Olympiad problems

2014 Ukraine Team Selection Test, 3

Let $ABCDEF$ be a convex hexagon with $AB=DE$, $BC=EF$, $CD=FA$, and $\angle A-\angle D = \angle C -\angle F = \angle E -\angle B$. Prove that the diagonals $AD$, $BE$, and $CF$ are concurrent.

Russian TST 2016, P3

Tags: incircle , geometry

The scalene triangle $ABC$ has incenter $I{}$ and circumcenter $O{}$. The points $B_A$ and $C_A$ are the projections of the points $B{}$ and $C{}$ onto the line $AI$. A circle with a diameter $B_AC_A$ intersects the line $BC$ at the points $K_A$ and $L_A$. [list=i] [*]Prove that the circumcircle of the triangle $AK_AL_A$ touches the incircle of the triangle $ABC$ at some point $T_A$. [*]Define the points $T_B$ and $T_C$ analogously. Prove that the lines $AT_A,BT_B$ and $CT_C$ intersect on the line $OI$. [/list]

2010 Mexico National Olympiad, 2

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

Let $ABC$ be an acute triangle with $AB\neq AC$, $M$ be the median of $BC$, and $H$ be the orthocenter of $\triangle ABC$. The circumcircle of $B$, $H$, and $C$ intersects the median $AM$ at $N$. Show that $\angle ANH=90^\circ$.

2014 Math Prize For Girls Problems, 9

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Let $abc$ be a three-digit prime number whose digits satisfy $a < b < c$. The difference between every two of the digits is a prime number too. What is the sum of all the possible values of the three-digit number $abc$?

2009 National Olympiad First Round, 22

Tags: modular arithmetic , geometry , 3d geometry

$ (a_n)_{n \equal{} 0}^\infty$ is a sequence on integers. For every $ n \ge 0$, $ a_{n \plus{} 1} \equal{} a_n^3 \plus{} a_n^2$. The number of distinct residues of $ a_i$ in $ \pmod {11}$ can be at most? $\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 6$

1954 AMC 12/AHSME, 10

Tags: algebra , polynomial , binomial theorem

The sum of the numerical coefficients in the expansion of the binomial $ (a\plus{}b)^8$ is: $ \textbf{(A)}\ 32 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 7$

1985 Swedish Mathematical Competition, 1

Tags: inequalities , algebra , radical

If $a > b > 0$, prove the inequality $$\frac{(a-b)^2}{8a}< \frac{a+b}{2}- \sqrt{ab} < \frac{(a-b)^2}{8b}.$$

2007 AMC 10, 15

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The angles of quadrilateral $ ABCD$ satisfy $ \angle A \equal{} 2 \angle B \equal{} 3 \angle C \equal{} 4 \angle D$. What is the degree measure of $ \angle A$, rounded to the nearest whole number? $ \textbf{(A)}\ 125 \qquad \textbf{(B)}\ 144 \qquad \textbf{(C)}\ 153 \qquad \textbf{(D)}\ 173 \qquad \textbf{(E)}\ 180$

2022 CCA Math Bonanza, L1.4

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Jongol and Gongol are writing calculus questions and grading tests. They want to write 90 calculus problems and they have 120 tests to grade. Jongol can write 3 questions per minute or grade 4 tests per minute. Gongol can write 1 question per minute or grade 2 tests per minute. Evaluate the shortest possible time, in minutes, for them to complete the two tasks. [i]2022 CCA Math Bonanza Lightning Round 1.4[/i]

2017 Finnish National High School Mathematics Comp, 2

Tags: algebra , system of equations

Determine $x^2+y^2$ and $x^4+y^4$, when $x^3+y^3=2$ and $x+y=1$

2019 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory

Determine positive integers $a$ and $b$ co-prime such that $a^2+b = (a-b)^3$ .

2014 AMC 8, 12

Tags: probability

A magazine printed photos of three celebrities along with three photos of the celebrities as babies. The baby pictures did not identify the celebrities. Readers were asked to match each celebrity with the correct baby pictures. What is the probability that a reader guessing at random will match all three correctly? $\textbf{(A) }\frac{1}{9}\qquad\textbf{(B) }\frac{1}{6}\qquad\textbf{(C) }\frac{1}{4}\qquad\textbf{(D) }\frac{1}{3}\qquad \textbf{(E) }\frac{1}{2}$

2017 Azerbaijan 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 $BDE$.

2025 Israel National Olympiad (Gillis), P4

Tags: geometry , rectangle , combinatorial geometry , combinatorics

A $100\times \sqrt{3}$ rectangular table is given. What is the minimum number of disk-shaped napkins of radius $1$ required to cover the table completely? [i]Remark:[/i] The napkins are allowed to overlap and protrude the table's edges.

2023-IMOC, C3

Tags: combinatorics

Graph $G$ has $n\geq 2$ vertices. Find the largest $m$ such that one of the following is true for always: 1. There exists a cycle with $k\geq m$ vertices. 2. There exists an independent set with $m$ vertices.

