Olympiad problems

2005 Estonia National Olympiad, 3

How many such four-digit natural numbers divisible by $7$ exist such when changing the first and last number we also get a four-digit divisible by $7$?

2000 Harvard-MIT Mathematics Tournament, 21

Tags:

How many ways can you color a necklace of $7$ beads with $4$ colors so that no two adjacent beads have the same color?

2019 Latvia Baltic Way TST, 4

Tags: algebra , polynomial , inequalities , chebyshev polynomial

Let $P(x)$ be a polynomial with degree $n$ and real coefficients. For all $0 \le y \le 1$ holds $\mid p(y) \mid \le 1$. Prove that $p(-\frac{1}{n}) \le 2^{n+1} -1$

2016 AMC 10, 25

Tags: number theory , least common multiple

How many ordered triples $(x,y,z)$ of positive integers satisfy $\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600$ and $\text{lcm}(y,z)=900$? $\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 64$

2017 China Team Selection Test, 1

Tags: inequalities

Let $n \geq 4$ be a natural and let $x_1,\ldots,x_n$ be non-negative reals such that $x_1 + \cdots + x_n = 1$. Determine the maximum value of $x_1x_2x_3 + x_2x_3x_4 + \cdots + x_nx_1x_2$.

2014 Contests, 2

Tags: pigeonhole principle , number theory

Prove that among any $16$ perfect cubes we can always find two cubes whose difference is divisible by $91$.

2017 Math Prize for Girls Problems, 10

Tags: geometry , 3d geometry

Let $C$ be a cube. Let $P$, $Q$, and $R$ be random vertices of $C$, chosen uniformly and independently from the set of vertices of $C$. (Note that $P$, $Q$, and $R$ might be equal.) Compute the probability that some face of $C$ contains $P$, $Q$, and $R$.

Indonesia MO Shortlist - geometry, g3

Tags: geometry , tangent circle

Given triangle $ABC$. A circle $\Gamma$ is tangent to the circumcircle of triangle $ABC$ at $A$ and tangent to $BC$ at $D$. Let $E$ be the intersection of circle $\Gamma$ and $AC$. Prove that $$R^2=OE^2+CD^2\left(1- \frac{BC^2}{AB^2+AC^2}\right)$$ where $O$ is the center of the circumcircle of triangle $ABC$, with radius $R$.

1957 Polish MO Finals, 2

Tags: trigonometry , geometry

Prove that between the sides $ a $, $ b $, $ c $ and the opposite angles $ A $, $ B $, $ C $ of a triangle there is a relationship $$ a^2 \cos^2 A = b^2 \cos^2 B + c^2 \cos^2 C + 2bc \cos B \cos C \cos 2A.$$

1958 February Putnam, B7

Tags: continuity , integral

Prove that if $f(x)$ is continuous for $a\leq x \leq b$ and $$\int_{a}^{b} x^n f(x) \, dx =0$$ for $n=0,1,2, \ldots,$ then $f(x)$ is identically zero on $a \leq x \leq b.$

2007 District Olympiad, 2

Tags: combinatorics

In an urn we have red and blue balls. A person has invented the next game: he extracts balls until he realises for the first time that the number of blue balls is equal to the number of red balls. After a such game he finds out that he has extracted 10 balls, and that there does not exist 3 consecutive balls of the same color. Prove that the fifth and the sixth balls have different collors.

2016 Tournament Of Towns, 1

Tags: logarithm , algebra

On a blackboard the product $log_{( )}[ ]\times\dots\times log_{( )}[ ]$ is written (there are 50 logarithms in the product). Donald has $100$ cards: $[2], [3],\dots, [51]$ and $(52),\dots,(101)$. He is replacing each $()$ with some card of form $(x)$ and each $[]$ with some card of form $[y]$. Find the difference between largest and smallest values Donald can achieve.

2018 Greece Team Selection Test, 1

Tags: inequalities

If $x, y, z$ are positive real numbers such that $x + y + z = 9xyz.$ Prove that: $$\frac {x} {\sqrt {x^2+2yz+2}}+\frac {y} {\sqrt {y^2+2zx+2}}+\frac {z} {\sqrt {z^2+2xy+2}}\ge 1.$$ Consider when equality applies.

1969 IMO Longlists, 69

Tags: inequality , three variable inequality , algebra

$(YUG 1)$ Suppose that positive real numbers $x_1, x_2, x_3$ satisfy $x_1x_2x_3 > 1, x_1 + x_2 + x_3 <\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}$ Prove that: $(a)$ None of $x_1, x_2, x_3$ equals $1$. $(b)$ Exactly one of these numbers is less than $1.$

2010 Today's Calculation Of Integral, 589

Tags: calculus , integration , calculus computations

Evaluate $ \int_0^1 \frac{x}{\{(2x\minus{}1)\sqrt{x^2\plus{}x\plus{}1}\plus{}(2x\plus{}1)\sqrt{x^2\minus{}x\plus{}1}\}\sqrt{x^4\plus{}x^2\plus{}1}}\ dx$.

