Olympiad problems

2001 Baltic Way, 3

Tags: combinatorics

The numbers $1, 2, \ldots 49$ are placed in a $7\times 7$ array, and the sum of the numbers in each row and in each column is computed. Some of these $14$ sums are odd while others are even. Let $A$ denote the sum of all the odd sums and $B$ the sum of all even sums. Is it possible that the numbers were placed in the array in such a way that $A = B$?

2012-2013 SDML (Middle School), 3

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Simplify $\frac{0.\overline{3}}{1.\overline{3}}$. $\text{(A) }\frac{3}{13}\qquad\text{(B) }\frac{1}{4}\qquad\text{(C) }\frac{3}{11}\qquad\text{(D) }\frac{1}{3}\qquad\text{(E) }\frac{3}{4}$

2018 Belarusian National Olympiad, 10.7

Tags: geometry , circumcircle

The square $A_1B_1C_1D_1$ is inscribed in the right triangle $ABC$ (with $C=90$) so that points $A_1$, $B_1$ lie on the legs $CB$ and $CA$ respectively ,and points $C_1$, $D_1$ lie on the hypotenuse $AB$. The circumcircle of triangles $B_1A_1C$ an $AC_1B_1$ intersect at $B_1$ and $Y$. Prove that the lines $A_1X$ and $B_1Y$ meet on the hypotenuse $AB$.

2023 Indonesia TST, 3

Tags: hyperbola , projective geometry , conic , geometry

Let $ABC$ be an acute triangle with altitude $\overline{AH}$, and let $P$ be a variable point such that the angle bisectors $k$ and $\ell$ of $\angle PBC$ and $\angle PCB$, respectively, meet on $\overline{AH}$. Let $k$ meet $\overline{AC}$ at $E$, $\ell$ meet $\overline{AB}$ at $F$, and $\overline{EF}$ meet $\overline{AH}$ at $Q$. Prove that as $P$ varies, line $PQ$ passes through a fixed point.

2008 F = Ma, 10

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Which is the best value for the mass of the block? (a) $\text{3 kg}$ (b) $\text{5 kg}$ (c) $\text{10 kg}$ (d) $\text{20 kg}$ (e) $\text{30 kg}$

2021 Azerbaijan Junior NMO, 3

Tags: inequality

$a,b,c $ are positive real numbers . Prove that $\sqrt[7]{\frac{a}{b+c}+\frac{b}{c+a}} +\sqrt[7]{\frac{b}{c+a}+\frac{c}{b+a}}+\sqrt[7]{\frac{c}{a+b}+\frac{a}{b+c}}\geq 3$

2022 Rioplatense Mathematical Olympiad, 5

Tags: geometry

Let $ABCDEFGHI$ be a regular polygon with $9$ sides and the vertices are written in the counterclockwise and let $ABJKLM$ be a regular polygon with $6$ sides and the vertices are written in the clockwise. Prove that $\angle HMG=\angle KEL$. Note: The polygon $ABJKLM$ is inside of $ABCDEFGHI$.

2024 Canada National Olympiad, 1

Tags: geometry , incenter , geometric transformation , reflection , angle bisector

Let $ABC$ be a triangle with incenter $I$. Suppose the reflection of $AB$ across $CI$ and the reflection of $AC$ across $BI$ intersect at a point $X$. Prove that $XI$ is perpendicular to $BC$.

2017 China Team Selection Test, 1

Tags: combinatorics

Given $n\ge 3$. consider a sequence $a_1,a_2,...,a_n$, if $(a_i,a_j,a_k)$ with i+k=2j (i<j<k) and $a_i+a_k\ne 2a_j$, we call such a triple a $NOT-AP$ triple. If a sequence has at least one $NOT-AP$ triple, find the least possible number of the $NOT-AP$ triple it contains.

2016 Purple Comet Problems, 2

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The trapezoid below has bases with lengths 7 and 17 and area 120. Find the difference of the areas of the two triangles. [center] [img]https://i.snag.gy/BlqcSQ.jpg[/img] [/center]

2015 Peru IMO TST, 11

Tags: number theory , frobenius , additive combinatorics , additive number theory , additive representation , induction

Let $n \ge 2$ be an integer, and let $A_n$ be the set \[A_n = \{2^n - 2^k\mid k \in \mathbb{Z},\, 0 \le k < n\}.\] Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_n$ . [i]Proposed by Serbia[/i]

