Галерея 3017089

Галерея 3017089

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High School Olympiads

Regional, national, and international math olympiads

Regional, national, and international math olympiads

High School Olympiads

Regional, national, and international math olympiads

Regional, national, and international math olympiads

Let be a point interior to triangle (with ) . The lines , and meet again its circumcircle at , , respectively . The tangent line at to meets the line at . Show that from follows .

Proposed by Marcin E. Kuczma, Poland

Source : Kvant Magazine No. 3 2022 M2690

Vasya has candies of several types, where . It is known that for any group of at least 145 candies, there is a type of candy which appears exactly 10 times. Find the largest possible value of .

Proposed by A. Antropov

Show that the only polynomial of odd degree satisfying for all is

Let with be the projection of onto . is midpoint of . Point is the intersection of and . Point is the Lemoine point of . Prove that are collinear

Monotonically decreasing number, game

The number is written on a blackboard. Two players Alice and Bob play a game in which two actions exist:

1. Divide a number on the board into integers and such that
2. If a number occurs twice, erase one or both.

The player who cannot make any move loses the game.

If Alice starts the game, who has a winning strategy?

Triangle ABC is 40-60-80 degree triangle,the angle bisector at A meets BC at D.let E be the midpoint of CD and let F be the foot of the altitude from C to AB. Prove that perpendicular bisector of EF intersects AC at its midpoint

There are people in a party. Assume that among any three of them some two know each other. Show that one can select at least people and arrange them at a round table so that each person sits between two of his/her acquaintances.

Source : Dürer Competition Finals 2023/E+ 2

a) Find all solutions of the equation , where are positive primes.
b) Show that for every positive integer , there exist three integers with .

Source : Dürer Competition Finals 2023/E+ 3 (modified)

and are integer sequences with , , and . It is known that and for all applicable and , and the numbers are pairwise different. Determine the total number of such pairs of sequences.

Source : Dürer Competition Finals 2023/E+ 6

Two players play a game on four piles of pebbles labeled with the numbers respectively. The players take turns in an alternating fashion. On his or her turn, a player selects integers and with , removes pebbles from pile , and places one pebble in each of the piles . A player loses the game if he or she cannot make a legal move. For each starting position, determine the player with a winning strategy.

Source : Dürer Competition Finals 2023/E+ 5

For an acute triangle , let be its circumcenter, and let be the circumcenter of respectively. Show that are concurrent.

Source : 2016 All-Russian Olympiad,Problem 9.3

Alexander has chosen a natural number and has written down in a line,and in increasing order,all his positive divisors (where and ) .For each pair of neighbouring numbers,he has found their greater common divisor.The sum of all these numbers (the greatest common divisors) is equal to . Find all possible values of .

Source : Dürer Competition Finals 2023/E+ 4

For a given integer , a pyramid of height if defined as a collection of stone cubes of equal size stacked in layers such that the cubes in the - th layer form a square with sidelength and every cube (except for the ones in the bottom layer) rests on four cubes in the layer below. Some of the cubes are made of sandstone, some are made of granite. The top cube is made of granite, and to ensure the stability of the piramid, for each granite cube (except for the ones in the bottom layer), at least three out of four of the cubes supporting it have to be granite. What is the minimum possible number of granite cubes in such an arrangement?

Source : IMO shortlist 2006 Algebra

May I ask how is Schur's inequalities applied here? I need some further explanation.

Covering perimeter of rectangle with disks

Source : Dürer Competition Finals 2023/E+ 1

Prove that for any real , one can cover the circumference of a rectangle with non-intersecting disks of unit radius.

Points of intersection of two unequal coplanar circles

Source : IMO 1983, Day 1, Problem 2

Let be one of the two distinct points of intersection of two unequal coplanar circles and with centers and respectively. One of the common tangents to the circles touches at and at , while the other touches at and at . Let be the midpoint of and the midpoint of . Prove that .

A security password is a number made up of 1's, 2's, 3's and 4's in which all the 4-digit numbers formed by these digits appear. What is the smallest number of digits that a password can have?
NOTE: for example, the number 123321 contains 123, 3321 and 32, among others; but 11 and 231 do not appear.

n is an integer greater than 3 and S is a set of 2^n + 1 elements. Let f be a function of binomial subsets of S up to {0,1,2,...,2^n -1}, so that for each member x,y,z of S one of f({x,y}), f({y,z}), f({z,x}) is equal to the sum of the other two. prove that there exist a,b,c of S that f({a,b}) = f({b,c}) = f({c,a}) = 0

N
Today at 1:24 AM
by Strudan_Borisov

Determine all the functions such that
for all real numbers and .

Non-Standard Functional Equation ft. Real Number c

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Today at 1:10 AM
by omeravci372742

Source : Turkey Olympic Revenge 2023 P1

Find all such that there exists a function satisfying for all .

Proposed by Kaan Bilge

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Today at 1:09 AM
by omeravci372742

Source : Turkey Olympic Revenge 2023 P3

Find all polynomials with integer coefficients such that for all .
Note: denotes the sum of digits of .

