt{34}$. What is the minimum possible volume of the octahedron?

2017 Romania Team Selection Test, P2

Tags: combinatorics

Consider a finite collection of 3-element sets $A_i$, no two of which share more than one element, whose union has cardinality 2017. Show that the elements of this union can be coloured with two colors, blue and red, so that at least 64 elements are blue and each $A_i$ has at least one red element.

VMEO III 2006 Shortlist, A1

Tags: algebra , functional equation , functional

Find all functions $f:R \to R$ such that $$f(x^2+f(y)-y) =(f(x))^2-f(y)$$ for all $x,y \in R$

1997 Tournament Of Towns, (532) 4

Tags: hexagon , angle , area , geometry

$AC' BA'C B'$ is a convex hexagon such that $AB' = AC'$, $BC' = BA'$, $CA' = CB'$ and $\angle A +\angle B + \angle C = \angle A' + \angle B' + \angle C'$. Prove that the area of the triangle $ABC$ is half the area of the hexagon. (V Proizvolov)

2021 Dutch Mathematical Olympiad, 3

Tags: combinatorics , lattice points

A frog jumps around on the grid points in the plane, from one grid point to another. The frog starts at the point $(0, 0)$. Then it makes, successively, a jump of one step horizontally, a jump of $2$ steps vertically, a jump of $3$ steps horizontally, a jump of $4$ steps vertically, et cetera. Determine all $n > 0$ such that the frog can be back in $(0, 0)$ after $n$ jumps.

2013 Singapore MO Open, 4

Tags: graph theory , combinatorics

Let $F$ be a finite non-empty set of integers and let $n$ be a positive integer. Suppose that $\bullet$ Any $x \in F$ may be written as $x=y+z$ for some $y$, $z \in F$; $\bullet$ If $1 \leq k \leq n$ and $x_1$, ..., $x_k \in F$, then $x_1+\cdots+x_k \neq 0$. Show that $F$ has at least $2n+2$ elements.

2012 Romanian Master of Mathematics, 1

Tags: induction , modular arithmetic , graph theory , combinatorics

Given a finite number of boys and girls, a [i]sociable set of boys[/i] is a set of boys such that every girl knows at least one boy in that set; and a [i]sociable set of girls[/i] is a set of girls such that every boy knows at least one girl in that set. Prove that the number of sociable sets of boys and the number of sociable sets of girls have the same parity. (Acquaintance is assumed to be mutual.) [i](Poland) Marek Cygan[/i]

May Olympiad L1 - geometry, 2002.2

Tags: geometry , paper , area , rectangle

A rectangular sheet of paper (white on one side and gray on the other) was folded three times, as shown in the figure: Rectangle $1$, which was white after the first fold, has $20$ cm more perimeter than rectangle $2$, which was white after the second fold, and this in turn has $16$ cm more perimeter than rectangle $3$, which was white after the third fold. Determine the area of the sheet. [img]https://cdn.artofproblemsolving.com/attachments/d/f/8e363b40654ad0d8e100eac38319ee3784a7a7.png[/img]

2012 South East Mathematical Olympiad, 1

Tags: modular arithmetic , number theory

A nonnegative integer $m$ is called a “six-composited number” if $m$ and the sum of its digits are both multiples of $6$. How many “six-composited numbers” that are less than $2012$ are there?

2011 Saudi Arabia Pre-TST, 3.4

Tags: number theory

Find all positive integers $n$ for which the equation $$x^3 + y^3 = n! + 4$$ has solutions in integers.

2007 Greece National Olympiad, 4

Tags: combinatorics

Given a $2007\times 2007$ array of numbers $1$ and $-1$, let $A_{i}$ denote the product of the entries in the $i$th row, and $B_{j}$ denote the product of the entries in the $j$th column. Show that \[A_{1}+A_{2}+\cdots +A_{2007}+B_{1}+B_{2}+\cdots +B_{2007}\neq 0.\]

2002 Turkey Team Selection Test, 3

Tags: inequalities

A positive integer $n$ and real numbers $a_1,\dots, a_n$ are given. Show that there exists integers $m$ and $k$ such that \[|\sum\limits_{i=1}^m a_i -\sum\limits_{i=m+1}^n a_i | \leq |a_k|.\]

2024 Belarusian National Olympiad, 9.8

Tags: combinatorics , geometry

Given right hexagon $H$ with side length $1$. On the sides of $H$ points $A_1$,$A_2$,$\ldots$,$A_k$ such that at least one of them is the midpoint of some side and for every $1 \leq i \leq k$ lines $A_{i-1}A_i$ and $A_iA_{i+1}$ form equal angles with the side, that contains the point $A_i$ (let $A_0=A_k$ and $A_{k+1}=A_1$. It is known that the length of broken line $A_1A_2\ldots A_kA_1$ is a positive integer Prove that $n$ is divisible by $3$ [i]M. Zorka[/i]