2024 Turkey Olympic Revenge, 1

Tags: combinatorics

Let $m,n$ be positive integers. An $n\times n$ board has rows and columns numbered $1,2,\dots,n$ from left to right and top to bottom, respectively. This board is colored with colors $r_1,r_2,\dots,r_m$ such that the cell at the intersection of $i$th row and $j$th column is colored with $r_{i+j-1}$ where indices are taken modulo $m$. After the board is colored, Ahmet wants to put $n$ stones to the board so that each row and column has exactly one stone, also he wants to put the same amount of stones to each color. Find all pairs $(m,n)$ for which he can accomplish his goal. Proposed by [i]Sena Başaran[/i]

2017 Spain Mathematical Olympiad, 5

Tags: algebra , inequalities , maximum

Let $a,b,c$ be positive real numbers so that $a+b+c = \frac{1}{\sqrt{3}}$. Find the maximum value of $$27abc+a\sqrt{a^2+2bc}+b\sqrt{b^2+2ca}+c\sqrt{c^2+2ab}.$$

1996 Bulgaria National Olympiad, 3

Tags: combinatorics

A square table of size $7\times 7$ with the four corner squares deleted is given. [list=a] [*] What is the smallest number of squares which need to be colored black so that a $5-$square entirely uncolored Greek cross (Figure 1) cannot be found on the table? [*] Prove that it is possible to write integers in each square in a way that the sum of the integers in each Greek cross is negative while the sum of all integers in the square table is positive. [/list] [asy] size(3.5cm); usepackage("amsmath"); MP("\text{Figure }1.", (1.5, 3.5), N); DPA(box((0,1),(3,2))^^box((1,0),(2,3)), black); [/asy]

PEN E Problems, 18

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Without using Dirichlet's theorem, show that there are infinitely many primes ending in the digit $9$.

2007 Regional Olympiad of Mexico Center Zone, 4

Tags: number theory , power of 2

Is there a power of $2$ that when written in the decimal system has all its digits different from zero and it is possible to reorder them to form another power of $2$?

1968 Putnam, A5

Tags: polynomial

Find the smallest possible $\alpha\in \mathbb{R}$ such that if $P(x)=ax^2+bx+c$ satisfies $|P(x)|\leq1 $ for $x\in [0,1]$ , then we also have $|P'(0)|\leq \alpha$.

2023 China Girls Math Olympiad, 6

Tags: algebra

Let $x_i\ (i = 1, 2, \cdots 22)$ be reals such that $x_i \in [2^{i-1},2^i]$. Find the maximum possible value of $$(x_1+x_2+\cdots +x_{22})(\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_{22}})$$

2003 Iran MO (2nd round), 1

Tags: number theory

We call the positive integer $n$ a $3-$[i]stratum[/i] number if we can divide the set of its positive divisors into $3$ subsets such that the sum of each subset is equal to the others. $a)$ Find a $3-$stratum number. $b)$ Prove that there are infinitely many $3-$stratum numbers.

2010 Contests, 3

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In a triangle $ABC$, let $M$ be the midpoint of $AC$. If $BC = \frac{2}{3} MC$ and $\angle{BMC}=2 \angle{ABM}$, determine $\frac{AM}{AB}$.

2019 Iran MO (3rd Round), 3

Tags: rectangle , combinatorics

Cells of a $n*n$ square are filled with positive integers in the way that in the intersection of the $i-$th column and $j-$th row, the number $i+j$ is written. In every step, we can choose two non-intersecting equal rectangles with one dimension equal to $n$ and swap all the numbers inside these two rectangles with one another. ( without reflection or rotation ) Find the minimum number of moves one should do to reach the position where the intersection of the $i-$th column and $j-$row is written $2n+2-i-j$.

2017 Brazil Team Selection Test, 2

Tags: cyclic quadrilateral , geometry , incenter , reflection , inversion

Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.

2023 Israel National Olympiad, P1

Tags: combinatorics

2000 people are sitting around a round table. Each one of them is either a truth-sayer (who always tells the truth) or a liar (who always lies). Each person said: "At least two of the three people next to me to the right are liars". How many truth-sayers are there in the circle?

2008 JBMO Shortlist, 4

Tags: algebra , system of equations

Find all triples $(x,y,z)$ of real numbers that satisfy the system $\begin{cases} x + y + z = 2008 \\ x^2 + y^2 + z^2 = 6024^2 \\ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{2008} \end{cases}$

2018 AMC 8, 5

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What is the value of $1+3+5+\cdots+2017+2019-2-4-6-\cdots-2016-2018$? $\textbf{(A) }-1010\qquad\textbf{(B) }-1009\qquad\textbf{(C) }1008\qquad\textbf{(D) }1009\qquad \textbf{(E) }1010$