2022 Iran MO (3rd Round), 2

Tags: geometry , dilation , homothety , orthocenter

Constant points $B$ and $C$ lie on the circle $\omega$. The point middle of $BC$ is named $M$ by us. Assume that $A$ is a variable point on the $\omega$ and $H$ is the orthocenter of the triangle $ABC$. From the point $H$ we drop a perpendicular line to $MH$ to intersect the lines $AB$ and $AC$ at $X$ and $Y$ respectively. Prove that with the movement of $A$ on the $\omega$, the orthocenter of the triangle $AXY$ also moves on a circle.

2005 Gheorghe Vranceanu, 1

Tags: combinatorics , number theory

Given a natural number $ n, $ prove that the set $ \{ -n+1,-n+2,\ldots ,-1,1,2,\ldots ,n-1,n\} $ can be partitioned into $ k $ subsets such that the sums of all elements of each of these subsets are equal, if and only if $ n $ is multiple of $ k. $

2009 Flanders Math Olympiad, 1

Tags: combinatorics

In an attempt to beat the Belgian handshake record come on $20/09/2009$ exactly $2009$ Belgians together in a large sports hall. Among them are Nathalie and thomas. During this event, everyone shakes hands with everyone exactly once other attendees. Afterwards, Nathalie says: “I have exactly $5$ times as many Flemish people shaken hands as people from Brussels.” Thomas replies with “I have exactly $3$ times as much Walloons and Brussels people shook hands”. From which region does Nathalie come and from which region comes Thomas?

2024 Bulgarian Autumn Math Competition, 12.2

Tags: geometry , circumcircle

Let $ABC$ be a triangle and let $X$ be a point in its interior. Point $S_A$ is the midpoint of arc $BC$ containing $X$ of the circumcircle of $BCX$. $S_B$ and $S_C$ are defined similarly. Prove that $S_A,S_B,S_C$ and $X$ are concyclic.

1981 AMC 12/AHSME, 9

Tags: geometry , 3d geometry , ratio

In the adjoining figure, $PQ$ is a diagonal of the cube. If $PQ$ has length $a$, then the surface area of the cube is $\text{(A)}\ 2a^2 \qquad \text{(B)}\ 2\sqrt{2}a^2 \qquad \text{(C)}\ 2\sqrt{3}a^2 \qquad \text{(D)}\ 3\sqrt{3}a^2 \qquad \text{(E)}\ 6a^2$

2022 AMC 12/AHSME, 22

Tags: geometry , trapezoid

Let $c$ be a real number, and let $z_1, z_2$ be the two complex numbers satisfying the quadratic $z^2 - cz + 10 = 0$. Points $z_1, z_2, \frac{1}{z_1}$, and $\frac{1}{z_2}$ are the vertices of a (convex) quadrilateral $Q$ in the complex plane. When the area of $Q$ obtains its maximum value, $c$ is the closest to which of the following? $\textbf{(A)}~4.5\qquad\textbf{(B)}~5\qquad\textbf{(C)}~5.5\qquad\textbf{(D)}~6\qquad\textbf{(E)}~6.5$

2007 IMO, 2

Tags: geometry , parallelogram , circumcircle , homothety

Consider five points $ A$, $ B$, $ C$, $ D$ and $ E$ such that $ ABCD$ is a parallelogram and $ BCED$ is a cyclic quadrilateral. Let $ \ell$ be a line passing through $ A$. Suppose that $ \ell$ intersects the interior of the segment $ DC$ at $ F$ and intersects line $ BC$ at $ G$. Suppose also that $ EF \equal{} EG \equal{} EC$. Prove that $ \ell$ is the bisector of angle $ DAB$. [i]Author: Charles Leytem, Luxembourg[/i]

2021 Simon Marais Mathematical Competition, A3

Tags: matrix , determinant , linear algebra

Let $\mathcal{M}$ be the set of all $2021 \times 2021$ matrices with at most two entries in each row equal to $1$ and all other entries equal to $0$. Determine the size of the set $\{ \det A : A \in M \}$. [i]Here $\det A$ denotes the determinant of the matrix $A$.[/i]

MathLinks Contest 2nd, 6.3

Tags: combinatorics

At a party there were some couples attending. As they arrive each person gets to talk with all the other persons which are [i]already [/i] in the room. During the party, after all the guests arrive, groups of persons form, such that no two persons forming a couple belong to the same group, and for each two persons that do not form a couple, there is one and only one group to which both belong. Find the number of couples attending the party, knowing that there are less groups than persons at the party.

2022 Regional Olympiad of Mexico West, 4

Tags: ratio , geometry , fixed , equal angles

Prove that in all triangles $\vartriangle ABC$ with $\angle A = 2 \angle B$ it holds that, if $D$ is the foot of the perpendicular from $C$ to the perpendicular bisector of $AB$, $\frac{AC}{DC}$ is constant for any value of $\angle B$.