Proposed by Şevket Onur YILMAZ

N
Today at 1:06 AM
by omeravci372742

Source : Turkey Olympic Revenge 2023 P4

Find all functions such that for all integers and , the number is a perfect square.

Proposed by Barış Koyuncu

KL >= PQ, locus of midpoints with ends on perpendicular lines

Source : Netherlands - Dutch NMO 1963 p2

De loodrechte snijlijn van k en l snijdt k in A en t in B

Let , , , prove that
Seems like Schur but I don't know how to apply it.

prove orthocenter for antipodal point

Source : Sharygin 2023 - P11 (Grade-8-10)

Let be the orthocenter of an acute-angled triangle ; , be points on respectively, such that is a parallelogram; be the common points of the line and the circumcircle of triangle ; be the point of opposite to . Prove that is the orthocenter of triangle .

Given a convex polygon with 9 vertices in the plane, prove that there are 2 diagonals that make between them an angle smaller than 7 degrees[

N
Yesterday at 11:52 PM
by VicKmath7

Source : Kvant Magazine No. 9 2022 M2716

Find all pairs of natural numbers such that for any natural the product is divisible by .

Proposed by P. Kozhevnikov

Let be given circles and be a given line. Construct a circle using ruler and compass such that it's center is on l and it is tangent to both .

P(s) and P(t) integers implies P(st) integer

N
Yesterday at 11:25 PM
by Inconsistent

Find all polynomials with integer coefficients such that for all real numbers and , if and are both integers, then is also an integer.

N
Yesterday at 11:21 PM
by WallyWalrus

The roots of a monic cubic polynomial P(x) are positive real numbers forming a G.P. Suppose that the sum of the roots is equal to 10. Under this conditions the largest possible value of |P(-1)| is written in the form m÷n where GCD of (m,n)=1. Find m+n.

Reflecting a ray wrt the sides of an equilateral triangle

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Yesterday at 11:19 PM
by Inconsistent

Let be an equilateral triangle. From the vertex we draw a ray towards the interior of the triangle such that the ray reaches one of the sides of the triangle. When the ray reaches a side, it then bounces off following the law of reflection, that is, if it arrives with a directed angle , it leaves with a directed angle . After bounces, the ray returns to without ever landing on any of the other two vertices. Find all possible values of .

N
Yesterday at 11:11 PM
by Inconsistent

A collection of squares on the plane is called tri-connected if the following criteria are satisfied:

(i) All the squares are congruent.
(ii) If two squares have a point in common, then is a vertex of each of the squares.
(iii) Each square touches exactly three other squares.

How many positive integers are there with , such that there exists a collection of squares that is tri-connected?

satisfy, for all real values of x, the following expression. Proof that is a periodic function.

Bounding difference of sum of fractions in an interval

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Yesterday at 11:07 PM
by Inconsistent

Let and be given by

.

Prove that for any non-integer real number satisfying .

N
Yesterday at 11:05 PM
by Inconsistent

Let be the orthocenter of the triangle . Let and be the midpoints of the sides and , respectively. Assume that lies inside the quadrilateral and that the circumcircles of triangles and are tangent to each other. The line through parallel to intersects the circumcircles of the triangles and in the points and , respectively. Let be the intersection point of and and let be the incenter of triangle . Prove that .

N
Yesterday at 10:59 PM
by VicKmath7

Source : Kvant Magazine No. 3 2022 M2691

There are points marked on the plane. Any three of them form a triangle, the values of the angles of which in are expressed in natural numbers (in degrees). What is the maximum for which this is possible?

Proposed by E. Bakaev

$$(f(x)+1)(f(y)+1)=f(x+y)+f(xy-1),\forall x,y \in \mathbb{R}$$

N
Yesterday at 10:54 PM
by cadaeibf

Source : Kvant Magazine No. 3 2022 M2693

Prove that there exists a natural number such that for any natural the sum of the digits of is not less than .

Proposed by D. Khramtsov

Source : Kvant Magazine No. 3 2022 M2692

In the circle the hexagon is inscribed. It is known that the point divides the arc in half, and the triangles and have a common inscribed circle. The line intersects segments and at points and and the line intersects segments and at points and respectively. Prove that the points and lie on the same circle.

Proposed by D. Brodsky

N
Yesterday at 10:29 PM
by Marinchoo

Source : Kvant Magazine No. 4 2022 M2694

Call a natural number interesting if any natural number not exceeding can be represented as the sum of several (possibly one) pairwise distinct positive divisors of .
[list=a]
[*]Find the largest three-digit interesting number.
[*]Prove that there are arbitrarily large interesting numbers other than the powers of two.
[/list]
Proposed by N. Agakhanov

Source : Kvant Magazine No. 2 2022 M2689

There are 1000 gentlemen listed in the register of a city with numbers from 1 to 1000. Any 720 of them form a club. The mayor wants to impose a tax on each club, which is paid by all club members in equal shares (the tax is an arbitrary non-negative real number). At the same time, the total tax paid by a gentleman should not exceed his number in the register. What is the largest tax the mayor can collect?