2022 Purple Comet Problems, 10

Tags:

Let $a$ be a positive real number such that $$4a^2+\frac{1}{a^2}=117.$$ Find $$8a^3+\frac{1}{a^3}.$$

Russian TST 2018, P3

Tags: combinatorics

Kirill has $n{}$ identical footballs and two infinite rows of baskets, each numbered with consecutive natural numbers. In one row the baskets are red, in the other they are blue. Kirill puts all the balls into baskets so that the number of balls in the either row of baskets does not increase. Denote by $A{}$ the number of ways to arrange the balls so that the first blue basket contains more balls than any red one, and by $B{}$ the number of arrangements so that the number of some blue basket corresponds with the number of balls in it. Prove that $A = B$.

2022 Taiwan TST Round 1, A

Tags: inequalities

Let $a_1, a_2, a_3, \ldots$ be a sequence of reals such that there exists $N\in\mathbb{N}$ so that $a_n=1$ for all $n\geq N$, and for all $n\geq 2$ we have \[a_{n}\leq a_{n-1}+2^{-n}a_{2n}.\] Show that $a_k>1-2^{-k}$ for all $k\in\mathbb{N}$. [i] Proposed by usjl[/i]

2018 Iran Team Selection Test, 6

Tags: geometry

Consider quadrilateral $ABCD $ inscribed in circle $\omega $. $P\equiv AC\cap BD$. $E$, $F$ lie on sides $AB$, $CD$ respectively such that $\hat {APE}=\hat {DPF} $. Circles $\omega_1$, $\omega_2$ are tangent to $\omega$ at $X $, $Y $ respectively and also both tangent to the circumcircle of $\triangle PEF $ at $P $. Prove that: $$\frac {EX}{EY}=\frac {FX}{FY} $$ [i]Proposed by Ali Zamani [/i]

2024 Ukraine National Mathematical Olympiad, Problem 6

Tags: geometry , cyclic quadrilateral

Cyclic quadrilateral $ABCD$ is such that $\angle BAD = 2\angle ADC$ and $CD = 2BC$. Let $H$ be the projection of $C$ onto $AD$. Prove that $BH \parallel CD$. [i]Proposed by Fedir Yudin, Anton Trygub[/i]

2005 Tuymaada Olympiad, 7

Tags: incenter , circumcircle , parallelogram , power of a point , radical axis , geometry

Let $I$ be the incentre of triangle $ABC$. A circle containing the points $B$ and $C$ meets the segments $BI$ and $CI$ at points $P$ and $Q$ respectively. It is known that $BP\cdot CQ=PI\cdot QI$. Prove that the circumcircle of the triangle $PQI$ is tangent to the circumcircle of $ABC$. [i]Proposed by S. Berlov[/i]

2018 Kazakhstan National Olympiad, 2

Tags: sequence , number theory , algebra

The natural number $m\geq 2$ is given.Sequence of natural numbers $(b_0,b_1,\ldots,b_m)$ is called concave if $b_k+b_{k-2}\le2b_{k-1}$ for all $2\le k\le m.$ Prove that there exist not greater than $2^m$ concave sequences starting with $b_0 =1$ or $b_0 =2$

1972 AMC 12/AHSME, 9

Tags:

Ann and Sue bought identical boxes of stationery. Ann used hers to write 1-sheet letters and Sue used hers to write 3-sheet letters. Ann used all the envelopes and had 50 sheets of paper left, while Sue used all of the sheets of paper and had 50 envelopes left. The number of sheets of paper in each box was \[ \begin{array}{rlrlrlrlrlrl} \hbox {(A)}& 150 \qquad & \hbox {(B)}& 125 \qquad & \hbox {(C)}& 120 \qquad & \hbox {(D)}& 100 \qquad & \hbox {(E)}& 80 & \end{array} \]

2016 China Team Selection Test, 5

Tags: geometry , cyclic quadrilateral

Refer to the diagram below. Let $ABCD$ be a cyclic quadrilateral with center $O$. Let the internal angle bisectors of $\angle A$ and $\angle C$ intersect at $I$ and let those of $\angle B$ and $\angle D$ intersect at $J$. Now extend $AB$ and $CD$ to intersect $IJ$ and $P$ and $R$ respectively and let $IJ$ intersect $BC$ and $DA$ at $Q$ and $S$ respectively. Let the midpoints of $PR$ and $QS$ be $M$ and $N$ respectively. Given that $O$ does not lie on the line $IJ$, show that $OM$ and $ON$ are perpendicular.