Kvant 2022, M2727

Tags: geometry , area

A convex quadrilateral $ABCD$ is given. Let $O_a$ be the circumcenter of the triangle $DBC$, and define $O_b,O_c$ and $O_d$ similarly. The points $O_a, O_b, O_c, O_d$ are the vertices of a convex quadrilateral. Prove that its area is equal to half of the absolute value of the difference between the areas of $AO_bCO_d$ and $BO_cDO_a$. [i]Proposed by V. Dubrovsky[/i]

2020 Regional Olympiad of Mexico Center Zone, 4

Tags: square , geometry , circles

Let $\Gamma_1$ be a circle with center $O$ and $A$ a point on it. Consider the circle $\Gamma_2$ with center at $A$ and radius $AO$. Let $P$ and $Q$ be the intersection points of $\Gamma_1$and $\Gamma_2$. Consider the circle $\Gamma_3$ with center at $P$ and radius $PQ$. Let $C$ be the second intersection point of $\Gamma_3$ and $\Gamma_1$. The line $OP$ cuts $\Gamma_3$ at $R$ and $S$, with $R$ outside $\Gamma_1$. $RC$ cuts $\Gamma_1$ into $B$. $CS$ cuts $\Gamma_1$ into $D$. Show that $ABCD$ is a square.

2023 239 Open Mathematical Olympiad, 1

Tags: combinatorics , winning strategy

There are $n{}$ wise men in a hall and everyone sees each other. Each man will wear a black or white hat. The wise men should simultaneously write down on their piece of paper a guess about the color of their hat. If at least one does not guess, they will all be executed. The wise men can discuss a strategy before the test and they know that the layout of the hats will be chosen randomly from the set of all $2^n$ layouts. They want to choose their strategy so that the number of layouts for which everyone guesses correctly is as high as possible. What is this number equal to?

Cono Sur Shortlist - geometry, 2020.G4

Tags: bisects segment , circumcircle , geometry

Let $ABC$ be a triangle with circumcircle $\omega$. The bisector of $\angle BAC$ intersects $\omega$ at point $A_1$. Let $A_2$ be a point on the segment $AA_1$, $CA_2$ cuts $AB$ and $\omega$ at points $C_1$ and $C_2$, respectively. Similarly, $BA_2$ cuts $AC$ and $\omega$ at points $B_1$ and $B_2$, respectively. Let $M$ be the intersection point of $B_1C_2$ and $B_2C_1$. Prove that $MA_2$ passes the midpoint of $BC$. [i]proposed by Jhefferson Lopez, Perú[/i]

1963 AMC 12/AHSME, 40

Tags: geometry , 3d geometry

If $x$ is a number satisfying the equation $\sqrt[3]{x+9}-\sqrt[3]{x-9}=3$, then $x^2$ is between: $\textbf{(A)}\ 55\text{ and }65 \qquad \textbf{(B)}\ 65\text{ and }75\qquad \textbf{(C)}\ 75\text{ and }85 \qquad \textbf{(D)}\ 85\text{ and }95 \qquad \textbf{(E)}\ 95\text{ and }105$

2003 Moldova National Olympiad, 12.2

Tags: calculus , derivative , polynomial , algebra

For every natural number $n\geq{2}$ consider the following affirmation $P_n$: "Consider a polynomial $P(X)$ (of degree $n$) with real coefficients. If its derivative $P'(X)$ has $n-1$ distinct real roots, then there is a real number $C$ such that the equation $P(x)=C$ has $n$ real,distinct roots." Are $P_4$ and $P_5$ both true? Justify your answer.

2002 Tournament Of Towns, 7

Tags: ceiling function , combinatorics

[list] [*] A power grid with the shape of a $3\times 3$ lattice with $16$ nodes (vertices of the lattice) joined by wires (along the sides of squares. It may have happened that some of the wires have burned out. In one test technician can choose any two nodes and check if electrical current circulates between them (i.e there is a chain of intact wires joining the chosen nodes) . Technicial knows that current will circulate from any node to another node. What is the least number of tests required to demonstrate this? [*] Previous problem for the grid of $5\times 5$ lattice.[/list]

2009 Croatia Team Selection Test, 4

Tags: quadratic , diophantine equation , number theory

Prove that there are infinite many positive integers $ n$ such that $ n^2\plus{}1\mid n!$, and infinite many of those for which $ n^2\plus{}1 \nmid n!$.

MathLinks Contest 5th, 6.3

Tags: algebra , inequalities

Let $x, y, z$ be three positive numbers such that $(x + y-z) \left( \frac{1}{x}+ \frac{1}{y}- \frac{1}{z} \right)=4$. Find the minimal value of the expression $$E(x, y, z) = (x^4 + y^4 + z^4) \left( \frac{1}{x^4}+ \frac{1}{y^4}+ \frac{1}{z^4} \right) .$$

2021 Bundeswettbewerb Mathematik, 3

Tags: geometry

Consider a triangle $ABC$ with $\angle ACB=120^\circ$. Let $A’, B’, C’$ be the points of intersection of the angular bisector through $A$, $B$ and $C$ with the opposite side, respectively. Determine $\angle A’C’B’$.