Proposed by I. Bogdanov

N
Yesterday at 10:20 PM
by IAmTheHazard

Source : Kvant Magazine No. 2 2022 M2686

At a two-round volleyball tournament participated 99 teams. Each played a match at home and a match away. Each team won exactly half of their home matches and exactly half of their away matches. Prove that one of the teams beat another team twice.

Proposed by M. Antipov

Source : Kvant Magazine No. 2 2022 M2688

Let and be the tangent points of the incircle of the triangle with the sides and respectively. Let and be points on the circle such that lies on the ray , lies on the ray and lies on the ray . Let be the intersection point of the segments and , and similarly and . Prove that the points and lie on the lines and , respectively.

Proposed by L. Shatunov (11th grade student)

Source : Kvant Magazine No. 2 2022 M2687

We have a regular - gon, with . We consider the arrangements of numbers on its vertices, each of which is equal to 1 or 2. For each such arrangement , we find the number of odd sums among all sums of numbers in several consecutive vertices. This number is denoted by .
[list=a]
[*]Find the largest possible value of .
[*]Find the number of arrangements for which takes this largest possible value.
[/list]
Proposed by P. Kozhevnikov

N
Yesterday at 10:06 PM
by IAmTheHazard

Find all pairs of positive integers so that

N
Yesterday at 9:50 PM
by StefanSebez

For each prime , construct a graph on , where are adjacent if and only if divides . Prove that is disconnected for infinitely many

Conditions for collinearity and cyclicity

Source : Kvant Magazine No. 4 2022 M2695

Let the circle and the line intersect at two different points and . For different and non-points. Let and be points on and and be points on , all of them different from and . Prove the following statements:
[list=a]
[*]The points and lie on the same line if and only if [*]The points and lie on the same circle if and only if [/list]
Note: In both points, the sign is selected in the right parts of the equalities if the points and lie on the same arc of the circle , and the sign if and lie on different arcs . By , we indicate the ratio of the lengths of and , taken with the sign or depending on whether the and vectors are co-directed or oppositely directed.

Proposed by M. Skopenkov

Source : Middle European Mathematical Olympiad 2022, problem I-1

Find all functions such that holds for all real numbers and .

angle bisector circumcircle tangency

Source : Sharygin 2023 - P10 (Grade-8-9)

Altitudes and of an acute-angled triangle meet at point . The perpendicular from to meets the line passing through and parallel to at point . The bisectors of two angles between and meet at points and . Prove that the circumcircles of triangles and are tangent.

Source : Kvant Magazine No. 4 2022 M2696

Does there exist a sequence of natural numbers such that the number has an even number of different prime divisors for any two different natural indices and ?

From the folklore

Source : Kvant Magazine No. 4 2022 M2697 (appeared on the 24th Kolmogorov Memorial Cup)

There are some gas stations on a circular highway. The total amount of gasoline in them is enough for two laps. Two drivers want to refuel at one station and starting from it, go in different directions, both of them completing an entire lap. Along the way, they can refuel at other stations, without necessarily taking all the gasoline. Prove that drivers will always be able to do this.

Proposed by I. Bogdanov

Suppose polynomial with Correct coefficients . if we have

prove thet

Source : Kvant Magazine No. 9 2022 M2714

Let and be polynomials with integers coefficients. The leading coefficient of is equal to 1. It is known that for infinitely many natural numbers the number is divisible by . Prove that is divisible by for all positive integers such that .

From the folklore

with , find all the polynomials such that

locus of coaxial circles intersection point

Source : Sharygin 2023 - P19 (Grade-10-11)

A cyclic quadrilateral is given. An arbitrary circle passing through and meets at points respectively. Find the locus of common points of circles and .

we have .
find sum of coefficients .

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Yesterday at 8:37 PM
by IAmTheHazard

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Yesterday at 8:28 PM
by ChemistLovesBashing

In the cyclic quadrilateral ABCD, the diagonal AC bisects the angle DAB. The side AD is extended beyond D to a point E. Show that CE = CA if and only if DE = AB

Sequence problem from 2018 Saudi Arabia GMO TST

Let be a sequence defined by and for . Prove that:
for all .

N
Yesterday at 8:06 PM
by ChemistLovesBashing

From a point O,two tangents are drawn to a given circle touching it at points A and B .the chord Ac is draw parallel to the tangent OB ,and OC intersects the circle at E.prove that the line AE bisects OB

Source : Kvant Magazine No. 6 2022 M2702

The bisectors of the triangle intersect at . The external bisectors of the angles and intersect at . The circle centered at passes through and touches the line at . The circle centered at passes through and touches the line
Голая индианка с гладко выбритой пиздой
Одев презик засандалил женщине
Ножки зрелых (Фиона) сет 4

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